Interntional Training Workshop on Groundwater Modeling in Arid and Semi Arid Areas

G-WADI 2007

Groundwater Modeling for Arid and Semi-Arid Areas

G-WADI Workshop, Lanzhou, China

11-15 June 2007

COURSE MATERIALS

UNESCO Beijing, New Delhi, Almaty and Tehran Offices

IHP-UNESCO Paris

Asian G-WADI Network

Cold and Arid Regions Environmental and Engineering Research Institute,

Chinese Academy of Sciences

Chapter 1

Workshop Introduction: Hydrological processes, groundwater recharge and surface water-groundwater interactions in arid and semi arid areas

Howard S. Wheater

Department of Civil and Environmental Engineering Imperial College of Science, Technology and Medicine,

London SW7 2BU, UK

1.1 Introduction to the workshop The management of water resources in arid and semi-arid areas has been

identified by UNESCO as a global priority. In arid and semi-arid areas, water resources are, by definition, limited, and these scarce resources are under increasing pressure. Population expansion and economic growth are generating increased demand for water - for domestic consumption, agriculture and for industry. At the same time, water resources are under threat. Point and diffuse pollution is increasing, due to domestic, industrial and agricultural activities, and over-abstraction of groundwater has associated risks of deterioration of water quality, particularly in coastal areas where saline water can be drawn into the aquifer. Ecosystems are fragile, and under threat from groundwater abstractions and the management of surface flows. Added to these pressures is the uncertain threat of climate change.

The hydrology of arid and semi-arid lands is distinct from that of humid areas, and raises particular challenges. Rainfall is highly variable, both in time and in space. Surface flows are generally ephemeral, and runoff is focused on stream (wadi) channels that provide a source of groundwater recharge through channel bed infiltration. Vegetation is dynamic, responding to rainfall, and affects runoff and recharge processes. The available resource, i.e. surface flows and/or groundwater recharge, is a small component (typically less than 5%) of the water balance. Scientific understanding of these processes is improving, but still limited, and, due to sparse populations, extreme climate and limited resources, available data are scarce.

UNESCO has recognized that there is a need to share knowledge and understanding of the hydrology of arid and semi-arid areas to provide access to state-of-the art information, and to share experience of water management, with respect to traditional methods and new technology. A global network, G-WADI, has been established to achieve this. G-WADI is placing emphasis on the sharing of information, including new data sources, and providing access to state-of-the-art modelling tools and case studies, to support integrated water resources management.

In March, 2005, a workshop was held in Roorkee, India, to focus on the problems of modelling surface water systems. The world’s leading experts were brought together to provide lectures, case studies and tutorial material to an invited

1

audience from the world’s arid regions. That material has been made available through the G-WADI web-site, and will shortly be published, with some additional material, as a book by Cambridge University Press.

This Lanzhou workshop aims to provide the same support for issues of groundwater modelling in arid and semi-arid areas. In many developed countries in humid climates, groundwater modelling is well established as a management tool, and recent research developments, building on the increase in available computing power to incorporate models within a framework of risk assessment, are rapidly finding their way into practice. For arid areas, there are particular challenges, associated with climate and data, and little information that addresses the specific problems of arid regions. The motivation for this workshop is to provide a guide for researchers and practitioners to the state of the art of groundwater modelling, as applied to the specific needs of arid and semi-arid areas. Once again, UNESCO has been able to bring together a set of the world’s leading experts, and key representatives of the major arid regions of the world, hosted on the edge of the Gobi desert by our friend and colleague Prof. Xin Li from Lanzhou, and with the support and assistance of UNESCO’s Asian Region. We are aiming, with the support and advice from the workshop participants, to produce a set of workshop outputs that will be of assistance to practitioners and managers world-wide.

1.2 Groundwater resources, groundwater modelling and the quantification of recharge.

The traditional development of water resources in arid areas has relied heavily on the use of groundwater. Groundwater uses natural storage, is spatially-distributed, and in climates where potential evaporation rates can be of the order of metres per year, provides protection from the high evaporation losses experienced by surface water systems. Traditional methods for the exploitation of groundwater have been varied, including the use of very shallow groundwater in seasonally-replenished river bed aquifers (as in the sand rivers of Botswana), the channelling of unconfined alluvial groundwater in afalaj (or qanats) in Oman and Iran, and the use of hand dug wells. Historically, abstraction rates were limited by the available technology, and rates of development were low, so that exploitation was generally sustainable.

However, in recent decades, pump capacities have dramatically increased, and hence agricultural use of water has grown rapidly, while the increasing concentration of populations in urban areas has meant that large-scale well fields have been developed for urban water supply. A common picture in arid areas is that groundwater levels are in rapid decline, and in many instances this is accompanied by decreasing water quality, particularly in coastal aquifers where saline intrusion is a threat. And associated with population growth, economic development and increased agricultural intensification, pollution has also become an increasing problem. The integrated assessment and management of groundwater resources is essential, so that aquifer systems can be protected from pollution and over-exploitation. This requires the use of groundwater models as a decision support tool for groundwater management.

Some of the most difficult aspects of groundwater modelling concern the

2

interaction between surface water and groundwater systems. This is most obviously the case for the quantification of long-term recharge, which ultimately defines sustainable yields. Quantification of recharge remains the major challenge for groundwater development world-wide, but is a particular difficulty in arid areas, where recharge rates are small – both as a proportion of the water balance, and in absolute terms. However, more generally, the interactions between surface water and groundwater systems are important. In arid areas, infiltration from surface water channels, as a ‘transmission loss’ for surface flows, may be a major component of groundwater recharge, and there is increasing interest in the active management of this process to focus recharge (for example in an extensive programme of construction of ‘recharge dams’ in Northern Oman). Conversely, the discharge of groundwater to surface water systems can be important in terms of valuable ecosystems. Hence the main thrust of this Chapter is to review hydrological processes in arid and semi-arid areas, to provide the context for surface-groundwater interactions, and their analysis and modelling.

1.3 Hydrological processes in arid areas

Despite the critical importance of water in arid and semi-arid areas, hydrological data have historically been severely limited. It has been widely stated that the major limitation of the development of arid zone hydrology is the lack of high quality observations (McMahon, 1979; Nemec and Rodier, 1979; Pilgrim et al., 1988). There are many good reasons for this. Populations are usually sparse and economic resources limited; in addition the climate is harsh and hydrological events infrequent, but damaging. However, in the general absence of reliable long-term data and experimental research, there has been a tendency to rely on humid zone experience and modelling tools, and data from other regions. At best, such results will be highly inaccurate. At worst, there is a real danger of adopting inappropriate management solutions which ignore the specific features of dryland response.

Despite the general data limitations, there has been some substantial and significant progress in development of national data networks and experimental research. This has given new insights and we can now see with greater clarity the unique features of arid zone hydrological systems and the nature of the dominant hydrological processes. This provides an important opportunity to develop methodologies for flood and water resource management which are appropriate to the specific hydrological characteristics of arid areas and the associated management needs, and hence to define priorities for research and hydrological data. The aim here is to review this progress and the resulting insights, and to consider some of the implications.

1.3.1 Rainfall Rainfall is the primary hydrological input, but rainfall in arid and semi-arid areas

is commonly characterized by extremely high spatial and temporal variability. The temporal variability of point rainfall is well-known. Although most records are of relatively short length, a few are available from the 19th century. For example, Table 1 presents illustrative data from Muscat (Sultanate of Oman) (Wheater and Bell,

3

1983), which shows that a wet month is one with one or two rain days. Annual variability is marked and observed daily maxima can exceed annual rainfall totals.

For spatial characteristics, information is much more limited. Until recently, the major source of detailed data has been from the South West U.S.A., most notably the two, relatively small, densely instrumented basins of Walnut Gulch, Arizona (150km2) and Alamogordo Creek, New Mexico (174km2), established in the 1950s (Osborn et al., 1979). The dominant rainfall for these basins is convective; at Walnut Gulch 70% of annual rainfall occurs from purely convective cells, or from convective cells developing along weak, fast-moving cold fronts, and falls in the period July to September (Osborn and Reynolds, 1963). Raingauge densities were increased at Walnut Gulch to give improved definition of detailed storm structure and are currently better than 1 per 2km2. This has shown highly localised rainfall occurrence, with spatial correlations of storm rainfall of the order of 0.8 at 2km separation, but close to zero at 15-20km spacing. Osborn et al. (1972) estimated that to observe a correlation of r2 = 0.9, raingauge spacings of 300-500m would be required.

Recent work has considered some of the implications of the Walnut Gulch data for hydrological modelling. Michaud and Sorooshian (1994) evaluated problems of spatial averaging for rainfall-runoff modelling in the context of flood prediction. Spatial averaging on a 4kmx4km pixel basis (consistent with typical weather radar resolution) gave an underestimation of intensity and led to a reduction in simulated runoff of on average 50% of observed peak flows. A sparse network of raingauges (1 per 20km2), representing a typical density of flash flood warning system, gave errors in simulated peak runoff of 58%. Evidently there are major implications for hydrological practice, and we will return to this issue, below.

The extent to which this extreme spatial variability is characteristic of other arid areas has been uncertain. Anecdotal evidence from the Middle East underlay comments that spatial and temporal variability was extreme (FAO, 1981), but data from South West Saudi Arabia obtained as part of a five-year intensive study of five basins (Saudi Arabian Dames and Moore, 1988), undertaken on behalf of the Ministry of Agriculture and Water, Riyadh, have provided a quantitative basis for assessment. The five study basins range in area from 456 to 4930 km2 and are located along the Asir escarpment (Figure 1), three draining to the Red Sea, two to the interior, towards the Rub al Khali. The mountains have elevations of up to 3000m a.s.l., hence the basins encompass a wide range of altitude, which is matched by a marked gradient in annual rainfall, from 30-100mm on the Red Sea coastal plain to up to 450mm at elevations in excess of 2000m a.s.l.

4

Tabl

e 1.

Sum

mar

y of

Mus

cat r

ainf

all d

ata

(189

3 -1

959)

(afte

r Whe

ater

and

Bel

l, 19

83)

Mon

thly

rain

fall

(mm

) Ja

n.Fe

b.M

ar.

Apr

. M

ayJu

neJu

lyA

ug.

Sept

.O

ct.

Nov

. D

ec.

Mea

n 31

.2

19.1

13

.1

8.0

0.38

1.

31

0.96

0.

45

0.0

2.32

7.

15

22.0

Stan

dard

dev

iatio

n 38

.9

25.1

18

.9

20.3

1.

42

8.28

4.

93

2.09

0.

0 7.

62

15.1

35

.1

Max

. 14

3.0

98.6

70

.4

98.3

8.

89

64.0

37

.1

14.7

0.

0 44

.5

77.2

17

1.2

Mea

n nu

mbe

r of r

aind

ays

2.03

1.

39

1.15

0.

73

0.05

0.

08

0.10

0.

07

0.0

0.13

0.

51

1.6

Max

. dai

ly fa

ll (m

m)

78.7

57

.0

57.2

51

.3

8.9

61.5

30

.0

10.4

0.

0 36

.8

53.3

57

.2

Num

ber o

f yea

rs re

cord

63

.0

64

62

63

61

61

60

61

61

60

61

60

5

Figure 1. Location of Saudi Arabian study basins

The spatial rainfall distributions are described by Wheater et al.(1991a). The extreme spottiness of the rainfall is illustrated for the 2869km2 Wadi Yiba by the frequency distributions of the number of gauges at which rainfall was observed given the occurrence of a catchment rainday (Table 2). Typical inter-gauge spacings were 8-10km, and on 51% of raindays only one or two raingauges out of 20 experienced rainfall. For the more widespread events, sub-daily rainfall showed an even more spotty picture than the daily distribution. An analysis of relative probabilities of rainfall occurrence, defined as the probability of rainfall occurrence for a given hour at Station B given rainfall at Station A, gave a mean value of 0.12 for Wadi Yiba, with only 5% of values greater that 0.3. The frequency distribution of rainstorm durations shows a typical occurrence of one or two-hour duration point rainfalls, and these tend to occur in mid-late afternoon. Thus rainfall will occur at a few gauges and die out, to be succeeded by rainfall in other locations. This is illustrated for Wadi Lith in Figure 2, which shows the daily rainfall totals for the storm of 16th May 1984 (Figure 2a), and the individual hourly depths (Figure 2b-2e). In general, the storm patterns appear to be consistent with the results from

6

the South West USA and area reduction factors were also generally consistent with results from that region (Wheater et al., 1989).

Figure 2a Wadi Al-Lith daily rainfall, 16th May 1994

7

Figure 2b Wadi Al-Lith hourly rainfall, 16th May 1994

8

Figure 2c Wadi Al-Lith hourly rainfall, 16th May 1994

9

Figure 2d Wadi Al-Lith hourly rainfall, 16th May 1994

10

Figure 2e Wadi Al-Lith hourly rainfall, 16th May 1994

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Table 2 Wadi Yiba raingauge frequencies and associated conditional probabilities for catchment rainday occurrence

Number of Gauges Occurrence Probability

1 88 0.372

2 33 0.141

3 25 0.106

4 18 0.076

5 10 0.042

6 11 0.046

7 13 0.055

8 6 0.026

9 7 0.030

10 5 0.021

11 7 0.030

12 5 0.021

13 3 0.013

14 1 0.004

15 1 0.004

16 1 0.005

17 1 0.004

18 1 0.004

19 0 0.0

20 0 0.0

TOTAL 235 1.000

The effects of elevation were investigated, but no clear relationship could be identified for intensity or duration. However, a strong relationship was noted between the frequency of raindays and elevation. It was thus inferred that once rainfall occurred, its point properties were similar over the catchment, but occurrence was more likely at the higher elevations. It is interesting to note that a similar result has emerged from an analysis of rainfall in Yemen (UNDP, 1992), in which it was concluded that daily rainfalls observed at any location are effectively samples from a population that is independent of

12

position or altitude.

It is dangerous to generalise from samples of limited record length, but it is clear that most events observed by those networks are characterized by extremely spotty rainfall, so much so that in the Saudi Arabian basins there were examples of wadi flows generated from zero observed rainfall. However, there were also some indications of a small population of more wide-spread rainfalls, which would obviously be of considerable importance in terms of surface flows and recharge. This reinforces the need for long-term monitoring of experimental networks to characterise spatial variability.

For some other arid or semi-arid areas, rainfall patterns may be very different. For example, data from arid New South Wales, Australia have indicated spatially extensive, low intensity rainfalls (Cordery et al., 1983), and recent research in the Sahelian zone of Africa has also indicated a predominance of widespread rainfall. This was motivated by concern to develop improved understanding of land-surface processes for climate studies and modelling, which led to a detailed (but relatively short-term) international experimental programme, the HAPEX-Sahel project based on Niamey, Niger (Goutorbe et al., 1997). Although designed to study land surface/atmosphere interactions, rather than as an integrated hydrological study, it has given important information. For example, Lebel et al. (1997) and Lebel and Le Barbe (1997) note that a 100 raingauge network was installed and report information on the classification of storm types, spatial and temporal variability of seasonal and event rainfall, and storm movement. 80% of total seasonal rainfall was found to fall as widespread events which covered at least 70% of the network. The number of gauges allowed the authors to analyse the uncertainty of estimated areal rainfall as a function of gauge spacing and rainfall depth.

Recent work in southern Africa (Andersen et al., 1998, Mocke, 1998) has been concerned with rainfall inputs to hydrological models to investigate the resource potential of the sand rivers of N.E.Botswana. Here, annual rainfall is of the order of 600mm, and available rainfall data is spatially sparse, and apparently highly variable, but of poor data quality. Investigation of the representation of spatial rainfall for distributed water resource modelling showed that use of convential methods of spatial weighting of raingauge data, such as Theissen polygons, could give large errors. Large sub-areas had rainfall defined by a single, possibly inaccurate gauge. A more robust representation resulted from assuming catchment-average rainfall to fall uniformly, but the resulting accuracy of simulation was still poor.

1.3.2 Rainfall-runoff processesThe lack of vegetation cover in arid and semi-arid areas removes protection of the

soil from raindrop impact, and soil crusting has been shown to lead to a large reduction in infiltration capacity for bare soil conditions (Morin and Benyamini, 1977). Hence infiltration of catchment soils can be limited. In combination with the high intensity, short duration convective rainfall discussed above, extensive overland flow can be generated. This overland flow, concentrated by the topography, converges on the wadi channel network, with the result that a flood flow is generated. However, the runoff generation

13

process due to convective rainfall is likely to be highly localised in space, reflecting the spottiness of the spatial rainfall fields, and to occur on only part of a catchment, as illustrated above.

Linkage between inter-annual variability of rainfall, vegetation growth and runoff production may occur. Our modelling in Botswana suggests that runoff production is lower in a year which follows a wet year, due to enhanced vegetation cover, which supports observations reported by Hughes (1995).

Commonly, flood flows move down the channel network as a flood wave, moving over a bed that is either initially dry or has a small initial flow. Hydrographs are typically characterised by extremely rapid rise times, of as little as 15-30 minutes (Figure 3). However, losses from the flood hydrograph through bed infiltration are an important factor in reducing the flood volume as the flood moves downstream. These transmission losses dissipate the flood, and obscure the interpretation of observed hydrographs. It is not uncommon for no flood to be observed at a gauging station, when further upstream a flood has been generated and lost to bed infiltration.

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Figure 3 Surface water hydrographs, Wadi Ghat 12 May 1984 – observed hydrograph and unit hydrograph simulation

As noted above, the spotty spatial rainfall patterns observed in Arizona and Saudi Arabia are extremely difficult, if not impossible, to quantify using conventional densities of raingauge network. This, taken in conjunction with the flood transmission losses, means that conventional analysis of rainfall-runoff relationships is problematic, to say the least. Wheater and Brown (1989) present an analysis of Wadi Ghat, a 597 km2

sub-catchment of wadi Yiba, one of the Saudi Arabian basins discussed above. Areal rainfall was estimated from 5 raingauges and a classical unit hydrograph analysis was undertaken. A striking illustration of the ambiguity in observed relationships is the relationship between observed rainfall depth and runoff volume (Figure 4). Runoff coefficients ranged from 5.9 to 79.8%, and the greatest runoff volume was apparently generated by the smallest observed rainfall! Goodrich et al. (1997) show that the combined effects of limited storm areal coverage and transmission loss give important

15

differences from more humid regions. Whereas generally basins in more humid climates show increasing linearity with increasing scale, the response of Walnut Gulch becomes more non-linear with increasing scale. It is argued that this will give significant errors in application of rainfall depth-area-frequency relationships beyond the typical area of storm coverage, and that channel routing and transmission loss must be explicitly represented in watershed modelling.

Figure 4 Storm runoff as a function of rainfall, Wadi Ghat

The transmission losses from the surface water system are a major source of potential groundwater recharge. The characteristics of the resulting groundwater resource will depend on the underlying geology, but bed infiltration may generate shallow water tables, within a few metres of the surface, which can sustain supplies to nomadic people for a few months (as in the Hesse of the North of South Yemen), or recharge substantial alluvial aquifers with potential for continuous supply of major towns (as in Northern Oman and S.W. Saudi Arabia).

The balance between localised recharge from bed infiltration and diffuse recharge from rainfall infiltration of catchment soils will vary greatly depending on local circumstances. However, soil moisture data from Saudi Arabia (Macmillan, 1987) and Arizona (Liu et al., 1995), for example, show that most of the rainfall falling on soils in arid areas is subsequently lost by evaporation. Methods such as the chloride profile method (e.g. Bromley et al., 1997) and isotopic analyses (Allison and Hughes, 1978)

16

have been used to quantify the residual percolation to groundwater in arid and semi-arid areas.

In some circumstances runoff occurs within an internal drainage basin, and fine deposits can support widespread surface ponding. A well known large-scale example is the Azraq oasis in N.E. Jordan, but small-scale features (Qaa’s) are widespread in that area. Small scale examples were found in the HAPEX-Sahel study (Desconnets et al., 1997). Infiltration from these areas is in general not well understood, but may be extremely important for aquifer recharge. Desconnets et al. report aquifer recharge of between 5 and 20% of basin precipitation for valley bottom pools, depending on the distribution of annual rainfall.

The characteristics of the channel bed infiltration process are discussed in the following section. However, it is clear that the surface hydrology generating this recharge is complex and extremely difficult to quantify using conventional methods of analysis.

1.3.3 Wadi bed transmission lossesWadi bed infiltration has an important effect on flood propagation, but also provides

recharge to alluvial aquifers. The balance between distributed infiltration from rainfall and wadi bed infiltration is obviously dependant on local conditions, but soil moisture observations from S.W. Saudi Arabia imply that, at least for frequent events, distributed infiltration of catchment soils is limited, and that increased near surface soil moisture levels are subsequently depleted by evaporation. Hence wadi bed infiltration may be the dominant process of groundwater recharge. As noted above, depending on the local hydrogeology, alluvial groundwater may be a readily accessible water resource. Quantification of transmission loss is thus important, but raises a number of difficulties.

One method of determining the hydraulic properties of the wadi alluvium is to undertake infiltration tests. Infiltrometer experiments give an indication of the saturated hydraulic conductivity of the surface. However, if an infiltration experiment is combined with measurement of the vertical distribution of moisture content, for example using a neutron probe, inverse solution of a numerical model of unsaturated flow can be used to identify the unsaturated hydraulic conductivity relationships and moisture characteristic curves. This is illustrated for the Saudi Arabian Five Basins Study by Parissopoulos and Wheater (1992a).

In practice, spatial heterogeneity will introduce major difficulties to the up-scaling of point profile measurements. The presence of silt lenses within the alluvium was shown to have important effects on surface infiltration as well as sub-surface redistribution (Parissopoulos and Wheater, 1990), and sub-surface heterogeneity is difficult and expensive to characterise. In a series of two-dimensional numerical experiments it was shown that “infiltration opportunity time”, i.e. the duration and spatial extent of surface wetting, was more important than high flow stage in influencing infiltration, that significant reductions in infiltration occur once hydraulic connection is made with a water table, and that hysteresis effects were generally small (Parissopoulos and Wheater,

17

1992b). Also sands and gravels appeared effective in restricting evaporation losses from groundwater (Parissopoulos and Wheater, 1991).

Additional process complexity arises, however. General experience from the Five Basins Study was that wadi alluvium was highly transmissive, yet observed flood propagation indicated significantly lower losses than could be inferred from in situ hydraulic properties, even allowing for sub-surface heterogeneity. Possible causes are air entrapment, which could restrict infiltration rates, and the unknown effects of bed mobilisation and possible pore blockage by the heavy sediment loads transmitted under flood flow conditions.

A commonly observed effect is that in the recession phase of the flow, deposition of a thin (1-2mm) skin of fine sediment on the wadi bed occurs, which is sufficient to sustain flow over an unsaturated and transmissive wadi bed. Once the flow has ceased, this skin dries and breaks up so that the underlying alluvium is exposed for subsequent flow events. Crerar et al., (1988) observed from laboratory experiments that a thin continous silt layer was formed at low velocities. At higher velocities no such layer occurred, as the bed surface was mobilised, but infiltration to the bed was still apparently inhibited. It was suggested that this could be due to clogging of the top layer of sand due to silt in the infiltrating water, or formation of a silt layer below the mobile upper part of the bed.

Further evidence for the heterogeneity of observed response comes from the observations of Hughes and Sami (1992) from a 39.6 km2 semi-arid catchment in S.Africa. Soil moisture was monitored by neutron probe following two flow events. At some locations immediate response (monitored 1day later) occurred throughout the profile, at others, an immediate response near surface was followed by a delayed response at depth. Away from the inundated area, delayed response, assumed due to lateral subsurface transmission, occurred after 21 days.

The overall implication of the above observations is that it is not possible at present to extrapolate from in-situ point profile hydraulic properties to infer transmission losses from wadi channels. However, analysis of observed flood flows at different locations can allow quantification of losses, and studies by Walters (1990) and Jordan (1977), for example, provide evidence that the rate of loss is linearly related to the volume of surface discharge.

For S.W. Saudi Arabia, the following relationships were defined:

LOSSL = 4.56 + 0.02216 UPSQ - 2034 SLOPE + 7.34 ANTEC

(s.e. 4.15)

LOSSL = 3.75 × 10-5 UPSQ0.821 SLOPE -0.865 ACWW0.497

(s.e. 0.146 log units (±34%))

LOSSL = 5.7 × 10-5 UPSQ0.968 SLOPE -1.049

(s.e. 0.184 loge units (±44%))

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Where:

LOSSL = Transmission loss rate (1000m3/km) (O.R. 1.08-87.9)

UPSQ = Upstream hydrograph volume (1000m3) (O.R. 69-3744)

SLOPE = Slope of reach (m/m) (O.R. 0.001-0.011)

ANTEC = Antecedent moisture index (O.R. 0.10-1.00)

ACWW = Active channel width (m) (O.R. 25-231)

and O.R. = Observed range

However, generalisation from limited experience can be misleading. Wheater et al. (1997) analysed transmission losses between 2 pairs of flow gauges on the Walnut Gulch catchment for a ten year sequence and found that the simple linear model of transmission loss as proportional to upstream flow was inadequate. Considering the relationship:

0 (1 )xxV V

where xV is flow volume (m3) at distance x downstream of flow volume 0V and represents the proportion of flow lost per unit distance, then was found to decrease with discharge volume:

0.710118.8( )V

The events examined had a maximum value of average transmission loss of 4076 m3

km-1 in comparison with the estimate of Lane et al. (1971) of 4800-6700 m3 km-1 as an upper limit of available alluvium storage.

The role of available storage was also discussed by Telvari et al. (1998), with reference to the Fowler’s Gap catchment in Australia. Runoff plots were used to estimate runoff production as overland flow for a 4km2 basin. It was inferred that 7000 m3 of overland flow becomes transmission loss and that once this alluvial storage is satisfied, approximately two-thirds of overland flow is transmitted downstream.

A similar concept was developed by Andersen et al. (1998) at larger scale for the sand rivers of Botswana, which have alluvial beds of 20-200m width and 2-20m depth. Detailed observations of water table response showed that a single major event after a seven weeks dry period was sufficient to fully satisfy available alluvial storage (the river bed reached full saturation within 10 hours). No significant drawdown occurred between subsequent events and significant resource potential remained throughout the dry season. It was suggested that two sources of transmission loss could be occurring, direct losses to the bed, limited by available storage, and losses through the banks during flood events.

It can be concluded that transmission loss is complex, that where deep unsaturated alluvial deposits exist the simple linear model as developed by Jordan (1977) and implicit in the results of Walters (1990) may be applicable, but that where alluvial storage is limited, this must be taken into account.

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1.3.4 Groundwater recharge

While the estimation of groundwater recharge is essential to determine the sustainable yield of a groundwater resource, it is one of the most difficult hydrological fluxes to quantify. This is particularly so in arid and semi-arid areas, where values are low, and in any water balance calculation groundwater recharge is likely to be lost in the uncertainty of the dominant inputs and outputs (i.e. precipitation and evaporation, together with another minor component of surface runoff).

It can be seen from the preceding discussion that groundwater recharge in arid and semi-arid areas is dependent on a complex set of spatial-temporal hydrological interactions that will be dependent on the characteristics of the local climate, the land surface properties that determine the balance between infiltration and overland flow, and the subsurface characteristics. Precipitation events in arid areas are generally infrequent, but can be extremely intense. Where convective rainfall systems dominate, precipitation is highly localised in space and time (e.g. the monsoon rainfall of Walnut Gulch, Arizona), but in other climates more widespread winter rainfall may be important, and may include snow. Precipitation may infiltrate directly into the ground surface or, if overland flow is generated, be focussed as surface runoff in a flowing channel. We have seen that for arid climates and convective rainfall, extensive overland flow can occur, enhanced by reduced surface soil permeabilities as a result of raindrop impact. The relationship between surface infiltration and groundwater recharge will depend on the subsurface properties, antecedent conditions, and the duration and intensity of the precipitation.

1.3.4.1 Groundwater recharge from ephemeral channel flows

The relationship between wadi flow transmission losses and groundwater recharge will depend on the underlying geology. The effect of lenses of reduced permeability on the infiltration process has been discussed and illustrated above, but once infiltration has taken place, the alluvium underlying the wadi bed is effective in minimising evaporation loss through capillary rise (the coarse structure of alluvial deposits minimises capillary effects). Thus Hellwig (1973), for example, found that dropping the water table below 60cm in sand with a mean diameter of 0.53mm effectively prevented evaporation losses, and Sorey and Matlock (1969) reported that measured evaporation rates from streambed sand were lower than those reported for irrigated soils.

Parrisopoulos and Wheater (1991) combined two-dimensional simulation of unsaturated wadi-bed response with Deardorff’s (1977) empirical model of bare soil evaporation to show that evaporation losses were not in general significant for the water balance or water table response in short-term simulation (i.e. for periods up to 10 days). However, the influence of vapour diffusion was not explicitly represented, and long term losses are not well understood. Andersen et al. (1998) show that losses are high when the alluvial aquifer is fully saturated, but are small once the water table drops below the surface.

Sorman and Abdulrazzak (1993) provide an analysis of groundwater rise due to

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transmission loss for an experimental reach in Wadi Tabalah, S.W. Saudi Arabia and estimate that on average 75% of bed infiltration reaches the water table. There is in general little information available to relate flood transmission loss to groundwater recharge, however. The differences between the two are expected to be small, but will depend on residual moisture stored in the unsaturated zone and its subsequent drying characteristics. But if water tables approach the surface, relatively large evaporation losses may occur.

Again, it is tempting to draw over-general conclusions from limited data. In the study of the sand-rivers of Botswana, referred to above, it was expected that recharge of the alluvial river beds would involve complex unsaturated zone response. In fact, observations showed that the first flood of the wet season was sufficient to fully recharge the alluvial river bed aquifer. This storage was topped up in subsequent floods, and depleted by evaporation when the water table was near-surface, but in many sections sufficient water remained throughout the dry season to provide adequate sustainable water supplies for rural villages. And as noted above, Wheater et al. (1997) showed for Walnut Gulch and Telvari et al. (1998) for Fraser’s Gap that limited river bed storage affected transmission loss. It is evident that surface water/groundwater interactions depend strongly on the local characteristics of the underlying alluvium and the extent of their connection to, or isolation from, other aquifer systems.

Very recent work at Walnut Gulch (Goodrich et al., 2004) has investigated ephemeral channel recharge using a range of experimental methods, combined with modelling. These included a reach water balance method, including estimates of near channel evapotranspiration losses, geochemical methods, analysis of changes in groundwater levels and microgravity measurements, and unsaturated zone flow and temperature analyses. The conclusions were that ephemeral channel losses were significant as an input to the underlying regional aquifer, and that the range of methods for recharge estimation agreed within a factor of three (reach water balance methods giving the higher estimates).

1.3.4.2 Spatial variability of groundwater recharge The above discussion of groundwater recharge from ephemeral channels highlighted

the role of subsurface properties in determining the relationship between surface infiltration and groundwater recharge. The same issues apply to the relationships between infiltration and recharge away from the active channel system, but data to quantify the process response associated with distributed recharge are commonly lacking.

An important requirement for recharge estimation has arisen in connection with the proposal for a repository for high level nuclear waste at Yucca Mountain, Nevada, and a major effort has been made to investigate the hydrological response of the deep unsaturated zone which would house the repository. Flint et al. (2002) propose that soil depth is an important control on infiltration. They argue that groundwater recharge (or net infiltration) will occur when the storage capacity of the soil or rock is exceeded. Hence where thin soils are underlain by fractured bedrock, a relatively small amount of

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infiltration is required to saturate the soil and generate ‘net infiltration’, which will occur at a rate dependent on the bedrock (fracture) permeability. Deeper soils have greater storage, and provide greater potential to store infiltration that will subsequently be lost as evaporation. This conceptual understanding was encapsulated in a distributed hydrological model, incorporating soil depth variability, and the potential for redistribution through overland flow. The results indicate extreme variability in space and time, with watershed modelling giving a range from zero to several hundred mm/year, depending on spatial location. The high values arise due to flow focussing in ephemeral channels, and subsequent channel bed infiltration. The need to characterise in detail the response of a deep unsaturated zone, led to the application at Yucca Mountain of a wide range of alternative methods for recharge estimation. Hence Flint et al. also review results from methods including analysis of physical data from unsaturated zone profiles of moisture and heat, and the use of environmental tracers. These methods operate at a range of spatial scales, and the results support the modelling conclusions that great spatial variability of recharge can be expected at the site scale, at least for comparable systems of fractured bedrock.

The Yucca Mountain work illustrates that high spatial variability occurs at the relatively small spatial scale of an individual site. More commonly, a broader scale of assessment is needed, and broader scale features become important. For example, Wilson and Guan (2004) introduce the term ‘mountain-front recharge’ (MFR) to describe the important role for many aquifer systems in arid or semi-arid areas of the contribution from mountains to the recharge of aquifers in adjacent basins. They argue that mountains have higher precipitation, cooler temperatures (and hence less evaporation) and thin soils, all contributing to greater runoff and recharge, and that in many arid and semi-arid areas this will dominate over the relatively small contribution to recharge from direct infiltration of precipitation from the adjacent arid areas.

It is clear from the above discussion that appropriate representation of spatial variability of precipitation and the subsequent hydrological response is required for the estimation of recharge at catchment or aquifer scale, and that even at local scale, these effects are important.

1.4 Hydrological modelling and the representation of rainfall The preceding discussion illustrates some of the particular characteristics of arid

areas which place special requirements on hydrological modelling, for example for flood, water resources or groundwater recharge estimation. One evident area of difficulty is rainfall, especially where convective storms are an important influence. The work of Michaud and Sorooshian (1994) demonstrated the sensitivity of flood peak simulation to the spatial resolution of rainfall input. This obviously has disturbing implications for flood modelling, particularly where data availability is limited to conventional raingauge densities. Indeed, it appears highly unlikely that suitable raingauge densities will ever be practicable for routine monitoring. However, the availability of 2km resolution radar data

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in the USA can provide adequate information and radar could be installed elsewhere for particular applications. Morin et al. (1995) report results from a radar located at Ben-Gurion airport in Israel, for example. Where convective rainfall predominates, rainfall variability is extreme, and raises specific issues of data and modelling. However, in general, as seen from the discussion of Mountain Front Recharge, the spatial distribution of precipitation is important.

One way forward for the problems of convective rainfall is to develop an understanding of the properties of spatial rainfall based on high density experimental networks and/or radar data, and represent those properties within a spatial rainfall model for more general application. It is likely that this would have to be done within a stochastic modelling framework in which equally-likely realisations of spatial rainfall are produced, possibly conditioned by sparse observations.

Some simple empirical first steps in this direction were taken by Wheater et al. (1991a,b) for S.W.Saudi Arabia and Wheater et al. (1995) for Oman. In the Saudi Arabian studies, as noted earlier, raingauge data was available at approximately 10km spacing and spatial correlation was low. Hence a multi-variate model was developed, assuming independence of raingauge rainfall. Based on observed distributions, seasonally-dependent catchment rainday occurrence was simulated, dependent on whether the preceding day was wet or dry. The number of gauges experiencing rainfall was then sampled, and the locations selected based on observed occurrences (this allowed for increased frequency of raindays with increased elevation). Finally, start-times, durations and hourly intensities were generated. Model performance was compared with observations. Rainfall from random selections of raingauges was well reproduced, but when clusters of adjacent gauges were evaluated, a degree of spatial organisation of occurrence was observed, but not simulated. It was evident that a weak degree of correlation was present, which should not be neglected. Hence in extension of this approach to Oman (Wheater et al., 1995), observed spatial distributions were sampled, with satisfactory results.

However, this multi-variate approach suffers from limitations of raingauge density, and in general a model in continuous space (and continuous time) is desirable. A family of stochastic rainfall models of point rainfall was proposed by Rodriguez-Iturbe, Cox and Isham (1987, 1988) and applied to UK rainfall by Onof and Wheater (1993,1994). The basic concept is that a Poisson process is used to generate the arrival of storms. Associated with a storm is the arrival of raincells, of uniform intensity for a given duration (sampled from specified distributions). The overlapping of these rectangular pulse cells generates the storm intensity profile in time. These models were shown to have generally good performance for the UK in reproducing rainfall properties at different time-scales (from hourly upwards), and extreme values.

Cox and Isham (1988) extended this concept to a model in space and time, whereby the raincells are circular and arrive in space within a storm region. As before, the overlapping of cells produces a complex rainfall intensity profile, now in space as well as

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time. This model has been developed further by Northrop (1998) to include elliptical cells and storms and is being applied to UK rainfall (Northrop et al., 1999).

Work by Samuel (1999) explored the capability of these models to reproduce the convective rainfall of Walnut Gulch. In modelling point rainfall, the Bartlett-Lewis Rectangular Pulse Model was generally slightly superior to other model variants tested. Table 3 shows representative performance of the model in comparing the hourly statistics from 500 realisations of July rainfall in comparison with 35 years from one of the Walnut Gulch gauges (gauge 44), where Mean is the mean hourly rainfall (mm), Var its variance, ACF1,2,3 the autocorrelations for lags 1,2,3, Pwet the proportion of wet intervals, Mint the mean storm inter-arrival time (h), Mno the mean number of storms per month, Mdur the mean storm duration (h).This performance is generally encouraging (although the mean storm duration is underestimated), and extreme value performance is excellent.

Table 3 Performance of the Bartlett-Lewis Rectangular Pulse Model in representing July rainfall at gauge 44, Walnut Gulch

Mean Var ACF1 ACF2 ACF3 Pwet Mint Mno Mdur

Mode 0.103 1.082 0.193 0.048 0.026 0.032 51.17 14.34 1.68

Data 0.100 0.968 0.174 0.040 0.036 0.042 53.71 13.23 2.38

Work with the spatial-temporal model was only taken to a preliminary stage, but Figure 5 shows a comparison of observed spatial coverage of rainfall for 25 years of July data from 81 gauges (for different values of the standard deviation of cell radius) and Figure 6 the corresponding fit for temporal lag-0 spatial correlation. Again, the results are encouraging, and there is promise with this approach to address the significant problems of spatial representation of convective rainfall for hydrological modelling.

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0

0.1

0.2

0.3

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0.6

0.7

0.8

0.9

1

0.75 0.8 0.85 0.9 0.95 1

Frequency

Cov

erag

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Data12510

Figure 5 Frequency distribution of spatial coverage of Walnut Gulch rainfall.

Observed vs. alternative simulations

0

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Figure 6 Spatial correlation of Walnut Gulch rainfall.

Observed vs. alternative simulations

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Where spatial rainfall variability is less extreme, and coarser time-scale modelling is appropriate (e.g. daily rainfall), recent work has developed a set of stochastic rainfall modelling tools that can be readily applied to represent rainfall, including effects of location (e.g. topographic effects or rainshadow) and climate variability and change (e.g. Chandler and Wheater, 2002, Yang et al., 2005). These use Generalised Linear Models (GLMs) to simulate the occurrence of a rainday, and then the conditional distribution of daily rainfall depths at selected locations over an area. Although initially developed and evaluated for humid temperate climates, work is currently underway to evaluate their use for drier climates, with application to Iran.

1.5 Integrated modelling for groundwater resource evaluationAppropriate strategies for water resource development must recognise the essential

physical characteristics of the hydrological processes. As noted above, groundwater is a resource particularly well suited to arid regions. Subsurface storage minimises evaporation loss and can provide long-term yields from infrequent recharge events. The recharge of alluvial groundwater systems by ephemeral flows can provide an appropriate resource, and this has been widely recognised by traditional development, such as the “afalaj” of Oman and elsewhere. There may, however, be opportunities for augmenting recharge and more effectively managing these groundwater systems. In any case, it is essential to quantify the sustainable yield of such systems, for appropriate resource development.

It has been seen that observations of surface flow are ambiguous, due to upstream tansmission loss, and do not define the available resource. Similarly, observed groundwater response does not necessarily indicate upstream recharge. Figure 7 presents a series of groundwater responses from 1985/86 for Wadi Tabalah which shows a downstream sequence of wells 3-B-96, -97, -98, -99 and -100 and associated surface water discharges. It can be seen that there is little evidence of the upstream recharge at the downstream monitoring point.

In addition, records of surface flows and groundwater levels, coupled with ill-defined histories of abstraction, are generally insufficient to define long term variability of the available resource.

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Figure 7 Longitudinal sequence of wadi alluvium well hydrographs and associated surface flows, Wadi Tabalah, 1985/6

To capture the variability of rainfall and the effects of transmission loss on surface flows, a distributed approach is necessary. If groundwater is to be included, integrated modelling of surface water and groundwater is needed. Examples of distributed surface water models include KINEROS (Wheater and Bell, 1983, Michaud and Sorooshian, 1994), the model of Sharma (1997, 1998) and a distributed model, INFIL2.0, developed by the USGS to simulate net infiltration at Yucca Mountain (Flint et al., 2000).

A distributed approach to the integrated modelling of surface and groundwater response following Wheater et al.(1995) is illustrated in Figure 8. This schematic framework requires the characterisation of the spatial and temporal variability of rainfall, distributed infiltration, runoff generation and flow transmission losses, the ensuing groundwater recharge and groundwater response. This presents some technical difficulties, although the integration of surface and groundwater modelling allows maximum use to be made of available information, so that, for example, groundwater response can feed back information to constrain surface hydrological parameterization. A distributed approach provides the only feasible method of exploring the internal response of a catchment to management options.

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DATA MODELS

RAINFALL DATA ANALYSIS

STOCHASTIC SPATIAL RAINFALL SIMULATION

DISTRIBUTED RAINFALL-RUNOFF MODEL

WABI BED TRANSMISSION LOSS

GROUNDWATER RECHARGE

DISTRIBUTED GROUNDWATER MODEL

RAINFALL RUNOFF CALIBRATION

GROUNDWATER CALIBRATION

Figure 8 Integrated modelling strategy for water resource evaluation

This integrated modelling approach was developed for Wadi Ghulaji, Sultanate of Oman, to evaluate options for groundwater recharge management (Wheater et al., 1995). The catchment, of area 758 km2, drains the southern slopes of Jebal Hajar in the Sharqiyah region of Northern Oman. Proposals to be evaluated included recharge dams to attenuate surface flows and provide managed groundwater recharge in key locations. The modelling framework involved the coupling of a distributed rainfall model, a distributed water balance model (incorporating rainfall-runoff processes, soil infiltration and wadi flow transmission losses), and a distributed groundwater model (Figure 9).

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WADI BED ALLUVIAL PLANE Water Balance Model Transmission Lose

Model

JEBEL PLANE Water Balance Model

Figure 9 Schematic of the distributed water resource model

The representation of rainfall spatial variability presents technical difficulties, since data are limited. Detailed analysis was undertaken of 19 rain gauges in the Sharqiyah region, and of six raingauges in the catchment itself. A stochastic multi-variate temporal- spatial model was devised for daily rainfall, a modified version of a scheme orgininally developed by Wheater et al., 1991a, b. The occurrence of catchment rainfall was determined according to a seasonally-variable first order markov process, conditioned on rainfall occurrence from the previous day. The number and locations of active raingauges and the gauge depths were derived by random sampling from observed distributions.

The distributed water balance model represents the catchment as a network of two-dimensional plane and linear channel elements. Runoff and infiltration from the planes was simulated using the SCS approach. Wadi flows incorporate a linear transmission loss algorithm based on work by Jordan (1977) and Walters (1990). Distributed calibration parameters are shown in Figure 10.

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Figure 10 Distributed calibration parameters, water balance model

Finally, a groundwater model was developed based on a detailed hydrogeological investigation which led to a multi-layer representation of uncemented gravels, weakly/strongly cemented gravels and strongly cemented/fissured gravel/bedrock, using MODFLOW.

The model was calibrated to the limited flow data available (a single event) (Table 4), and was able to reproduce the distribution of runoff and groundwater recharge within the catchment through a rational association on loss parameters with topography, geology and wadi characteristics. Extended synthetic data sequences were then run to investigate catchment water balances under scenarios of different runoff exceedance probabilities (20%, 50%, 80%), as in Table 4, and to investigate management options.

Table 4 Annual catchment water balance, simulated scenarios

Scenario Rainfall Evaporation Groundwaterrecharge Runoff %Runoff

Wet 88 0.372 12.8 4.0 4.6

Average 33 0.141 11.2 3.5 4.0

Dry 25 0.106 5.5 1.7 3.2

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This example, although based on limited data, is not untypical of the requirements to evaluate water resource management options in practice, and the methodology can be seen as providing a generic basis for the assessment of management options. An integrated modelling approach provides a powerful basis for assimilating available hard and soft surface and groundwater data in the assessment of recharge, as well as providing a management tool to explore effects of climate variability and management options.

1.5 ConclusionsThis Chapter has attempted to illustrate the hydrological characteristics of arid and

semi-arid areas, which are complex, and generally poorly understood. For many hydrological applications, including the estimation of groundwater recharge and surface/groundwater interactions, these characteristics present severe problems for conventional methods of analysis. Much high quality experimental research is needed to develop knowledge of spatial rainfall, runoff processes, infiltration and groundwater recharge, and to understand the role of vegetation and of climate variability on runoff and recharge processes. Recent data have provided new insights, and there is a need to build on these to develop appropriate methods for water resource evaluation and management, and in turn, to define data needs and research priorities.

Groundwater recharge is one of the most difficult fluxes to define, particularly in arid and semi-arid areas. However, reliable quantification is essential to maximise the potential of groundwater resources, define long-term sustainable yields and protect traditional sources. This requires an appropriate conceptual understanding of the important processes and their spatial variability, and the assimilation of all relevant data, including surface water and groundwater information. It is argued that distributed modelling of the integrated surface water/groundwater system is a valuable, if not an essential tool, and a generic framework to achieve this has been defined, and a simple application example presented. However, such modelling requires that the dominant processes are characterised, including rainfall, rainfall-runoff processes, infiltration and groundwater recharge, and the detailed hydrogeological response of what are often complex groundwater systems.

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Groundwater is a strategic resource due to its usuallyhigh quality and perennial availability. However,groundwater management all over the world often lackssustainability as evidenced by falling water tables, dry-ing wetlands, increasing sea-water intrusion and gen-eral deterioration of water quality, As groundwater can-not be renewed artificially on a large scale, sustainablemanagement of this resource is vital. A number of scien-tific tools are available to assist in his task. Three itemsare discussed here. They include methods for the deter-mination of groundwater recharge, groundwater model-ing including the estimation of its uncertainty, and theinterfacing to the socio-economic field. Generally thequality of water management work can be largelyenhanced with new tools available, including remotesensing, digital terrain models, differential GPS, envi-ronmental tracers, automatic data collection, modelingand the coupling of models from different disciplines

Introduction

Ground water contributes worldwide about 20% of people’s freshwater. Despite this relatively small proportion its role is importantfor two reasons: On the one hand, ground water is well suited for thesupply of drinking water due to its usually high quality. On the otherhand, ground water basins are important long-term storage reser-voirs, which in semi-arid and arid countries often constitute the onlyperennial water resource. The storage capacity is evident if one com-pares the volumes of surface and ground water resources. Globallythe volume of fresh water resources in rivers and lakes is about100,000 km3. With about 10,000,000 km3, the volume of groundwater is two orders of magnitude larger (e.g. Gleick, 1993, Postel etal., 1996). For sustainable water management, however, the renewalrate is more relevant, and for this quantity the situation is reversed.The renewal rate of surface water resources is 30,000 km3/a, that ofground water only about 3,000 km3/a. Worldwide, about 800 km3 ofground water are utilized by mankind annually. This number stilllooks considerably smaller than the yearly renewal rate. However,the global comparison does not do justice to the real situation. Aver-age figures hide the fact that of the yearly withdrawal rate about onequarter is supplied by non-renewable fossil ground water reserves(Sahagian et al., 1994).

Similarly, in a globally averaged analysis the average pollutantconcentrations of ground water become negligible. The natural bal-ance unit for water resources is the catchment of a river, a spring ora well. And on that scale things look very different on a case-by-casebasis. Ground water resources are degraded in the long term by over-pumping and pollution, and the local and regional utilization ofground water is impaired or even has to be abandoned. As ground

water cannot be renewed artificially on a large scale its non-sustain-able use is a serious danger. This will be illustrated in the followingwith various examples.

Sustainable water management

For an assessment of the ground water situation in a catchment, adefinition of sustainable resource utilization is necessary as a startingpoint. In the following we will define it as a set of management prac-tices, which avoids an irreversible or quasi-irreversible damage tothe resource water and the natural resources depending on it such assoil and ecosystems. Such management allows the resource water toextend its service, including ecological service, over very long peri-ods of time.

The abstraction from a ground water reservoir should in thelong term not be larger than the long-term average recharge. Thestorage property of course allows temporary overpumping. As thequantities abstracted may be used consumptively (e.g. by evapotran-spiration in agriculture) and reduce the downstream flows, sustain-able management with respect to quantity requires that abstraction islimited to a fraction of recharge in order to guarantee a minimumavailability of water in the downstream. These principles are violatedin many aquifers all over the world.

Overpumping of an aquifer

Steadily falling ground water levels are an indicator of over-pumping. A famous example is the Ogallala aquifer in the UnitedStates. Up to 1990, 162 km3 of water above the natural rechargewere abstracted from this aquifer. Finally, the pumping had to bereduced drastically as the pumping costs rose to a level which madeirrigation with ground water unfeasible (HPUWC, 1998). In North-ern China, Eastern India and North- and South Africa ground waterlevels drop at a speed of 1 to 3 m/yr. The abstractions in the Saharaand in Saudi Arabia empty aquifers, which since the last ice age havenot had any recharge worth mentioning. It is estimated that by 2010the water reserves of the deeper aquifers of Saudi Arabia will containless than one half of what there was in 1998.

The cause of overpumping is in all cases the large-scale irriga-tion with ground water. The quantities pumped for drinking waterare small in comparison. Globally, the water use in households isonly 8% of the total consumption, while irrigation accounts for 70%.In arid and semi-arid regions, the proportion of irrigation in totalwater use may be as large as 90%.

Long before an aquifer is dried up by overpumping other phe-nomena occur, which indicate a non-sustainable situation.

Consequences of ground water table decline

Large drawdowns of the ground water table or pressure headlead to increased pumping costs, which will limit pumping rates foreconomic reasons. But increased cost is not the only consequence ofdrawdowns. Pressure reduction may lead to land subsidence in softstrata, as happens in Bangkok and Mexico City for example. Long

by Wolfgang Kinzelbach, Peter Bauer, Tobias Siegfried, and Philip Brunner

Sustainable groundwater management —problems and scientific toolsInstitute for Hydromechanics and Water Resources Management, ETH Zürich, Switzerland

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before an issue of water supply develops, the stability of buildingsrequires the reduction or even halt of ground water abstraction.

If a ground water table is declining too quickly or to a too-deeplevel the roots of trees relying on ground water (Phreatophytes) maynot be able to follow, which leads to their dying off. This is criticalin dry areas such as the Sahel, where the loss of the capability of treesto withstand wind leads to an increase in aeolic soil erosion. In wet-lands and swamps, in which the visible water table is often a mani-festation of a ground water table above ground, a drawdown willlead to their drying up.

Sea water intrusion and upconing of salt water

A special consequence of ground water level decline is shownin coastal areas. Due to the density difference between fresh water onthe land side and salt water on the sea side a salt water wedge devel-ops, which progresses inland until the pressure equilibrium at thesalt-fresh water interface is reached. Every perturbation of this equi-librium by reducing the fresh water flow will lead to a further pro-gression of the salt water wedge inland, until it eventually reachesand destroys the pumping wells. Sea water intrusion is notoriousalong the coasts of India, Israel, China, Spain and Portugal, to men-tion just a few. In the Egyptian Nile Delta, the zone where groundwater quality is impaired by sea water, reaches up to 130 km inland.On islands in the sea the fresh water lens formed by recharge fromprecipitation is often used for drinking water supply. In this situationthe drawdown due to pumping can cause a rise of the salt-fresh waterinterface (upconing). If the interface reaches the well, the well has tobe abandoned.

Similarly, saline water from salt lakes (Chotts) in Tunisia andAlgeria, in the vicinity of which ground water is pumped, can beattracted to the wells. A ground water level decline inland can lead tothe rise of saline water from underlying aquifers. This tendency isseen in Brandenburg, Germany. It can also not be excluded in theUpper Rhine Valley.

Degradation of ground water quality

The discussion of seawater intrusion shows, that the definitionof sustainability must include both quantity and quality aspects. Theavailable resources are diminished by pollution. Sea water intrusionis certainly the most widely spread pollution of ground water world-wide. It is followed by pollution by nitrates and pesticides from agri-culture. In industrial areas, chlorinated hydrocarbons and mineral oilhydrocarbons are still the main problem.

Ground water pollution is in principle not irreversible. In thecase of sea water intrusion, a diminishing or abandoning of abstrac-tions will in the long term let the system return to the original state.In other pollution cases eradication of the source will lead to clean-ing up within the typical renewal time of the resource. But this timespan may be so long that the aquifer cannot be used for drinkingwater purposes for several generations, or alternatively money has tobe spent on the processing of this water to make it suitable for drink-ing.

The experience of industrialized nations shows that aquifer pol-lution is a lengthy affair even if active remediation measures aretaken. The remaining option of treating the water to bring it up todrinking water quality is not available for small and decentralizedground water users in the third world.

Soil salination

A further non-sustainable practice, which must be mentionedhere due to its connection with ground water, is soil salination. It iscaused by too high ground water tables. If in arid climates the groundwater table rises (e.g. due to irrigation) to a distance of 1 to 2 mbelow the ground surface evaporation of ground water through cap-illary rise starts. The dissolved salts precipitate and accumulate in thetopsoil, finally leading to infertility of the land. In regions with verysmall gradients and little permeable soil, this problem can usually

not be solved by artificial drainage. Of the irrigated 230 Mio. ha agri-cultural land worldwide around 80 Mio. ha are in one way or theother afflicted by soil salination.

Soil salination is not irreversible, but the time spans for rehabil-itation of the soil can be very long. Soil salination is a problem in allirrigated areas. Strongly affected countries are Iraq, Egypt, Australia,the USA, Pakistan and China, to mention only a few.

Scientific methods

The implementation of water resources management strategies is inthe end a political and economic question. Still, in the analysis ofsustainability and the search for solutions, modern methods of waterresearch can make a contribution. Three areas are discussed further:1. The quantification of ground water recharge2. The use of ground water models including the estimates of

uncertainty; and3. The incorporation of economical aspects of sustainability

Determination of ground water rechargerates

Ground water recharge is the most important parameter for sustain-ability in arid and semi-arid regions. Despite that fact, it is not yetpossible to measure this important hydrological quantity with suffi-cient accuracy. Recharge is not directly measurable, and indirectmethods introduce various uncertainties. To make things worse, rainand recharge events in dry climates are of an extremely erraticnature. Integral methods such as the Wundt method, based on lowflow analysis of rivers are very useful in humid climates, but do notwork in arid zones, where the low flow is usually zero. The methodof hydrological water balance is notoriously inaccurate: In the long-term, average and under neglect of surface runoff recharge is the dif-ference between precipitation and evapotranspiration. As thesequantities are both of the same magnitude and inaccurately known,their difference is even more inaccurate. Only if long time series ofbalance components are available is the balance method feasible, asin this case systematic errors accumulate, and can be identified assuch.

The methods based on Darcy's law, in which the flow Q throughan aquifer cross-section is computed from the cross-sectional area A,the hydraulic conductivity K and the hydraulic gradient I as

Q = A*K*I (1)

are very inaccurate, as it is extremely difficult to find a representa-tive effective value on the large scale for the usually abnormally dis-tributed hydraulic conductivity K.

Both methods can easily be a factor of 10 to 50 off. Better esti-mates can be obtained at least in sand-gravel aquifers using environ-mental tracers. The classical tracer for recharge estimation is Tritiumfrom the nuclear bomb experiments in the 1960s. The atmosphericpeak of those years reaches the springs and pumping wells with adelay. The interpretation of the concentration observed in the wellrequires a model for the residence time distribution in the aquifer.Often, simple black box model concepts are used. In the simplestcase a plug flow is assumed. If more information on the geometry ofthe aquifer is available, more realistic residence time distributionscan be derived by means of numerical ground water flow models,which are able to take into account a more realistic aquifer structure.It has to be remembered however that the interpretation leads to porevelocities and not specific fluxes. Both are connected by the effec-tive porosity, a quantity, which in sand gravel aquifers is reasonablywell known and much less variable than the hydraulic conductivity.

In principle, every non-reactive trace substance in the atmos-phere with a well-known concentration history and distribution can

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be used in a similar fashion as Tritium. This is of great interest, asthe Tritium peak has lost much of its usefulness by decay. A newgroup of tracers are the freons and SF6, the atmospheric concentra-tions of which have been growing exponentially since the middle ofthe last century. The freon concentrations have been stagnating ordeclining for the last few years due to the ban on their production.These molecules are however not a complete replacement for Tri-tium, as Tritium moves as part of the water molecule, while freons asgases diffuse much faster through the unsaturated zone and only inthe ground water are they transported with the velocity of the waterin which they are dissolved.

In Southern Botswana we used freon tracers for a crude test. Inthis desert-like region one can occasionally find fresh water in themiddle of a very saline environment. The question is now whetherthis water has been formed recently or whether it is a just a nest offossil fresh water, which would not survive a prolonged pumpingtest. Freons were found only in wells in which another method — thechloride method — which is described below, also indicated rela-tively large recharge. But while the chloride method cannot say any-thing about the time of recharge, the presence of freons confirmedthat these aquifers have had recent recharge, and pumping themmakes sense.

In combination with its decay product Helium-3, Tritium can beemployed as a stop watch. The ratio of the concentration of both iso-topes allows a direct age determination over a time of up to 40 years.In the beginning, there is no Helium-3 as long as it can escape as gasto the atmosphere. As soon as a water drop reaches the saturatedzone, Helium-3 can no longer escape, and is enriched relative to Tri-tium. This method is independent of source strength. Corrections arenecessary due to oversaturation of ground water with air and possi-ble inputs of Helium from mantle and crust (Schlosser et al., 1989).

Stable isotopes such as oxygen 18 and Deuterium can also yieldinformation on recharge and relevant processes. Their concentrationin water is determined by phase transitions. In the project inBotswana we could have, for example, very reliably determined theproportion of water in a well coming from the nearby river or fromthe native ground water as the river water has a clear evaporation sig-nature. Stable isotopes can be employed as mixing tracers as well astransient tracers. For very old water, carbon 14 can be used as an agetracer. The progress in mass spectroscopy will in future make manymore tracer groups available for hydrology e.g. isotope ratios of rareearths or metals.

A method often employed successfully in arid regions is thechloride method. This method uses the deposition of wind-blown seasalt aerosol. The resulting chloride concentration in soil waterincreases by evaporation and transpiration. If the surface runoff isnegligible, the concentration increase is proportional to the evapo-transpiration rate and the recharge rate can be derived from

R = (cP*P+D)/(cB) (2)

where cP is the chloride concentration in rain water, P the pre-cipitation and D the dry deposition of chloride. cB is the chlorideconcentration below the zero upward flux plane. It can be approxi-mated by the concentration in shallow ground water. The methodrequires the long term observation of wet and dry deposition of chlo-ride, and of course there must not be any other source of chloridesuch as evaporates in the subsoil.

Contrary to the inaccurate large-scale water balance methods,the chloride method yields more accurate but very local estimates ofrecharge rates. Because of this complementarity a combination of thetwo methods is promising. In northern Botswana the differencebetween precipitation and evapotranspiration was determined usingsatellite images over 10 years (Brunner et al., 2002). This differenceindicated a recharge potential. The precipitation can be obtainedfrom METEOSAT data, and the evapotranspiration can be estimatedfrom NOAA-AVHRR data using for example the SEBAL algorithm(Bastiaanssen et al., 1998). The result (on the basis of 97 utilizableNOAA images) its shown in Figure 1. While the numerical valuesthemselves are unreliable the pattern is stable, and shows up every

year in similar fashion. This can be proved by principal componentanalysis. The spatial information from satellite images thereforealready allows a more realistic spatial distribution of recharge thanwould be arrived at with the assumption of homogeneity over thewhole region. But the absolute value must be adjusted, and it can beobtained from an average of all chloride values of the region. Thatthis procedure makes sense can also be seen from a comparison ofrecharge rates obtained from remote sensing data and the rechargerates obtained from well clusters in the same pixels. These show acorrelation coefficient of about 70%. The correlation can be used tocalibrate the recharge map obtained from remote sensing. It isencouraging to see that the sub-region with the highest computedrecharge rates is also the region with the highest observed groundwater levels.

Generally, remote sensing, be it from an airplane or a satelliteplatform, opens up new avenues for water resources managementespecially in large countries with weak infrastructure. Local hydro-logical research can be interpolated and completed into more arealdata sets. Generally, remote sensing data are always of interest ifthey allow reduction of the degrees of freedom of a model by anobjective zonation of the whole region considered. We use satelliteimages not only in the determination of precipitation and evapotran-spiration; from the vegetation distribution and the distribution ofopen water surfaces after the rainy season hints of the spatial distrib-ution of recharge can be abstracted. Multi-spectral satellite imagescan show the distribution of salts at the soil surface. For large areas,the GRACE mission will allow the contribution of soil water bal-ances calculated from variations in the gravitational field of theearth. New developments in geomatics allow us to obtain accuratedigital terrain models on small scales from laser scanning orstereophotometric methods and on a large scale from radar satelliteimages. In both cases, checkpoints on the surface are required whichcan be obtained with differential GPS. Digital terrain models areimportant in all cases where the distance to ground water table or aflooded area at varying water surface elevation play a role. The com-bination of remote sensing, automatic sensors with data loggers,GPS, and environmental tracers in combination with the classicalmethods of hydrology and hydrogeology allows us to introduce intowater resource studies a new quality, especially in those cases inwhich in earlier times lack of infrastructure and poor accessibilitywere limiting.

Ground water modeling

Modeling is indispensable in the field of ground water, as in the geo-sciences in general, as the accessibility of the objects of study is verylimited. Because of the rather slow reactions of a ground water sys-tem to changes in external conditions, a tool for prediction and plan-ning is necessary. Ground water models allow us to bring all avail-able data together into a logical holistic picture on a quantitative

Episodes, Vol. 26, no. 4

Figure 1 10-yearly mean of the difference between precipitationand evapotranspiration in (mm/a) from remotely sensed data.

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basis. As the first step, the system with all its dominant processesmust be understood. In this calibration phase observation data areintegrated and parameters identified. Only a calibrated model can beutilized for predictions. In the prognostic application the strength ofmodels lies in the ease with which scenarios can be compared to eachother. Finally, the integrative effect a model can develop in a projectshould not be underestimated. It forces the participants to undertakedisciplined and goal-oriented cooperation.

An example of a model application done in cooperation withChinese colleagues is given in Figure 2 (Li, 1994). A ground watermodel is used to study the sustainability of ground water abstraction.On the island of Weizhou, off the South China coast, an increase ofground water pumping for the extension of tourism is planned.About a third of the available recharge is required for the additionalwater supply. In principle, shallow or even horizontal wells would beadvisable in order to avoid saltwater upconing. But due to the unfa-vorable conductivity of the upper strata, the freshwater lens on Weizhou is best pumped in a deeperaquifer layer. If the pumping is concentrated in onewell, within a few years, the salinity of the pumpedwater becomes unacceptable due to saltwater upcon-ing. If the pumping is distributed over two wells at a4 km distance, an acceptable final salinity isreached. It the pumping rate is distributed over fourwells sitting on a square with side length 1 km, thechloride concentration stays below 200 mg/l all ofthe time (Figure 3). The model in this case allows usto find a technical solution to the problem, as thetotal pumping rate stays well below the totalrecharge. If the pumping rate is higher than totalrecharge the best model in the world can not find apumping strategy that avoids salt water intrusion.

Despite the widespread use of models, care is advisable. Mod-els are notoriously uncertain, and their prognostic ability is oftenoverestimated. The necessary input data are usually known only par-tially or only at certain points, instead of in their spatial distribution.Lacking input parameters have to be identified during calibration,which, as a rule, does not yield a unique solution. Ground watermodels are usually overparameterized, and good model fits are fea-sible with different sets of parameters, which in a prognostic appli-cation of the model however would yield different results. A respon-sible use of models acknowledges their uncertainty and takes it intoaccount when interpreting results. For the estimation of modeluncertainty, several methods are available today, which go farbeyond the classical engineering philosophy of best case-worst caseanalysis. The spectrum of stochastic modeling methods containserror propagation, as do Monte Carlo simulation and many others.

Let us look for example at the often-used formula (1) whichestimates the flow through an aquifer cross-section with Darcy's law.None of the three factors is known exactly, and would therefore beconsidered a stochastic variable with uncertainty. The hydraulic con-ductivity for example is usually lognormally distributed and even ifwe know the value of this variable at several points in the aquifer thespatial average value estimated from those still allows for a consid-erable estimation variance. The measured hydraulic gradients arealso not exact numbers, not so much due to the measurement accu-racy of the dipper, but due to inaccuracies in the leveling of the welland temporal variations of heads. Finally there is some leeway in thecross-sectional area, which might be obtained from a geophysicalmethod. The product of three stochastic variables is again a stochas-tic variable into which the uncertainties of the factors propagate. In anumerical model this process takes place in every cell of the dis-cretized model domain, as every flux across a finite difference cellfor example is also calculated by (1). It is therefore advisable to showmodesty when heading towards prognostic use of models. How a

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Figure 2 Computed horizontal and vertical distributions of thefreshwater lens under Weizhou Island (In the vertical section theupconing caused by the central well is visible).

Figure 3 Development of chloride concentration in the pumpingwells for different configurations of wells with the same totalabstraction rate.

Figure 4 Catchment area of a well: deterministic and stochastic approaches, a)Result of deterministic modeling, b) Unconditional stochastic modeling, c) Stochasticmodeling with conditioning by 11 long term observed heads.

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more appropriate stochastic procedure could look like is discussedwith the example of the determination of catchment areas of wells(Vassolo et al., 1998, Stauffer, et al., 2002).

It is reckless to draw the catchment of a well by a precise line,as shown in Figure 4a. It will never be possible to determine a catch-ment with this precision. A stochastic approach calculates all catch-ments which result from parameter combinations within the uncer-tainty limits of the parameters (e.g. transmissivities of n zones andrecharge rates of m zones). In the simplest procedure, this is possibleby sampling the parameter combination from the individual distribu-tions. Then, for each set of parameters a catchment is computed,yielding a set of catchments which can be analyzed for their statisti-cal properties. By superimposing all catchments, every point on thesurface can be assigned a probability of belonging to the real catch-ment of the well. This distribution is usually rather broad (Figure4b). It does not yet take into account measured data for state vari-ables such as heads, spring flows or tracer concentrations. Throughthis information the distribution can be conditioned. Only the subsetof catchments is selected which is compatible with the observations.Incorporating 11 long-term observation data of piezometric headsthe distribution of Figure 4b is narrowed considerably (Figure 4c).This is the typical way we gain knowledge in the earth sciences. Inthe beginning, almost everything is conceivable. Measurementsallow us to filter the realistic cases which are compatible with thedata from the set of imaginable cases. This procedure corresponds tothe Bayes principle. From an a priori distribution, an a posteriori dis-tribution is obtained by conditioning with data. Data are valuable ifthey lead to a reduction of the width of the stochastic distribution. Tobe fair, it has to be mentioned that stochastic modeling needs data toyield reasonably narrow distributions. These data are however of aslightly different nature than usual. What is required besides theusual scarce observation data of state variables are distribution char-acteristics of parameters.

The usefulness of models need not suffer from the admissionthat they are uncertain. The resource engineer is not interested inevery detail of nature; he wants to propose a robust solution whichwill function also if the underground does not exactly look as antici-pated. For this type of decision the stochastic approach is adequate.It allows even the quantification of the risk to take a wrong decision.

Socio-economic aspects

Decisions on sustainability take place in the economic-politicalsphere. Models are therefore more useful the more they manage tointerface with socio-economic aspects. The aquifer exploitationproblem is a typical common pool problem. Consider the situation ofn users of an aquifer, which receives a total recharge QN.

As long as the pumping rate of the n users is smaller than QN,the difference between recharge and consumptive use will flow outof the basin to a receiving stream. It is clear that in the long term thetotal abstraction cannot surpass the recharge rate. Sooner or later, theabstraction has to adapt to the availability of water. It is howeverwise to get to that equilibrium at an early stage, i.e. at a relativelyhigh water table. First, the higher water table allows larger short-term overpumping to overcome a drought, for example. Secondly,the higher water table reduces the specific pumping cost for all users.If every user goes for a large drawdown and a deep well, all userswill in the end have to shoulder higher specific pumping costs.Ground water is a resource which rewards partnership and punishesegotism. Negri showed (Negri, 1989) how a collective decision inthe common pool problem leads to a better solution than the sum ofall independent, individual decisions. In the soil salination problem,the situation is similar, only with the opposite sign in the water tablechange. All irrigators infiltrate water and cause the ground watertable rise. At a depth to water table of less than 2 m, the alreadysaline ground water starts to evaporate, leading to rapid salinization.Collective self-restraint will again lead to a better solution. The soilsalination problem in the Murrumbidgee Irrigation District in Aus-tralia is of this type. The irrigation of rice leads to considerable per-colation, and results in ground water table rise. If the present prac-tices are continued, up to another 35% of the arable land could besalinized before an equilibrium between the input of irrigation waterfrom the river, the outflow from drains and the output by evaporationis reached. With model calculations it has been shown (Zoellmann,2001) that reducing the rice growing area by one half will basicallyhalt the salination process at present levels. Figure 5 shows a com-parison between the ground water table rise after 30 years for thepresent rice growing area and for a 50% reduced rice growing area.

In a project of our institute in Xinjiang, China, it was investi-gated how far the pumping of ground water for irrigation could keepthe water table at a suitable depth in order to prevent salination.Again, this is primarily an economic problem, as the pumped groundwater costs about 10 times as much as river water, which has beenused exclusively up to now and which has led to considerable groundwater table rise. The choice of the system boundary for optimizationis crucial. If downstream ecological benefits are taken into account,the higher cost of ground water may be bearable.

The soil salination problem illustrates another basic economicaspect. If one applies the classic economic principle of optimizingdiscounted net benefits, the discount rate plays an important role. Ata discount rate of 5%, costs and benefits in 40 years' time are alreadyso small that they have no impact on the optimization. The short-term gain has much more weight. For slow processes such as salina-tion, the time scale of which can be easily 40 years or more, the usualeconomic optimization is unable to take those future penalties into

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Figure 5 Modeled groundwater table rise over 30 years in Murrumbidgee Irrigation District, Australia, a) at present rice growing area, b)at 50% reduction of rice growing area.

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account. Alternatives to the classical optimization are requiredwhich give a greater value to the future, e.g., by constraints on thefinal value of the agricultural land and its salinity. Of course the statecan repair this deficiency by regulations, which enter the optimiza-tion problem as constraints.

Conclusions

The sustainable management of aquifers is a burning problem inmany countries. New scientific tools can assist in its solution, both inthe analysis of whether the present management is sustainable and inthe definition of strategies and measures to achieve sustainability.Models will always play a role in this task. A new model generationcan use new data sources such as environmental tracers, remote sens-ing data and geophysical data. It must however not only simulate anaverage deterministic situation but also analyze the uncertainty of itspredictions by a stochastic approach in order to remain credible.Finally, the natural and engineering sciences have to interface muchmore with economics and politics to be of real practical use.

References

Bastiaanssen, W.G.M., Menenti, M. Feddes, R. A., Holtslag, A. A. M.(1998). A remote sensing surface energy balance algorithm for land(SEBAL). 1. Formulation. Journal of Hydrology, 212-213:213-229.

Brunner, P., M. Eugster, P. Bauer, W. Kinzelbach, Using Remote Sensing toregionalize local precipitation recharge rates obtained from the ChlorideMethod, Journal of Hydrology, accepted, 2003.

Gleick P.H. (editor), 1993. Water in Crisis, A Guide to the World's FreshWater Resources, Oxford University Press, New York.

HPUWC, 1998. High Plains Underground Water Conservation District No.1,The Ogallala Aquifer, http://www.hub.of the.net/hpwd/ogallala.html.

Li, G.-M., 1994. Saltwater Intrusion in Island Aquifers, Dissertation (in Chi-nese), China University of Geosciences, Wuhan.

Negri, D.H., 1989. The common property aquifer as a differential game.Water Resources Research 25, 9-15.

Postel S. L., Daily G. C., Ehrlich P. R., 1996. Human Appropriation ofRenewable Fresh Water, SCIENCE, Vol. 271.

Sahagian D.L., Schwartz F. W., Jacobs D.K., 1994. Direct AnthropogenicContributions to Sea Level Rise in the Twentieth Century, NATURE,Vol. 367.

Schlosser, P. Stute M., Sonntag, C. und Münnich, K. O., 1989. Tritiogenic3He in shallow ground water. Earth Planet. Sci. Lett. 94, p. 245-256.

Stauffer, F., S. Attinger, S. Zimmermann, W. Kinzelbach, 2002. Uncertaintyestimation of well catchments in heterogeneous aquifers, WRR, 38(11),p. 20.1-20.8.

Vassolo, S., W. Kinzelbach, W. Schäfer, 1998. Determination of a well headprotection zone by stochastic inverse modelling. Journal of Hydrology206, p. 268-280.

Zoelmann, K., 2001. Nachhaltige Bewässerungslandwirtschaft in semi-ari-den Gebieten, Diss. ETH Zürich, Nr. 1399

December 2003

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Peter Bauer is developing modelconcepts for the sustainable watermanagement in the Okavango Delta,Botswana. He received his Diplomaat the Swiss Federal Institute ofTechnology thesis in 1999.

Wolfgang Kinzelbach was born inGermersheim, Germany, in 1949.Educated as a physicist he turned toenvironmental engineering anddevote his research to flow andtransport phenomena in the aquaticenvironment. He received his doc-toral degree from Karlsruhe Univer-sity in 1978. After professorships atKassel University and HeidelbergUniversity he is presently the headof the Institute of Hydromechanicsand Water Resources Managementat the Swiss Federal Institute ofTechnology (ETH) in Zurich. Hisresearch interests are in environ-mental hydraulics, nuclear wastedisposal and sustainable waterresources management.

Philip Brunner studied environ-mental engineering at the SwissFederal Institute of Technology(ETH). He currently works at theInstitute of Hydromechanics andWater Resources Management as aPhD student. His research focuseson the modelling of the water andsalt fluxes through an agriculturallyused basin in Xinjiang, China. Thefinal goal of the PhD- project is tounderstand and reproduce theprocess of salination and to developsustainable strategies of water- andsalt management.

Tobias Siegfried studied environ-mental sciences at Swiss FederalInstitute of Technology (ETH). Hereceived his master degree for thedevelopment of a groundwater flowand transport model of a small basinin Botswana. He deepened his stud-ies at the London School of Econom-ics in the field of International Rela-tions as a postgraduate. In his cur-rent PhD work at the Institute ofHydromechanics and WaterResources Management at ETH, heis concerned with optimal utilisationof a large-scale non-renewablegroundwater resource in NorthernAfrica.

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Coinceptual Models For Recharge Sequences In Arid And Semi-Arid Regions Using Isotopic And Geochemical Methods

W.M. Edmunds1

ABSTRACT

The application of hydrogeochemical techniques in understanding the water quality problems of semi-arid regions is described and follows the chemical pathway of water from rainfall through the hydrological cycle. In this way it becomes possible to focus on the question of modern recharge, how much is occurring and how to recognise it as well as the main processes controlling water composition. It is vital to be able to recognise the interface between modern water and palaeowaters as a basis for sustainable management of water resources in such regions. The relative advantages of chemical. isotopic and related methods are introduced and it is emphasized that a conjunctive multitracer approach is usually required. Chemical and especially isotopic methods help to distinguish groundwater of different generations. In fact, the development of water quality may be viewed as an evolution series over time: i) undisturbed evolution under natural conditions; ii) the development phase with some disturbance of natural conditions, especially stratification; iii) development with contamination, and iv) artificially managed systems. A number of case studies from Sudan, Nigeria, Senegal as well as North African counties and China are used to illustrate the main principles.

INTRODUCTION

Groundwater quality issues now assume major importance in semi-arid and arid zones, since scarce reserves of groundwater (both renewable and non-renewable) are under threat due to accelerated development as well as a range of direct human impacts. In earlier times, settlement and human migration in Africa and the Middle East region were strictly controlled by the locations and access to fresh water mainly close to the few major perennial rivers such as the Nile, or to springs and oases, representing key discharge points of large aquifers that had been replenished during wetter periods of the Holocene and the Pleistocene. This is a recurring theme in writings of all the early civilisations where water is valued and revered (Issar 1990). The memory of the Holocene sea level rise and pluvial periods is indicated in the story of the Flood (Genesis 7:10) and groundwater quality is referred to in the book of Exodus (15:20). More recent climate change may also be referred to in the Koran (Sura 34; 16).

In present times groundwater forms the primary drinking water source in arid and semi-arid regions, since river flows are unreliable and large freshwater lakes are 1 Oxford Centre for Water Research, School of Geography and the Environment, South Parks Road,,Oxon. OX1 3TB, UK. E-mail: [email protected].

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either ephemeral (e.g. Lake Chad) or no longer exist. The rules of use and exploitation have changed dramatically over the past few decades by the introduction of advanced drilling technology for groundwater (often available alongside oil exploitation), as well as the introduction of mechanised pumps. This has raised expectations in a generation or so of the availability of plentiful groundwater, yet in practice falling water levels testifies to probable over-development and inadequate scientific understanding of the resource, or failure of management to act on the scientific evidence available. In addition to the issues related to quantity, quality issues exacerbate the situation. This is because the natural groundwater regime established over long time scales has developed chemical (and age) stratification in response to recharge over a range of climatic regimes and geological controls. Borehole drilling cuts through the natural quality layering and abstraction may lead to deterioration in water quality with time as water is drawn either from lower transmissivity strata, or is drawn down from the near-surface where saline waters are commonplace.

Investigations of water quality are therefore needed alongside, and even in advance of widespread exploitation. The application of hydrogeochemical techniques are needed to fully understand the origins of groundwater, the timing of its recharge and whether or not modern recharge is occurring at all. The flow regime at the present day may differ from that at the time of recharge due to major changes in climate from the original wetter recharge periods; this may affect modelling of groundwater in semi-arid and arid regions and geochemical investigation may help to create and to validate suitable models.

Study is needed of the extent of natural layering and zonation as well as chemical changes taking place along flow lines due to water-rock interaction (predicting for example if harmful concentrations of certain elements may be present naturally). The superimposed impacts of human activity and the ability of the aquifer system to act as a buffer against surface pollution can then be determined against natural baselines.

Typical landscape elements of (semi-)arid regions are shown in Figure 1, using the example of North Africa, where a contrast is drawn between the infrequent and small amounts of modern recharge as compared with the huge reserves of palaeowaters recharged during wetter climates of the Pleistocene and Holocene. Broad wadi systems and palaeo-lakes testify to former less arid climates. Against this background the main water quality issues of arid and semi-arid regions can be considered both for North Africa and other similar regions. Geochemical techniques can be applied for defining and mitigating several of the key water issues in (semi-)arid areas. These issues include salinisation, recharge assessment, residence time estimation and the definition of natural (baseline) water quality, as a basis for studying pollution and human-induced changes more generally. In this context it is noted that the mainly continental sandstones indicated in the diagram are highly oxidised and that dissolved O2 can persist to considerable depths in many basins. Under oxidising conditions several elements (Mo, Cr, As) or species such as NO3

remain stable and may persist or build up along flow pathways.

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Figure 1. Landscape hydrogeology and hydrogeochemical processes in arid and semi-arid regions. The generalised cross-section illustrates the geological environment

found in North Africa, where sedimentary basins of different ages overstep each other unconformably and may be in hydraulic continuity with each other.

This paper describes the application of hydrogeochemical techniques in conceptualising groundwater quality in semi-arid regions, especially in Africa, and follows the chemical pathway of water from rainfall through the hydrological cycle. In this way it becomes possible to focus on the question of modern recharge, how much is occurring and how to recognise it. It is vital to be able to recognise the interface between modern water and palaeowaters as a basis for sustainable management of water resources in such regions. Chemical and especially isotopic methods can help to distinguish groundwater of different generations and to understand water quality evolution.

METHODS OF INVESTIGATION

Hydrogeochemical investigations consist of two distinct steps – sampling and analysis. Special care is needed with sampling because of the need for representativeness of sample, the question of mixtures of water (especially groundwaters which may be stratified), the need to filter or not to filter, as well as the stability of chemical species. For groundwater investigations a large range of tools is available, both chemical and isotopic, for investigating chemical processes and overall water quality in (semi-)arid regions. However, it is essential that unstable variables such as pH, temperature, Eh and DO be measured in the field. Some details on field approaches are given by Appelo & Postma (1993) and also by Clark & Fritz (1997).

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Table 1. Principal geochemical tools for studies of water quality in (semi-)arid zones. Examples and references to the literature are given in the text

Geochemical/

isotopic tool

Role in evaluating water quality and salinity

Cl Master variable: Inert tracer in nearly all geochemical processes, use in recharge estimation and to provide a record of recharge history.

Br/Cl Use to determine geochemical source of Cl.

36Cl

Half-life 3.01 105 years. Thermonuclear production - use as tracer of Cl cycling in shallow groundwater and recharge estimation. Potential value for dating over long time spans and also for study of long-term recharge processes. However, in situ production must be known.

37Cl/35Cl Fractionation in some parts of the hydrological cycle, mainly in saline/hypersaline environments, may allow finger printing.

Inorganic tracers:

Mg/Ca

Sr, I, etc.

Mo,Cr,As,U,NO3,

Fe2+

Diagnostic ratio for (modern) sea water.

Diagenetic reactions release incompatible trace elements and may provide diagnostic indicators of palaeomarine and other palaeowaters.

Metals indicative of oxidising groundwater. Nitrate stable under aerobic conditions.

Indicative of reducing environments.

Nutrients: (NO3,K,PO4 – also DOC)

Nutrient elements characteristic of irrigation return flows and pollution.

18O, 2HEssential indicators, with Cl, of evaporative enrichment and to quantify evaporation rates in shallow groundwater environments. Diagnostic indicators of marine and palaeomarine waters.

87Sr,11B Secondary indicators of groundwater salinity source, especially in carbonate environments.

34S Indicator for evolution of sea-water sulphate undergoing diagenesis. Characterisation of evaporite and other SO4 sources of saline waters.

CFC’s, SF6Recognition of modern recharge: past 50 years. Mainly unreactive. CFC-12, CFC-11 and CFC-113 used conjunctively.

3H Recognition of modern recharge (half-life 12.3 years).

14C, 13C Main tool for dating groundwater. Half-life 5730 years. An understanding of carbon geochemistry (including use of 13C) is essential to interpretation.

A summary of potential techniques for the hydrogeochemical study of groundwaters in arid and semi-arid areas is given in Table 1. Chloride can be regarded as a master variable. It is chemically inert and is therefore conserved in the groundwater system - in contrast to the water molecule, which is lost or fractionated during the physical processes of evapotranspiration. The combined use of chloride and the stable isotopes of water ( 18O, 2H) therefore provide a powerful technique

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for studying the evolution of groundwater salinity as well as recharge/discharge relationships (Fontes 1980; Clark & Fritz 1997; Coplen et al. 1999). A major challenge in arid zone investigations is to be able to distinguish saline water of different origins including saline build-up from rainfall sources, formation waters of different origins, as well as relict sea water. The Br/Cl ratio is an important tool for narrowing down different sources of salinity (Rittenhouse 1969; Edmunds 1996; Davis et al. 1998), discriminating specifically between evaporite, atmospheric and marine Cl sources. The relative concentrations of reactive tracers, notably the major inorganic ions, must be well understood, as they provide clues to the water-rock interactions which give rise to overall groundwater mineralisation. Trace elements also provide an opportunity to fingerprint water masses; several key elements such as Li and Sr are useful tools for residence time determinations. Some elements such as Cr, U, Mo and Fe are indicative of the oxidation status of groundwater. The isotopes of Cl may also be used: 36Cl to determine the infiltration extent of saline water of modern origin (Phillips 1999) and 37Cl to determine the origins of chlorine in saline formation waters. The measurement precision (better than ±0.09‰) makes the use of chlorine isotopes a potential new tool for studies of environmental salinity (Kaufmann et al. 1993). In addition, several other isotope ratios: 15N (Heaton 1986), 87Sr(Yechieli et al. 1992) and 11B (Bassett 1990; Vengosh & Spivak 1999) may be used to help determine the origins and evolution of salinity. Accumulations of 4He may also be closely related to crustal salinity distributions (Lehmann et al. 1995).

For most investigations it is likely that conjunctive measurement by a range of methods is desirable (e.g. chemical and isotopic; inert and reactive tracers). However, several of the above tools are only available to specialist laboratories and cannot be widely applied, although it is often found that the results of research using detailed and multiple tools can often be interpreted and applied so that simple measurements then become attractive. In many countries advanced techniques are not accessible and so it is important to stress that basic chemical approaches can be adopted quite successfully. Thus major ion analysis, if carried out with a high degree of accuracy and precision, can prove highly effective; the use of Cl mass balance and major element ratios (especially where normalised to Cl) are powerful investigative techniques.

TIME SCALES AND PALAEOHYDROLOGY

Isotopic and chemical techniques are diagnostic for time scales of water movement and recharge, as well as in the reconstruction of past climates when recharge occurred. In this way the application of hydrogeochemistry can help solve essentially physical problems and can assist in validating numerical models of groundwater movement.

A palaeohydrological record for Africa has been built up through a large number of palaeolimnological and other archives which demonstrate episodic wetter and drier interludes throughout the late Pleistocene and Holocene (Servant & Servant-Vildary 1980; Gasse, 2000). The late Pleistocene was generally cool and wet although at the time of the Last Glacial Maximum much of North Africa was arid, related to the much

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lower sea surface temperatures. The records show, however, that the Holocene was characterised by a series of abrupt and dramatic hydrological events, mainly related to monsoon activity, although these events were not always synchronous over the continent. Evidence (isotopic, chemical and dissolved gas signatures) contained in dated groundwater forms important and direct proof of the actual occurrence of the wet periods inferred from the stratigraphic record as well as the possible source, temperature and mode of recharge of the groundwater. Numerous studies carried out in North Africa contain pieces of a complex hydrological history.

There is widespread evidence for long-term continuous recharge of the sedimentary basins of North Africa based on sequential changes in radiocarbon activities (Gonfiantini et al. 1974; Edmunds & Wright 1979; Sonntag et al. 1980). These groundwaters are also distinguishable by their stable isotopic composition; most waters lie on or close to the global meteoric water line (GMWL) but with lighter compositions than at the present day. This signifies that evaporative enrichment was relatively unimportant and that recharge temperatures were lower than at present. This is also supported by noble gas data, which show recharge temperatures at this time typically some 5-7 oC lower than at present (Fontes et al. 1991; 1993; Edmunds et al. 1999). Following the arid period at the end of the Pleistocene, groundwater evidence records episodes of significant recharge coincident with the formation of lakes and large rivers, with significant recharge from around 11 kyr BP. Pluvial episodes with a duration of 1-2000 years are recorded up until around 5.5 kyr BP, followed by onset of the aridity of the present day from around 4.5 kyr BP.

In Libya, during exploration studies in the mainly unconfined aquifer of the Surt Basin, a distinct body of very fresh groundwater (<50 mg L-1) was found to a depth of 100 m which cross-cuts the general NW-SE trend of salinity increase in water dated to the late Pleistocene (Edmunds & Wright 1979). This feature (Figure 2) is around 10 km in width and may be traced in a roughly NE-SW direction for some 130 km. The depth to the water table is currently around 30-50 m. Because of the good coverage of water supply wells for hydrocarbon exploration a three-dimensional impression could be gained of the water quality. It is clear that this feature is a channel that must have been formed by recharge from a former ancient river system. No obvious traces of this river channel were found in the area, which had undergone significant erosion, although Neolithic artefacts and other remains testify that the region had been settled in the Holocene. Whereas the regional, more mineralised groundwater gave values of 0.7-5.4% modern carbon, indicating a late Pleistocene age, the fresh water gave values from 37.6-51.2% modern carbon (corresponding ages ranging from 5000-7800 years) and were also distinctive in their hydrogeochemistry. Evidence from shallow wells in the vicinity, however, proved that fresh water of probable Holocene age was also present more widely at the water table, indicating that direct recharge was simultaneously occurring at a regional scale. Similar channels, preserving the Holocene/ Pleistocene interface, have been described by Pearson & Swarzenski (1974) from Kenya. In Mali the records of Holocene floods from northward migration of the Niger River inland delta are also well recorded (Fontes et al. 1991).

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Figure 2 Chloride concentration contours in groundwater from the Surt Basin, Libya, define a freshwater feature cross-cutting the general water quality pattern. This feature

is younger water from a former wadi channel, superimposed on the main body of palaeowater.

In comparison with the distinctive light isotopic composition of late Pleistocene recharge which lies close to the GMWL, Holocene groundwater is characterised by heavier, more evaporated compositions (Edmunds & Wright 1979; Darling et al. 1987; Dodo & Zuppi 1997). Extrapolation of these light isotopic compositions along lines of evaporation on the delta diagrams to the GMWL allow identification of the composition of the parent rains. It is proposed (Fontes et al. 1993) that these systematically light isotopic compositions were caused by an intensification of the monsoon coincident with northward movement of the ITCZ, resulting in convective heavy rains with low condensation temperatures. From the distribution of such groundwaters over the Sahara it is implied that the ITCZ moved some 500 km to the north during the Holocene.

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RAINFALL CHEMISTRY

Information on the conservative chemical components of rainfall is required by hydrologists to perform chemical mass balance studies of river flow and recharge. Rainfall may also be considered as the 'titrant' in hydrogeochemical processes, since it represents the initial solvent in the study of water-rock interaction. A knowledge of rainfall chemistry can also contribute to our fundamental understanding of air mass circulation, both from the present day synoptic viewpoint, as well as for past climates. In fact, very little information is available on rainfall composition. This is particularly true for Africa, where only limited rainfall chemical data are available. However, a reasonable understanding of the isotopic evolution of rainfall in (semi-) arid areas is emerging, for example within the African monsoon using isotopes (Taupin et al. 1997; 2000). The stable isotope ( 18O and 2H) composition of precipitation can aid identification of the origin of the precipitated water vapour and its condensation history, and hence give an indication of the sources of air masses and the atmospheric circulation (Rozanski et al. 1993). Several chemical elements in rainfall (notably Cl) behave inertly on entering the soil and unsaturated zone and may be used as tracers (Herczeg & Edmunds 1999). The combined geochemical signal may then provide a useful initial tracer for hydrologists, as well as providing a signal of past climates from information stored in the saturated and unsaturated zones (Cook et al. 1992; Tyler et al. 1996; Tyler & Edmunds 2001). In fact, over much of the Sahara-Sahel region rainfall-derived solutes, following concentration by evapotranspiration, form a significant component of groundwater mineralisation (Andrews et al. 1994).

Rainfall chemistry can vary considerably in both time and space, especially in relation to distance from the ocean. This is well demonstrated in temperate latitudes for North America (Junge & Werby 1957); the basic relationship between decreasing salinity and inland distance has also been demonstrated for Australia (Hingston & Gailitis 1976). The chemical data for Africa are very limited and it is not yet possible to generalise on rainfall chemistry across the continent. The primary source of solutes is marine aerosols dissolved in precipitation, but compositions may be strongly modified by inputs from terrestrial dry deposition. Because precipitation originates in the ocean, its chemical composition near the coast is similar to that of the ocean. Aerosol solutes are dissolved in the atmospheric moisture through release of marine aerosols near the sea surface (Winchester & Duce 1967). This initial concentration is distinctive in retaining most of the chemical signature of sea water, for example the high Mg/Ca ratio as well as a distinctive ratio of Na/Cl. As rainfall moves inland towards the interior of continents, sulphate and other ions may increase relative to the Na and Cl ions.

The more detailed studies of African rainfall have used stable isotopes in relation to the passage of the monsoon from its origins in the Gulf of Guinea (Taupin et al. 2000) towards the Sahel, where the air masses track east to west in a zone related to the position and intensity of the Inter-tropical convergence zone (ITCZ). While the inter-annual variations and weighted mean compositions of rainfall are rather similar, consistent variations are found at the monthly scale which are related to temperature

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and relative humidity. This is found from detailed studies in southern Niger, where the isotopic compositions reflect a mixture of recycled vapour (Taupin et al. 1995).

Chloride may be regarded as an inert ion in rainfall, with distribution and circulation in the hydrological cycle taking place solely through physical processes. A study of rainfall chemistry in northern Nigeria has been made by Goni et al. (2001). Data collected on an event basis throughout the rainy seasons from 1992 to 1997 showed weighted mean Cl values ranging from 0.6 to 3.4 mg L-1. In 1992, Cl was measured at five nearby stations in Niger with amounts ranging from 0.3 to 1.4 mg L-1.The spatial variability at these locations is notable. However, it seems that apart from occasional localized convective events, the accumulation pattern for Cl is temporally and spatially uniform during the monsoon; chloride accumulation is generally proportional to rainfall amount over wide areas. This is an important conclusion for the use of chloride in mass balance studies.

Bromide, like chloride, also remains relatively inert in atmospheric processes, though there is some evidence of both physical and chemical fractionation in the atmosphere relative to Cl (Winchester & Duce 1967). The Br/Cl ratio may thus be used as a possible tracer for air mass circulation, especially to help define the origin of the Cl. The ratio Br/Cl is also a useful palaeoclimatic indicator (Edmunds et al. 1992), since rainfall ratio values may be preserved in groundwater in continental areas. Initial Br enrichment occurs near the sea surface with further modifications taking place over land. The relative amounts of chloride and bromide in atmospheric deposition result initially from physical processes that entrain atmospheric aerosols and control their size. A significant enrichment of Br over Cl, up to an order of magnitude, is found in the Sahelian rains (Goni et al. 2001), when compared to the marine ratio. Some Br/Cl enrichment results from preferential concentration of Br in smaller sized aerosol particles (Winchester & Duce 1967. However, the significant enrichment in the Nigerian ratio is mainly attributed to incorporation of aerosols from the biomass as the monsoon rains move northwards, either from biodegradation or from forest fires. What is clear is that atmospheric dust derived from halite is unimportant in this region, since this would give a much lower Br/Cl ratio.

THE UNSATURATED ZONE

Diffuse recharge through the unsaturated zone over time scales ranging from decades to millennia is an important process in controlling the chemical composition of groundwater in (semi-)arid regions. In this zone fluctuations in temperature, humidity and CO2 create a highly reactive environment. Below a certain depth (often termed the zero-flux plane) the chemical composition will stabilise, and in homogeneous porous sediments near steady-state movement (piston flow) takes place towards the water table. It is important that measurements of diffuse groundwater recharge only consider data below the zero-flux plane. Some vapour transport may still be detectable below these depths at low moisture fluxes, however, as shown by the presence of tritium at the water table in some studies (Beekman et al. 1997). The likelihood of preferential recharge via surface runoff (see below) is also important in many (semi-)arid regions.

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In indurated or heterogeneous sediments in (semi-)arid systems in some terrains, by-pass (macropore or preferential) flow may also be an important process. In older sedimentary formations joints and fractures are naturally present. In some otherwise sandy terrain where carbonate material is present, wetting and drying episodes may lead to mineralisation in and beneath the soil zone, as mineral saturation (especially calcite) is repeatedly exceeded. This is strictly a feature of the zone of fluctuation above the zero-flux plane, however, where calcretes and other near-surface deposits may give rise to hard-grounds with dual porosities. Below a certain depth the pathways of soil macropore movement commonly converge and a more or less homogeneous percolation is re-established. In some areas, such as the southern USA, by-pass flow via macropores is found to be significant (Wood & Sandford 1995; Wood 1999). In areas of Botswana it is found that preferential flow may account for at least 50% of fluxes through the unsaturated zone (Beekman et al. 1999; De Vries et al. 2000).

Four main processes influence soil water composition within the upper unsaturated zone:

i) The input rainfall chemistry is modified by evaporation or evapotranspiration. Several elements such as chloride (also to a large extent Br, F and NO3) remain conservative during passage through this zone and the atmospheric signal is retained. A build-up of salinity takes place, although saline accumulations may be displaced annually or inter-annually to the groundwater system.

ii) The isotopic signal ( 2H, 18O) is modified by evaporation, with loss of the lighter isotope and enrichment in heavy isotopes with a slope of between 3 to 5 relative to the meteoric line (with a slope of 8). Transpiration by itself, however, will not lead to fractionation (Fontes 1982).

iii) Before passage over land, rainfall may be weakly acidic and neutralisation by water-rock interaction will lead rapidly to an increase in solute concentrations.

iv) Biogeochemical reactions are important for the production of CO2, thus assisting mineral breakdown. Nitrogen transformations are also important (see below), leading frequently to a net increase in nitrate input to the groundwater.

These processes may be considered further in terms of the conservative solutes that may be used to determine recharge rates and recharge history. The reactive solutes provide evidence of the controls during reaction, tracers of water origin and pathways of movement, as well as an understanding of the potability of water supplies.

Tritium and 36Cl

Tritium has been widely used in the late 20th century to advance our knowledge of hydrological processes, especially in temperate regions (Zimmerman et al. 1967). It has also been used in a few key studies in (semi-)arid zones where it has been applied, in particular, to the study of natural movement of water through unsaturated zones. In several parts of the world including the Middle East (Edmunds & Walton 1980; Edmunds et al. 1988), North Africa (Aranyossy & Gaye 1992; Gaye & Edmunds

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1996) and Australia (Allison & Hughes, 1978), classical profiles from the unsaturated zone show well-defined 1960s tritium peaks some metres below surface, indicating homogeneous movement (piston flow) of water through profiles at relatively low moisture contents (2-4 wt%). These demonstrate that low, but continuous rates of recharge occur in many porous sediments. In some areas dominated by indurated surface layers, deep vegetation or very low rates of recharge, the tritium peak is less well defined (Phillips 1994), indicating some moisture recycling to greater depths (up to 10 m), although overall penetration of modern water can still be estimated. Some problems have been created with the application of tritium (and other tracers) to estimate recharge, through sampling above the zero-flux plane, where recycling by vegetation or temperature gradients may occur (Allison et al. 1994).

The usefulness of tritium as a tracer has now largely expired due to weakness of the signal following cessation of atmospheric thermonuclear testing and radioactive decay (half-life 12.3 years). It may still be possible to find the peak in unsaturated zones, but this is likely to be at depths of 10-30 m based on those areas where it has been successfully applied. Other radioisotope tracers, especially 36Cl (half-life 301,000 years), which also was produced during weapons testing, still offer ways of investigating unsaturated zone processes and recharge at a non-routine level. However, in studies where both 3H and 36Cl have been applied, there is sometimes a discrepancy between recharge indications from the two tracers due to the non-conservative behaviour of tritium (Cook et al. 1994; Phillips 1999). Nevertheless, the position and shape of the tritium peak in unsaturated zone moisture profiles provides convincing evidence of the extent to which 'piston displacement' occurs during recharge, as well as providing reliable estimates of the recharge rate.

An example of tritium profiles (with accompanying Cl profiles) sampled in 1977 from adjacent sites in Cyprus is shown in Figure 3, which shows peaks at between 9 and 13 m depth. These peaks correspond with rainfall tritium (uncorrected) for the period from the 1950s to 1970s. The profiles approximate piston displacement and the peaks align well. A vertical moisture movement of 0.9 m yr-1 is calculated and the recharge rates are 52 and 53 mm yr-1; these figures compare well with estimates using Cl (Edmunds et al. 1988). A further example of tritium as a timescale indicator is shown in Figure 4.

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Tritium (TU)0 100 200

0 100 200

Dep

th (m

)

0

5

10

15

20

25

300 100 200

Cl (mg l-1)0 100 200

Dep

th (m

)0

5

10

15

20

25

30

Tritium (TU)

C l (mg l-1)

Water Table

19631963

AK3AK 2

Cs = 119

Cs =120

Figure 3. Tritium and chloride profiles from Akrotiri Cyprus with corresponding rainfall tritium.

Moisture Content0 2 4 6 8 10

Dep

th (m

)

0

5

10

15

20

25

30

35

40

18O (per mil)

-5 0 5

Chloride (mg l-1)0 20 40 60 80 100

2H (per mil)

-60 -40 -20 0 20 40

Cl18O H

3H (Tritium Units)0 20 40 60 80 100

3H

NO3-N (mg l-1)0 10 20 30 40

NO3MC

1963SAHELDROUGHT

Figure 4. Profiles of tritium, stable isotopes, chloride and nitrate in the unsaturated zone from the same location – profile L18, Louga, Senegal. This profile records the

impact of the Sahel drought from 1969 to 1989.

Stable isotopes

Stable isotopes have been used in the study of recharge but in general only semi-quantitative recharge estimates can be obtained. In (semi-)arid regions, however, the

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fractionation of isotopes between water and vapour in the upper sections of unsaturated zone profiles has been used successfully to calculate rates of discharge (Barnes & Allison 1983; Fontes et al. 1986) and to indicate negligible recharge (Zouari et al. 1985).

At high rainfall, the recharging groundwater undergoes seasonal fractionation within the zone of fluctuation (Darling & Bath, 1988), but any seasonal signal is generally smoothed out with the next pulse of recharge and little variation remains below the top few metres. In (semi-) arid zones, however, where low recharge rates occur, the record of a sequence of drier years may be recorded as a pulse of 18O-enriched water, as recorded for example from Senegal (Gaye & Edmunds 1996). Isotopic depletion in deep unsaturated zones in North America has been used to infer extended wet and dry intervals during the late Pleistocene (Tyler et al 1996). Extreme isotopic enrichment in the unsaturated zone accompanies chloride accumulation over intervals when recharge rates are zero (Darling et al. 1987).

Chloride

Numerous studies using Cl as a conservative tracer in recharge calculations have been reported, and Cl mass-balance methods probably offer the most reliable approach to recharge estimation for semi-arid and arid regions (Allison et al. 1994). In addition, Cl analysis is inexpensive and is widely applicable, bringing it within the budgets of most water resource organisations, although the capacity for accurate measurements at low Cl concentrations is required. The methods of investigation are straightforward and involve the recovery of dry samples by augur, percussion drilling, or dug wells followed by the measurement of moisture content and the elution of Cl. A number of criteria must be satisfied or taken into account for successful application of the technique: That no surface runoff occurs, that Cl is solely derived from rainfall, that Cl is conservative with no additions from within the aquifer, and that steady-state conditions operate across the unsaturated interval where the method is applied (Edmunds et al. 1988; Herczeg & Edmunds 1999; Wood 1999). As with tritium, it is important that sampling is made over a depth interval which passes through the zone of fluctuation. For the example shown in Figure 3, from Cyprus, the mean (steady-state) concentrations of Cl in the pore waters (119 and 122 mg L-1 respectively) provide recharge estimates of 56 and 55 mm yr-1. These estimates are derived using mean Cl data over three years at the site for rainfall chemistry and the measured moisture contents. Macropore flow is often present in the soil zone, but in the Cyprus and northern African profiles, in the absence of large trees this is restricted to the top 2-4 m of the profile.

Nitrate

Groundwater beneath the arid and semi-arid areas of the Sahara is almost exclusively oxidising. The continental aquifer systems contain little or no organic matter and dissolved oxygen concentrations of several mg L-1 may persist in palaeowater dated in excess of 20,000 years old (Winograd & Robertson 1982,

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Heaton et al. 1983). Under these conditions nitrate also is stable and acts as an inert tracer recording environmental conditions. Nitrate concentrations in Africa are often significantly enriched and frequently exceed 10 mg L-1 NO3-N (Edmunds & Gaye 1997; Edmunds 1999). High nitrate concentrations may be traced in interstitial waters through the unsaturated zone to the water table and are clearly the result of biogeochemical enrichment. This enrichment is well above concentrations from the atmosphere, allowing for evapotranspiration, and the source is most likely to be naturally occurring N-fixing plants such as acacia, though in the modern era some cultivated species may also contribute. In Senegal (Edmunds & Gaye 1997), the NO3-N/Cl ratio can even exceed 1.0. A record of nitrate in the unsaturated zone can therefore be used as an indicator of vegetation changes, including land clearance and agriculture.

Reactive tracers and water-rock reactions in the unsaturated zone

The main process of groundwater mineralisation takes place by acid-base reactions in the top few metres of the earth's crust. Natural rainfall acidity is between pH 5 and 5.5. Atmospheric CO2 concentrations are low, but may increase significantly due to microbiological activity in the soil zone. In warmer tropical latitudes the solubility of CO2 is lower than in cooler mid-high latitude areas and weathering rates will be somewhat lower (Tardy, 1970; Berner & Berner, 1996). The concentration of soil CO2 (pCO2) is nevertheless of fundamental importance in determining the extent of a reaction; an open system with respect to the soil/atmosphere reservoir may be maintained by diffusion to a depth of several metres.

Investigation of the geochemistry of pore solutions in the unsaturated zone is usually hampered by low moisture contents. Elutriation using distilled water is possible for inert components but not for reactive solutes, since artefacts may be created in the process. Where moisture contents exceed about 5 wt% it may be possible to extract small volumes by immiscible liquid displacement (Kinniburgh & Miles 1983). Results for one profile from Nigeria (see Edmunds et al. 1999) illustrate the trends for some elements in relation to Cl (Figure 5). An increase in Na/Cl above that in rainfall indicates that some reaction with the rock is taking place (probably feldspar dissolution). Concentrations of Fe and Al show strong gradients, probably related to low pH in the top 10 m of the profile.

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mgl-1

0 4 8 12 16

mMg/Cl

0.0 0.2 0.4 0.6 0.8 1.0

mSr/Ca.103

0 5 10 150.0 0.2 0.4 0.6

mg l-10 5 10 15 20

mNa/Cl

0.0 0.5 1.0 1.5 2.0

NO3-N

mMg/Cl Sr/Ca.103

Fe (total)

Al

mgl-1

0 20 40 60 80 100

Dep

th (

m)

0

2

4

6

8

10

12

14

16

18

Cl

water table mNa/Cl

rain

mgl-1

rain

Figure 5. Geochemistry of interstitial waters in the unsaturated zone – profile MD1 from northern Nigeria with concentrations and ratios for a number of elements. The

molar N/Cl ( 10) is also plotted on the nitrate diagram.

Recharge history

In deep unsaturated zones where piston flow can be demonstrated and where recharge rates are very low it may be possible to reconstruct climate and recharge history. Such profiles have been used to examine the recharge history over the decadal to century scale (or longer) in several parts of the world, as well as to reconstruct the related climate change (Edmunds and Tyler, 2002). More recently global estimates of recharge have been made based mainly on Cl mass balance and these are examined in relation to climate variability (Scanlon et al. 2006). Records up to several hundreds of years have been recorded from China (Ma and Edmunds 2005) and for several millennia from USA (Tyler et al. 1996)

An example of an integrated study from Senegal using tritium, Cl and stable isotopes is given in Figure 4, based on Gaye & Edmunds (1996) and Aranyossy & Gaye (1992). Samples were obtained from Quaternary dune sands where the water table was at 35 m and where the long-term (100 year) average rainfall is 356 mm yr-1

(falling by 36% to 223 mm since 1969 during the Sahel drought). The tritium peak clearly defines the 1963 rainfall as well as demonstrating that piston displacement is occurring. A recharge rate of 26 mm yr-1 is indicated and little if any by-pass flow is taking place.

The profile has a mean Cl concentration (23.6 mg L-1) that corresponds to a recharge rate of 34.4 mm yr-1, calculated using a value for rainfall Cl of 2.8 mg L-1

and an average rainfall of 290 mm yr-1. This rate is in close agreement with the tritium-derived value, indicating homogeneous movement with little dispersion of both water and solute. Oscillations in Cl with depth are due to variable climatic conditions, and the prolonged Sahel drought is indicated by the higher Cl concentrations. These profiles have been used to create a chronology of recharge events (Edmunds et al. 1992). The stable isotope values are consistent with the trends in recharge rates indicated by Cl, the most enriched values corresponding to high Cl

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and periods of drought, but the stable isotopes are unable to quantify the rate of recharge, unlike Cl and 3H.

Spatial Variability

The use of the chloride mass balance in the unsaturated zone is now being widely adopted for diffuse recharge estimation in semi-arid regions (Allison et al., 1994; Gaye and Edmunds, 1996; Wood and Sandford, 1995). Conservative rainfall solutes, especially chloride, are concentrated by evapotranspiration and the consequent build-up in salinity is inversely proportion to the recharge rate. This method is therefore ideally suited to the measurement of low recharge rates; it is also attractive since, in sharp contrast to instrumental approaches, decadal or longer average rates of recharge, preserved in thick unsaturated zone profiles may be obtained. The results of recent applications of the chloride mass balance approach using profiles to derive recharge assessments in different parts of the northern hemisphere (Africa and Asia) are shown in Figure 6. Some of these are single data points but for some areas multiple sites are represented for the same area; only those profiles are shown which are 8m in length, where steady state moisture and chloride are recorded, indicating net downward flux of moisture. The sources of data are given in the caption. Several interesting features emerge:

i) There is a cut off point around 200 mm mean annual rainfall which signifies the lower limit for effective recharge.

ii) Above this threshold there is the possibility for recharge as in Cyprus where 100 mm recharge is measured from 400 mm rainfall. Nevertheless, the technique records small but finite recharge rates in some areas with rainfall as high as 550 mm.

iii) At any one site a large range of recharge may occur in a small area implying that spatial variability is likely to be considerable, due to soil, geology or vegetation controls, since seasonal rainfall variability is likely to be evened out over the decadal timescales of most profiles.

iv) Finite, if low, recharge rates still occur even in those areas with mean annual rainfall below 200mm (eg. Tunisia, Saudi Arabia). However these low rates also signify moderate salinity in the profiles; with the absence of recharge, salinity accumulation will be continuous as found in southern Jordan.

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Figure 6. Recharge estimates from different countries using the Cl mass balance

Most of the profiles are from sandy terrain (sandstones or unconsolidated dune sands). Few examples are shown of clay-rich sediments. An exception is the Damascus Basin profile which is predominantly clay-loam (Abou Zakhem and Hafez, 2001) and which shows very low recharge. Two profiles from China are from loess sediments and these show modest rates of recharge. The recharge phenomenon is examined further with reference to the data from Senegal where a wide range of recharge rates has been measured across the Sahel region with rainfall ranging from 290-720 mm/yr.

One way to determine the spatial variability of recharge using geochemical techniques is to use data from shallow dug wells. Water at the immediate water table in aquifers composed of relatively homogeneous porous sediments represents that which is derived directly from the overlying unsaturated zone. Recharge estimates may then be compared with those from unsaturated-zone profiles to provide robust long term estimates of renewable groundwater, taking into account the three dimensional moisture distributions and based on timescales of rainfall from preceding decades. A series of 120 shallow wells in a 1600 km2 area of NW Senegal (Figure 7) were used to calculate the areal recharge from the same area from which the point source recharge estimates were obtained (Edmunds and Gaye, 1994). Average chloride concentration for seven separate profiles at this site is 82 mg L-1, giving a spatially averaged recharge of 13 mm yr-1.

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Figure 7. Spatial distribution of Cl measured in dug wells in NW Senegal and regional recharge estimates. Unsaturated zone profiles were measured at the research sites

indicated and gave good correlation for diffuse recharge rates

The topography of this area has a maximum elevation of 45m and decreases gently towards the coast and to the north; the natural groundwater flow is to the north-west. The dug wells are generally outside the villages and human impacts on the water quality are regarded as minimal; the area is either uncultivated savannah or has been

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cleared and subjected to rainfed agriculture. The Cl concentrations in the shallow wells have been contoured, and range from <50 to >1000 mg/l Cl (Figure 3a). These are proportional to recharge amounts which vary from >20 to <1mm a-1 (Figure 3b) The spatial variability may be explained mainly by changes in sediment grain size between the central part of the dune field and its margins where the sediments thin over older, less permeable strata. Over the whole area the renewable resource may be calculated at between 13 000 and 1100 m3/ km2/yr.

Focused recharge

The paper has focused so far on diffuse recharge where regional resource renewal takes place through mainly homogeneous sandy terrain, typical of much of North Africa. In areas of hard rock or more clay-rich terrain or where there is a well developed drainage system, it is likely that recharge will be focused along wadi lines and some recharge will occur via storms and flash floods, even where the mean annual average rainfall is below 200mm.

One example of this has been described from Sudan (Darling et al. 1987; Edmunds et al. 1992). The wadi system (Wadi Hawad) flows as a result of a few heavy storms each year in a region with rainfall around 200mm/yr. The regional terrain (the Butana Plain) is mainly clay sediments and it can be shown that there is no net diffuse recharge, demonstrated by fact that Cl is accumulating in the unsaturated zone. The deep groundwater in the region is clearly a palaowater with recharge having occurred in the Holocene. Shallow wells at a distance of up to 1 km from the wadi line contain water with tritium, all of which indicated a large component of recent recharge (24-76 TU). Samples of deep wells from north of the area, away from the main Wadi Hawad, all gave values 9 T.U. Thus the wadi acts as a linear recharge source and sustains the population of small villages and one small town through a freshwater lens that extends laterally from the wadi.

HYDROCHEMISTRY OF GROUNDWATER SYSTEMS IN (SEMI-)ARID REGIONS

From the foregoing discussion it is evident from geochemical and isotopic evidence that significant recharge to aquifer systems does not occur at present in most arid and semi-arid regions. In the Senegal example illustrated above, recharge rates to sandy areas with long-term average rainfall of around 350 mm are between <1 and 20 mm. It is likely that these results could be extrapolated to other regions with similar landscape, geology and climate over much of northern Africa, though preferential recharge along drainage systems is also important, providing areas with small but sustainable supplies. These present-day low recharge conditions will have an impact on the overall chemistry due to lower water-rock ratios and larger residence times, leading to somewhat higher salinities and evolved hydrogeochemistry.

Africa is characterised by several large overstepping sedimentary basins (Figure 1) which contain water of often excellent quality. Much of this water is shown to be

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palaeowater recharged during the late Pleistocene or Holocene pluvial periods. Hydrogeochemical processes observed in these aquifers have principally been taking place along flow-lines under declining heads towards discharge points in salt lakes or at the coast. Present rates of groundwater movement in the large sedimentary basins are likely to be less than one metre per year; in the Great Artesian Basin of Australia rates as low as 0.25 mm yr-1 are recorded (Love et al. 2000). Piezometric decline has generally been occurring, since fully recharged conditions and transient conditions are likely to be still operating at the present day as adjustment continues towards modern recharge conditions. Groundwater in large basins is never ‘stagnant’ and water-rock interaction will continue to occur. The evolution of inorganic groundwater quality is considered here in the light of the isotopic evidence, which provides information on the age and different recharge episodes in the sedimentary sequences.

Input conditions – inert elements and isotopic tracers

The integrated use of inert chemical and isotopic techniques provides a powerful means of determining the origin(s) of groundwater in semi-arid regions. To a large extent these tracers (Cl, 18O, 2H) are the same as used in studies of the unsaturated zone and may be adopted to clearly fingerprint water from past recharge regimes. However, the longer time scales of groundwater recharge require the assistance of radiocarbon dating and, if possible, other radioisotopic tools such as 36Cl, 81Kr and 39Ar (Loosli et al. 1999). Noble gas isotopic ratios also provide evidence of recharge temperatures which are different from those of the present day (Stute & Schlosser 2000).

Several elements, especially Br and Br/Cl ratios, remain effectively inert in the flow system and may be used to follow various input sources and the evolution of salinity in aquifers (Edmunds 1996a; Davis et al. 1998). An example is given from the Continental Intercalaire in Algeria (Edmunds et al. 2003) along a section from the Saharan Atlas to the discharge area in the Tunisian Chotts (Figure 6). The overall evolution is indicated by Cl, which increases from around 200 to 800 mg L-1. The sources of salinity are shown by the Br/Cl ratio; the increase along the flow-lines for some 600 m is due to the dissolution of halite from one or more sources, whilst in the area of the salt lakes (chotts) some influence of marine formation waters is indicated (possibly from water flowing to the chotts from a second flow-line). Fluorine remains buffered at around 1 mg L-1 controlled by saturation with respect to fluorite and iodine follows fluorine behaviour, being released from the same source as F possibly from organic rich horizons. The Cl and Br concentrations and ratio are related to inputs and differ from F and I, which are principally controlled by water-rock interaction.

Reactions and evolution along flow lines

The degree of groundwater mineralisation in the major water bodies in semi-arid and arid zones, recharged during the Holocene and the late Pleistocene, is likely to be a reflection of atmospheric/soil chemical inputs, the relatively high recharge rates during the pluvial episodes, the aquifer sedimentary facies and mineralogy. Most of

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North Africa south of latitude 28oN comprises variable thicknesses of continental sediments overlying crystalline basement, whereas to the north, marine facies may occur containing residual formation waters and intra-formational evaporites which lead to high salinities. Similar relationships are found in the coastal aquifers of West Africa, as well as on the Gulf of Guinea. In the Surt Basin of Libya the change of facies to marine sediments is marked by a progressive change in chemistry (especially Sr increase) as groundwater moves north of latitude 28o30 (Edmunds 1980). For many of the large basins of the Sahara and Sahel, away from coastal areas, the groundwater chemistry for inert solutes reflects, to a significant extent, inputs from the atmosphere (Fontes et al 1993). Superimposed on these inputs, the effects of water-rock interactions in mainly silicate dominated rock assemblages are recorded.

Important changes in reactive tracers (major and trace elements) also occur, which can be followed by (time-dependent) water-rock reactions. In addition to the chemical signature derived from atmospheric inputs, the chemistry of groundwater is determined to a significant extent by reactions taking place in the first few metres of the unsaturated or saturated zone and reflecting the predominant rock type. Any incoming acidity will be neutralised by carbonate minerals or, if absent, by silicate minerals. In the early stages of flow the groundwater will approach saturation with carbonates (especially calcite and dolomite) and thereafter will react relatively slowly with the matrix in reactions where impurities (e.g. Fe, Mn and Sr) are removed from these minerals, and purer minerals are precipitated under conditions of dynamic equilibrium towards saturation limits with secondary minerals (e.g. fluorite and gypsum).

During this stage, other elements may accumulate with time in the groundwater and indicate if the flow process is homogeneous or not; discontinuities in the chemistry are likely to indicate discontinuities in the aquifer hydraulic connections. An example from the Continental Intercalaire aquifer in Algeria/Tunisia (Edmunds et al. 2003) shows how the major elements may vary along the flow line. The ratios of reactive versus an inert tracer are used to indicate the evolution. The very constant Na/Cl ratio (weight ratio of 0.65) throughout the flow path as salinity increases indicates the dissolution of halite with very little reaction; at depth a different source is indicated. The Mg/Ca ratio indicates that saturation with respect to calcite (0.60) is approached after a short distance but that this is disturbed at depth by water depleted in Mg, probably from the dissolution of gypsum. The K/Na ratio increases along the flow-line from 0.05 to 0.18, suggesting a time-dependent release of K from feldspars or other silicate sources.

Redox reactions

Under natural conditions, groundwater undergoes oxidation-reduction (redox) changes moving along flow-lines (Champ et al. 1979; Edmunds et al. 1984). The solubility of oxygen in groundwater at the point of recharge (around 10-12 mg L-1)reflects the ambient air temperature and pressure. The concentration of dissolved oxygen (DO) in newly recharged groundwater may remain high (8-10 mg L-1),indicating relatively little loss of DO during residence in the soil or unsaturated zone

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(Edmunds et al. 1984). Oxygen slowly reacts with organic matter and/or with Fe2+

released from dissolution of impure carbonates, ferromagnesian silicates or with sulphides. Oxygen may persist, however, for many thousands of years in unreactive sediments (Winograd & Robertson 1982). Complete reaction of oxygen is marked by a fall in the redox potential (Eh) by up to 300 mV. This provides a sensitive index of aquifer redox status. A decrease in the concentration of oxygen in pumped groundwater therefore may herald changes in the input conditions. An increase in dissolved iron (Fe2+) concentration as well the disappearance of NO3 may be useful as secondary indicators.

The Continental Intercalaire aquifer illustrates redox conditions for a typical continental aquifer system in an arid zone. The redox boundary corresponds with a position some 300 km along the flow path in the confined aquifer and in waters of late Pleistocene age (Edmunds et al. 2003). Neither Eh nor O2 measurements were possible in the study, but the position of the redox change is clearly marked by the disappearance of NO3, the increase in Fe2+, as well as in the sharp reduction in concentrations of several redox-sensitive metals (Cr, U for example), which are stabilised as oxy-anions under oxidising conditions.

Salinity generation

Salinity build up in groundwater in (semi-)arid regions has several origins, some of which are referred to in the preceding sections. Edmunds & Droubi (1998) review the topic and discuss the main issues and techniques for studies in such regions. Three main sources are important – atmospheric aerosols, sea water of various generations and evaporite sequences, all of which may be distinguished using a cocktail of chemical or isotopic techniques (Table 1).

Atmospheric inputs that slowly accumulate over geological time scales are of great importance as a source of salinity. The impact on groundwater composition will be proportional to the inputs (including proximity to marine or playa source areas), climate change sequences and the turnover times of the groundwater bodies. The accumulation of salinity from atmospheric sources is most clearly demonstrated from the Australian continent (Hingston & Galitis 1976), where the landscape and groundwaters reflect closely the deposition of aerosols over past millennia.

Formation waters from marine sediments are important as a salinity source in aquifers near to modern coastlines. Different generations of salinity may be recognised, either from formation waters laid down with young sediments or from marine incursions arising from eustatic or tectonic changes. Evaporites containing halite and/or gypsum are an important cause of quality deterioration in many aquifers in present day arid and semi-arid regions. Formation evaporites are usually associated with inland basins of marine or non-marine origin. The different origins and generations of salinity in formation waters may be characterised by a range of isotopic and chemical tracers, such as 87Sr and 11B (Table 1).

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CONCLUSIONS

This paper has focused on water quality from a number of different viewpoints: a) as a diagnostic tool for understanding the occurrence evolution of groundwater; b) as a means for determining the recharge amounts; and c) as a means for studying the controls on groundwater quality dependent on geology and time scales. Emphasis has been placed on natural processes rather than direct human impacts such as pollution. Nevertheless, in arid regions the deterioration of groundwater quality due to human activity, such as excessive pumping, may lead to loss of resources through salinisation and the approach adopted here will assist in a better understanding of what may be going on. Salinity increase is also a consequence of point source (single well/village) or regional (town and city) pollution, and any overall quality deterioration may be monitored using simple parameters such as specific electrical conductance (SEC) or chloride.

The approach adopted here has been to recognise that water contains numerous items of chemical information that may be interpreted to aid water management. Measurement in the field and in the laboratory of the basic chemistry can inexpensively provide additional information during exploration, development and monitoring of groundwater resources. Too often only physical parameters are seen as important. For example, it is important to understand not only that water levels are falling, but why they are falling. Some key recommendations that emerge from this chapter may be summarised as:

Data requirements

Useful information may be obtained through the measurement and careful interpretation of a few key parameters. In the field, the measurement of T, pH and SEC are thus recommended. Temperature is essential, since it provides a proxy for the depth of sample origin in cases where details of production are unknown. Field measurement of pH is essential if meaningful interpretation of the results, especially for groundwater modelling, is to be made, since loss of CO2 will take place between the well head and the laboratory, leading to a pH rise; laboratory pH is therefore meaningless. SEC is a key parameter in both surveys and monitoring, enabling the hydrogeologist to quickly detect spatial or temporal changes in the groundwater. In the field, two filtered (0.45 m) samples (one acidified to and the other not acidified) can then be taken for laboratory analysis (Edmunds 1996b).

In hydrochemical studies the measurement of chloride is particularly important, since Cl behaves as an inert tracer which allows it to be used as a reference element to follow reactions in the aquifer and to interpret physical processes such as groundwater recharge, mixing and the development of salinity. Chloride should always be measured, and it is recommended that rainfall Cl also be measured in collaboration with meteorologists.

Most central laboratories are able to cope with at least major ion analyses and these should be measured as the basic data set for water quality investigations, after first establishing that there is an ionic balance (Appelo & Postma 1993). Minor and

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trace elements described above are helpful in the diagnosis of processes taking place in the aquifer as well as in determining potability criteria. The more detailed measurements described in this chapter are desirable, but are not essential for routine investigations. Nevertheless, the implications of case studies where combined isotopic and chemical data have been used should be digested and extrapotated for local studies. The results of expensive and specialised case studies are often of considerable value, since at practical and local level these then allow simple tools such as nitrate or chloride to be subsequently used for monitoring.

Groundwater resources assessment

In (semi-)arid regions the main resources issue is the amount of modern recharge. In this paper stress has been placed on how to determine whether modern recharge is taking place and, if so, how much. In sandy terrain, or other areas with unconsolidated sediments, it is recommended that attention be paid to information contained in the unsaturated zone. The use of Cl to measure the direct recharge component is recommended. However, the distribution of Cl on a regional scale is also important, since salinity variations will be proportional to the amount of recharge. The use of isotopes and additional chemical parameters will then be needed to help confirm whether this recharge belongs to the modern cycle or is palaeowater.

Groundwater exploration and development

Important information can be obtained during the siting and completion of new wells or boreholes. Chemical information obtained at the end of pumping tests provides the 'initial' baseline chemistry against which subsequent changes may be monitored. It is recommended that analyses of all major ions, minor ions and also stable isotopes of oxygen and hydrogen if possible, be carried out during this initial commissioning phase.

In more extensive groundwater development it is recommended that exploration of the quality variation with depth be made. One or more boreholes should be regarded as exploration wells for this purpose. Depth information can be obtained during the drilling using packer testing, air-lift tests or similar, which can then help determine the optimum screen design or well depth, for example to minimise salinity problems. Alternatively, single wells can be drilled to different depths in a well field for purposes of studying the stratification in quality or in age (e.g. whether modern groundwater overlies palaeowater). It should be remembered that subsequent pumped samples are inevitably going to be mixtures in terms of age and quality, and this initial testing will enable assessment of changes to be made as the wells are used.

Groundwater quality and use

Most water sampling programmes are heavily oriented towards suitability for drinking use, having regard to health-related problems. In this chapter emphasis has been placed on understanding the overall controls on water quality evolution. Natural

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geochemical reactions taking place over hundreds or thousands of years will thus give rise to distinctive natural properties. It is important to recognise this natural baseline, without which it will be impossible to identify if pollution from human activity is taking place. It has also been shown how salinity distribution is related to natural geological and climatic factors. In addition, the oxidising conditions prevalent in many (semi-)arid regions may give rise to enhanced concentrations of metals such as Cr, As, Se and Mo. Prolonged residence times may also lead to high fluoride and manganese concentrations. Under reducing conditions, high Fe concentrations may occur.

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Allison, G.B., Gee, G.W. & Tyler, S.W. (1994): Vadose-zone techniques for estimating groundwater recharge in arid and semi-arid regions. Soil Sci. Soc. Am. J. 58: 6-14.

Andrews, J.N., Fontes, J.Ch., Edmunds, W.M., Guerre, A. & Travi, Y. (1994): The evolution of alkaline groundwaters in the Continental Intercalaire aquifer of the Irhazer Plain, Niger. Water Resour. Res. 30: 45-61.

Appelo, C.A.J. & Postma, D. (1993): Geochemistry, Groundwater and Pollution.Balkema, Rotterdam.

Aranyossy, J.F. & Gaye, C.B. (1992): La recherche du pic du tritium thermonucléaire en zone non-saturée profonde sous climat semi-aride pour la mesure de recharge des nappes: première application au Sahel. C.R. Acad. Sci. Paris,315 (ser.II): 637-643.

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Distance (km)

0 100 200 300 400 500 600 700 800

F (m

g l-1

)

0.0

0.5

1.0

1.5

2.0

I/Cl

(wt)

0.00000.00010.00020.00030.00040.0005

Br/C

l (w

t)

0.0000.0010.0020.0030.0040.0050.006

30Cl (

mg

l-1)

0200400600800

1000

Br (

mg

l-1)

012345678

seawater

halite

(seawater - 3.1x10-6

)

ATLAS MTS(ALGERIA)

CHOTTS(TUNISIA)

Cl

Br

Br/Cl

I/Cl

F

Figure 8 Down-gradient hydrogeochemical profile (west-to-east) in the Continental Intercalaire aquifer from the Atlas mountains of Algeria to the discharge area in the Chotts of southern Tunisia. The chloride profile indicates a progressive increase in

salinity: The other halogen elements Br, I, F help to explain the origins of salinity and the general chemical evolution down-gradient.

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Numerical Modeling of Groundwater Flow in Heihe River Basin

at Middle Reaches: A Review

Xu-Sheng Wang

School of Water Resources & Environment, China University of Geosciences, Beijing PR China

1. Introduction

Heihe River Basin is located in the arid west-north part of China with area of land surface about 13 104 km2. Heihe River, the primary stream in the basin with a length about 821 km, runs down from Qilian Mountain (upper reaches)to the sedimentary basin on Hexi Corridor(middle reaches), travels to the Ejina Basin in inner Mongolia (lower reaches) and then disappears when it reaches Juyan Lakes. The middle segment of Heihe River is controlled by two outlets from upper reaches to lower reaches: Yingluo Gorge and Zhengyi Gorge.

In the middle reaches of Heihe River Basin, oasis is strongly developed around Zhangye City and Jiuquan City by introducing stream flow into channels to irrigate farm land in this arid area near by Badain Jaran Desert. Since average annual precipitation is less than 200 mm locally on the contrast, annual potential evaporation is greater than 2000 mm the oasis is strictly dependent on the surface water resources came from Qilan Mountain. Together with Heihe River, there are more than twenty streams developed in Qilan Mountain which provide about 30 108 m3 of water resources annually to the north area. 2/3 or more of the water resources is expended at middle reaches. With increasing of population and strength of economic activities, demand of water resources in the middle reaches area increases rapidly and reduces the residual stream flow. It is a disadvantage to the oasis at lower reaches. In 1961, West Juyan Lake was dried up. Thirty years later, the same fate fell on East Juyan Lake. To prevent further deteriorating of ecological environment at Ejina oasis, Chinese government started a project from 2001 to control export of surface water at middle reaches of Heihe River. It leaded to a good result: water level in East Juyan Lake was rising and has covered 35 km2 until 2004. However, West Juyan Lake is still drying.

Although surface water is the initial water resources in the middle reaches area, groundwater also plays an important role in terrestrial water cycle and utilization of water resources. Firstly, more than half of water flow out from Qilian Mountain leaks down as recharge of groundwater through beds of rivers and channels. Secondly, groundwater flows out through springs or directly discharges into Heihe River at the lower north part of the middle reaches area, provides water resources to irrigation.

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The amount of discharge flow is about 10 108 m3 annually. In addition, groundwater is directly abstracted, by over 6000 wells, from aquifers to irrigated farm lands. The amount of annual groundwater pumping flow is about 3 108 m3 recently.

An aquifer system is composed of Quatemary sediments with sandy gravel or sand with calyey interbeds under the middle reaches area. Underlying Tertiary sediments of sand-stones and shale-stones is relatively weak permeable and is normally regarded as the impermeable bed of the groundwater basin. Thickness of the aquifer system varies from 20 m to 1800 m. Groundwater table can be deeper than 200 m under the slopy plain (pluvial fans) near Qilian Mountain and comes to be shallow or overflow in the flat fluvial plain on the north area.

To research the behavior of groundwater and its contribution to water resources at middle reaches of Heihe River Basin, numerical modeling has been developed in the last two decades.

East Juyan Lake

West Juyan Lake

Zhengyi Gorge

Yingluo Gorge

Zhangye

Jiuquan

Heihe River

Heihe River

Heihe River

Scalekm

Middle

Qilian

Mountain

ReachesArea

Figure 1 Plan view of Heihe River Basin and the area of middle reaches

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2. Browsing on the numerical models developed

Groundwater flow in the aquifer system in the middle reaches area of Heihe River Basin has been studied through numerical modeling since 1990. Zhou X.Z., et. al. (1990) treated the aquifer system as a single-layer unconfined aquifer and built a two-dimensional numerical model with finite-difference method on a polygonal grid. It is called Model-1990 in this presentation. Control equation of groundwater flow in the aquifer is written as follow

yHSW

yHbK

yxHbK

x yyx (1)

Where: H is the groundwater level; Kx and Ky are the hydraulic conductivity along X-axis and Y-axis respectively (in fact, the aquifer is treated as homogenous porous medium in which Kx=Ky); b is the thickness of saturated zone from groundwater table to the bottom of the aquifer; Sy is the specific yield; W is the recharge rate of groundwater. In this numerical model, transmissivity T=Kxb=Kyb is applied in fact and treated as constant parameter identified by statistic analysis of pumping testes. The transmissivity varies from 200 m2/d to 6000 m2/d in five parameter-zones and increases with increasing of the thickness of Quatemary deposit. The value of transmissivity and specific yield were adjusted to make the transient numerical modeling of groundwater flow satisfied to phreatic water table changes in 65 observation wells from 1987 to 1989. The result shown that total discharge of groundwater is 11.7~12.5 108 m3/a through springs and 4.7~5.1 108 m3/a through evaporation in 1987~1989.

Two numerical model of groundwater flow in the middle reaches area were developed by Zhang G.X., et. al. (2004) with finite-element method: a single-layer model (Model-2004-1) and a double-layers model (Model-2004-2). Model-2004-1 is similar to Model-1990. However, 60 parameter zones are arranged for distribution of transmissivity and specific yield. Purpose of the model is to evaluate the regional groundwater balance under changes of hydrological and agricultural conditions since 1990. In Model-2004-2, the aquifer system is treated as an unconfined aquifer (with shallow groundwater) overlies on a confined aquifer (with deep groundwater). Leaky condition is simulated to configure interaction of shallow groundwater and deep groundwater. This double-layers model can show a three-dimensional pattern of groundwater flow: in the recharge area near Qilian Mountain a downward flow is developed; in the discharge area far away Qilian Mountain a upward flow is developed. However, it is a quasi-three-dimensional model.

The numerical model developed by Hu L.T. and Chen C.X. (2006, 2007), which is called Hu-Chen Model in this presentation, is a three-dimensional model with 8-layers. Finite difference method with random polygon grid is applied in the model.

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The governing equation for groundwater flow is given in three-dimensional pattern as follows

tHSw

zHK

zyHK

yxHK

x svhh (2)

Where Kh and Kv are horizontal and vertical hydraulic conductivity respectively, w is the source-sink term, Ss is the specific storage of the aquifer medium. Kh is 0.01~43.75 m/d and Kv is 0.002~11.0 m/d as assigned in 20 parameter zones of 8 model-layers. Relationships between groundwater and surface water such as rivers and springs are treated. Also multi-layer wells are simulated with coupling seepage-pipe flow model. Both the trial-and-error method and the Fibonacci optimization method (Chen and Tang, 1994) are applied to calibrate the model parameters against observed groundwater level (1995-2000), flow rate of springs and baseflow of Heihe River along the lower-part in the middle reaches area. The results of Hu-Chen Model show that, in the main middle reaches area (without the West part of Jiuquan sub-basin) of Hiehe River Basin, 9.65 108 m3/a of surface water leaks from river-channel beds and irrigated farm lands as recharge of groundwater, however, 9.93 108 m3/a of groundwater flow out through springs and discharges as baseflow of lower-part Heihe River. Taking into account the exploitation of groundwater from wells, 2.25 m3/a, the total discharge rate of groundwater is significantly greater than the recharge rate. Consequently, a continued withdrawal of groundwater storage was arising before 2000 and leaded to decreasing of flow rate of springs and residual flow of Heihe River across Zhengyi gorge.

A double-layers model was also developed by Jia Y.W., et al, (2006) as a part of the hydrological model WEP-Heihe. It is called WEP-GW Model in this presentation. The structure of WEP-GW Model is similar to Model-2004-2. Exchange of groundwater and river water is treated. However, arrangement of aquifer-parameters and processing of groundwater-surface water interactions are not reported in detail.

3. Issues

While numerical models are developed for groundwater flow in Heihe River Basin at middle reaches, the modelers face the complexity of terrestrial water cycle in the studied area. Surface water run down from Qilian Moutain to the sedimentary basin, recharges to groundwater through river-channel leakage and irrigated farm land or losses to atmosphere by evaportranspiration. A thick vadose zone always exists over groundwater table in the recharge area. Almost all of natural river-beds are dried up early before arriving the lower reaches of Heihe River. At the other part of the same sedimentary basin, groundwater flows out the aquifer system from springs or directly discharge to the Heihe River as baseflow across Zhengyi gorge for the lower reaches area. A large number of wells are spread in agriculture districts abstracting

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groundwater. Also lumped pumping wells are developed at several locations to supply water for industrial purpose.

These characteristics of terrestrial water cycle make difficulties in numerical modeling of groundwater flow with conventional approaches. Modelers try to handle the main features of this groundwater system with different methods.

3.1 Vadose Zone

In traditional groundwater models, recharge of groundwater is input to the phreatic surface directly (as done in Model-1990, Model-2004-1 and Model-2004-2). However, it is well known that when infiltrated water reaches to groundwater table, it should travel through vadose zone. Recharge patterns would be quite different from the patterns of infiltration. In the middle reaches area of Heihe River Basin, groundwater table can be deeper than 100 m under slopy pluvial fans in the vicinity of Qilian Moutain. Then thick vadose zone exists. It do significant effect to the recharge of groundwater.

In considering of the effect of vadose zone to groundwater recharge in numerical modeling of groundwater flow, Chen (1998) present a method called Delayed Recharge Function (DRF) Method. It assumed that infiltration rate P at the time i(month) is distributed in recharge rate at the time i, i+1, i+2, and so on. The contribution in recharge at the time j (month, j >i), Rij, of infiltration at the time i(month), Pi, is given as

iij PijR )( (3)

Where, (j-i) is the distribution power. Chen (1998) gave an empirical function of (j-i) as follows

)/exp()!()/()( BDij

BDijij

(4)

Where D is the depth of groundwater table, B is a parameter dependent on soil properties in the vadose zone. This method provides a simple way to treat the effect of vadose zone to groundwater recharge with only an additional parameter.

In Hu-Chen Model, DRF Method is applied. The factor B is identified in several parameter-zones to make the numerical results match the observed response of groundwater level to recharge from surface water.

In WEP-GW Model, a transition layer is hypothetically arranged between top-soil layer and the underlying unconfined aquifer to consider the thick vadose zone. An experiential function is introduced to configure the flow and storage of this transition layer. However, it is not described in detail in publications

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3.2 Interactions between groundwater and surface water

3.2.1 Rivers

River is usually treated as boundary condition in groundwater system. The type of boundary is dependent on the relation between groundwater and river. In the middle reaches area of Heihe River Basin, Heihe River can not be regarded as a single type of boundary for groundwater flow.

When Heihe River and the other rivers run out of Qilian Moutain, they lie on a thick vadose zone of the aquifer system. Surface water within them can not touch groundwater directly but leaks as groundwater recharge. In 1967 and 1985, investigation of leakage of Heihe River was conducted. It was found that 0.32~1.52 % of stream flow leaks along per kilometer of the sharp river beds just leave Qilian Moutain, 4~10 % of stream flow leaks along per kilometer of the wide river beds before arriving the flat fluvial plain. In Model-1990, Model-2004-1 and Model-2004-2, this kind of leakage is calculated previously as well as leakage from channels, and prepared as linear sources for groundwater flow model. However, in Hu-Chen Model, a semi-permeable riverbed is considered. When groundwater level is less than the bottom of a riverbed, the hydraulic connection of groundwater and the river is called “infiltration type” and leakage from surface water is determined as follow

)( ssrz

r ZHAMKQ (5)

Where: Qr is the leakage flux to a node-zone where the river goes through in the discrete grid; K z and M are the hydraulic conductivity and thickness of the riverbed respectively; Ar is the area controlled by river node; Hs and Zs are the river stage and bottom elevation of the riverbed respectively. In addition, Qr is not treated as direct recharge of groundwater but DRF Method is applied to consider the effect of vadose zone. In WEP-GW Model, a similar treatment of river leakage is selected and delay of recharge is handled by a transition layer.

In the flat fluvial plain far away Qilian Moutain where groundwater level is shallow buried and river beds cuts down the aquifer in directly touching with groundwater, general head boundary can be arranged for Heihe River as done in Model-1990, Model-2004-1 and Model-2004-2. In Hu-Chen Model, exchange of water between river and the aquifer in this condition is determined as follow

hssrz

r QHHAMKQ )( 1 (6)

Where Qr is the exchange rate of water (it is positive when river recharges to the aquifer), Hs is the river stage and Hs+1 is the hydraulic head of aquifer-layer lower than the riverbed-within layer, Qh is the discharge of groundwater at the node duce to

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horizontal flow among the riverbed-within layer. However, the river stage, Hs, is given previously according to observed water level within wells in the vicinity of Heihe River. In WEP-GW Model, Darcy law is also applied to calculate the exchange of water between river and the aquifer through riverbed, a stream flow model is coupled with the groundwater flow model in the regional hydrological model.

3.2.2 Springs

In numerical modeling of groundwater flow, a spring could be regarded as flux boundary with constant discharge rate, or head boundary with constant head equal to its elevation, or drain boundary similar to a segment of river discharge groundwater. In the middle reaches area of Heihe River Basin, springs are widely developed in the fluvial plain where local low-lying land release the phreatic surface. They can not bee treated as constant flow rate boundary because the flow rate is varying with time associated to local or regional groundwater level.

In Model-1990, a two steps method is applied for a spring: at the first step, it is treated as constant head boundary and a flow rate is calculated through local water balance; at the second step, if the calculated flow rate is zero or less than zero, the spring-node switches to a normal node. However, this model is a single-layer unconfined aquifer model with a large thickness. How groundwater flows to a point-sink (a spring) at the top of the aquifer can not been configured successfully in this type of two-dimensional model.

In Model-2004-1 and Model-2004-2, no single spring is identified but the springs are hypothetically distributed in average in a overflow region. The overflow region is defined as the location where depth of groundwater table under normal surface ground is less than 3.5 m. Elevation of all the nodes in overflow region is artificially lowered (abstract 3.5 m) to produce spring-nodes (lowered elevation is less than groundwater level). A spring-node is treated as drain boundary in which discharge of groundwater is proportional to the difference of groundwater level and elevation of drain bottom.

In Hu-Chen Model, the elevation of a spring, Zs, is a constraint to spring flow occurrence. A semi-permeable layer is considered under the spring and the discharge rate, Qsp, is determined as follow

hssspz

sp QZHAMKQ )( 1 , (7) ss ZH 1

0spQ , ss ZH 1 (8)

Where K z , M , Asp, Qh and Hs+1 are factors similar to that in considering of river-groundwater interaction as shown in Eq.(6). Two springs (S3 and S6) with long-time measured data are typically studied to identify the parameters of spring-nodes. However, the other springs are lack of observation data and have been

81

treated as normal spring nodes in the mode.

In WEP-GW Model, it is not clear how springs are processed.

3.3 Wells

There are two types of wells drilled in the middle reaches area of Heihe River Basin: shallow agricultural wells that are spread in irrigation districts; deeper industrial wells to supply water for a factory or drinking water for a city.

It is impossible to simulate each agricultural wells since the number of agricultural wells is very large (more than 6000). Hence in numerical modeling, they are usually assumed uniform distributed in an irrigation district and the total pumping rate is apportioned to each node in the area. A trouble for multi-layer model (Hu-Chen Model) is to apportion the total pumping rate for every associated layer. Depth of agricultural wells was roughly investigated statistically. The data is used in determining the proportion of pumping rate for a model-layer with a certain range of depth.

Industrial wells are located in several pumping sites. Each pumping site has several wells pumping groundwater at the same time. However, distance between the wells is small and can not be distinguished in a regional numerical model. So a single-well is normally arranged at a node within the pumping site and is assumed to be equal to the group of wells. Another problem is, these wells are drilled deeply (100~200 m) across several aquifer layers that are called multi-layer wells. In multi-layer model, the total pumping rate of a multi-layer well is a result of groundwater contribution for each layer. It is typically considered in Hu-Chen Model with coupling seepage-pipe flow model that developed by Chen, et al (1999). An equivalent vertical hydraulic conductivity is calculated for a multi-layer well dependent on the flow rate and is introduce into the integrated vertical conductance of the aquifer-well system. Also this coupling seepage-pipe flow model is applied to obtain groundwater level within several deep observation wells that can be regarded as multi-layer pumping well with zero pumping rate.

4. Expectations of Further Development

Recently, it is a challenge for the middle reaches area of Heihe River Basin to find a sustainable way in utilizing water resources. This way should take care of water demand in the lower reaches area to maintain a well condition of ecological environment. To achieve the goal, it is necessary to understand the role of groundwater as well as the relationship of surface water and subsurface water.

The behavior of groundwater has been studied at the basin scale via numerical modeling that developed previously. A principle finding is that the vadose zone, the rivers and channels, the springs and the pumping wells do effect on and be affected by the regional groundwater flow in an integrated way. It indicates the further

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development of groundwater flow model in the future. Physical based variable saturation model may give a more reasonable understand of flow in thick vadose zone instead of empirical functions. Coupling of groundwater flow model with models of surface flow system such as rivers and channels is required while the whole hydrological processes should be quantitative evaluated. When pumping wells and springs provide irrigation water at the same time, an inner water cycle may produce and will increase the complexity of groundwater flow locally. Numerical model should be carefully designed to configure the feature. Currently, limited data is a difficulty in built up an basin-scale integrated hydrological model in which a three-dimensional groundwater model is coupled with models of surface water system and soil-vegetation-atmosphere system, especially with a reasonable parameterized scheme of human activities. Solving of the problem is dependent on the development of regional field investigation and data collection of hydrological and ecological conditions.

Acknowledgement

The author is grateful to the support of Cold and Arid Regions Environmental and Engineering Research Institute, Chinese Academy of Sciences, Lanzhou. The author also appreciates Prof. Chen C.X. and Dr. Hu L.T for suggestions in understanding their work on Heihe River Basin.

References

[1] Chen CX, Jiao JJ. 1999. Numerical simulation of pumping test in multilayer wells with non-Darcian flow in the wellbore. Ground Water, 37(3): 465–474.

[2] Chen CX, Tang ZH. 1994. Numerical Modeling of Groundwater Flow. Wuhan: China University of Geosciences Publishing House. [in Chinese]

[3] Chen CX. 1998. Delayed recharge function: A method to treat delay recharge of rainfall to unconfined aquifer. Hydrogeology and Engineering Geology, (6): 22-24. [in Chinese]

[4] Hu L.T, Chen C.X., Jiao J.J, et. al., 2007. Simulated groundwater interaction with rivers and springs in the Heihe river basin. Hydrological Processes, in Press, DOI: 10.1002/hyp.6497.

[5] Hu Li Tang, Chen Chong Xi, 2006. Dynamical Simulation for Multilayer Aquifer System at Middle Reach of Heihe River Basin. Journal of System Simulation, 18(7): 1966-1975. [in Chinese]

[6] Jia Yang Wen, Wang Hao, Yan Deng Hua, 2006. Distributed model of hydrological cycle system in Hihe River Basin I: Model development and verification. Shuili Xuebao, 37(5): 534-543. [in Chinese]

[7] Jia Yang Wen, Wang Hao, Yan Deng Hua, 2006. Distributed model of hydrological cycle system in Hihe River Basin II: Applications. Shuili Xuebao, 37(6): 655-662. [in Chinese]

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[8] Zhang Guang Hui, Liu Shao Yu, Xie Yue Bo, et. al., 2004, Water Cycle and Development of Groundwater in the Inland Heihe River Basin in West-North China. Beijing: Geology Publication House. [in Chinese]

[9] Zhou Xing Zhi, Zhao Jian Dong, Wang Zhi Guang, et. al., 1990, Survey of groundwater resources and analysis on its application in the middle reaches area of Heihe River in Gansu. Zhangye: The Second Hydrogeoloical and Engineering Geology Team, Gansu Bureau of Geology and Mineral Exploitation and Development, Technical Report-1990. [in Chinese]

84

Variable density groundwater flow:

From modelling to applications

Craig T. Simmons1, Peter Bauer-Gottwein, Thomas Graf, Wolfgang Kinzelbach, Henk

Kooi, Ling Li, Vincent Post, Henning Prommer, Rene Therrien, Clifford Voss, James

Ward & Adrian Werner

1School of Chemistry, Physics and Earth Sciences

Flinders University, GPO Box 2100, Adelaide, Australia

([email protected])

DRAFT Book Chapter for GWADI Workshop, Beijing, China, June 11-15, 2007.

85

1. Introduction

Arid and semi-arid climates are mainly characterised as those areas where precipitation is

less (and often considerably less) than potential evapotranspiration. These climate regions are

ideal environments for salt to accumulate in natural soil and groundwater settings since

evaporation and transpiration remove essentially freshwater from the system, leaving residual

salts behind. Similarly, the characteristically low precipitation rates reduce the potential for salt

to be diluted by rainfall. Thus arid and semi-arid regions make ideal “salt concentrator”

hydrologic environments. Indeed, salt flats, playas, sabkhas and saline lakes, for example, are

therefore ubiquitous features of arid and semi-arid regions throughout the world [Yechieli and

Wood, 2002]. In such settings, variable density flow phenomena are expected to be important,

especially where hypersaline brines overlie less dense groundwater at depth. In contrast, seawater

intrusion in coastal aquifers is a global phenomenon that is not constrained to only arid and

semi-arid regions of the globe and is inherently a variable density flow problem by its very

nature. These two examples make it clear that variable density flow problems both occur in, but

importantly extend beyond, arid and semi-arid regions of the globe. The intention of this book

chapter is therefore not to limit ourselves to modelling of arid zone hydrological systems, but

rather to present a more general treatment of variable density groundwater flow and solute

transport phenomena and modelling. The concepts presented in this chapter are therefore not

climatologically constrained to arid or semi-arid zones of the world, although they do apply

equally there. The objective of this book chapter is to illustrate the state of the field of variable

density groundwater flow as it applies to both current modelling and applications. Throughout,

we highlight approaches, current resolutions, and explicitly point to future challenges in this area.

86

2. Importance of variable density flow

It has long been known that the density of a fluid can be modified by changing its solute

concentration, temperature and, to a lesser extent, its pressure. Indeed, researchers in the area of

fluid mechanics have studied how fluid density affects flow behaviour for well over a century,

both in fluids only and in fluid-filled porous media. It was understood that areas of application

for these studies was wide and varied including applications to astrophysics, chemical reactor

engineering, energy storage and recovery, geophysics, geothermal reservoirs, material science,

metallurgy, nuclear waste disposal and oceanography, to name just a few. More recently, there

has been a massive explosion in the field because of worldwide concern about the future of our

energy resources and the pollution of our environment. These have been a principal catalyst for

the application of earlier developments made in traditional fluid mechanics to the study of

groundwater hydrology. In the preface to their popular text Convection in Porous Media, Nield and

Bejan [1999] state that “Papers on convection in porous media continue to be published at a rate of over 100

per year”. In groundwater hydrology in particular, Table 1 summarises some key areas in variable

density flow research. The listed references are intended to be illustrative rather than exhaustive.

This table clearly demonstrates the widespread importance, diversity, extent and interest in

applications of variable density flow phenomena in groundwater hydrology. As noted in

Simmons [2005] some key areas include seawater intrusion, fresh-saline water interfaces and

saltwater upconing in coastal aquifers, subterranean groundwater discharge, dense contaminant

plume migration, DNAPL studies, density driven transport in the vadose zone, flow through salt

formations in high level radioactive waste disposal sites, heat and fluid flow in geothermal

systems, palaeohydrogeology of sedimentary basins, sedimentary basin mass and heat transport

and diagenesis, processes beneath sabkhas and salt lakes and buoyant plume effects in applied

tracer tests. The modelling of these physical systems is central to hydrogeologic analyses.

Simmons [2005] presented an overview of variable density flow phenomena, including

current challenges, and future possibilities for this field of groundwater investigation. The

widespread importance and interest in variable density flow phenomena in groundwater

hydrology are also illustrated in exhaustive review articles on this topic by Diersch and Kolditz

[2002] and Simmons et al. [2001]. In addition, the text by Holzbecher [1998] is an excellent

source of material relating to the modelling of density dependent flows in porous media. The

reader is referred to these texts for an exhaustive review of this subject matter. In particular,

Diersch and Kolditz [2002] recently reviewed the state of the art in modelling of density-driven

flow and transport in porous media. They discuss conceptual models for density driven

convection systems, governing balance equations, critical phenomenological laws and

87

constitutive relations for fluid density and viscosity, and the numerical methods employed for

solving the resulting nonlinear problems. Furthermore, these authors discuss the major

limitations of current model-verification (benchmarking) test cases used in the testing of

numerical models and methods. Diersch and Kolditz [2002] and Simmons [2005] provide

detailed treatments of the current challenges that face the field of variable density groundwater

flow phenomena, from both physical and modelling viewpoints. The reader is referred to those

discussions as an accompaniment to this text.

3. A brief historical perspective

Nield and Bejan [2006] provide an excellent discussion of the topic of convection in

porous media including historical and new conceptual developments within this field. Some of

the earliest work in this field was carried out in the field of traditional fluid mechanics. A logical

trend emerged in the way this field evolved. Earliest work in the early 1900’s involved heated fluids

only (Benard, [1900]; Rayleigh, [1916]), followed by a move to combined heat and porous media

studies in the 1940’s (Horton and Rogers [1945]; Lapwood, [1948]), solute and porous media studies

in the 1950’s and 1960’s (Wooding [1969, 1963, 1962, and 1959]) and finally combined

thermohaline studies in porous media in the 1960’s [Nield, 1968]. Most of these studies were

designed to discover the most fundamental aspects of the behaviour of variable density flow

behaviour and were typically either laboratory or analytically based. In comparison, as groundwater

hydrologists, our interest in energy and solute transport is relatively recent (post 1950’s in general).

Therefore, it is challenging to find much evidence of interest and exploration in variable density

flow groundwater applications at all throughout the earlier historical development of groundwater

hydrology. But, there is one clear exception – the problem of seawater intrusion. Some of the

earliest pioneering work traces back to that of Badon-Ghyben [1888] and Herzberg [1901] whose

analyses could be used to determine the steady position of saltwater-freshwater interfaces in coastal

aquifers. Some of the earliest analytical solutions for the sharp saltwater-freshwater interface in an

infinitely thick confined aquifer emerged in the 1950’s [Glover, 1959; Henry, 1959]. In the problem

of seawater intrusion, Hele-Shaw cell analogs [Bear, 1972] were used in some of the earliest

exploratory studies of seawater intrusion and clearly predated numerical solutions to these

problems. Some of the earliest numerical analyses emerged in studies by Pinder and Cooper [1970],

Segol et al., [1976], Huyakorn and Taylor [1976] and Frind, [1982a,b]. Table 1 illustrates that the

interest in variable density groundwater flow applications has grown to include many other

important areas of hydrogeology. In more recent times, groundwater modelling codes have begun

to employ a range of new and improved numerical techniques and this coupled to faster computers

88

with larger memories has allowed for the simulation and solution of more complex problems, with

fully-coupled flow and solute transport in even 3D cases. Recent developments in modelling efforts

have included simultaneous heat and transport in thermohaline convection problems [Diersch and

Kolditz, 2002; Graf and Therrien, 2007b], adaptation of these to heterogeneous systems including

fractured rock hydrology [Graf and Therrien, 2005; Graf, 2005] and even chemically reactive

transport [Freedman and Ibaraki 2002; Post and Prommer, 2007]. Indeed, it may be argued that one

major limitation of the current field of variable density flow is not necessarily the availability of

numerical models with the inherent capacity to solve these sorts of very complex problems, but is

perhaps the availability of real data to occupy, test and verify the emerging new generation of

computer models.

4. Variable density flow physics

4.1 Overview

Field, laboratory and modelling studies demonstrate that fluid density gradients caused by

variations in concentration and temperature (and to a much lesser degree fluid pressure) are

recognised to play a significant role in solute and/or heat transport in groundwater systems.

Figures 1 and 2 show how fluid density varies as a function of concentration and temperature. It

is easy to accept that fluid density should be at least one factor influencing groundwater flow

processes because in many systems, groundwater concentrations can vary by several orders of

magnitude (e.g., salt lakes, seawater intrusion) and in all cases, the earth’s geothermal gradient has

the potential to set up circulations where warm fluid at depth can rise and shallower cooler fluids

can sink. Simmons [2005] considered a typical groundwater hydraulic gradient of say a 1m

hydraulic head drop over a lateral distance of a kilometre i.e., a gradient of 1 in a 1000 and

showed that the equivalent “driving force” in density terms would be a density difference of 1

kg/m3 relative to a reference density of freshwater whose density is 1000 kg/m3. This is a

solution whose concentration is only about 2g/L (about 5% of seawater!) and this is quite dilute

in comparison to many plumes one may encounter in groundwater studies.

4.2 Possible manifestations of variable density flow

Whether density driven flow is important in a physical hydrogeologic setting is

determined by a number of competing factors. These involve a complex interplay between fluid

properties, the interaction within mixed convective flow of fluid motion driven by both density

differences (free convection) and hydraulic gradients (advection or forced convection) and

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finally, the properties of the porous medium. The ability to maintain a density gradient and a

convective flow regime is dissipated by dispersive mixing processes. The density contrast and its

stable or unstable configuration (e.g., light fluid overlying dense fluid as in a stable case of

seawater intrusion, or dense fluid overlying light fluid as in the potentially unstable case of a

leachate plume migrating from a landfill) is also of critical importance. These will directly affect

solute transport processes and the way in which variable density flow processes are subsequently

manifested. For example, in a steady state coastal setting, density effects create a seawater wedge

that penetrates inland for some distance. Although tidal effects do create mixing, a thin

interfacial mixing-layer is maintained by a slow flow of less saline groundwater over the wedge.

This sort of rotational flow associated with seawater intrusion phenomena is the manifestation of

variable density flow physics and may be referred to as the ‘‘Henry circulation’’ [Henry, 1964]. In

some specialised cases, where there are large unstable density inversions, transport can be

characterised by rapid instability development where lobe-shaped instabilities (fingers) form and

sink under gravitational influence resulting in enhanced solute transport and greater mixing

compared to diffusion alone [Simmons et al., 2001]. The complexity of the problems involved

generally increases as one moves from situations where the light fluid overlies the dense fluid to

potentially unstable situations where the dense fluid overlies the less dense fluid. The challenges

for numerical simulation appear to grow in the same way, as will be discussed in later sections of

this chapter.

4.3 Dimensionless Numbers

It is important to note that, in all cases, the propensity for density driven flow is largest

where there are large density gradients and the porous media is of high permeability. In contrast,

dispersion acts to reduce the density gradient and therefore to dissipate density driven flow. Text

books such as Nield and Bejan [2006] and Holzbecher [1998] provide a comprehensive overview

of the theoretical treatment of density driven flow in porous media. Critical dimensionless

numbers such as the Rayleigh Number (Ra) and Mixed Convection ratio (M), are important

parameters used in the characterisation of such systems.

In simple free convective systems where mechanical dispersion is assumed independent of convective flow velocity, the onset of instability is determined by the value of a

nondimensional number called the Rayleigh number (Ra). This dimensionless number is the

ratio between buoyancy driven forces and resistive forces caused by diffusion and dispersion.

Ra is defined by (e.g., Simmons et al., [2001]):

90

DispersionandDiffusionnGravitatioandBuoyancy=

D)HC-C(gk=

DHU=Ra

00

minmax

o

c (1)

where Uc is the convective velocity, H is the thickness of the porous layer, D0 is the molecular

diffusivity, g is the acceleration due to gravity, k is the intrinsic permeability, C is

the linear expansion coefficient of fluid density with changing fluid concentration, Cmax and Cmin

are the maximum and minimum values of concentration respectively, is the aquifer porosity,

0= is the kinematic viscosity of the fluid. Simmons et al., [2001] and Simmons [2005]

discuss some of the difficulties encountered in the application of the Rayleigh number to

practical field based settings. For sufficiently high Rayleigh numbers greater than some critical

Rayleigh number Rac, gravity induced instability will occur in the form of waves in the boundary

layer that develop into fingers or plumes. This critical Rayleigh number defines the transition

between dispersive/diffusive solute transport (at lower than critical Rayleigh numbers) and

convective transport by density-driven fingers (at higher the critical Rayleigh numbers).

In many cases, the system we are studying is a mixed-convective system where both free

and forced convection operate to control solute distributions. In those cases, it is interesting to

examine the relative strengths of each process in controlling the resultant transport process. The

ratio of the density-driven convective flow speed to advective flow speed determines the

dominant transport mechanism. In the case of an isotropic porous medium the mixed

convection ratio may be written as (see for example, Bear [1972]):

speedconvectiveForcedspeedconvectiveFree

Lh

M o (2)

If M>>1, then free convection is dominant. If M<<1, then forced convection is dominant.

Where M~1 they are of comparable magnitude. Here, is the density difference, is the

lower reference density [both free convection parameters], h is the hydraulic head difference

measured over a length L [both forced convection parameters].

4.4 The use of Equivalent Freshwater Head in variable density analyses

Despite being one of the seemingly simplest methods of analysing variable density flow

systems, the correct application of the equivalent freshwater head is actually more complicated

than is often understood. Post et al., [in press] present a detailed analysis on the use of hydraulic

91

head measurements in variable density flow analyses. By way of introduction, freshwater head at

point i may be defined as:

(3)

where zi (elevation head) represents the (mean) level of the well screen, Pi is pressure of the

ground water at the well screen, f is fresh water density and g is acceleration due to gravity. A

common error that occurs in practice is for equivalent freshwater head to be used in the standard

non-density dependent version of Darcy’s Law that does not include a buoyancy gradient term

critical in vertical flow calculations. This leads to errors in predicting flow rates and directions

where vertical flow effects are of interest in vertically stratified layers (with respect to density) of

fluid. In a recent issue paper by Post et al., [in press], it is noted that the use of EFH without

care, can be particularly problematic. Indeed, Lusczynski [1961] who originally formulated the

concept recognized that equivalent freshwater heads cannot be used to determine the vertical

hydraulic gradient in an aquifer with water of non-uniform density. The term Environmental

Water Head (EWH) was established for this purpose [Lusczynski, 1961]. However, it can be

demonstrated that this Environmental Water Head is equivalent to using Equivalent Freshwater

Head where the appropriate density dependent version of Darcy’s Law is used (as was actually

shown in Appendix 3 of the original Lusczynski study). Post et al., [in press] argue that, provided

that the proper corrections are taken into account, fresh water heads can be used to analyze both

horizontal and vertical flow components. To avoid potential confusion, they recommended that

the use of the so-called environmental water head, which was initially introduced to facilitate the

analysis of vertical groundwater flow, be abandoned in favour of properly computed freshwater

head analyses.

Indeed, some variable density groundwater flow simulators such as FEFLOW and

SEAWAT (see Table 2) employ the equivalent freshwater head formulation in the variable

density version of Darcy’s Law in the familiar form (for the vertical flow component):

(4)

Where Kf is the fresh water hydraulic conductivity, hf is the freshwater head, is fluid density and

f is fresh water density. Here the term f

f , represents the relative density contrast, and

accounts for the buoyancy effect on the vertical flow. In the case of horizontal flow

f

fffz z

hKq

gPzhf

iiif ,

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components, this term is neglected in the respective horizontal flow velocity formulations in

Eqn. 3.

5. Modelling Variable Density Flow Phenomena

An excellent treatment on this subject was presented in a review article by Diersch and

Kolditz [2002] and the reader is referred to that material for an exhaustive discussion on this

topic. The limited number of analytical solutions for variable density flow problems creates a

heavy dependence on numerical simulators. A large and growing number of numerical simulators

now exist for the simulation of variable density flow phenomena, as illustrated in Table 2. Several

of these codes will be demonstrated in later sections of this book chapter where it will be seen

that there are numerous formulations that are used for governing equations and their numerical

solution. In this section, we demonstrate the approach to variable density flow modelling by

using the specific case of the widely used SUTRA [Saturated-Unsaturated TRAnsport Model]

numerical model [Voss, 1984] to demonstrate key issues in variable density flow simulation.

SUTRA is one of the earliest codes that emerged for the simulation of variable density flow

phenomena, and is in wide use today. A summary of the history of SUTRA and its use may be

found in Voss [1999].

5.1 Physical Processes, Governing Equations and Numerical Solutions

Using the example of the SUTRA (Saturated-Unsaturated-TRAnsport) code [Voss,

1984], we demonstrate the general form of the governing equations used by variable density

numerical models. The SUTRA code is a numerical solver of two general balance equations for

variable-density single-phase saturated-unsaturated flow and single-species (solute or energy)

transport based on Bear [1979]. Further details on this code may be found in Voss [1984]. The

summary presented below is originally found in the texts by Voss [1999] and Voss and Provost

[2003].

5.1.1. SUTRA Processes

Voss and Provost [2003] provide a detailed discussion on this subject. Simulation using

SUTRA is in two or three spatial dimensions. A pseudo-3D quality is provided for 2D, in that

the thickness of the 2D region in the third direction may vary from point to point. A 2D

simulation may be done either in the areal plane or in a cross sectional view. The 2D spatial

coordinate system may be either Cartesian (x,y) or radial-cylindrical (r,z). Areal simulation is

93

usually physically unrealistic for variable-density fluid and for unsaturated flow problems. The

3D spatial coordinate system is Cartesian (x,y,z).

Ground-water flow is simulated through numerical solution of a fluid mass-balance

equation. The ground-water system may be either saturated, or partly or completely unsaturated.

Fluid density may be constant, or vary as a function of solute concentration or fluid temperature.

SUTRA tracks the transport of either solute mass or energy in flowing ground water through a

unified equation, which represents the transport of either solute or energy. Solute transport is

simulated through numerical solution of a solute mass-balance equation where solute

concentration may affect fluid density. The single solute species may be transported

conservatively, or it may undergo equilibrium sorption (through linear, Freundlich, or Langmuir

isotherms). In addition, the solute may be produced or decay through first- or zero-order

processes. Energy transport is simulated through numerical solution of an energy-balance

equation. The solid grains of the aquifer matrix and fluid are locally assumed to have equal

temperature, and fluid density and viscosity may be affected by the temperature.

Most aquifer material, flow, and transport parameters may vary in value throughout the

simulated region. Sources and boundary conditions of fluid, solute and energy may be specified

to vary with time or may be constant. SUTRA dispersion processes include diffusion and two

types of fluid velocity-dependent dispersion. The standard dispersion model for isotropic media

assumes direction-independent values of longitudinal and transverse dispersivity. A flow-

direction-dependent dispersion process for anisotropic media is also provided. This process

assumes that longitudinal and transverse dispersivities vary depending on the orientation of the

flow direction relative to the principal axes of aquifer permeability.

5.1.2 SUTRA Governing Equations

The general fluid mass balance equation that is usually referred to as the 'ground-water flow

model' is:

pww

p0w QpktU

US

tp

pSSS gk r (5)

where:

wS is the fractional water saturation [1]

is the fractional porosity [1]

p is the fluid pressure [kg/m•s2]

94

t is time [s]

U is either solute mass fraction, C [kgsolute/kgfluid], or temperature, T [°C]

k is the permeability tensor [m2]

rk is the relative permeability for unsaturated flow [1]

is the fluid viscosity )T( [kg/ m•s]

g is the gravity vector [m/s2]

pQ is a fluid mass source [kg/m3•s].

Additionally,

is the fluid density [kg/m3], expressed as:

00 UUU

(6)

where:

0U is the reference solute concentration or temperature

0 is fluid density at U0

U is the density change with respect to U (assumed constant)

and,

p0S is the specific pressure storativity [kg/m•s2]-1

1S p0 (7)

where:

is the compressibility of the porous matrix [kg/m•s2]-1

is the compressibility of the fluid [kg/m•s2]-1.

Fluid velocity, v [m/s], is given by:

gkv r pSk

w (8)

95

The solute mass balance and the energy balance are combined in a unified solute/energy balance

equation usually referred to as the 'transport model':

s0s

w0ws

s1s

w1w

*wp

swwwwwssww

1SU1USUUcQ

U1Sc-UcStUc1cS IDIv

(9)

where:

wc is the specific heat capacity of the fluid [J/kg·°C]

sc is the specific heat capacity of the solid grains in porous matrix [J/kg·°C]

s is the density of the solid grains in porous matrix [kg/m3]

w is the diffusivity of energy or solute mass in the fluid (defined below)

s is the diffusivity of energy or solute mass in the solid grains

(as defined below)

I is the identity tensor [1]

D is the dispersion tensor [m2/s]

*U is the temperature or concentration of a fluid source (defined below)

w1 is the first-order solute production rate [s-1]

s1 is the first-order sorbate production rate [s-1]

w0 is an energy source [J/kg·s], or solute source [kgsolute/kgfluid·s],

within the fluid (zero-order production rate)

s0 is an energy source [J/kg·s], or solute source [kgsolute/kgfluid·s],

within the fluid (zero-order production rate)

In order to cause the equation to represent either energy or solute transport, the following

definitions are made:

for energy transport

TU ** TUw

ww c w

ss c

0s1

w1

96

for solute transport

CU ss CU ** CU mw D 0s 1sc

1cw

where:

w is the fluid thermal conductivity [J/s·m·°C]

s is the solid-matrix thermal conductivity [J/s·m·°C]

T is the temperature of the fluid and solid matrix [°C]

C is the concentration of solute (mass fraction) [kgsolute/kgfluid]

*T is the temperature of a fluid source [°C]

*C is the concentration of a fluid source [kgsolute/kgfluid]

sC is the specific concentration of sorbate on solid grains [kgsolute/kggrains]

mD is the coefficient of molecular diffusion in porous medium fluid [m2/s]

1 is a sorption coefficient defined in terms of the selected equilibrium sorption isotherm

(see Voss (1984) for details).

The dispersion tensor is defined in the classical manner:

2jT

2iL

2ii vdvdD v (10a)

jiTL2

ij vvddD v (10b)

for y,xi and y,xj but, ji

with:

vLLd (10c)

vTTd (10d)

where:

dL is the longitudinal dispersion coefficient [m2/s]

dT is the transverse dispersion coefficient [m2/s]

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L is the longitudinal dispersivity [m]

T is the transverse dispersivity [m]

A very useful generalization of the dispersion process is included in SUTRA allowing the

longitudinal and transverse dispersivities to vary in a time-dependent manner at any point

depending on the direction of flow. The generalization is:

kv2

maxLkv2

minL

minLmaxLL sincos

(11a)

kv2

maxTkv2

minT

minTmaxTT sincos

(11b)

where:

maxL is the longitudinal dispersivity

for flow in the maximum permeability direction [m]

minL is the longitudinal dispersivity

for flow in the minimum permeability direction [m]

maxT is the transverse dispersivity

for flow in the maximum permeability direction [m]

minT is the transverse dispersivity

for flow in the minimum permeability direction [m]

kv is the angle from the maximum permeability direction

to the local flow direction [degrees].

These relations (11a,b) provide dispersivity values that are equal to the square of the radius of an

ellipse that has its major axis aligned with the maximum permeability direction. The radius

direction is the direction of flow.

In the case of variable density saturated flow with non-reactive solute transport of total dissolved

solids or chloride, and with no internal production of solute, equations (5), (8) and (9), are greatly

simplified. The fluid mass balance is:

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pp0 QptU

UtpS gk (12)

The fluid velocity is given by:

gkv p (13)

and the solute mass balance is:

CCQUD-CtU *

pm DIv (14)

5.1.3 SUTRA Numerical Methods

Again, using the example code SUTRA, we demonstrate how one popular variable

density numerical model solves the above governing equations. A previous summary of the

discussion by Voss [1999] on this subject is repeated below.

The numerical technique employed by the SUTRA code to solve the above equations

presented in Section 5.1 uses a modified two-dimensional Galerkin finite-element method with

bilinear quadrilateral elements. Solution of the equations in the time domain is accomplished by

the implicit finite-difference method. Modifications to the standard Galerkin method that are

implemented in SUTRA are as follows. All non-flux terms of the equations (e.g. time derivatives

and sources) are assumed to be constant in the region surrounding each node (cell) in a manner

similar to integrated finite-differences. Parameters associated with the non-flux terms are thus

specified nodewise, while parameters associated with flux terms are specified elementwise. This

achieves some efficiency in numerical calculations while preserving the accuracy, flexibility, and

robustness of the Galerkin finite-element technique. Voss [1984] gives a complete description of

these numerical methods as used in the SUTRA code.

An important modification to the standard finite-element method that is required for

variable-density flow simulation is implemented in SUTRA. This modification provides a

velocity calculation within each finite element based on consistent spatial variability of pressure

gradient, p , and buoyancy term, g , in Darcy’s law, equation (13). Without this ‘consistent

velocity’ calculation, the standard method generates spurious vertical velocities everywhere there is

a vertical gradient of concentration within a finite-element mesh, even with a hydrostatic

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pressure distribution [Voss, 1984; Voss and Souza, 1987]. This has critical consequences for

simulation of variable density flow phenomena. For example, the spurious velocities make it

impossible to simulate a narrow transition zone between fresh water and seawater with the

standard method, irrespective of how small a dispersivity is specified for the system.

The two governing equations, fluid mass balance and solute mass (or energy) balance, are

solved sequentially on each iteration or time step. Iteration is carried out by the Picard method

with linear half-time-step projection of non-linear coefficients on the first iteration of each time

step. Iteration to the solution for each time step is optional. Velocities required for solution of

the transport equation are the result of the flow equation solution (i.e. pressures) from the

previous iteration (or time step for non-iterative solution).

SUTRA is coded in a modular style, making it convenient for sophisticated users to

modify the code (e.g. replace the existing direct banded-matrix solver) or to add new processes.

Addition of new terms (i.e. new processes) to the governing equations is a straightforward

process usually requiring changes to the code in very few lines. Also to provide maximum

flexibility in applying the code, all boundary conditions and sources may be time-dependent in

any manner specified by the user. To create time-dependent conditions, the user must modify

subroutine BCTIME. In addition, any desired unsaturated functions may be specified by user-

modification of subroutine UNSAT.

Two numerical features of the code are not often recommended for practical use:

upstream weighting and pinch nodes. Upstream weighting in transport simulation decreases the

spatial instability of the solution, but only by indirectly adding additional dispersion to the

system. For full upstream weighting, the additional longitudinal dispersivity is equivalent to one-

half the element length along the flow direction in each element. Pinch nodes are sometimes

useful in coarsening the mesh outside of regions where transport is of interest by allowing two

elements to adjoin the side of a single element. However, pinch nodes invariably increase the

matrix bandwidth possibly increasing computational time despite fewer nodes in the mesh.

5.2 Testing variable-density aspects of variable density numerical codes

Diersch and Kolditz [2002] provide a very comprehensive summary of model test cases

and benchmarks, including details on the successes and limitations of each case. As summarised

from that article, the major test cases that are used for numerical simulators currently include:

1. The Hydrostatic Test: This type of benchmark is quite simple, but is very instructive. It is

a test of the velocity consistency under hydrostatic and sharp density transition

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conditions as originally proposed by Voss and Souza [1987]. A closed rectangular region

containing fresh water above highly saline water, separated by a horizontal transition

zone one element wide and with a hydrostatic pressure distribution [Voss and Souza,

1987]. No simulated flow may occur in the cross section under transient or steady state

conditions.

2. The Henry seawater intrusion problem: This test case has been performed on coarse

meshes [Voss and Souza, 1987] and on a very fine mesh [Segol, 1993]. Note that until

Segol [1993] improved Henry’s approximate semi-analytical solution [Henry, 1964], this

test was merely an inter-comparison of numerical code results. No published results had

matched the Henry solution. Segol’s result provides a true analytic check on a variable-

density code. Some confusion has existed in the literature over the correct choice of

diffusion to use in the model test case and apparently two diffusivity values have been

employed that differ by a factor of porosity. In addition to this problem, the Henry

problem has some other deficiencies. An unrealistically large amount of diffusion is often

introduced which results in a wide transition zone. This assists with the numerical

solution, making the solution smooth and less problematic. However, as pointed out by

Diersch and Kolditz [2002], the Henry problem is not appropriate for testing purely

density-driven flow systems. This is because a large component of the flow system

response is actually driven by forced convection and not free convection. As a result,

additional benchmark tests are necessary to test numerical models for free convection

problems (such as the Elder problem) and for cases with very narrow transition zones

(such as the salt dome problem).

3. The Elder natural convection problem: [Voss and Souza, 1987], an unstable transient

problem consisting of dense fluid circulating downwards under buoyancy forces from a

region of high specified concentration along the top boundary. The Elder problem

serves as an example of free convection phenomena, where the bulk fluid flow is driven

purely by fluid density differences. Elder [1967a, 1967b] presented both experimental

results in a Hele-Shaw cell and numerical simulations concerning the thermal convection

produced by heating a part of the base of a porous layer (what we often call now the

“Short Heater Problem”). The Elder problem is a very popular benchmark case and its

convective flow circulation patterns (see Figure 3) are still the subject of scientific

discussion. No exact or qualified measurements exist for the Elder problem and as such

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the test case relies on an intercode comparison between numerical simulators. The main

issue that has been discussed widely in relation to this problem is the nature of both the

number of fingers and either the central upwelling or downwelling patterns, which are

now recognised to be highly dependent on the nature of the grid discretisation (i.e.,

coarse, fine, extremely fine grids all lead to very different spatiotemporal patterns of

behaviour). It is likely that convergence has been achieved at extremely fine grid

resolutions (15,000-1,000,000 nodes) as reported in Diersch and Kolditz [2002].

4. The HYDROCOIN salt-dome problem: [e.g. see Konikow et al., 1997], a steady state

and transient problem, initially consisting of a simple forced fresh-water flow field

sweeping across a salt dome (specified concentration located along the central third of

the bottom boundary). This generates a separated region of brine circulating along the

bottom. The nature of the circulation patterns has been shown to be sensitive to the way

in which the bottom boundary condition is represented and implemented, and the

importance choice of mechanical dispersion. These are seen to affect the circulation

patterns that arise in the system.

5. The Salt Lake problem: First introduced by Simmons et al., [1999], this is a very

complicated, transient system in which an evaporation-driven density driven fingering

process evolves downwards in an area of upward-discharging ground water in a Hele-

Shaw cell experiment (see Figure 4). Numerical results are compared with laboratory

measurements (see Figure 5). There has been much discussion on this test case in recent

literature. In particular, the number of fingers and their temporal evolution appear to be

intimately related to the choice of numerical discretisation and the numerical solution

procedure. Interestingly, and paradoxically, several authors have now found that better

agreement between numerical results and experimental results is obtained using a coarser,

rather than finer, mesh [Mazzia et al., 2001; Diersch and Kolditz, 2002]. Numerical grid

convergence has not been achieved for this problem. The higher Rayleigh number of this

test case problem (about 10 times larger than for the Elder problem) is clearly in a range

where oscillatory convection flow regimes are expected. In comparison to the Elder

problem, the salt lake problem is much more complicated. Grid convergence for the

Elder problem required extremely fine numerical resolution, and we therefore expect this

to be increased further again for the case of the salt lake problem. The major challenge

here is that this problem is extremely dynamic and that it is unstable. Moreover,

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numerical perturbations and dispersion can both trigger fingering and control associated

dispersion (creating either fatter or skinner fingers!) and these are all intimately related to

both the patterns of finger formation as well as their subsequent growth/decay

processes. It is clear that the numerical simulation of such phenomena is extremely

challenging and appears to be currently unresolved.

6. The Salt Pool Problem: The saltpool problem was introduced by Johannsen et al., [2002].

It represents a three dimensional saltwater upconing process in a cubic box under the

influence of both density driven flow and hydrodynamic dispersion. A stable layering of

saltwater below freshwater is considered in time for both a low density case (1% salt

mass fraction) and a high density case (10% salt mass fraction). A laboratory experiment

was used as part of the model testing process. As freshwater sweeps over the stable

saltier water at the bottom of the cube, the breakthrough curves at the outlet point are

measured. The position of the salt water – freshwater interface was measured using

nuclear magnetic resonance (NMR) techniques. Numerous authors have attempted to

model this problem and have achieved variable success. The challenge appears to lie in

the small dispersivity values and the large density contrasts, particularly in the high

density cases. The experiment and test case are resolved for low density contrasts and for

early time behaviour. Some questions do remain about the simulation of high density

contrasts and late time behaviour. Previous studies have emphasised the need for a

consistent velocity approximation in the accurate simulation of the salt pool problem. As

noted by Diersch and Kolditz [2002], the saltpool experiment does provide reliable

quantitative results for a three-dimensional saltwater mixing process under density effects

which were not available prior to its introduction. It is seen that extreme mesh

refinement is required in order to accurately simulate the problem.

More recently than the publication of the work by Diersch and Kolditz [2002], a very new test

case suite has been proposed by Weatherill et al., [2004] that has an exact and irrefutable

solution based upon an analytic stability theory. This is:

7. Convection in infinite, finite, and inclined porous layers: A body of work was developed

in the 1940’s originally by Horton and Rogers [1945] and independently by Lapwood

[1948] to examine the convective stability of an infinite layer with an unstable density

configuration applied across it. The work has an exact analytical solution for both the

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onset of convection (a critical Rayleigh number of 4 2) and the wavelength of the

convection pattern. Unlike all previous test cases, the availability of analytically based

stability criteria offers an excellent way of testing a numerical code, which does not rely

on a comparison with other numerical simulators. Additionally, these stability criteria

have been applied in traditional convection problems in the field of fluid mechanics for

many decades both within numerical and laboratory based frameworks – from a

scientific viewpoint they are well understood, robust and irrefutable. They are therefore

ideal for testing numerical models. In their study, Weatherill et al., [2004] compared

numerical results using SUTRA with these previous studies and showed that SUTRA

results (for both the onset conditions for instability and its resultant wavelength) were in

excellent agreement with the traditional stability criteria. Numerical solutions to

extensions of this original problem, namely cases of finite layers and inclined layers

(where analytical stability criteria also exist), were seen to be in excellent agreement with

stability criteria and previous results reported in the traditional fluids mechanics

literature. Given the availability of exact and irrefutable stability criteria for these “box

problems”, it is expected that they will become commonplace in future numerical model

testing. They also have solutions which can be extended into 3D benchmark test cases.

5.3 Future challenges for numerical modelling

Simmons [2005] and Diersch and Kolditz [2002] highlighted some current challenges that

face the field of variable density groundwater flow and these clearly have critical implications for the

modelling of such phenomena. Simmons [2005] identified a number of emerging challenges and

suggested that there is little, if any, evidence in current literature that points to the future extinction

of this field of hydrogeology. Future drivers for both research and practical application in this area

will drive numerical modelling needs. Just some future drivers for variable density flow research,

and hence modelling, identified by Simmons [2005] included, amongst others, : (i) the need to better

understand the relationship between variable density flow phenomena and dispersion, (ii) better

geological constraints on variable density flow analyses using lineaments, sedimentary facies data

and structural properties of fractured rock aquifers, (iii) improving the resolution of geophysical,

geoprospecting and remote sensing tools for non-invasive characterisation of dense plume

phenomena and heterogeneity, (iv) linking the fields of tracer and isotope hydrogeology and variable

density flow phenomena, (v) double-diffusive and multiple species transport problems in variable

density flow phenomena, (vi) links between climate change phenomena and the response of variable

density flow phenomena, such as sea level and coastline position change, and the impact of

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transgression and/or regression cycles on aquifer salinisation will be critical, (vii)the gas-liquid phase

chemistry of carbon sequestration processes and its efficiency, safety and long term viability as a

storage option will necessarily involve an understanding of variable density transport dynamics with

multiple phases, (viii) links between variable density flow phenomena and a number of

hydroecological applications (e.g., how does variable density flow effect baseflow accessions and

other surface-groundwater interactions, or salt budge profiles under vegetation, or the

spatiotemporal distribution of stygofauna and biota in subsurface groundwater ecosystems?), (ix)

Some of the above issues point to the need for more detailed and accurate coupling of surface

water, vadose zone and groundwater models – and which will drive an inherent increase in the

complexity of the modelling approach. The numerous future possibilities outlined above suggest

both short term and long term prognoses for variable density flow research and its applications are

very good. It is also therefore logical to conclude that there are many issues that remain largely

unexplored and poorly understood in the numerical modelling of variable density phenomena.

Some major future challenges for the field of variable density flow phenomena are

summarised below. Here, those matters of particular importance to the modelling of variable

density flow systems are highlighted. This extended analysis is based on the treatment presented

recently by Simmons [2005]. Some major challenges identified in that study include:

1. Large mix of spatial and temporal scales: Variable density flow phenomena may be triggered,

grow and decay over a very large mix of different spatial and temporal scales. The particular

challenge lies in the fact that information on very small spatial scales and short time scales is

needed to feed into long time and large spatial scale processes and simulations. Measuring field

scale parameters across this large range of spatial and temporal scales as input for modelling

approaches is a major challenge. Additionally, many current groundwater systems are highly

transient and steady assumptions may be an oversimplification, especially where solute transport

is concerned. For example, tidal oscillations and sealevel rise create the need for transient

analyses in seawater intrusion phenomena. In unstable problems, the onset and growth/decay

of instabilities requires a truly transient analysis of the system. Furthermore, the source of dense

plumes (e.g., leachate from a landfill) are often represented by simplified constant head, flux or

concentration boundary conditions and yet the style of loading is expected to be important

[Zhang and Schwartz, 1995]. There are challenges in better understanding how transient

boundary conditions will affect variable density flow phenomena (e.g., continuous or

intermittent sources).

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2. Heterogeneity and dispersion: It is only more recently that heterogeneity has been considered in

the study of dense plume migration [e.g., Schincariol and Schwartz, 1990; Schincariol et al.,

1997]. Schincariol et al., [1997] explain that heterogeneity in hydraulic properties can perturb

flow over many length scales (from slight differences in pore geometry to larger

heterogeneities at the regional scale), triggering instabilities in density stratified systems. This

is particularly true for physically unstable situations but is also important in other situations.

For example, the mixing process along a saltwater-freshwater interface can be significantly

modified by nonuniform velocities. This can lead to local physical instabilities resulting in a

corrugated interface that at larger scales is interpreted as increased dispersion. Modelling

narrow transition zones and low dispersion situations is challenging, most often requiring

very fine numerical grids and increased computational power. Current research is suggesting

that heterogeneity may serve as a physical perturbation in fingering processes and is therefore a

critical feature controlling the onset, growth and/or decay in variable density flow processes.

For example, low-permeability lenses can effectively dampen instability growth or even

completely stabilize a weakly unstable plume. Dense plume migration, particularly at high

density differences and in highly heterogeneous distributions, is therefore not easily

amenable to prediction [Schincariol et al., 1997; Simmons et al., 2001; Nield and Simmons,

2006]. Whilst some studies have examined dense plume migration in fractured rock using

numerical models [Shikaze et al., 1998; Graf and Therrien, 2005], there is still a need to explore

how more complex and realistic fracture geometries affect variable-density flow processes and

to better understand the links with macroscopic dispersion [e.g., Welty et al., 2003; Kretz et al.,

2003; Schotting et al., 1999].

3. Fluids, solutes and fluid-matrix interaction: Previous studies have typically been limited to single

species conservative transport but some have examined heat and solute transport

simultaneously within thermohaline convection regimes [see Diersch and Kolditz, [2002] for a

discussion on this topic in their exhaustive review of variable density modelling] which may be

important in understanding many processes including mineralization and ore formation in

sedimentary basins, geothermal extraction processes and flow near salt domes. The

simultaneous interaction of heat and solute creates challenges of its own and this complexity is

exemplified where multiple species with differing diffusivities interact. In a subset of these

“double-diffusive” processes, instabilities can occur even where the net density is increasing

downwards. Here, it is the difference in the diffusivities of the heat and solute that drives

transport and it should be realised that the simultaneous interaction of heat and solute creates

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many challenges of its own and that this complexity will be exemplified where multiple species

with differing diffusivities interact. Multiple species studies (beyond the two species heat and

salt) are fairly limited in groundwater literature. However, more recent studies [e.g., Zhang and

Schwartz, 1995] suggest that the chemical composition and reactive character of a plume can

greatly influence plume dynamics. Other studies have begun to examine how chemical reactions

can be coupled with density-dependent mass transport. For example, Freedman and Ibaraki

[2002] incorporated equilibrium reactions for the aqueous species, kinetic reactions between the

solid and liquid phases, and full coupling of porosity and permeability changes that result from

precipitation and dissolution reactions in porous media and showed that complex concentration

distributions result in the variable density flow system. Recently, Ophori [1998] suggested that

the variable viscosity nature of fluids is typically ignored in variable density groundwater flow

problems and suggest that plume migration pathways and rates are inaccurately predicted in the

absence of the variable-viscosity relationship. Other studies have suggested that dense plume

migration is significantly affected by the extent of porous medium saturation. Thorenz et al.,

[2002] and Boufadel et al., [1997] demonstrate that significant lateral flows and coupled density-

driven flow may take place in the partially saturated region above the water table and at the

interface between the saturated and partially saturated zones. All these studies suggest that

further investigation in relation to multiple species transport, non-conservative solute transport,

fluid-matrix interactions and a more complex representation of the chemical and physical

properties of the dense fluid than has previously been assumed is warranted. The coupling of

variable density flow phenomena with more complex chemical and fluid property

characterisation is an area that is still in its infancy and warrants further exploration. As

additional complexity is added to the numerical simulator in this area, there will be a need to

identify new test cases that actually test those newly added features of the code.

4. Numerical modelling: In general, variable density flow problems do not lend themselves easily

to analytical study except under the most simplified of conditions. However, a number of

numerical codes such as those outlined in Table 2 now exist to simulate variable density flow

phenomena. There has been good success simulating variable-density flow at low to

moderate density contrasts and in benchmarking the codes that are used. As was seen in the

analysis of test case problems, numerical modelling efforts were most successful in the cases

of stable problems (e.g., the Henry salt water intrusion case), but increasing problems are

encountered in the case of unstable problems (e.g., confusion over downwelling/upwelling

patterns and the massive computational power required for grid convergence in the case of

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the Elder problem, and the currently unresolved nature of the Salt Lake problem where grid

convergence has not currently been achieved). But it appears that even seemingly simple

matters in numerical modelling remain unresolved and appear to be currently overlooked.

For example, Kooi [submitted], noted that Fickian-type dispersive flux equations are

implemented in models where the dispersive flux is linearly related to the gradient of solute

concentration. He argued that this classical approach, in combination with solute boundary

conditions, fails to properly represent back-dispersion at outflow boundaries and produces

spurious, unphysical dispersive boundary layers. It is clear from Kooi’s [submitted] analysis

that greater insight in the exact nature of dispersive transport at outflow boundaries and

practical ways to represent this in models.

It is clear that variable density problems increase in modelling complexity as one moves

from stable to unstable configurations. The modelling complexity and computational

demand is compounded for large density difference in both stable (e.g., Salt Pool problem)

and unstable problems (e.g., Salt Lake Problem) and in cases where physical dispersion is

very small. Additionally, artefacts of the numerical scheme itself can mimic physically

realistic but perhaps completely inaccurate results (e.g., dispersion enhancing the transition

zones in stable problems, or enhancing the width and reducing their speed of unstable

fingers). The problem is compounded at much higher (unstable) density differences and

hence Rayleigh numbers where the physical problem itself is characterised by chaotic and

oscillatory regimes such as that seen in the recently proposed “Salt Lake Problem” [Simmons

et al., 1999]. What is particularly problematic in the simulation of such unstable phenomena

is that small differences in dispersion parameters or spatial and temporal discretization can

cause very different types of instabilities to be generated [Diersch and Kolditz, 2002]. The

nature of physical perturbations (c.f. numerical perturbations) is almost impossible to

measure and simulate. When using standard numerical approaches, one may attempt to

control numerical perturbations by reducing numerical errors through the use of very fine

meshes and small timesteps but the nature of what physical perturbation is present in the

system in order to generate particular patterns of instabilities is never known. It is almost

impossible to resolve, therefore, what are numerical artifacts and what are physical realities.

Furthermore, as noted by Diersch and Kolditz [2002], one must then ask what perturbation

is no longer significant for a convection process, and what error in the numerical scheme can

be tolerated. These authors note that there are two possible dangers in simulating such highly

nonlinear convection problems within a numerical framework. Firstly, a scheme which is

only conditionally stable, but of higher accuracy, is prone to create non-physical

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perturbations due to small perturbations in the solutions unless the element size and the

timestep are sufficiently small. Conversely, an unconditionally stable numerical scheme but

with lower accuracy, smooths out physically induced perturbations, unless the element size

and the timestep are sufficiently small. It is clear that a very fine spatial and temporal

resolution are essential if high density contrast convection phenomena are to be modelled.

One may conclude that the simulation of unstable phenomena, particularly highly unstable

cases, remains a challenge for further research.

In addition to the lack of control over numerical perturbations and the inability to

quantify the physical perturbation in real convective systems, it is also questionable whether

oscillatory or chaotic regimes such as those encountered at high Rayleigh numbers can be

simulated and predicted by a deterministic modelling approach. In more general terms, it

can be seen that variable density flow problems are currently limited by the need to simulate

large density contrast and low dispersion systems. Both are massively computationally

demanding. As noted by Diersch and Kolditz [2002], large-scale variable density simulations

have an inherent need for expensive numerical meshes. Some of the most promising

developments in the numerical modelling of real-field based problems include adaptive

techniques, unstructured meshes and parallel processing. Diersch and Kolditz [2002] note

that adaptive and unstructured irregular geometries are prone to uncontrollable perturbations

and that robust, efficient and accurate strategies are required, particularly for three-

dimensional applications. Given these numerical complexities and demands, a critical matter

that warrants attention in modelling of such systems is the level of simplification that is

possible without loss of physical information and accuracy. In many cases the science in

relation to this question is ambiguous, unresolved or is simply ignored. For example, when can

homogeneous equivalents for heterogeneous systems be employed? Or, when is a 2D analysis

of a 3D system permissible? What are the implications of such a simplification? The complexity-

simplicity matter must be considered carefully in the modelling of any variable density flow

system. In many cases, it appears that the decision is made implicitly, often with little or no

justification. Without a systematic evaluation of modelling complexity and the subsequent

implementation of a parsimonious modelling framework, it is readily apparent that the

implication of taking either a complex or simple approach may often not be properly

understood. Finally, and perhaps most importantly, there is also a real challenge for greatly

improved field measurements and observations for field truthing of models, minimising their

non-uniqueness and to inspire greater confidence in model predictive output.

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6. Applications and case studies

In this section, we demonstrate a range of case studies and applications, where variable

density flow is considered important. These examples include: saltwater intrusion and tidally

induced phenomena, salinisation associated with transgression and regression cycles, variable density

processes in aquifer, storage and recovery, flow and transport modelling in fractured rock aquifers,

and, chemically reactive transport modelling. Many of these examples are relevant to arid zone areas

but importantly extend beyond them. These applications and case studies are designed to illustrate

important areas where the field of variable density flow modelling is developing. The applications

and case studies are designed to be illustrative rather than exhaustive.

6.1 Seawater Intrusion and Tidally Induced Phenomena

6.1.1 Background

Seawater intrusion (or saltwater intrusion) is the encroachment of saline waters into

zones previously occupied by fresh groundwater. Under natural conditions, hydraulic gradients

in coastal aquifers are towards the sea and a stable interface between the discharging

groundwater and seawater exists, notwithstanding climatic events or sea-level rise [e.g.

Cartwright et al., 2004; Kooi et al., 2000]. Persistent disturbances in the hydraulic equilibrium

between the fresh groundwater and denser seawater in aquifers connected to the sea produce

movements in the position of the seawater-freshwater interface, which can lead to degradation of

freshwater resources. The impacts of seawater intrusion are widespread, and have lead to

significant losses in potable water supplies and in agricultural production [e.g. Barlow, 2003;

Johnson and Whitaker, 2003; FAO, 1997].

Seawater intrusion is typically a complex three-dimensional phenomenon that is

influenced by the heterogeneous nature of coastal sediments, the spatial variability of coastal

aquifer geometry and the distribution of abstraction bores. The effective management of coastal

groundwater systems requires an understanding of the specific seawater intrusion mechanisms

leading to salinity changes, including landward movements of seawater, vertical freshwater-

seawater interface rise or “up-coning”, and the transfer of seawater across aquitards of multi-

aquifer systems [e.g., van Dam, 1999; Sherif, 1999; Ma et al., 1997]. Other processes, such as relic

seawater mobilisation, salt spray, atmospheric deposition, irrigation return flows and water-rock

interactions may also contribute to coastal aquifer salinity behaviour, and need to be accounted

for in water resource planning and operation studies [e.g., Werner and Gallagher, 2006; Kim et

al., 2003; FAO, 1997].

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The investigation of seawater intrusion is a challenging undertaking, and requires careful

consideration of groundwater hydraulics and abstraction regimes, the influence of water density

variability, the hydrochemistry of the groundwater and the inherent dynamics of coastal systems

(i.e. tidal forcing, episodic ocean events such as storm surges and tsunamis, salinity and water

level variation in estuaries and other tidal waterways, amongst others). The challenges of

seawater intrusion investigation have led to the development of specific analytical methods and

mathematical modelling tools that enhance the interpretation of field-based observations, which

are invariably too sparse to provide a complete representation of seawater intrusion processes

and trends.

6.1.2 Analytical Solutions

The simplest analyses of seawater intrusion adopt the assumption of an sharp seawater-

freshwater interface, giving rise to analytical solutions to interface geometry. The Ghyben-

Herzberg approximation (developed independently by B. Ghyben in 1888 and by A. Herzberg in

1901; FAO, 1997) for the depth of the interface below the watertable was the first such solution

and was based on hydrostatic conditions, given as:

ffs

f hz (15)

where z is the depth below sea level of the interface (m), f and s are the freshwater and

saltwater densities (kgm-3), and hf is the height of the watertable above sea level (m). Ghyben and

Herzberg, working independently, showed that sea water actually occurred at depths below sea

level equivalent to approximately 40 times the height of freshwater above sea level. Glover

(1964) modified the solution to account for a submerged seepage face, as:

Kxq

Kqz

fs

f

fs

f '2' (16)

where q’ is the discharge per unit width of aquifer (m2/s), K is the hydraulic conductivity (ms-1)

and x is the distance from the ocean boundary (m). Further advances in analytical solutions to

sharp interface problems include the works of Kacimov and Sherif [2006], Dagan and Zeitoun

[1998], Naji et al. [1998], amongst others.

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Sharp interface approaches are only applicable to settings where interface thicknesses are

small in proportion to the total aquifer depth. Alternately, the dispersive nature of salt transport

needs to be considered and numerical modelling approaches are therefore ultimately required in

most practical applications in order to solve the coupled groundwater flow and solute transport

equations.

6.1.3 Numerical Models

A major limitation of the sharp-interface approach is that the movement of brackish

groundwater (i.e. diluted seawater) is not able to be analysed in the transition zone. Where the

diffusive behaviour of solute transport is a necessary component of seawater intrusion analyses,

models are applied that adopt numerical methods to resolve the groundwater flow and solute

transport equations. The computational effort of simulating density-coupled groundwater flow

and solute transport is usually an imposition that influences the spatial and temporal resolution

of simulations, and model grid design usually involves trade-offs between model run-times and

the resolution of simulation. This is particularly important for investigations of large areas

and/or where simulations are carried out in a three-dimensional domain.

The U.S. Geological Survey’s finite-element SUTRA code [Voss, 1984] is arguably the

most widely applied simulator of seawater intrusion and other density-dependent groundwater

flow and transport problems in the world [Voss, 1999]. Several other finite-element codes are

capable of seawater intrusion modelling, including the popular FEFLOW model [Diersch, 2005],

FEMWATER [Lin et al., 1997], etc. SUTRA has been applied to a wide range of seawater

intrusion problems, ranging from regional-scale assessments of submarine groundwater

discharge (SGD) [e.g., Shibuo et al., 2006] to riparian-scale studies of estuarine seawater intrusion

under tidal forcing effects (e.g. Werner and Lockington, [2006]). Gingerich and Voss [2005]

demonstrate the application of SUTRA to the Pearl Harbour aquifer, southern Oahu, Hawaii in

analysing the historical behaviour of the seawater front during 100 years of pumping history.

The list of seawater intrusion simulators includes hybrids of the popular finite-difference

MODFLOW [Harbaugh, 2005] including the groundwater flow code SEAWAT [Langevin et al.,

2003] and MODHMS [HydroGeoLogic Inc., 2003]. The advantage of these codes is that they

can be applied to existing MODFLOW models through relatively straightforward modifications

to MODFLOW input data files, thereby reducing model development effort. Further, existing

MODFLOW pre- and post-processors can be utilised. Robinson et al. [2007] applied SEAWAT

to the beach scale problem of groundwater discharge under tidal forcing, and predicted the

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temporal behaviour of submarine groundwater discharge and the geometry of salt plumes at the

coastal boundary.

Werner and Gallagher [2006] demonstrate the application of MODHMS to seawater

intrusion at the regional scale, by adapting an existing hydro-dynamically calibrated MODFLOW

model of the Pioneer Valley coastal plain aquifers. They used the model to demonstrate that

seawater intrusion had not reached an equilibrium state, and that alternative salinisation

processes (e.g., basement dissolution, relic seawater mobilisation, sea salt spray, irrigation salinity)

produced observed salinity patterns in some parts of the study area. Figure 6 shows the original

groundwater model design, Figure 7 provided results from the calibrated model process and

finally concentration distributions are provided in Figure 8. The modelling suggested that the

salinity in the Pioneer Valley is not all caused by classic seawater intrusion (i.e. seawater entering

the aquifer from the ocean). Werner and Gallagher [2006] described the development of a three

dimensional seawater intrusion model from a two dimensional groundwater flow (MODFLOW)

model, and describe some of the practical modelling problems that were encountered along the

way. Some practical problems included: 1. converting from 2D planar to 3D heterogeneous

model frameworks (while maintaining the same depth-averaged groundwater hydraulics between

both 2D and 3D representations), 2. The adoption of 3D boundary conditions (i.e. the

representation of partially penetrating estuaries, amongst others) and their influence on the

results compared to 2D groundwater flow; 3. Converting from the Dupuit assumption to

variably saturated flow (and the conversion from MODFLOW to MODHMS modelling

frameworks). The modelling was used to provide an analysis of the “susceptibility to seawater

intrusion” throughout the aquifer. Susceptibility was assessed using various information sources

– the modelling provided an indication of areas susceptible to future seawater intrusion by

running long-term simulations (80+ years) that used predicted water resource operational

scenarios. Various simulations of water resource management, including “same as existing”,

“worst case”, “severe cutbacks”, were tested. The overall study described a detailed analysis of

seawater intrusion – including model conceptualisation, numerical modelling, and the final

management options for seawater intrusion.

6.1.4 Future Challenges

In recent times, seawater intrusion modelling has become a more straightforward

undertaking, given advancements in computer processing capabilities, improvements in

modelling codes, and the development of user-friendly pre- and post-processors. Further, the

advent and evolution of MODFLOW-based seawater intrusion simulators facilitates a relatively

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uncomplicated transition from MODFLOW model to seawater intrusion simulator. However,

the interpretation of modelling results is still an extremely challenging task, especially where the

aim of the investigation is to develop practical solutions to the mitigation and remediation of

seawater intrusion.

Despite an abundance of bench-marking studies for variable-density codes, the results of

these have not translated into guidelines for interpreting and applying seawater intrusion models.

The reasons for this are probably because of differences in scale and in vertical-to-horizontal

aspect ratios between benchmark problems and real field-scale seawater intrusion problems. For

example, both the Henry problem and the benchmark problem recently described by Goswami

and Clement [2007] involve horizontal-to-vertical ratios of about 2, while the seawater intrusion

problem of Werner and Gallagher [2006] involved a horizontal-to-vertical ratio of about 500.

Another significant challenge of practical seawater intrusion modelling is determining the degree

that aquifer heterogeneities are represented in simulations of real-world systems. It is generally

accepted that preferential flow paths play an important role in the behaviour of contaminant

plumes; however the degree of representation of the variability in aquifer properties has not been

adequately explored for the seawater intrusion problem. The problem of selecting the level of

representation of aquifer heterogeneities needs to be considered in combination with the

selection of the appropriate grid resolution. Other potential issues include the inclusion of

transient information in models (irrigation cycles, groundwater extraction regimes) and over large

spatial scales that influence intrusion phenomena. All of these challenges point to the growing

need for further 3D variable density and transient analyses, often covering very large spatial and

temporal scales. The additional inclusion of heterogeneity (whether it be in boundary conditions

in space or time or in geologic properties of the aquifer) poses additional complexity. Resolving

narrow transition zones further compounds the need for greater computational effort. Increased

efficiency in numerical solvers and other numerical efficiencies will be required to meet these

growing computational demands. Additionally, the large computational effort associated with

seawater intrusion modelling of regional-scale systems generally precludes the application of

automated calibration techniques, such as the gradient methods adopted by standard

groundwater model calibration codes like PEST [Doherty, 2004]. Further research is needed to

develop methods of calibrating seawater intrusion models, and to analyse the model uncertainty

of making predictions with these models.

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6.2 Transgression-Regression salinisation of coastal aquifers

6.2.1 Importance of transgression-regression cycles

Tidal forcing, seasonal changes in recharge and groundwater abstraction are key factors

responsible for variability and change in coastal groundwater systems. These drivers act on

human time scales. However, when viewed on longer time scales from centuries to millions of

years, coastal zones are even more dynamic. Sea-level change, erosion and sedimentation have

caused coastlines to shift back (i.e., seaward: regression) and forth (i.e., landward: transgression)

over distances of up to several hundreds of kilometres. Groundwater systems responded to these

conditions by adjustment of flow fields and redistribution of fresh and saline water. Although

the geological changes in boundary conditions generally tend to occur at a slower pace than more

recent man-induced changes or diurnal and seasonal forcing, their impact on current

groundwater conditions is often enormous.

More noticeable, or even catastrophic, retreats of the coastline occur in arid regions

when droughts, river diversion or irrigation schemes decrease the inputs of (fresh) water to

inland seas or lakes. Well documented cases include the Dead Sea, the Aral Sea, Lake

Corangamite and Lake Chad [Nihoul et al., 2004]. Conversely, shoreline advance is also

associated with decreased river discharge to coastal zones when decrease in sediment loads

causes land loss due to erosion and subsidence. Decreased discharge of the River Murray in

Australia and the Nile in Egypt are key examples [Frihy et al., 2003]. Besides enormously adverse

economic and ecological consequences, the hydrogeological effects of such coastline shifts are of

paramount importance as well. These include the relocation of discharge and recharge zones,

changes in groundwater-surface interaction processes and water quality degradation. Variable

density flow also plays a crucial role in these settings.

6.2.2 Observational evidence and conceptual models

It is essential to realize that the textbook conception of the fresh and saline groundwater

distribution, which is classically conceived of as a fresh water lens overlying a wedge of saline

groundwater, is seldom encountered in real field settings due to the dynamic nature of

shorelines. The most conspicuous manifestations of transient effects are offshore occurrences of

fresh groundwater and onshore occurrences of salt water.

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Offshore fresh and brackish paleo-groundwater

Over the years, evidence has been growing in support of the observation that sub-

seafloor fresh and brackish groundwater are common features of continental shelves and shallow

seas around the world. Figure 9 shows the known global occurrences of offshore fresh and

brackish groundwater. Studies from New Jersey, USA [Hathaway et al., 1979; Kohout et al.,

1988], New England, USA [Kohout et al., 1977], Suriname, South America [Groen, 2002] and

the Dutch sector of the North Sea [Post et al., 2000] have shown that groundwater with salt

contents between 1 and 50% of seawater occur up to 150 km from the present coastline and at

depths up to 400 m below the seafloor. Additional, less conclusive cases have been reported for

Port Harcourt, Nigeria, Jakarta, Indonesia and for the Chinese Sea [Groen, 2002].

In many instances these waters occur too far offshore to be explained by active sub-sea

outflow of fresh water due to topographic drive. Moreover, lowest salinities often occur at

substantial depths beneath the seafloor and are overlain by more saline pore waters, suggesting

absence of discharge pathways. These waters therefore are considered paleo-groundwaters that

were emplaced during glacial periods with low sea level. During subsequent periods of sea-level

rise, salinization was apparently slow enough to allow relics of these fresh waters to be retained.

Onshore saline groundwater

In many flat coastal and delta areas the coastline during the recent geologic past was

located further inland than today. As a result, vast quantities of saline water were retained in the

subsurface after the sea level retreated. Such occurrences of saline groundwater are sometimes

erroneously attributed to seawater intrusion, i.e., the inland movement of seawater due to aquifer

over-exploitation (see section 6.1). Clearly, effective water resource management requires proper

understanding of the various forcing functions on the groundwater salinity distribution on a

geological timescale [Kooi and Groen, 2003].

High salinities are maintained for centuries to millennia, or sometimes even longer, when

the presence of low-permeability deposits prevents flushing by meteoric water [e.g., Groen et al.,

2000; Yechieli et al., 2001]. Rapid salinization due to convective sinking of seawater plumes

occurs when the transgression is over a high-permeability substrate. This process is responsible

for the occurrence of saline groundwater up to depths of 400 m in the coastal area of the

Netherlands [Post and Kooi, 2003].

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6.2.3 Numerical modelling

In the analysis of transient effects of coastline migration on the salinity distribution in

groundwater, variable density groundwater flow and solute transport modelling plays a vital role

in developing comprehensive conceptual models.

The large spatial and time scales involved pose special challenges. In particular, the high

resolution model grid required to capture convective flow features imposes a severe

computational burden that limits the size of the model domain. Other complications include the

lack of information on past boundary conditions, insufficient data for proper parameterization,

especially for the offshore domain, and unresolved numerical issues with variable-density codes.

Mathematical modelling applications addressing offshore paleo-groundwater have shown a

typical progression in model complexity. Indeed, all modelling studies to date are cross-sectional

only (2D). Earliest steady-state, sharp interface, homogeneous permeability models [Meisler,

1984] could not capture the marked disequilibrium conditions that are so apparent. The Meisler

[1984] model was only able to simulate the steady-state position of the (sharp) seawater-fresh

groundwater interface. Different simulations were carried out for different sea level low stands.

The model was not capable of simulating the effect of the subsequent sea level rise. With a non-

steady, sharp interface, single aquifer model, Essaid [1990] made a convincing case that the

interface offshore Santa Cruz Country, California, likely has not achieved equilibrium with

present-day sea level conditions, but is still responding to the Holocene sea-level rise. Kooi et al.,

[2000] used a variable density flow model that included a moving coastline to study the transient

behaviour of the salinity distribution in coastal areas during transgression (Figure 10). They

found that Holocene transgression rates were often sufficiently high to cause the fresh-salt

transition zone to lag behind coastline migration.

Person et al., [2003] and Marksammer et al., [2007] have conducted variable density

modelling for the New England continental shelf, simulating sea-level change over several

millions of years. Their results indicate that the first-order continental shelf topographic gradient

is insufficient to cause freshening of aquifer systems to several hundreds of metres depth far out

on the continental shelf. Continental ice sheet recharge [Marksammer, 2007] and/or secondary

flow systems associated with incised river systems in the continental shelf during low-stands

therefore appear to be required to account for distal (several tens of kms off the coast, well

beyond the reach of active, topography driven fresh water circulation) offshore paleo-waters.

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6.2.4 Future challenges

Knowledge of the “anomalous” salinity distributions discussed above is not only relevant

from an academic perspective. Offshore fresh and brackish groundwater occurrences are

expected to become a viable alternative water resource, notably in arid climates and in places

with high population densities. Moreover, in the development of conceptual hydrogeologic

models of coastal aquifers, the relevance of transient phenomena should be considered more

routinely than at present. Such an analysis has major implications for the numerical models based

on these conceptualizations, such as the selection of appropriate boundary conditions and the

choice of the timescales that need to be considered.

The much needed improvements in our understanding of the impact of coastline

migration on hydrogeologic systems rely on more and better field observations. Age dating of

groundwater, and especially coastal groundwater, is extremely difficult but is an essential

component of paleohydrologic analyses. Exploration techniques needed to better delineate the

subsurface salinity distribution require refinement and more versatility, especially in order to

investigate offshore aquifers.

Finally, numerical models need to be improved as well as has been outlined in an earlier

section. Difficulties currently arise in the accurate and physically realistic simulation of density

driven fingering due to grid discretization requirements. Resolving the nature of the relationship

between numerical features of a code and the physical fingering process is crucial since this

unstable fingering process appears to be of prime importance in many hydrogeologic settings.

Greater flexibility in boundary conditions are also required to allow for both heads and

concentrations changes to occur due to changes in sea level. Land subsidence and sedimentation

are among the additional complications that currently remain largely unexplored and poorly

understood. The addition of these additional physical processes into a quantitative modelling

framework will undoubtedly provide challenges for numerical models.

6.3 Aquifer storage and recovery

6.3.1 Importance of Aquifer Storage & Recovery (ASR)

Aquifer storage and recovery (ASR) simply involves the injection and storage of

freshwater in an aquifer. The theory was introduced by Cederstrom (1947) as an alternative to

surface water storage such as dams and reservoirs. The main advantages of ASR over surface

water storage include that it requires a relatively small area of land, potentially reduces the water

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lost to evaporation (e.g., from the surface of a dam), and to a certain extent the aquifer can

behave as a water purification medium.

The original, idealised concept of ASR involved the injection of a cylindrical “bubble” of

freshwater out of a well, into a confined aquifer containing groundwater of a poorer quality (such

that the groundwater itself would typically not be used). The bubble of water would then be

stored in the aquifer until needed, at which point it would be pumped back out of the same well.

In this situation, the volume of recoverable freshwater could be reduced by mixing between the

injected and ambient waters, and by the injected plume migrating down-gradient (in the case

where the water was injected into an aquifer where a background hydraulic gradient existed as is

typical in most cases). In both cases, the recoverable volume will be smaller than the injected

volume. Situations with minimal mixing and little migration down-gradient can allow almost

100% of the injected volume to be recovered. However, situations with large amounts of mixing

and/or a significant migration of the plume down-gradient can lead to a very small (even zero)

percentage of recoverable freshwater. In each specific situation, the percentage of recoverable

freshwater (called the “recovery efficiency”) will determine whether an ASR operation will be

economically viable, and whether it actually offers any advantages over surface water storage

technology.

6.3.2 Variable density flow in ASR

As well as mixing and down-gradient movement, density effects can cause a reduction in

ASR recovery efficiency. It appears that this phenomenon was first documented by Esmail and

Kimbler [1967]. If freshwater is injected into a saltwater aquifer, the density contrast between

injected and ambient water leads to an unstable interface at the edge of the injected plume. The

denser ambient water tends to push towards the bottom of the interface and the lighter injected

water tends to float towards the top. The result is tilting of the interface and a transition from

the idealised cylindrical plume to a conical-shaped plume (see Figure 11). Upon recovery, the

ambient water is much closer to the well at the bottom, and breakthrough of salt occurs, causing

premature termination of the recovery before all of the freshwater has been extracted. In this

way, it is possible to have situations with minimal mixing and minimal background hydraulic

gradient, but still low recovery efficiency due to density effects.

6.3.2 Analytical modelling

Bear & Jacobs [1965] presented an analytical method for predicting the position of the

freshwater/saltwater sharp interface under the influence of pumping (either injection or

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extraction) and a background hydraulic gradient. The analytical solution assumed a homogeneous

aquifer and neglected density effects. Esmail & Kimbler [1967] produced an analytical/empirical

formula to address the density-induced tilting of a freshwater/saltwater “gradient” interface (i.e.

accounting for a mixed zone) in radial geometry. The presence of a mixed zone was found to be

significant. The influence of a background hydraulic gradient was neglected by Esmail &

Kimbler [1967], and the process assumed that the interface would only tilt during storage, not

during either the injection or extraction phase. Peters [1983] presented a semi-analytical solution

for an injected bubble under the influence of density effects, but the presence of a mixed zone

was neglected and the background hydraulic gradient was not considered. When considering the

influence of pumping (especially multiple cycles), background hydraulic gradients, heterogeneity

and density-dependent flow phenomena, analytical solutions become far too complicated to

derive and is often the case, a numerical model is required.

6.3.3 Numerical modelling

Numerical modelling of ASR has been described by Merritt [1986], Missimer

[2002], Pavelic et al., [2006], Yobbi [1996] and Ward et al., [2007]. Brown [2005] provides a

comprehensive summary of ASR modelling (p. 170). Merritt [1986], Missimer et al., [2002] and

Ward et al., [2007] specifically included results of numerical modelling with density effects

explicitly accounted for. Yobbi [1996] included density effects in a modelling study but did not

report on the specific influence and effects of density on the simulation results. Ward et al.,

[2007] performed numerical modelling of density dependent ASR and quantified the influence of

various parameters on the tilting interface.

6.3.3.1 Choice of model

An ASR process can be modelled in 2 dimensions as a horizontal plane [Merritt, 1986;

Pavelic et al., 2004]. Being a radial flow system, it can also be modelled as a 2-dimensional

“radial-symmetric” projection, that is, a 2-dimensional vertical projection that is symmetrical

about the injection well [Merritt, 1986; Ward et al., 2007]. Of course, ASR can also be modelled

in 3 dimensions but this obviously increases computational burden. Consequently, 2D

simplifications (in radial flow coordinates) are often preferred where symmetrical flow can be

assumed.

Largely hypothetical ASR cases where there is no background hydraulic gradient, and

where the aquifer structure is homogeneous or consists of infinite horizontal layers, can be

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modelled as a 2-dimensional radial-symmetric projection [Ward et al., 2007]. Sample model

output comparing density-invariant, small and large density contrasts are shown in Figure 12.

These results were obtained using the FEFLOW model (Diersch, 2005). FEFLOW is a

numerical finite element system that allows various two-dimensional (2D) projections as well as

three-dimensional (3D) flow and transport processes in porous media. For this study a 2D

axisymmetric projection was chosen to model the ASR system. A transient, density-dependent

flow and mass transport regime was chosen. Cases involving a background hydraulic gradient

(more realistic) are inherently asymmetrical and must be modelled either as a 2-dimensional

horizontal or 3-dimensional simulation.

To simulate ASR with variable density, flow must be considered in the vertical

dimension, meaning that a 2-dimensional horizontal plane model will not suffice. One concludes

that density-dependent ASR systems must be modelled either as a 2-dimensional radial-

symmetric projection (which is incapable of simulating background hydraulic gradient or

asymmetrical geological structures) or a fully 3-dimensional model. Because in real systems a

background hydraulic gradient is typical, a simple 2-dimensional radial-symmetric projection is

generally insufficient. However, the time and computational power required to establish and

execute a 3-dimensional model that is sufficiently discretized to accurately simulate the density-

dependent flow of an injected freshwater plume can lead without robust quantification to the

simpler choice of a density-invariant simulation model. This decision should not be made lightly,

as judging the likely influence of density effects is rather complicated [Ward et al., 2007].

6.3.3.2 When can density effects be ignored?

The concentration difference between injected and ambient water is the most obvious

parameter driving the density-induced tilting of the interface. Indeed, in many cases the

concentration difference is (explicitly or implicitly) assumed to be the only significant parameter

that influences variable-density flow, and modellers make the decision to use a density-

dependent or density-invariant numerical model based on this parameter alone. However, Ward

et al. [2007] demonstrated that an ASR system must be considered as a mixed-convective regime

and derived the following mixed convection ratio M as an approximate ratio between the

respective influences of free and forced convection in a simple (radial-symmetric) ASR system.

In the case of an ASR system, they showed that the appropriate formulation of the mixed

convection ratio (equation 2) is given by:

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0

02VK

QrBM (17)

where r is the hypothetical injected bubble radius (L), B is the aquifer thickness, Q is the injection

or extraction flow rate (L3/T), and KV is the vertical hydraulic conductivity (L/T). When M <<

1, forced convection dominates the system (e.g., fast pumping at small radii) and interface tilting

is expected to be relatively insignificant. As M approaches 1, the relative influences of free and

forced convection are comparable (e.g. slow pumping at large radii) and interface tilting is

expected to become significant in this transition zone. Clearly during the storage phase, Q = 0

and M is infinite, indicating that the flow in system is totally dominated by free convection [Ward

et al., 2007]. As can be seen from the mixed convection ratio give in equation 17, a low density

contrast in a highly permeable medium may have the same “density effect” (i.e., the same rate of

tilting) as a higher density contrast in a less permeable medium. Equation 17 brings together

several other potentially significant parameters, such as aquifer thickness, bubble radius and

pumping rate. A highly anisotropic aquifer (in which the vertical hydraulic conductivity is a very

small proportion of the horizontal) may retard the vertically-acting density effects, thereby

attenuating density-induced tilting. After injection finishes, the duration of the storage period is

critical: depending on the density contrast and hydraulic conductivity, a short period of storage

(e.g., <1 month) may lead to minimal tilting, but longer periods may cause significant tilting, and

therefore significant reduction in recoverable freshwater. If ASR is to be promoted as a long-

term drought mitigation strategy then long storage durations may be expected [Bloetscher and

Muniz, 2004]. In such cases, the effects of variable density flow are likely to be important in

general hydrogeologic analyses and in numerical modelling efforts.

6.3.3.3 Density effects in layered aquifers

Heterogeneous aquifers that consist of sedimentary strata can lead to density-induced

fingering. Injected freshwater will readily flush ambient saltwater out of layers of high

permeability, but saltwater will remain in layers of low permeability as the freshwater will not be

able to penetrate those layers during injection. Then, during storage, the dense saltwater can

migrate downwards out of the low permeability layers and “bleed” into the freshwater, and

buoyant freshwater can also migrate upwards, flushing salt out of low conductivity layers (see

Figure 13). This can potentially lead to a very significant reduction in recovery efficiency as large

volumes of freshwater stored in the high permeability layers essentially become contaminated

with salt from low permeability layers above and below.

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When attempting to simulate ASR processes, one is faced with several problems. First, to

model a realistic system with a background hydraulic gradient and density effects, a 3D model is

required. Setting up a detailed 3D density-dependent model can be time-consuming and is

generally computationally expensive. Furthermore, if the aquifer is heterogeneous, then the finite

element mesh or grid may need to be highly discretized (perhaps in all 3 dimensions) at

interfaces between regions of different hydraulic conductivity and the run-times are likely to

become excessively large (several days or more). The additional requirement to place model

boundaries sufficiently far away from the injection/extraction activity will also increase

simulation run-times. Large field scale modelling applications of variable density flow in 3D are

therefore limited because of heavy computational requirements.

Ward et al., [2007] described a method for simulating the injection/extraction well that

considered the injection and extraction well to be a column of highly conductive elements (like a

fracture), rather than a uniform flux across the boundary. The reason for this method of

implementation was that the uniform flux method would fail to account for the dense water

initially in the aquifer, which may in fact lead to a non-uniform velocity profile (low velocities at

the bottom of the well and large velocities at the top of the well). Figures 12 and 13 demonstrate

that the density-induced tilting of the interface can actually drive saltwater back up the well

during the storage phase (when the well is not operational) where it can then circulate upwards

into the upper part of the aquifer. The likelihood of observing this phenomenon in the field is

unknown and requires further research.

6.4 Fractured rock flow and transport modelling

6.4.1 Importance of variable density flow in fractured rock aquifers

Fractures may have a major impact on variable density flow because they represent

preferential pathways where a dense plume may migrate at velocities that are several orders of

magnitude faster than within the rock matrix itself. Studying dense plume migration in fractured

rock is especially important in the context of hazardous waste disposal in low-permeability rock

formations. Variable density flow in a vertical fracture has previously been studied by Murphy

[1979], Malkovsky and Pek [1997, 2004] and Shi [2005], showing that two-dimensional

convective flow with a rotation axis normal to the fracture plane can occur. Shikaze et al., [1998]

numerically simulated variable-density flow and transport in a network of discrete orthogonal

fractures. They found that vertical fractures with an aperture as small as 50 m significantly

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increase contaminant migration relative to the case where fractures are absent. It was also shown

that dense solute plumes develop in a highly irregular fashion and are extremely difficult to

predict. However, Shikaze et al., [1998] represented fractures by one-dimensional segments, thus

inhibiting convection within the fracture. Importantly, Shikaze et al., [1998] limited their studies

to a regular fracture network consisting of only vertical and horizontal fractures, embedded in a

porous matrix. Recently, Graf and Therrien [2005] included inclined fractures in the simulation

of dense plume migration and showed that variable density flow in a 1D inclined fracture

initiates convection in the 2D porous matrix and that convection cells are independent and

separated by the high-permeability fracture (Figure 14). In a recent study, Graf and Therrien

[2007a] investigated dense plume migration in irregular 2D fracture networks. Simulations

indicated that convection cells form and that they overlap both the porous matrix and fractures.

Thus, transport rates in convection cells depend on matrix and fracture flow properties. A series

of simulations in statistically equivalent networks of fractures with irregular orientation showed

that the migration of a dense plume is highly sensitive to the geometry of the network. If

fractures in a random network are connected equidistantly to the solute source, few equidistantly

distributed fractures favor density-driven transport while numerous fractures have a stabilizing

effect.

6.4.2 Verification of variable-density flow in fractured rock

Few test cases exist to verify a new code which simulates variable-density flow in

fractures. Results presented by Shikaze et al. [1998] can be used to verify dense plume migration

in a set of vertical fractures by comparing isohalines at certain simulation times. The inclined-

fracture problem introduced by Graf and Therrien [2005] can be used in both a qualitative (using

isohalines) and a quantitative (using detailed information on breakthrough curves, mass fluxes,

maximum velocities etc.) manner to verify variable density flow in an inclined fracture.

Caltagirone [1982] has presented the analytical solution for the onset of convection in

homogeneous media. The solution can be applied to vertical and inclined fractures by assuming

constant fracture aperture (homogeneity) and by introducing a cosine weight to account for

fracture incline [Graf, 2005]. Although the solution by Caltagirone [1982] is the only available

analytical solution on variable-density flow applied to box type problems of finite domain

dimensions, it is not truly an analytical solution, but rather uses analytical procedures to derive

the critical Rayleigh number for different aspect ratios in order to determine the conditions that

govern the stable and unstable states of convection in box type problems. It is these solutions

that also form the basis for newly proposed benchmark test cases presented in Weatherill et al.,

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[2004]. We conclude that, to date, a robust and well accepted benchmark problem for variable

density flow in fractures oriented in 3D and embedded in a 3D porous matrix does not exist.

Thus, there is a clear need to further explore how to best test and benchmark 3D variable density

flow in fractured media.

6.4.3 HydroGeoSphere code capability

The HydroGeoSphere (HGS) model simulates variable-density flow in 3D porous media

and in 2D fractured media where fractures are planar and parallel to at least one coordinate axis

[Therrien and Sudicky, 1996; Therrien et al., 2004; Graf and Therrien, 2005; 2007a; 2007b].

There are, in principal, two methods to calculate density: (i) Using Pitzer’s ion interaction model

in the VOPO model [Monnin, 1989; 1994], which is physically based but slow because it iterates

between density and molality. The VOPO algorithm calculates the density of water from the

concentrations of the solutes Na+, K+, Ca2+, Mg2+, SO24 , HCO3 and CO2 , based on Pitzer’s

ion interaction model. It allows for the accurate calculation of the density of natural waters over

a wide range of salinities (i.e., from fresh water to brine). However, the speed of the computation

often makes it impractical for numerical modeling. (ii) Using an approximate but faster empirical

model. HGS can use both methods and salinity is calculated from concentrations of individual

ions found in natural water.


Assuming 2D spatial dimensions is a common simplification made in numerical

simulations. Because density-driven convection can naturally occur in multiple ways in 3D,

representing non-planar fractures in 3D is essential. Prior attempts have shown that this is a

tedious task but has been addressed successfully [Graf and Therrien, 2006; 2007c]. Variable-

density flow in inclined fractures embedded in a 3D porous matrix is a great future challenge and

subject to ongoing investigation (Figure 15). Subsequently, benchmark problems for variable-

density flow in 3D fractured porous media are important and should be defined, followed by

further 3D analyses.

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6.5 Chemically reactive transport modelling

6.5.1 Importance of chemical reactions in variable density flow

Mixing of different waters, whether driven by the existence of salinity/temperature

gradients or other forces, may trigger geochemical reactions. Well-documented salt-water related

problems include the dissolution of calcite in fresh-salt water mixing zones [e.g. Wigley and

Plummer, 1976; Sanford and Konikow, 1989] and cation exchange due to salt water intrusion or

freshening [Valocchi et al., 1981; Appelo and Willemsen, 1987; Appelo et al., 1990; Appelo,

1994]. Precipitation of (carbonate) minerals has also been extensively studied in geothermal

problems due to its relevance for aquifer thermal energy storage [e.g., Brons et al., 1991].

Moreover, ore formation in relation to thermal convection is a topic that has received ample

attention in basin-scale geological studies [e.g., Person et al., 1996].

6.5.2 Governing Equations Reactive transport of dissolved solutes in groundwater is described by the advection-

diffusion equation, which is a statement of the conservation of mass of a chemical component i:

reaciii rqCCnD

tnC )()( (18)

where Ci denotes the chemical concentration of component i (M/L3), n is the porosity

(dimensionless), D is the dispersion coefficient (L2/T) and q is the specific discharge (L/T), i.e.

the volumetric flow rate per unit of cross-sectional area. The third term on the right-hand side,

rreac, accounts for the change in solute concentration due to chemical reactions. A similar

expression holds for the temperature in heat-transport problems. Darcy’s law in a variable

density formulation is given in Eqn (14), and the general equation of state is given in Eqn (6).

The concentration depends on the flow field (through q in equation 18), the flow field depends

on the density (through in equation 13) and is influenced by C and hence also by q.

Therefore, in variable-density flow problems, the governing equations for groundwater flow and

solute transport need to be solved simultaneously.

For reactive transport modelling, it is important to pay careful attention to the equation of state.

This expression relates fluid density and dissolved solute concentration can take different forms,

for example a simple linear expression:

126

C0 (19)

where 0 is the density when C = 0, and is the slope of the -C curve. In multicomponent

systems, is not a function of a single solute but depends on all the different solutes that are

present in the solution. A more appropriate form of equation (19) then becomes:

niiiC

,10 (20)

where n is the number of components and the subscripts i refer to the individual components.

Relations of this type were employed by Zhang and Schwartz [1995] and Mao et al. [2006].

Alternatively fluid density might be computed (slightly) more accurately on the base of a

thermodynamic framework [Monnin, 1994; Freedman and Ibaraki, 2002; Post and Prommer,

2007].

6.5.3 Development and applications of numerical models

The development of numerical codes that address reactive transport under variable

density conditions was driven by a range of different motivations. On one hand, for example,

large-scale transport phenomena over geological time-scale are simulated to understand the

evolution and spatial distribution of ore-bodies [e.g., Raffensperger and Garven, 1995] while, on

the other hand, models and simulations that focus on the movement of groundwater and the

related water quality evolution are typically dealing with much smaller spatial- and shorter time-

scales. The latter, for example, was the case in the studies by Zhang and Schwartz [1995], who

have developed a numerical simulator to assess the fate of contaminants in a dense leachate

plume and by Christensen et al. [2001], who studied hydrochemical processes during horizontal

seawater intrusion. Furthermore, Mao et al. [2006] have developed a model and applied it to

simulate the results of a tank simulation involving chemical reactions in an intruding seawater

wedge. Sanford and Konikow [1989], Freedman and Ibaraki [2002] and Rezaeia et al., [2005]

developed codes and used them for generic simulations that evaluated the effect of chemical

reactions on dynamic changes of the hydraulic properties of the aquifer material. Post and

Prommer [2007] developed a variable-density version of the reactive multicomponent transport

model PHT3D [e.g., Prommer et al., 2003; Prommer and Stuyfzand, 2005; Greskowiak et al.,

2005; Prommer et al., 2006]. Their model combines the previously existing and widely used tools

SEAWAT, which accounts for multicomponent transport under variable-density conditions, and

127

the geochemical reaction simulator PHREEQC-2. They used the model to study the relative

importance of reaction-induced density changes (resulting from cation exchange and calcite

dissolution) on the groundwater flow field during free convection. To assess and quantify the

potential effects, they reformulated the classic, well-studied Elder problem, in which flow is

solely driven by density gradients (free convection), into a reactive multicomponent transport

problem. The main outcome of the modelling study was that the flow field appears to be altered

only in cases when the fluid density is strongly affected by changes in solute concentration due to

chemical reactions (Figure 16). Perturbations smaller than approximately 10% did not result in

differences in the flow pattern that could visually be detected, but an effect was noticed in the

rate of salt plume descent that became higher as the density of the downward migrating plumes

increased. The critical result here is that when low density-contrasts drive groundwater flow,

chemical reactions are more likely to be important and hence need to be incorporated in the

analysis, unlike when density contrasts are high, such as in seawater intrusion type of problems.

The PHT3D model was also applied in the study of Bauer-Gottwein et al., [2007],

simulating shallow, unconfined aquifers underlying islands in the Okavango Delta (Botswana).

These islands are fascinating natural hydrological systems, for which the interplay between

variable density flow, evapotranspiration and geochemical reactions has a crucial effect:

Astonishingly, the combination of these mechanisms keeps the Okavango delta fresh, as salts are

transported from the freshwater body towards the island’s centre, where evapoconcentration

triggers (i) mineral precipitation and (ii) convective, density-driven downward transport of the

salt into deeper aquifer systems. The model simulations evaluated the conceptual model and

quantified, using measured hydrochemical data as constraints, the individual processes and

elucidated their relative importance for the system. They found that the onset of density-driven

flow was affected by mineral precipitation and carbon-dioxide degassing as well as by

complexation reactions between cations and humic acids.

6.5.4. Future challenges

Numerical modelling of reactive transport under variable density conditions is a complex

undertaking. As with transport under density-invariant conditions, numerical approximations

introduce artificial dispersion and oscillations which pose special difficulties for chemical

calculations. These are especially felt when redox problems are simulation as these require a high

accuracy. The need therefore remains to search for mathematical techniques that combat these

undesirable artefacts. In many cases, it should be noted that model results are extremely

sensitive to spatial discretization and that grid convergence can be a problem. There are also

128

limitations faced by the assumption of local chemical equilibrium and how this relates to the

choice of spatial and temporal discretisation (e.g., the time taken to equilibrium could be longer

that the time step size).

The capabilities of the codes developed so far are already quite impressive but existing

limitations warrant further improvement and refinement. Multiphase systems, interaction with

hydraulic properties and deforming porous media can already be handled by some codes but the

development of more comprehensive models is warranted due to the complexity of geological

systems.

Despite these limitations, the level of sophistication of the reactive transport codes has

already outrun our capability to collect the appropriate input data. Parameterization of even the

simplest models, such as well controlled laboratory column experiments, is by no means trivial.

For example, challenges relate to the translation of thermodynamic constants for ideal minerals

to more amorphous phases in natural sediments or batch-determined kinetic laws to field

conditions. Unless new techniques emerge (e.g., nanotechnology?) that allows us to change the

way we collect field data, the reliable application of sophisticated codes will be seriously limited.

129

Variable density groundwater system Relevant papers Sea water intrusion, fresh-saline water interfaces in coastal aquifers

Yechieli et al., (2001) Kooi et al., (2000) Post and Kooi (2003) Underwood et al., (1992) Voss and Souza (1987) Huyakorn et al., (1987) Pinder and Cooper (1970) Werner and Gallagher (2006)

Subterrenean groundwater discharge Langevin (2003) Kaleris et al., (2002)

Infiltration of leachates from waste disposal sites, dense contaminant plumes

Liu and Dane (1996) Zhang and Schwartz (1995) Oostrom et al., (1992a,b) Koch and Zhang (1992) Schincariol and Schwartz (1990) Pashcke and Hoopes (1984) Le Blanc (1984) Frind (1982)

DNAPL flow and transport Li and Schwartz (2004) Lemke et al., (2004) Oostrom et al., (2003)

Density driven transport in the vadose zone Ying and Zheng (1999) Ouyang and Zheng (1999)

Flow through salt formations in high level disposal sites, heat and solute movement near salt domes

Jackson and Watson (2001) Williams and Ranganathan (1994) Hassanizadeh and Leijnse (1988)

Heat and fluid flow in geothermal systems Oldenburg and Pruess (1999) Gvirtzman et al., (1997)

Sedimentary basin mass and heat transport processes, diagenesis processes

Garven et al., (2003) Sharp et al., (2001) Raffensperger and Vlassopoulous (1999)Wood and Hewett (1984)

Palaeohydrogeology of sedimentary basins Senger (1993) Gupta and Bair (1997)

Processes beneath playas, sabkhas and playa lakes Yechieli and Wood (2002) Sanford and Wood (2001) Simmons et al., (1999) Wooding et al., (1997a,b) Duffy and Al-Hassan (1988)

Operation of saline (and irrigation) water disposal basins Simmons et al., (2002) Density affects in applied tracer tests Barth et al., (2001)

Zhang et al., (1998) Istok and Humphrey (1995) Le Blanc et al., (1991)

Table 1. Some key physical systems where variable density groundwater flow

phenomena are important in groundwater hydrology.

130

Simulator References for code development and verification

CFEST Gupta et al., [1987]

FAST-C Holzbecher [1998]

FEFLOW Diersch [1988], Diersch [2005]

FEMWATER Lin et al., [1997]

HEATFLOW Frind [1982]

HST3D Kipp [1987]

HYDROGEOSPHERE Therrien et al., [2004]

METROPOL Leijnse and Hassanizadeh [1989]

MITSU3D Ibaraki [1998]

MOCDENSE Sanford and Konikow [1985]

MODHMS HydroGeoLogic Inc., [2003]

NAMMU Herbert et al., [1988]

PHT3D/SEAWAT-2000 Post and Prommer [2007]

ROCKFLOW Krohn [1991], Kolditz et al. [1995]

SEAWAT-2000 Langevin and Guo [2006]

SUTRA Voss [1984]

SWIFT Reeves et al., [1986]

TOUGH2 Oldenburg and Pruess [1995]

VapourT Mendoza and Frind [1990]

VARDEN Kuiper [1983, 1985], Kontis & Mandle [1988]

Table 2. Some numerical simulators for variable-density groundwater flow and solute

transport problems. This list is illustrative not exhaustive.

131

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Figure 1. Brine density as a function of NaCl mass fraction, computed for the conditions of

atmospheric pressure at temperature of 25oC. Source: Adams and Bachu [2002].

Figure 2. Brine density calculated at conditions characteristic of some typical sedimentary basins.

Source: Adams and Bachu [2002].

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Figure 3. Effect of spatial discretization on the computed salinity distribution at 4, 10, 15, and

20 years simulation time. Positions of the 20% and 60% isochlors are shown for: (a) coarse mesh

(1170 nodes, 1100 quadrilateral elements), comparable with the discretization used originally by

Elder [1967] and Voss and Souza [1987], (b) fine mesh (10,108 nodes, 9900 quadrilateral

elements). It is clear that the nature of the central upwelling and downwelling is very sensitive

to grid discretisation. Source: Diersch and Kolditz [2002].

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Figure 4. Developmental stages of unstable plume behaviour in Hele-Shaw cell laboratory experiment. The effective Rayleigh number for this system is Ra=4870. Elapsed times (a,b,....,h) given in dimensionless times are T=2.02, 4.03, 8.00, 10.01, 12.02, 13.98, 15.99, 18.00, measured from a virtual starting time of 12 hours and 15 minutes (the actual experimental starting time was 11 hours and 32 minutes). One unit in dimensionless time is equivalent to 21.4 minutes real time [after Wooding et al., 1997b; Simmons et al., 1999].

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Figure 5. Developmental stages of unstable plumes in the SUTRA solute transport model. Contours represent the concentration of the plumes. Contour interval is 2,000 mgL-1. The effective Rayleigh number for this system is Ra=4870. Elapsed times (a,b,...,h) given in dimensionless times are T=2.01, 4.02, 8.03, 10.05, 12.06, 14.06, 16.07,18.08, measured from start of numerical simulation. One unit in dimensionless times is equivalent to 21.4 minutes real time. Source: Simmons et al., [1999].

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Figure 6. Numerical model of salt water intrusion in a regional scale 3D model, Pioneer Valley,

Australia. Based upon Kuhanesan [2005] groundwater flow model. [Source: Werner and

Gallagher, 2006].

Figure 7. A comparison between observed salinity contours (grey lines) and predicted salinity

contours (black lines) from the a uncalibrated model and b calibrated model. The coastline is not

shown. [Source: Werner and Gallagher, 2006].

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Figure 8. Profiles of salinity distributions (relative salinity units used, whereby 1.0 is sea water).

The vertical axis scale is m AHD. [Source: Werner and Gallagher, 2006].

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Figure 9. Inferred occurrences of offshore fresh to brackish paleo-groundwater (modified from

Groen, 2002) and continental shelf areas that were exposed during the last glacial maximum (25-

15 ka BP).

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Figure 10. Variable density flow simulation showing salinization lags behind coastline migration

during transgression on a gently sloping surface and development of offshore brackish

groundwater (after Kooi et al., 2000). Highly unstable convective fingering is seen as dominant

vertical salinization mechanism.

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Figure 11. Basic concept of interface tilting leading to conical plume in the case of variable

density analysis of aquifer storage and recovery.

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Figure 12. An illustrative comparison between FEFLOW numerical model results for the cases

of no density, small density and large density contrasts. Results are given for: end of injection

phase, end of storage phase, and end of recovery phase. Each image is symmetrical about its left-

hand boundary, which represents the injection/extraction well where fresh water is injected into

the initially more saline aquifer. The effects of variable density flow are clearly visible. Increasing

the density contrast results in deviation from the standard cylindrical “bubble” most, as seen in

the no density difference case.

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50 days

150 days

200 days

250 days

300 days

350 days

100 days

Injection

Storage

Figure 13. Time series plot of a 100-day injection and 250-day storage, in an aquifer consisting of

6 even horizontal layers, with the hydraulic conductivity varying by a ratio of 10 between high

and low K layers. Note the fingering processes.

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Figure 14. Variable-density flow in an inclined fracture embedded in a 2D porous matrix. Red

colors refer to high concentration and blue colors refer to low concentration. Arrows represent

groundwater flow velocities.

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Figure 15. Variable density flow in an inclined fracture embedded in a 3D porous matrix. Arrows

represent groundwater flow velocities.

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Figure 16. Chemically reactive adaptation of the Elder short heater natural convection problem.

Width of the model domain is 600 m. Top: Plume configuration after 2920 days for the adapted

Elder problem modelled by Post and Prommer [2007] when no chemical reactions are

considered. Bottom: Plume configuration after 2920 days for the adapted Elder problem when

cation exchange and calcite equilibrium are included in the simulation. Darker shading represents

higher density.

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Chapter 1

Development and Application of Seawater Intrusion Models: theory, analytical, benchmarking and numerical developments, applications

Abdelkader Larabi

*Professor, LIMEN, Ecole Mohammadia d’Ingenieurs, B.P. 765, Agdal, Rabat – Morocco, E-Mail : [email protected]

Abstract

The management of groundwater resources involves the allocation of groundwater supplies and water quality to competing water demands and uses. The resource allocation problem is characterized by conflicting objectives and complex hydrologic and environmental constraints, especially in coastal aquifers. The development of mathematical simulation models provides groundwater planners with quantitative techniques for analyzing alternatives groundwater resources management. Coastal aquifers involve varying conditions in time and space, owing to the occurrence of seawater intrusion, leading to upconing due to intensive pumping near the coast. An overview of modelling seawater intrusion in coastal aquifers is presented, including the physical approaches that assume the occurrence of a sharp interface or solute transport mixing zone. Some analytical solutions have been also outlined as to serve benchmark problems for testing developed numerical algorithms for sharp and solute transport interfaces. Based on the two physical approaches, numerical models have been developed and using conventional finite differences, finite elements and boundary integral techniques. Results of some field applications are presented to show the utility and the capability of these tools of predicting accurate information with regard to seawater intrusion, its extent, interface upcoming, critical pumping rates and other information on management issues.These information are required to assist managers, planners and decision makers for an integrated water resources management as demonstrated by the regional models developed for some coastal aquifers.

Keywords: Coastal aquifers, Seawater intrusion, Modelling, Groundwater management.

1. Introduction

Management of a groundwater system means making such decisions as to the total quantity of water to be withdrawn annually, the locations of wells for pumping and for artificial recharge and their rates and control conditions at aquifer boundaries in addition to decisions related to groundwater quality. In order to assist the planner, or the decision maker, to compare alternative models of action and to ensure that the

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constraints are not violated, a tool is needed that will provide information about the response of the system to various alternatives. Good management requires the ability to predict the aquifer’s response to planned operations, that may take the form of changes in the water levels, changes in water quality, or land subsidence (Bear and Veruijt, 1987).

The necessary information about the response of the system can be provided by a model that describes the behavior of the considered system in response to excitation. During the past 40 years, the field of aquifer management modeling has developed as a distinct discipline. It has provided a framework which replaces trial and error simulations. Modern aquifer management models combine simulation tools with optimization techniques. This combination of the optimization and aquifer simulation provides a framework that forces the engineer or hydrogeologist to formulate carefully the groundwater management problem. The problem must contain a series of constraints on heads, drawdowns, pumping rates, hydraulic gradients, groundwater velocities, and solute concentrations. In addition, an objective, usually involving costs, cost surrogates, or risk, must be seated.

This is the case of aquifers in islands and coastal areas which are prone to seawater intrusion. Because seawater is denser than freshwater, it will invade aquifers which are hydraulically connected to the sea. Under natural conditions, freshwater recharge forms a lens that floats on top of a base of seawater. This equilibrium condition can be disturbed by changes in recharge and/or induced conditions of pumping and artificial recharge. Seawater intrusion must be addressed in managing groundwater resources in islands and coastal areas. The freshwater lens is very susceptible to contamination, and once contaminated, it can be very difficult to restore to pristine conditions. To plan counter measures, it is necessary to predict the behaviour of groundwater under various conditions, such as to determine the extent to which ingress of seawater occurs. As seawater intrusion progresses, existing pumping wells, especially close to the coast, become saline and have to be abandoned, thus reducing the value of the aquifer as source of freshwater. Also, the area above the intruding seawater wedge is lost as source of water.

The contact between freshwater and seawater in islands and coastal aquifers, requires special techniques of management, because of varying conditions in time and space, such as pumping near the coast. In response to pumping from a well in the fresh-water zone, the fresh-saltwater interface moves vertically toward the well causing the upconing phenomenon. The problem of saltwater intrusion can be analysed by two methods. The first method considers both fluids miscible and takes into account the existence of transition zone (Voss and Souza, 1984). The second method is based on an abrupt interface approximation (Bear and Verruijt, 1987) where it is assumed that (i) the freshwater and saltwater do not mix and (ii) are separated by a sharp interface. This method is considered as appropriate approximation in the case where the extent of the transition zone is relatively small compared to the aquifer thickness. The first attempts to solve the saltwater intrusion were based on the assumption of the existence of a sharp interface between freshwater and saltwater (Strack, 1976; Collins and Gelhar, 1971).

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In this paper, we discuss different aspects of seawater intrusion in coastal aquifers, since the famous works of Badon-Ghyben (1888). Some analytical solutions available for both sharp interface and transition zone approaches are also reviewed, as they serve as benchmark examples for testing developed numerical models and codes. Also, some 3-D numerical models are presented which enables the prediction of the fresh-saltwater interface displacement in response to varying conditions in time and space. Applications of these codes to some field cases are also given assuming both approaches of the abrupt interface and the mixing zone. The freshwater-saltwater interface is determined, including its location, shape and extent. Examples of the regional models, presented in this work, are based on the Galerkin finite element approach (Larabi and De Smedt, 1994) and the finite difference technique (Mcdonalds and Harbaugh, 1988). These models are capable to simulate groundwater flow, solute transport and the transient interface displacement due to groundwater exploitation, especially under upconing conditions due intensive pumping. Although simulation models provide the resource planner with important tools for managing the groundwater system, the predictive models do not identify the optimal groundwater development, design, or operational policies for an aquifer system. Hence, they should be combined to groundwater optimization models to identify the optimal groundwater planning or design alternatives in the context of the system’s objectives and constraints (Willis and Yeh, 1987).

2. Approaches to modeling seawater intrusion

2.1 Occurrence of seawater intrusion

The occurrence of seawater intrusion in coastal aquifers can be visualised with a simplified conceptual model (Figure 1). The essential elements are freshwater recharge from inland sources, seawater intrusion from the ocean and an interface or transition zone between the two types of water. The transition zone is a zone of mixing between the freshwater and the saltwater (Figure 2). Under equilibrium conditions, the interface will remain fixed, and freshwater will discharge along the seepage face. In coastal aquifers, recharge is predominantly due to lateral inflow, whereas in island aquifers, it is due to vertical recharge. These conceptual models are analysed in different ways. Coastal aquifers are usually modelled in the vertical plane. Island systems are also analysed in the vertical plane when the thickness of the freshwater lens is a significant fraction of the horizontal width of the lens, a situation common in atoll islands. When the lens thickness is small compared with its horizontal extent, the system can often be treated in the horizontal plane.

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Figure 1. Schematic sketch of saltwater intrusion, sharp interface model.

2.2 Approaches for sharp interface and miscible flow models

The first question in developing a simulation model is whether it is appropriate to deal with the problem as miscible or immiscible flow. If the transition zone is thin relatively to the thickness of the freshwater lens and it is immobile, then it is appropriate to assume that the freshwater and saltwater don’t mix (immiscible), and the transition zone is considered to be a sharp interface.

Figure 2. Salt/fresh water transition zone in multi-layered aquifer.

There are several methods for dealing with seawater intrusion as two-phase, immiscible problem. These methods are generally known as interface methods. Under very dynamic conditions, or in case where the freshwater lens is relatively thin compared with the transition zone, it may be necessary to adopt miscible flow models. Reilly and Goodman (1985) presented a historical perspective of quantitative analysis of the saltwater freshwater relationship in groundwater systems. Custodio (1992) reported that both approaches produce good results if applied to the right

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circumstance. Pinder and Stothoff (1988) pointed out that the sharp interface assumption is appropriate for steady state conditions. Additionally, if conditions of variable recharge and/or significant pumping exist, the interface will move in response to the non equilibrium conditions. For this problem, the hydraulic head and fluid velocity in both the fresh and salt portions of the aquifer must be considered. Since these quantities are unknown, the problem becomes one of coupling differential equations for motion of the freshwater and saltwater, and solving for the unknown position of the interface between them.

In fact when the sharp interface approximation is inappropriate owing to the large thickness of the transition zone relative to the freshwater lens, it is advisable to treat the freshwater and saltwater as miscible. There are two elements to the problem: 1) a single fluid phase with variable density; and 2) a solute representing the total dissolved solids in seawater, which affects the fluid density. As a result, there are four unknown dependent variables: fluid specific discharge, fluid pressure, salt concentration, and fluid density, requiring the solution of four equations (Frind and Mangata, 1985). Often the most disruptive influence is the effect of pumping activities. Withdrawal of water from the freshwater lens via pumping wells creates saltwater upconning, which is the migration of the fresh-salt interface towards a well in response to pumping withdrawals. However, Bear (1979) has given some examples of real aquifers in which experimental measurements have shown the salt concentration to vary sharply at a determined location, clearly establishing a region of low salt concentration (freshwater) and a region of high salt concentration (saltwater).

3. Analytical solutions and benchmarking

3.1 Key role of analytical solutions

In fact, analytical solutions can not solve ‘real-world’ problems, due to their simplified physical assumptions and geometry. But they play an important role to serve a number of important purposes such as instructional tools to present fundamental insights to clearly understand the mechanical trend of the groundwater flow in coastal aquifers, while numerical solutions are often difficult to provide it.

In addition and despite their simplifying assumptions, analytical solutions can also be used as a tool for first-cut engineering analysis in a feasibility study. More sophisticated models normally require hydrological and hydrogeological information that is either not available or not known to the required solution at the time of initial investigation. Hence, without the support of reliable input data, these sophisticated models cannot provide more accurate results. That is why before a large scale site study, or a comprehensive numerical modelling, it is often interesting to perform some preliminary calculations based on basic information and simplifying assumptions.

However, still the most important contribution of the analytical solutions is to serve as benchmark problems for testing numerical algorithms. Numerical models, especially newly developed ones based on numerical techniques such as FE, FD, BE

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or mixed techniques, need to be verified and validated against programming and other errors before distribution for use and/or for real world applications.

3.2 Interface models

An exact mathematical statement of the saltwater intrusion problem, assuming that an abrupt interface separates fresh water and saltwater, was presented by Bear (1979). However, in order to simplify the problem, the Ghyben-Herzberg relationship can be introduced. The vertical position of the saltwater interface measured from the sea level, zi, is given by

z=z-

= ww

fs

fiz (1)

where = ( s - f)/ f and zw is the position of the water table measured above sea level. Assuming a sea water density s of 1025 kg/m3, and a fresh water density f of 1000 kg/m3, this relationship predicts that there will be approximately 40 times as much fresh water in the lens beneath sea level as there is fresh water above sea level.

Bear and Dagan (1964) investigated the validity of the Ghyben-Herzberg relationship. The hodograph method was used to derive an exact solution and it was shown that close to the coast the depth of the interface is greater than that predicted by the Ghyben-Herzberg relationship.

Analytical solutions for the interface problem based on the Dupuit assumption can be found in the works of Strack (1987), Van Dam (1983), and Bear (1979). These analytical models give reasonable results if the aquifer is shallow, but lack accuracy in the region of significant vertical flow components. This assumption is also incapable to handle anisotropy and nonhomogeneity of the aquifer. Only a few analytical solutions are available for layered aquifers (Mualem and Bear, 1974; Collins and Gelhar, 1971).

The simplest stationary interface solution based on this assumption was proposed by Glover (1959) for confined flow, later extended by Van der Veer (1977) to include phreatic flow. In the Glover solution the aquifer is assumed to be confined, with a bounding layer positioned at sea level. The origin of the coordinate system is chosen at the contact point of the sea and the continent. The saltwater interface can be considered as a streamline and, its position can be determined as a function of the distance from the shoreline, (Glover, 1959)

x K

2Q+KQ=

2

iz (2)

where Q is the constant total freshwater flow per unit length of shoreline (L2T-1). The width of the region through which freshwater discharges to the ocean is given by

K2Q-=0x (3)

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If the fresh water flow Q can be estimated or measured, then equation (2) provides a simple method to calculate the depth to saltwater in the coastal zone. Alternatively x0 can be estimated.

Detournay and Strack (1988) derived an approximate analytical solution technique to solve groundwater flow which involves the determination of a phreatic surface, an interface separating flowing fresh water from saltwater at rest, and seepage and straight outflow faces where fresh water discharges into the sea, by application of conformal mapping techniques using the hodograph method. Results are presented in a table, where the coordinates of the phreatic surface and the interface are calculated for the flow case in which the coast and the sea bottom slopes have angles of respectively 30o and 15o with the x-axis.Van der Veer (1977) has proposed an analytical solution for the steady interface flow in coastal aquifer flow systems involving recharge on the top phreatic surface.

Strack (1976), developed an analytical solution for regional interface problems in coastal aquifers based on single-valued potentials and the Dupuit assumption and the Ghyben-Herzberg formula, in the context of steady state. The critical pumping rate has been also computed in the case of a fully penetrating well. Motz (1992) proposed an analytical solution for calculating the critical pumped flow rate in an artesian aquifer for a critical interface rise supposed to be 0.3(b – l), where b denotes the aquifer thickness and l the distance from top of aquifer to bottom of well screen (Figure 2). Bower et al.(1999) modified the critical interface rise based on an analytical solution, which allows the critical pumping rate to be increased.

Figure 3. Saltwater upconing beneath a well

Solutions for the miscible fluid model with dispersion have also been obtained by a number of investigators. Henry (1964) used a Fourrier-Galerkin method, but was restricted in the choice of the parameters by convergence requirements of the method. This was the first solution for predicting the steady state salt distribution in a cross section of a confined coastal aquifer.

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3.3 Benchmark models

For the sharp interface benchmarking, a 3-D sand model (Sugio, 1992) has been designed to study saltwater intrusion into freshwater in a sand box. This laboratory model consists of a 3-D sand box, 163.8cm long and 47.5cm high, while the width of the model has two values : 63.2cm from the downstream end until the length 82.3cm, and 30cm in the remaining part. Salt was added to the fresh water and thoroughly mixed up to the density of 1030kg/m3 to model sea water, and then was colored by dye in order to observe the position of the interface separating fresh water and saltwater. Sand with 0.76mm mean diameter was used for which the hydraulic conductivity was obtained from measurements (Sugio, 1992) as 1.293cm/s. The upstream and downstream water levels are respectively h1 = 44.15cm and zs = 40.67cm. Fresh water flows through the sand model for about three hours to achieve steady state conditions. The behaviour of the fresh water-saltwater interface was measured on the front, side and bottom sections of the box model. This sand model is used to test the numerical model capability to accurately simulate 3-D groundwater flow problems with a fresh water-saltwater interface as it was demonstrated by the GEO_SWIM code (Larabi, 1994).

Three well known benchmark problems in the literature have been selected to be presented for validation purposes of the developed numerical models based on variable density dependent flow and solute transport in coastal aquifers:

The Henry’s problem: This test (Henry, 1964) deals with the advance of a salt water front in a uniform isotropic aquifer limited at the top and the bottom by impermeable boundaries recharged at one side by a constant freshwater i0nflux, and faced the sea on the other side with hydrostatic pressure distribution. For the mass fraction conditions, the influx side is at c= 0, the top part of the sea side is treated in a manner to permit convective mass transport out of the system (outflow face) and the remained part of the seaside is at c=1. Many numerical codes have been evaluated and tested with the Henry’s solution. Segol (1993) showed, however that the Henry’s solution was not exact because Henry (1964) eliminated, for computational reasons, mathematical terms from the solution that he thought to be insignificant. When Segol (1993) recalculated Henry’s solution with the additional terms, the improved answer was slightly different from the original solution. With the new solution, Segol (1993) showed that numerical codes, such as SUTRA (Voss, 1984), could reproduce the correct answer for the Henry problem.

The Elder’s problem: This case serves as an example of free convection phenomenon originally used for numerical analysis of thermally driven convection. After adaptation, this test benchmark is used to verify the accuracy of a model in representing fluid flow purely driven by density differences. The large density contrast (1.2 to 1) constitutes a strong test for coupled flow problems. The domain is represented by a rectangular box with a source of solute at the top taken as a specified concentration boundary (c=1), whereas the bottom is maintained at zero concentration

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(c = 0). For the flow conditions, zero value of equivalent freshwater head is attributed to the upper corners. Several authors studied this case, among others, Voss and Souza (1987), Oldenburg and Pruess (1995), Kolditz et al.(1998) Aharmouch and Larabi (2004). This problem was originally designed for heat flow (Elder, 1967), but Voss and Souza (1987) recast the problem as a variable density groundwater problem in which fluid density is a function of salt concentration.

The salt dome (HYDROCOIN level 1, case 5): The purpose of the Hydrologic Code Intercomparison (HYDROCOIN) project was to evaluate the accuracy of selected groundwater modelling codes. One of the problems used for testing is the HYDROCOIN problem. This test is used to model variable density groundwater over a salt dome. The boundary conditions are: the hydraulic head varies linearly on the top of the aquifer whereas the other sides are impervious. The mass fraction (c = 0) on the top at the inflow part while the central base of the aquifer is at c =1. For this problem, Herbert et al.(1988) presented a steady state solution based on the parameter stepping technique.

More literature on these benchmark examples can be found in several papers such as those of Voss and Souza (1987), Oldenburg and Pruess (1995) and Kolditz et al. (1998), among other authors.

4. Numerical models

4.1 Numerical methods for groundwater management

Numerical modeling has emerged over the past 40 years as one of the primary tools that hydrologists use to understand groundwater flow and saltwater movement in coastal aquifers. Numerical models are mathematical representations (or approximations) of groundwater systems in which the important physical processes that occur in the systems are represented by mathematical equations. The governing equations are solved by mathematical numerical techniques (such as finite-difference, finite-element, boundary finite element, or finite volume methods) that are implemented in computer codes. The primary benefit of numerical modeling is that it provides a means to represent, in a simplified way, the key features of what are often complex systems in a form that allows for analysis of past, present, and future groundwater flow and saltwater movement in coastal aquifers. Such analysis is often impractical, or impossible, to perform by field studies alone. Numerical models have been developed to simulate groundwater flow solely or groundwater flow coupled to solute transport (the movement of chemical species through an aquifer). For a number of reasons, numerical models that simulate groundwater flow and solute transport are more difficult to develop and to solve than those that simulate groundwater flow alone. Coastal aquifers are particularly difficult to simulate numerically because the density of the water and the concentrations of chemical species dissolved in the water can vary substantially throughout the modeled area. Numerical models based on the sharp interface approximation assumption are reported in Mercer et al. (1980), Polo and

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Ramis (1983), Essaid (1990), Larabi and De Smedt (1994a, 1994b, 1997). Huyakom et al. (1987), Galeati et al. (1992), Das and Datta (1995, 1998), Putti and Paniconi (1995), Langevin and Guo (2002), and Aharmouch and Larabi (2004) present some of the recent developments in modeling of seawater intrusion in coastal aquifers for density dependent miscible flow and transport.

Two and Three-dimensional, dynamic numerical models handle more complex situations under realistic representation of the mechanisms involved and account for variability in physical situations. Three-dimensional models are more realistic and yields better insight. They are particularly effective in handling irregularities in coastal aquifer geometry, water use; heterogeneity in aquifer formations; temporal variations in boundary conditions; and spatial variations in salt concentrations. Therefore, these models can provide greater detail and accurate results. The only drawback is related to the intensive computational cost (memory and CPU) and the amount of data needed to feed up these models.

4.2 Governing equations for groundwater flow and solute transport models

Analysis of saltwater intrusion, assuming that mixing occurs at the transition zone between seawater and freshwater, because of hydrodynamic dispersion, requires the solution of two partial differential equations representing the mass conservation principle for variable-density fluid (flow equation) and for the dissolved solute (transport equation). The mathematical model of density-dependent flow and transport in groundwater is expressed here in terms of an equivalent freshwater total head (for more details see Langevin, 2003): The variable density groundwater flow is described by the following partial differential equation:

( )f ff f f

f

Chz n qh SK t C t(4)

Where:

x,y, and z are coordinate directions; z is aligned with gravity (L). is fluid density (ML-3).

Kf is the equivalent fresh water hydraulic conductivity (LT-1).hf is the equivalent fresh water head (L).

f is the density of fresh water (ML-3).Sf is equivalent fresh water storage coefficient (L-1).t is time (T). n is porosity (L0). C is the concentration of the dissolved constituent that affects fluid density (ML-3).

is the fluid density of a source or a sink (ML-3).

q is the flow rate of the source or sink (T-1).

To solve the variable density ground water flow equation, the solute-transport equation also must be solved because fluid density is a function of solute

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concentration, and concentration may change in response to the groundwater flow field. For dissolved constituents that are conservative, such as those found in seawater, the solute transport equation is:

ss

qC D C C Ct n (5)

Where:

D is the dispersion coefficient (L2T-1).

is the groundwater flow velocity (LT-1).

qs is the flux of a source or sink (T-1).

Cs is the concentration of the source or sinks (ML-3).

4.3 Recent Numerical Codes

Currently, several solute transport models suitable for the simulation of seawater intrusion and up-conning of saline water beneath pumping sites are available. These include SUTRA (Voss, 1984), which is a two-dimensional, vertical cross-sectional model; SEAWAT (Guo and Bennett, 1998), CODESA3D (Gambolati et. al., 1999), GEO_SWIM (Aharmouch and Larabi, 2004) and FEFLOW (Diersch, 2004) are examples of three-dimensional models. These models provide solution of two simultaneous, non-linear, partial differential equations that describe the conservation of “mass of fluid” and “conservation of mass of solute” in porous media; as described above.

SUTRA

SUTRA (which is named from the acronym Saturated Unsaturated TRAnsport) was published by the United States Geological Survey (Voss, 1984). The model is two-dimensional and can be applied either aerially or in a cross-section to make a profile model. The equations are solved by a combination of finite element and integrated finite difference methods. Although SUTRA is a two-dimensional model, a three-dimensional element is provided since the thickness of the two-dimensional region may vary over the solution domain. The coordinate system may be either Cartesian or radial which makes it possible to simulate phenomena such as saline up-coning beneath a pumped well. SUTRA permits sources, sinks and boundary conditions of fluid and salinity to vary both spatially and with time. The dispersion processes available within SUTRA are particularly comprehensive. They include diffusion and a velocity-dependent dispersion process for anisotropic media. This means that it is possible to model the variation of dispersivity when the flow direction is not along the principal axis of aquifer transmissivity.

SEAWAT

The original SEAWAT code was written by Guo and Bennett (1998) to simulate ground water flow and salt water intrusion in coastal environments. SEAWAT uses a modified version of MODFLOW (McDonald and Harbaugh 1988) to solve the variable density, ground water flow equation and MT3D (Zheng, 1990) to solve the

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solute-transport equation. To minimize complexity and runtimes, the SEAWAT code uses a one-step lag between solutions of flow and transport. This means that MT3D runs for a time step, and then MODFLOW runs for the same time step using the last concentrations from MT3D to calculate the density terms in the flow equation. For the next time step, velocities from the current MODFLOW solution are used by MT3D to solve the transport equation. For most simulations, the one-step lag does not introduce significant error, and the error can be reduced or evaluated by decreasing the length of the time step.

CODESA3D

CODESA-3D (COupled variable DEnsity and SAturation 3-Dimensional model) can solve the density-dependent miscible flow and transport equations and predict solute migration. It is a three-dimensional finite element simulator for flow coupled to solute transport in variably saturated porous media on unstructured domains (Gambolati et. al., 1999; Lecca, 2000). The numerical model is a standard finite element Galerkin scheme, with tetrahedral elements and linear basis functions, and weighted finite differences are used for the discretization of the time derivatives. Model parameters that are spatially dependent are considered constant within each tetrahedral element. Parameters that depend on pressure head and/or concentration are evaluated using values averaged over each element and are also element-wise constant. The CODESA-3D code is born from the integration and extension of parent codes developed, during the last ten years, by the Department of Applied Mathematics of the University of Padova in collaboration with the Environment Group of CRS4 (Center for Advanced Studies, Research and Development).

GEO_SWIM

GEO_SWIM (GEOprofessional SaltWater Intrusion Model) is a 3-dimensional finite element model for variable density flow and solute transport in variably saturated porous media. The Galerkin technique is used for the spatial discretization, using hexahedral elements, with a finite difference for time discretization. The mathematical model is made of a set of nonlinear coupled PDE’s of flow and solute transport to be solved for hydraulic head and mass fraction of the solute component. The flow equation simulates the water movement in the porous medium whereas the transport equation calculates the displacement of the dissolved salt. The accuracy of the model has been checked via the most commonly used benchmark tests (Aharmouch and Larabi, 2004 and 2005), either for seawater intrusion (Henry’s problem) or brine transport (Elder’s problem and Hydrocoin test). Application of the model had been realized in the case of seawater intrusion in some coastal aquifers situated in Morocco (Aharmouch and Larabi, 2004). A version of the code also exists for handling the sharp interface case (Larabi and De Smedt, 1997).

FEFLOW

FEFLOW (Diersch, 2004) is a finite element groundwater modelling commercial code developed by WASY AG (Germany). The code solves groundwater flow, solute transport, density dependent solute transport, coupled heat and flow, saturated and unsaturated flow. Different types of licenses are available at different prices

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depending on the type of problems that the user needs to solve. The cheapest version includes only 2D groundwater flow, while the most expensive includes 3D flow, transport, and heat processes. The code includes different types of elements that can be combined. This allows for example to model 2D planar structures such as a fault plane within a porous matrix or linear elements such as a drain or a stream in which the flow equation may be different than within the porous matrix. The code solves both steady state and transient equations.

Other codes have been also developed in USA such as SHARP, which is a quasi-three-dimensional, numerical finite-difference model to simulate freshwater and saltwater flow separated by a sharp interface in layered coastal aquifer systems; and FEMWATER which is a 3D finite element, saturated / unsaturated, density driven flow and transport model.

5. Verification of the models and Applications (GEO_SWIM)

5.1 The Glover problem:

To test the validity and the accuracy of the proposed numerical scheme, several benchmark problems with available analytical solutions (Glover, 1959; Strack, 1976; Van der Veer, 1977a; Motz, 1992) and numerical solutions (Larabi and De Smedt, 1997) are solved and compared.

Figure 4. Glover (1959) problem.

This example treats a steady state two-dimensional seawater intrusion as depicted in Figure 4. Glover (1959) presented an analytical solution for this case. For the numerical analysis, an arbitrary length of the flow domain is taken as 420 m in the x -direction. The flow domain is discretized with hexahedral elements of size 4 4 3m3. This yields grids with 106, 2 and 10 nodes respectively in the x -, y - and z -directions. The domain contains 2,120 nodes and 945 elements. For the upstream freshwater outflow values, it is taken to be 3.9 m2/day for case 1 and 18.8 m2/day for case 2. Other parameters of the problem are 31 g/cmf , 31.029 g/cms , and

69 m/dayK .

In the present problem, the aquifer is confined; hence there is no unsaturated zone. In Figure 5, we present the interface location found from the numerical solution

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in solid lines for the two cases with different outflow rate. For comparison, Glover’s analytical solution is plotted as dash lines. We observe excellent agreement.

Figure 5. Comparison of numerical solution and Glover (1959) analytical solution.

5.2 Upconing beneath a pumping well in an unconfined aquifer (Strack, 1976)

This example deals with an unconfined aquifer, meaning that there exist a phreatic surface (water table). Also, there is a pumping well located in the freshwater zone, which draws down the water table, causing the saltwater/freshwater interface to rise. This phenomenon is called upconing. This problem is modeled after Strack (1976), which not only uses the Ghyben-Herzberg assumption, but also the Dupuit assumption of nearly horizontal flow. The problem becomes two-dimensional in the horizontal plane. In the present work, however, the Dupuit assumption is not taken and the problem is solved as a three-dimensional one.

The domain is of the size 1,000m 4,000m 21m respectively along the x , yand z directions. Due to the symmetry with respect to the y -axis, only half of the domain is discretized. A non-uniform grid is used for the xy -plane, where a minimum grid spacing of 5 m along the x- and y-axis is used to account for the steep hydraulic gradients in the vicinity of the pumping well, whereas a maximum of 100 m is used elsewhere (39 columns 31 rows). The vertical axis is subdivided into 11 planes. A constant spacing is used from mean sea level to the aquifer bottom and another plane is taken at z = 0.89 m, above the sea level. The total number of nodes is 13,299 with 11,400 elements.

Table 1 shows the parameters used in the simulation, which follows those in Huyakorn, et al. (1996). For the boundary condition, the imposed flux 21 m /dayq is for two-dimensional planar simulation, which is modified to a specific discharge of

0.048 m/dayq for the present case. For the coastal boundary condition we use a fixed hydraulic head ( 0h at 0x ) as taken by Strack (1976).

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Table 1. Parameters used for the Strack (1976) problem.

Parameters Values

Fresh water flux fQ 1 m2/day ( at x = 1000 m)

Well pumping rate wQ 400 m3/day

Well location ,( )w wx y (600 m, 0)

Hydraulic conductivity K 70 m/day

Depth from mean sea level to aquifer base 20 m

Figure 6. Interface locations provided by GEO-SWIM Model

and the analytical solution, for the Strack’s problem.

5.3. Saltwater intrusion in an unconfined aquifer with recharge (Van Der Veer, 1977)

Van Der Veer (1977a) proposed an analytical solution for steady state flow in an unconfined coastal aquifer where both freshwater-saltwater interface and water table are present. This solution differs from Strack (1976) solution in that the flow in vertical direction is fully considered. This consideration of the vertical direction

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allows the existence of a freshwater outflow face at where the aquifer meets the sea (Figure 7).

The solution region is 2,000 m in length and 410 m in thickness. For the numerical solution, 202 2 51 hexahedral elements are used. The portion above the sea level is refined more than the portion below, in order to determine the water table location more accurately. The spacing in the x-direction is 10 m for the first 1960 m and 8 m for the remainder, and 5 m for the y-direction. In the z-direction, it varies for from 1 m above sea level to 10 m below it. The other characteristics of the problem are 31 g/cmf , 31.020 g/cms and 10 m/dayK . The boundary conditions used are as follows: at x=2000, where the aquifer is exposed to the sea, the constant head condition due to hydrostatic sea water is applied. On the top of aquifer, there is a replenishment of 0.004 m/day .

Figure 7 shows the steady freshwater-saltwater interface and the phreatic surface for both the Van der Veer (1997a) analytical solution and the present numerical solutions.

0 500 1000 1500 2000X (m)

-400

-300

-200

-100

0

Z(m

)

Dash: Present model solution (run 1)Solid Present model solution (run 12)Dash-dot: Analytical solution (Van der Veer, 1977a)

Figure 7. Water table and interface positions produced by GEO-SWIM and Van der Veer (1997a) analytical solution (dash-dot line).

5.4. Upconing in a confined aquifer overlain by an aquitard (Motz, 1992)

This example involving a confined aquifer overlain by an aquitard with a pumping well in the aquifer was investigated by Motz (1992, 1997). The data is based on a field case in Pasco County, Florida (Motz, 1992), and is summarized in Table 2. For the numerical solution, the aquifer is represented by a square as described by Huyakorn, et al. (1996). Due to symmetry, only the first quadrant of the three-dimensional domain was simulated, with pumping well located at 0x y (Figure 6). The size of the x-y plane is 15000 m 15000 m . The mesh is non-uniformly spaced:

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from the origin until ( , ) (2000 m,0 m)x y and ( , ) (0 m,2000 m)x y , a regular spacing of 50 m is used. Beyond these two points, greater spacing of 1000 m is used. Thus the mesh consists of 54 rows × 54 columns. The z -direction is subdivided into 21 irregularly spaced planes. The first two planes are located at 3 mz and

39 mz respectively. Constant spacing ( 24.56 mz ) is used along the well screen ( 39 mz to 260 mz ). Another constant spacing ( 17.9 mz ) is used from 260 mz to 439 mz . The hexahedral element mesh is composed of 61,236 nodes with 56,180 elements. Several steady state simulations are performed for different pumping rates: 25, 50, 75,100,120,140 Q and 180 kg/s.

Figure 8 shows the saltwater-freshwater interface below the pumping well for both the numerical solution and the Motz (1992) analytical solution. For low pumping rates (25, 50 and 75 kg/s), the comparison between the two solutions shows very good agreement. With increasing pumped rates (100, 120 and 140 kg/s), the agreement is still good, except in the vicinity of the pumping well. For these pumping rates, the differences between the numerical and analytical solution for the maximum interface rise are 1.8%, 4.1%, 6.8%, 9.8%, 12.3%, and 14.8%, respectively, with reference to the freshwater lens thickness of 439 m.

It should be mentioned that the above flow rates are chosen to be less or equal to the critical pumping rate 140cQ kg/s as found by Motz (1992), which allows the interface to rise up a distance about 30% between the pumping well and the undisturbed interface. For the last pumping rate (180 kg/s) simulated here, the numerical and the analytical solutions show significant difference. As pointed out in Motz (1992), the analytical solution is limited to small interface rises; hence the discrepancy with the present three-dimensional solution is anticipated.

Table 2. Parameters for the Motz (1997) problem.

Parameters Value

Initial elevation of the saltwater-freshwater interface

Initial freshwater head

Horizontal intrinsic permeability xk , yk

Leakance of the confining bed /zk b

Freshwater head in overlying constant head source bed

Freshwater head at constant head boundary nodes

Bottom elevation of well screen

Length of well screen

Distance from well bottom to the initial interface

-438.8 m

10.9 m -129.26 10 m2

-31.2 10 /day

10.97 m

10.97 m

260.0 m

221.0 m

178.8 m

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0 500 1000 1500 2000X (m)

-450

-425

-400

-375

-350

-325

-300

Z(m

)

Solid line: Present model solutionDashdot line: Analytical solution (Motz, 1992)

Initial interface

Figure 8. Interface position for both numerical and Motz (1992) analytical solutions, respectively for pumping rates of 25, 50, 75, 100, 120, 140 and 180 kg/s.

5.5 The Henry’s problem:

This test (Henry, 1964) deals with the advance of a saltwater front in a uniform isotropic aquifer, limited at the top and the bottom by impermeable boundaries, recharged at one side by a constant freshwater influx, and faced the sea on the other side. For the flow conditions the seaside is considered with hydrostatic pressure distribution whereas an imposed flux of sm1066 5

0 /.q is applied to the upstream side. For mass fraction conditions, the influx side is at .0c , the top part of the seaside is treated in a manner to permit advective mass transport to flush out of the system (outflow face condition) and the remained part is at .1c The results in steady state is given in Figure 10, 11 and 12 respectively for the isochlor distribution, the streamlines and the equivalent freshwater heads and velocity field.

The boundary conditions used in this test are the same used previously by Frind (1982); Putti and Paniconi (1995). The studied domain is represented by a two-dimensional refined mesh consisting of 41 columns and 21 rows ( cm5zx )yielding 861 nodes and 800 quadrilateral finite elements.

Figure 9. Description and boundary condition for the Henry’s problem

Simulation runs are performed with either isotropic dispersivity ( 0350.TL ) or constant dispersion tensor ( secm1066 26 /.D ) for the cases of density ratio of 0.0245, 0.0500 and 0.1000. The results in steady state are given in

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Figure 10, 11 and 12 respectively for the isochlor distribution, the streamlines and the equivalent freshwater heads and velocity field.

Figure 10. Steady state solution of the Henry’s problem. The lines indicate the isochlors.

Figure 11. Steady state solution of the Henry’s problem. The stream lines show the re-circulating saltwater.

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Figure 12. Steady state solution of the Henry’s problem. Equivalent freshwater heads and Velocity field.

5.6 Elder’s problem:

This case serves as an example of free convection phenomenon. The large density contrast (1.2 to 1.) constitutes a strong test for coupled flow problems.

The domain is represented by a rectangular box with a source of solute at the top taken as a specified concentration boundary (c=1.), whereas the bottom is maintained at zero concentration (c=0.). For the flow conditions, zero value of equivalent freshwater head is attributed to the upper corners. Parameters necessary for running this test are the same used previously by Voss and Souza (1987) and Kolditz et al.(1998).

Figure 13. Description and boundary conditions of Elder’s problem

The mesh used is relatively fine and consists of 80 elements horizontally and 50 elements vertically (Kolditz et al., 1998). For time discretization we used dynamic stepping strategy, starting with low time steps which are increased or decreased depending on the convergence behaviour, in conjunction with fully implicit Euler method. Simulations are performed for salinity patterns at 2, 4, 10, 15 and 20 years for which Figure 14 presents 2 and 4 years salinity patterns.

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Figure 14. Salinity patterns for 2 and 4 years elapsed time

5.7 Hydrocoin test:

This test is used to model variable density groundwater over a salt dome. The modeled region is a vertical section of 300m height and 900m width. The salt dome is located at the central section base at 300m depth. The boundary conditions used and the parameters necessary to this test case are those taken by (Herbert et al., 1988; Oldenburg and Pruess, 1995; Kolditz et al., 1998). A fine mesh consisting on 91 columns and 31 rows uniformly spaced is used

Two simulation sets are performed until dynamic equilibrium was reached. These sets consider purely dispersive ( L = 20 m, T =2 m, D0=0 m²s-1) and purely diffusive ( L = T =0 m, D0=5 10-7 m²s-1) systems. For each system, simulations are conducted for salinity patterns around the salt dome respectively for the two systems of flow are determined (Figure 15 and 16)

For the case of the purely dispersive system, the salinity patterns (Figure 15) seems to reproduce those obtained by Kolditz et al. (1998) and more or less those of Herbert et al. (1988). In this case, the flushing water, coming from the aquifer top, seems to be unable to carry away the dissolved salt. Instead, a stagnation zone is taking place.

Figure 15. Results for purely-dispersive flow ( L =20 m, T =2 m, D0 = 0 m²s-1)

In the case of a purely diffusive system (constant dispersion tensor), we perform long term transient simulations until reaching the equilibrium. The obtained results

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show practically the same salinity patterns (Figure 16).

Figure 16. Results for purely diffusive flow (D0 = 5 10-7 m²s-1)

6. Case studies

6.1 The Martil coastal aquifer (GEO_SWIM code)

The Martil aquifer is situated in the North-Western part of Morocco on the Mediterranean coast (Figure 17). It is the important local aquifer for potable water supply, irrigation and industry in the Martil region. In accordance with geology, the aquifer system consists of two aquifer units, separated by an aquitard. The upper unit is a Quaternary alluvial deposits of Martil river, the lower unit is composed of sandstone-limestone Pliocene formation, whereas the aquitard is generally marly and clayey. The aquifer thickness is very irregular in different directions. Figure 18a shows a cross-section from West to East.

Figure 17. Location map of the Martil aquifer

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The conceptual model used in this study is a multi-layer aquifer system of variable thickness. The finite element mesh used for the numerical simulation of the Martil aquifer, shown in Figure 3b, is adjusted to fit the structure and the extension of the hydrogeological layers and the wells location. The 3-D mesh contains 56232 nodes, and 48288 hexahedral elements. The saturated hydraulic conductivity for different aquifers are Ks = 3.5 m/day in the alluvial material (upper), Ks = 4.5 m/day for the sandstone-limestone (lower), and Ks = 0.05 m/day for the marl/clay aquitard [9]. The other parameters are held constant ( s = 0.45, r = 0.1).

Figure 18a. Martil West-East cross sectional view

Figure 18b. Martil 3-D structured mesh.

The boundaries are treated as impervious, except at the eastern part, which is in direct contact with the sea (outflow face). Its extent is not known and constitutes a part of the problem. We also consider Dirichlet conditions at nodes belonging to the Martil and Alila rivers. In order to take into account the dry seasons these values are not set necessarily to be equal to the elevation. For the effective rainfall, we use a non-uniformly distribution taking into account the variably material distribution on the top layer. In a first step steady state simulations are performed under natural conditions such that natural groundwater flow can be reproduced efficiently. The

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purpose is to adjust conductivities and effective recharge. The obtained results are compared to observed piezometric map of 1966. The calibration is done using the trial and error technique. This procedure is repeated until a satisfactory pattern is found. Computed groundwater potential heads, shown in Figure 19, are compared to recorded values at the observation wells in 1966. The correspondence is relatively good. The 1966 interface location is also illustrated in Figure 20.

Figure 19. Computed (left) and observed (right) steady state groundwater potentials in the phreatic aquifer for 1966.

A second set of simulations is performed starting from the 1966 head distribution, to predict the seawater intrusion in the aquifer. A transient simulation is made of over 30 years, considering variable pumping rates used for water public supply shown in Table 4. For irrigation, an amount of 7949 m3/day is pumped from several wells to irrigate 300 ha of agricultural area. The recharge is maintained constant along the simulation time.

Table 4. ONEP pumping rates

Well number Pumping rate (m3/day)

992/2 780

994/2 1351.56

995/2 1181

996/2 418.75

1153/2 1260

1174/2 248.4

The results shown in Figure 21 show that the saltwater interface starts to move until 1982 and then remains stable along the time, although pumping wells withdraw upstream in the aquifer. This simulation scenario shows that there is no significant displacement of the interface due to the ONEP pumping wells upstream of the Martil river.

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Figure 20. Plane view of the 1966 interface to the location.

Figure 21. Cross-sectional view of the moving interface positions from 1966 to 2014 (a new steady state regime is reached since 1982).

6.2 The Oued-Laou coastal aquifer (GEO_SWIM code)

The Oued-Laou coastal aquifer is located 48 km to the southeast of Tétouan city, in the northwestern part of Morocco, on the Mediterranean coast. This 18 km² alluvial aquifer is the region’s major groundwater source for potable water supply and irrigation. The aquifer is widely opened on the sea and it is composed of plio-quaternary material (El Amrani et al., 2004).

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Y

X

Z

Medi

terr

anea

nSe

a

Figure 22. 3-D structured mesh of Oued-Laou aquifer

The conceptual model used in this study is a one-layer aquifer system of variable thickness. Figure 22 shows the 3-D structured mesh of Oued-Laou aquifer. The mesh is adjusted to fit the aquifer geometry and the well locations. The GEO_SWIM code (abrupt interface version) was applied to the aquifer system to describe groundwater flow to determine the seawater intrusion interface behavior.

Figure 23. Plan view of the aquifer with pumping well locations.

Figure 24. Position of the interface toe (scenario 2).

Present model (dashed and solid lines for sets 1 and 2 respectively)

Two wells are located as shown in Figure 27, one close to the coastline with pumping rate of 661.3 m3/day for potable water supply, and one inland with pumping rate 148.4m3/day for irrigation. For management purposes, two scenarios are investigated for the extent of the interface location. The first scenario considers the same set of data as described above, except that the rainfall rate is reduced to half ( 0 0.0006 m / dayq ). In the second scenario, in addition to the reduction of rainfall,

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the pumping rates are amplified to 2,800 m3/day for the water supply well and 600 m3/day for the irrigation well. The simulated results of these two scenarios are compared to those published by Larabi and De Smedt (1997). Figures 25 and 26 show the interface position in cross section and plane views. The numerical results obtained by the two numerical models are in good agreement. For scenario 1, the pumped rates don’t represent a threat for the freshwater inland, instead the reduced replenishment. For the second scenario however, the supposed pumping rates constitute the critical exploitation feature. Indeed, upconing is developed beneath the pumping well w1 near the coast, however, the interface don’t reach the well bottom. On the other hand, freshwater can be safely pumped from well W2.

-75

-50

-25

0

25

6500 528000 529500 531000.

Figure 25. Vertical cross-section through pumping well W1 showing the interface position (scenario 2). Present model (dashed and solid lines for sets 1 and 2

respectively)

Transient simulations have been also performed to analyse the solute transport interface motion. The initial condition for our simulations is static saturated domain without salt. The two wells implanted near the coastline are also considered. Figures 26 and 27 show the concentration distribution respectively on a vertical section close to the pumping wells and on the bottom of the aquifer, at time simulation t = 8 years.

Figure 26. Concentration profile (c=0.1 ; 0.3 ; 0.5 ; 0.7 ; 0.9)

Figure 27. Isochlors and velocity field on the bottom of the aquifer (convergence of velocity areas indicate the pumping well locations).

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6.3 The Rmel coastal aquifer (SEAWAT code)

Introduction

The R’mel coastal aquifer is located north of (Morocco, and considered among the main sub-atlantic coastal aquifers in Morocco with an area of 240 km2. the availability of ground water and the good hydrodynamic parameters of the aquifer have favourably increased irrigation by pumping, and contributed to the agriculture development in this plain; in addition to supplying potable water to tree large urban centers (Larache, Ksar lakbir, etc.) and rural agglomerations.

But, this development of pumping has induced environmental problems, such as: over-exploitation of the aquifer with a groundwater table decrease; salt water intrusion; degradation of the groundwater quality. The aim of the study is to improve Understanding of its hydrodynamic, to estimate the water balance taking into account the contribution of the invaded seawater; to characterize the evolution of seawater intrusion in time and space and the use of the model as a tool for sustainable quantitative and qualitative management of the R' Mel aquifer.

For this, a mathematical model in transient conditions, based on the SEAWAT code, has been developed to study groundwater flow and salt water intrusion in this aquifer. Different management scenarios have been simulated to provide the managers with a prediction tool that could help in decision making. The simulation results with respect to development of water resources in this coastal aquifer showed that surface water is required to protect the irrigated area and to improve water quality in the area which is contaminated by seawater intrusion, on the other hand.

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Figure 28. Location map of the Rmel aquifer

The Rmel plain is caracterised by a moderated climate with oceanic influence. The average monthly temperature ranges from 12 in January to 24 in August; the average annual precipitation is 695 mm. (90% are recorded between November and April) and the average evapotranspiration is 467 mm (DRPE, 1987).

Sea water intrusion simulation in steady state and transient conditions

In order to study the seawater intrusion effect on the aquifer management, a mathematical model has been designed, calibrated in study state and in transient conditions. The objective of this model is to simulate the extent of the saltwater intrusion. The model results will be of great importance for the management policy at the local level.

Figure 29 shows the calibrated, simulated groundwater heads in the Rmel aquifer for the reference 1972. The results show good agreement between calculated and measured heads, and the established freshwater-saltwater interface was far away from inland. However, the calibration is more reliable in the North-western sectors and

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center-East where we have piezometric data. Table 5 shows also the resulting water balance with different components, including the discharge to the sea.

Figure 29. Calibration of the Rmel aquifer

The results of the transient calculations are prescribed in Figure 30 which shows satisfactory agreement between measured and calculated heads in different target observation wells for the considered simulation period. These transient simulations have led to answer some questions raised by the decision makers and local managers such as:

- To determine the date of the beginning of the seawater intrusion in the aquifer and to follow up its evolution;

- To identify invaded zones by this intrusion, as well as their degree of contamination; and

- To quantify the seawater intrusion volume, as well as the other components of the hydraulic mass balance.

Figure 30 illustrates the sea water intrusion evolution versus abstractions and recharge of the aquifer. This shows that seawater intrusion started since 1992 following the increased development of ONEP’s withdrawals for the drinking water supply of Larache.

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Table 5. Calculated mass balance in 1960 under the steady state conditions

Figure 30. Quantitative evolution of seawater intrusion, natural recharge and withdrawals in the R’mel aquifer.

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Table 6 illustrates the results of different water balance terms between 1972 and 2003, as obtained from the transient simulations. This table shows a reserve increase between 1972-1980: 2.93 Mm3/year between 1972-1975 and of 1.615 Mm3/yearbetween 1975-1980. As a consequence, seawater intrusion advances inland and continues to progress until 1995.A pronounced destocking of the aquifer is developed between 1980-1985 of 4.22 Mm3/year, which continued until 1990 but with an average of 0.2 M m3/year.

Table 6. Calculated mass balance for the transient simulations.

Figure 31. Calculated salinity map for 1996 and 2000

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Figure 31 shows the areas intruded by sea water which is located immediately close to the ONEP pumping well field for which abstractions increased since1992.It shows also that the salinity exceeds 2 g/l in layer 2 where the well filters are located. Extension of the invaded zone by seawater intrusion becomes 2 times between 1996 and 2000.

Simulating provisional scenarios for water resources management of the aquifer

3 scenario schemes have been applied based on the water authority planning in order to assess quantitative and qualitative impact on the aquifer:

Scenario 1 consists on maintaining the present aquifer exploitation, which means that the exploitation volume stands as 13.24 M m3/year until 2020.

Scenario 2 consists on satisfying the regional water demand until 2020. This demand should come from the ONEP’s withdrawals from the existing wells field supplying Larache and its agglomerations projected for 2020;

Scenario 3 is expected to satisfy the maximum demands projected for the drinking water and supply for Larache and Ksar El Kebir cities until 2020.

Table 7 gives predicted volumes of seawater intrusion for the 3 scenarios. It also shows that sea water intrusion volume would decrease in scenario 1 since 2003 to reach a steady state beyond the year 2020, while it continues to increase for scenario 2 and 3.

Table 7. Predicted volumes of seawater intrusion for the 3 scenarios

The predicted results for scenario 1 are illustrated in Figure 32.This shows that the contaminated zone by seawater intrusion continues to extend more (Figure 32a and b). The water salinity is much higher on the level of layer 2 than layer 4 and this is due to the overexploitation in layer 2 (location of the filters). Also the ONEP’s well field will be reached by seawater intrusion and the concentration would be of 1g/l (Figure 32c).

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Figure 32. Predicted salinity distribution for Scenario1

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Figure 33. Predicted salinity distribution for Scenario 2

Figure 34. Predicted salinity distribution for Scenario 3

Figure 33 shows considerable increase of salinity in layer 2 (values generally range between 15 and 25 g/l) for scenario 2.The ONEP’s well field would be already reached by the seawater intrusion front in 2010 and the concentration would be of 1.5 g/l;

Figure 34 shows that the plume distribution is almost identical for scenario 2 and 3, but the RADEEL’s well field, located at the south of the ONEP wells, would not be

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contaminated by sea water intrusion, although with the pumping increase, because it is far from the coast.

In conclusion, for scenario 1, the ONEP well field would be already contaminated by seawater around 2020; and the groundwater quality varies badly in the West Northern coastal part of the aquifer. In all, the impact of scenarios 2 and 3 on the groundwater quality would accelerate seawater intrusion advance, which would reach the ONEP well field in 2010; and would more accentuate the deterioration of groundwater quality to higher concentrations.

6.4 The Gaza Coastal aquifer (Palestine) (SEAWAT)

The Gaza coastal aquifer is a dynamic groundwater system, with continuously changing inflows and outflows. The equilibrium condition that once may have existed between fresh and saline water has been disturbed by large scale pumping (Figure35). The aquifer has been overexploited for the past 40 years; this has induced seawater flow towards the major pumping centers in the Gaza Strip to the north of the Gaza City and near Khan-Younis City. A typical hydrogeological section of the Gaza strip aquifer is given by Figure 36.

Figure 35. Spatial distribution of the existing pumping wells

Figure 36. Typical hydrogeological section of the Gaza strip aquifer

The coupled flow and transport finite difference code (SEAWAT) was applied to examine how far inland seawater transition zone has moved since intrusion began.

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Across section finite difference grid along the first 5 km of Khan-Younis is illustrated in Figure 37. The SEAWAT model result was compared to the previous SUTRA model result which are in a good agreement (Figure 38).

Figure 37. Finite difference grid along the first 5 km of Khan-Younis cross section

Figure 38. Calculated salinity by SEAWAT and SUTRA codes versus observed values

The model gave good results for evolution of salinity in the aquifer. The preliminary model results suggested that the seawater intrusion began in the 1960s which is in agreement with the available information about general pumping and well information. Most of the seawater intrusion is happened to the north of Gaza city and also near Khan-Younis city in the south as it is demonstrated in Figure39, which illustrates the isolines of calculated TDS concentration (kg/m3) by SEAWAT in year 1996 along Deir El Balah geoelectrical cross section; the isolines of calculated TDS concentration (kg/m3) by SEAWAT in year 2003 along Jabalya cross section; and

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isolines of calculated TDS concentration (kg/m3) by SEAWAT in year 2003 along Khan Younis cross section.

Figure 39. Isolines of calculated TDS concentration (kg/m3) for Deir El Balah, Jabalya and Khan Younis cross sections

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The numerical model is applied to test the overall regional impact on the aquifer with two future scenarios of pumping. The first scenario is pumping from the aquifer continuously until year 2020 to reach 200 Mm3/yr; and the second scenario is to decrease the pumping from the aquifer year 2020 to reach 110 Mm3/yr from the existing pumping wells. It is predicted that between years 2003 and 2020, the first scenario will induce a considerable quantity of seawater intrusion especially in the northern part (Figure 40). Model results indicate that the extent of the isoline (TDS concentration = 2.0 kg/m3) at the base of sub-aquifer A will move about an additional 1.5 km in the northern part. On the other hand, the results of comparison indicate that the second scenario will prevent any further seawater intrusion after year 2003. In year 2020 the total inflow from the sea is estimated to be 72 Mm3/yr and 32 Mm3/yrfor the first scenario and second scenario respectively, where the discharge to the sea for the same year is estimated to be about 3 Mm3/yr and 18 Mm3/yr for the first and second scenarios.

Figure 40. Simulated extent of TDS concentration (less than 2.0 kg/m3) in sub-aquifers A in 2003 and 2020

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Figure 41. Predicted contour lines of groundwater levels for second scenario in 2020

Given the uncertainties in the available data, additional refinement of the model grid at this stage does not provide more accuracy. However, in general the model reasonably simulates the position of saltwater transition zone, particularly near the coast. The current model is a reasonable representation of the aquifer in an overall regional context. In future as new data become available, the model should be updated periodically to refine estimates of input parameters, and simulate new management options

6.5 Numerical modeling of seawater intrusion in the Sahel aquifer of the Atlantic coast of Morocco (CODESA-3D code)

The Sahel region, of about 100 km in length, is an important agricultural plain where groundwater is heavily used as an irrigation supply. In 1994 a study to evaluate the impact of pumping on the salinization of the aquifers and to define a proper exploitation scheme, was completed by the regional hydraulic department in collaboration with FAO. A hydrogeochemical database was built up using data collected from a network of 30 wells. These data were used to implement a 2D groundwater flow model using the USGS MODFLOW software (McDonald and Harbaugh, 1988). To overcome some of the limitations of this earlier study, among which the assumption of a static salt- freshwater interface coinciding with the aquifer bottom which over-estimated the freshwater outflow to the sea, simulations using CODESA-3D model was undertaken. The model implementation allows a more realistic representation of the hydrodynamics of the aquifer system, considering also the presence of a mixing zone between the freshwater and the seawater.

This study describes simulations being conducted on a series of cross-sections perpendicular to the coast and corresponding to the geophysical profiles performed in

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a previous study, have been analyzed. Along these profiles measures of the depth of the salt/freshwater interface were available. The first simulation, covering the period 1950-1992, is calibrated to the water table levels and to the observation data and flow simulations. Figure 42 shows the comparison results in 1950. The Darcy velocity field is also shown, which denotes at the ocean side on the left boundary of the domain an upper outlet of freshwater into the sea and, correspondingly, a lower inlet of seawater into the aquifer. Figure 43 compares the measured saltwater-freshwater interface with the computed relative concentration isocontours of a 42-year transient coupled flow and transport simulations.

The study on these selected cross-sections has pointed out the important role of an adequate grid density (20 m) and a good estimation of the unsaturated zone and transport dispersivity parameters for the characterization of the saltwater intrusion in the Sahel aquifer system.

Figure 42. Comparison between observed water table level (black dashed line) and computed pressure head (m) isocontours for profile 4w. The computed Darcy

velocity field (m/s) is also shown (blue arrows).

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Figure 43. Measured salt freshwater interface (black line, from geophysical investigations) and computed isosaline c [/] contours.

7. Conclusion

An overview of modelling seawater intrusion in coastal aquifers is presented. Some analytical solutions and laboratory model, either based on sharp or variable density approaches, have been also outlined as to serve benchmark problems for testing developed numerical codes. However, the question of which type of physical approach is appropriate to deal with seawater intrusion in coastal aquifer, must be clarified, especially when studying field cases. Sharp interface model or the disperse interface model? This depends on the scale of interest: if near-coast, the effects and variations in salt concentration are important, then dispersion features of the interface are important; if far-coast and the regional monitoring aspects are important, then more interested to investigate the distance and depth of the interface, and so the sharp interface approximation is preferred.

Recent numerical codes based on a 3-D finite elements and finite differences, developed to predict the location, the shape and the extent of the moving interfaces in coastal aquifers involving unconfined or confined flow have been also presented. Some field applications have been studied using recent numerical codes in order to assess groundwater resources and to test the groundwater system response to planning scenarios to help the managers in making decision for rational groundwater management. Examples are taken from Morocco and Palestine, where models were applied on some coastal aquifers to simulate groundwater flow and to study the impact of irrigation and increasing pumping well rates used for water public supply. The site description and the simulation results are presented in separate papers published previously (Aharmouch and Larabi 2004, Hilali and Larabi 2004, Lakfifi et al., 2004, Qahman and Larabi 2006).

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Lecca, G., (2000). Implementation and testing of the CODESA-3D model for density-dependent flow and transport problems in porous media, CRS4-TECH-REP-00/40, Cagliari, Italia.

McDonald M.G., and Harbaugh. A.W. (1988) A modular three dimensional finite-difference ground-water flow model. U.S. Geological Survey Techniques of Water Resources Investigations, Book 6.

Motz, L. H., (1992). Saltwater upconing in an aquifer overlain by a leaky confining bed. J. Ground water 30(2), 192-198.

Mualem, Y., Bear, J.(1974), The shape of the interface in steady flow in a stratified aquifer. Water Resour. Res., 10(6), 1207-1215.

Mercer, J. W., . Larson S. P, and Paut C. R., (1980). Simulation saltwater interface motion, Groundwater, 18(4), 374-385.

Oldenburg C. M. & Pruess K. (1995), Dispersive transport dynamics in a strongly coupled groundwater-brine flow system. Water Resour. Res., 31(1995), 289-302.

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Pinder G. and S. Stothoff, Can the sharp interface salt-water model capture transient behaviour? Proc. Computational Methods in Water Resources, MA. Celia, L.A. Ferrand, CA Brebbia, WG. Gray and GF Pinder (eds.), Vol.1, pp. 217-222, 1988.

Polo, J. F., Ramis F. J. R. (1983) Simulation of salt water fresh water interface motion. Water Resour. Res. 19(1): 61-68.

Putti, M., Paniconi C. (1995) Finite element modeling of saltwater intrusion problems.In Springer-Verlag (ed) Advanced Methods for Groundwater Pollution Control. Int. Centre for Mechanical Sciences, New York, 364, pp.65-84.

Reilly T.E and A.S. Goodman, Quantitative analysis of saltwater-freshwater relationship in groundwater systems – A historical perspectives, J. Hydrol., 80, pp. 125-160, 1985.

Segol, G. (1993), Classic groundwater simulations: Proving and improving numerical models/

Englewood, Cliffs, N.J., PTR, Prentice Hall, 531pp. Strack, O. D. L. (1976), A single-potential solution for regional interface problems in coastal

aquifers, Water Resour. Res., 12(6), 1165-1174. Strack, O. D. L. (1987), Groundwater Mechanics. Prentice Hall, Englewood Cliffs.NJ. Sugio, S., Abrupt interface model of seawater intrusion in coastal aquifer, in HYDROCOMP'92,

International Conference on Interaction of Computational methods and Measurements in Hydraulics and Hydrology, edited by J. Gayer, O. Starosolszky and C. Maksimovic, Water Resource Centre, Budapest, pp. 81-88, 1992.

Van Dam, J. C. (1983), The shape and position of the salt water wedge in coastal aquifers. In: Proc.Hamburg Symposium of Relation of Groundwater Quantity and quality. IAHS, Publication No. 146.

Van der Veer, P. (1977), Analytical solution for steady interface flow in a coastal aquifer involving a phreatic surface with precipitation, J. of Hydrol., 34, 1-11.

Voss, C.I. (1984), A finite element simulation model for saturated-unsaturated, fluid density dependent groundwater flow with energy transport or chemically reactive single species solute transport : USGS Water Resources Investigation Report 84-4369, 409 pp.

Voss, C.I. and WR. Souza, Variable density flow and solute transport simulation of regional aquifers containing a narrow freshwater-saltwater transition zone: Water Resou. Res., V. 23, no. 10, pp. 1851-1866, 1987.

Voss, C.I. and W. R. Souza, AQUIFE-SALT : A finite element model for aquifer containing a seawater interface. Water Resources Investigations Report 84-4369, US Geological Survey (1984) 37.

Willis, R. and Yeh W. W-G (1987). Groundwater Systems Planning and Management. Prentice-Hall, Inc, New Jersey.

Zheng, C. (1990). MT3D: A modular three-dimensional transport model for simulation of advection, dispersion and chemical reactions of contaminants in groundwater systems. Report to the U.S. Environmental Protection Agency, Ada, Oklahoma.

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Chapter 2

Coastal Aquifer Management: Management Model For Seawater Intrusion In The Chaouia Coastal Aquifer (Morocco)

Abdelkader LARABI* and Latifa LAKFIFI**

*LIMEN, Ecole Mohammadia d’Ingénieurs, B.P. 765, Rabat, e-mail: [email protected]

**Direction de la Recherche et de la Planification des Eaux, Rabat

Abstract:

Management of coastal aquifers involves seawater intrusion problem, which is a special issue in groundwater resources management, and needs a special treatment to study the seawater-freshwater interface. A full detailed field case has been studied, and a particular attention has been done to seawater intrusion problem and how to control it, using mathematical modelling. The Chaouia coastal aquifer is located south of the Casablanca city (Morocco), and considered among the main sub-Atlantic coastal aquifers in Morocco with an area of 1200 km2. The only water resource available in the Chaouia plain comes from the shallow groundwater. This favourable situation has increased irrigation by pumping, and contributed to the agriculture development in this plain. But, this development of pumping has induced environmental problems, such as: over-exploitation of the aquifer with a groundwater table decrease of 0.6 m/yr; progressive dry out of some aquifer levels; salt water intrusion; degradation of the groundwater quality; and significant increase of the abandoned pumping wells. For this, a mathematical model in transient conditions, based on the SEAWAT code, has been developed to study groundwater flow and salt water intrusion in this aquifer. Different management scenarios have been simulated to provide the managers with a prediction tool that could help in decision making for quantitative and qualitative groundwater management in this groundwater system. The simulation results with respect to development of water resources in this coastal aquifer showed that surface water is required to protect the irrigated area and to restore the abandoned exploitation on one hand, and to improve water quality in the area which is already contaminated by seawater intrusion, on the other hand.

Keywords: Coastal aquifer, groundwater, modeling, seawater intrusion, over-pumping. Morocco.

1. INTRODUCTION

Morocco is extended on a coast of more than 3500 km, on the Atlantic Ocean and the Mediterranean Sea. As for the remainder of the world, the majority of urban

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and farming agglomeration activities (e.g. fishing, industry, harbors, agriculture and tourism) are located on the coasts and the inshore plains. These intensive socioeconomic developments have leaded to extensive use of freshwater resources in the coastal zones, and especially the groundwater resources. Groundwater has some advantages if it is compared to the surface water, because of its spatial distribution, regularity, easiness of exploitation and low cost of mobilization.

The coastal aquifers in Morocco are considered very important source for water supply and very essential for socioeconomic development. High rates of urbanization and increased agricultural and economical activities have required more water to be pumped from the aquifers. This pumping has continually increased the risk of seawater intrusion and deterioration of freshwater quality of the coastal aquifers. The contamination of groundwater by seawater intrusion threatens the agricultural development. In recent years, the effect of seawater intrusion is more remarkable on many coastal areas in Morocco, especially, in some aquifers located on the Atlantic coast.

Currently, the agricultural activity is in a critical situation, because of the progressive water resources shortness and due to the deterioration of groundwater quality, due to the overexploitation of the aquifer and to successive drought seasons.

Groundwater quality in the Chaouia aquifer is generally poor. Over-exploitation has resulted in saltwater intrusion and upconing. In some parts of the study area, a slow, continuing decline in groundwater levels has been observed. Saltwater intrusion presently poses important threat to water supply, and especially with future water demand as it is expected to increase. Hence, it is necessary to develop efficient and fast tools for predicting the response of coastal aquifers to different pumping schemes while taking into account as constraint the risk of saltwater intrusion. In practice, the spatial relationship between freshwater and saline groundwater in coastal and inland aquifers is complex, and management of the freshwater resources can be difficult. The aquifer system is rarely near equilibrium, and the fresh and saline water bodies are normally separated by a transition zone, the result of chemical diffusion and mechanical mixing. Under these conditions, the response of the saline water body to pumping is difficult to predict and depends on various factors, including aquifer geometry and properties; abstraction rates and depths; recharge rate; and distance of pumping wells from the coastline. Efficient tools such as numerical groundwater models are required to quantify the aquifer response to these excitations. This is the case of the Chaouia coastal aquifer, where a mathematical model that considers seawater intrusion has been implemented to help the managers in sustainable water resources planning and management. Hence, the objectives of this study are:

1- To analyze the phenomenon of seawater intrusion and the definition of conditions that govern the behavior of freshwater/saltwater transition zone in the coastal aquifer under different physical approaches.

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2- To test management scenarios based on various economic projects; and select the best scenario for the regional water resources authorities in order to implement the corresponding economic projects which improve water resources development.

For this, a specific hydrogeological study of seawater intrusion must be done, focusing on the hydrodynamics and hydrochemical behavior of the seawater intrusion in the aquifer and to assess its impacts on freshwater resources using the variable-density SEAWAT code. The model output will show the seawater intrusion occurrence and predict its future behavior along the Chaouia coastal aquifer.

2. DESCRIPTION OF THE CHAOUIA AQUIFER

The Chaouia is a part of the Moroccan coastal plain located to the south-west of Morocco. Its area is about 1200 km2 and its length is approximately 25 km along the Atlantic coast and located between the cities of Casablanca to the North and Azemmour near the city of El Jadida to the South-West (Figure 1).

Figure 1. Location map of the Chaouia aquifer.

The average mean temperature ranges from 25 °C in summer and 10 °C in winter. The climate is semi-arid, but influenced by the Atlantic Ocean. The summer is hot; especially on July and August; and this leads to make more pressures on groundwater abstractions for irrigation from the aquifer system. The coldest months on the year are December and January. The average annual rainfall varies from 500 mm/yr in the north at Casablanca to 300 mm/yr in the south near Azemmour. Most of the rainfall occurs in the period from October to April, the rest of the year being completely dry.

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The annual mean of the evapotranspiration in the study area is estimated to 840 mm/yr.

The hydrogeology of the Chaouia coastal aquifer consists of Paleozoic shale which forms the aquifer bottom, but the aquifer system is filled with the following formations:

altered shale in the part between Tnine Chtouka and Casablanca; limestone and Cenomanian marine limestone with 60 m of average thickness, in the south-western zone; sandy dunes of 10 m thickness at the coastal strip, chalky hills that constitute mainly the quaternary deposits and cover the wadi area;conglomerates, alluvia and silts of small thickness, which crop out just on the wadi Oum Er Rbia banks.

The altered Paleozoic shale constitutes the main aquifer, with 90% of the total area. Toward the south-west of the area, the Paleozic bottom rises up and allows a water divide to occur, separating circulations of water in the chalky horizons of the Cenomanian to the west of the one of the quaternary deposits and altered shale to the East. The Quaternary deposits generally have a maximum saturated thickness of 10 m except, in the inshore zone where it can reach about 20 m.

Schematization of a hydrogeological cross section of the Chaouia aquifer system is shown in Figure 2.

Figure 2. Schematic hydrogeological cross section of the Chaouia aquifer.

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The groundwater system characteristics

The regional groundwater flow is mainly SE-NW discharging towards the Atlantic Ocean, except in the south-western where a part of the outflow discharges toward the Oum Er Rbia river. To the southwest the hydraulic gradients range between 0.1 and 1‰. Between Tnine Chtouka and Casablanca they vary from 2 to 3‰, with lower values toward the uphill limit of the zone, of the order of 0.5‰. This difference in the hydraulic gradient is due to the variation of the permeability. The depth of water table varies between 10 m along the coastline and can exceed 40m along the south-west part in the Cenomanian marine-chalk. The comparison of the piezometric situations of 1971 and 1995 shows that the decrease of the groundwater level reached 10 m in the coastal strip and 15 m in the rest of the aquifer.

The groundwater productivity of the aquifer varies from 0.5 to 2 L/s in the coastal zone; 1 to 2 L/s in the zone between Tnine Chtouka and Casablanca and 2 to 4 L/s in the part between Tnine Chtouka and Azemmour.

From a hydrological point of view, the Chaouia plain is not crossed by any permanent river. Wadi Oum Er Rbias and Wadi Bousskouras, which are located respectively on the western and eastern limits of the study area, don't contribute to the replenishment of the aquifer. Hence, the major source of renewable groundwater in the aquifer is rainfall. The total rainfall recharge to the aquifer is estimated to be approximately 53 Mm3/yr, while the lateral inflows to the aquifer is estimated to 6.6 Mm3/yr. The groundwater abstraction from the pumping wells was estimated to 34 Mm3/yr, this is based on a field investigation in 1995. This investigation has also shown that 54% of the total pumping wells used for irrigation have been abandoned, mainly due to the high salinity in the pumped groundwater, which exceeds 3 g/L. The outflow towards the Atlantic Ocean is estimated to approximately 41 Mm3/yr in 1949.

The recent results of a hydrogeochemical analysis show that the groundwater salinity values are very high in some inshore sectors, exceeding 6 g/L. However, in 1971 the salinity reached only 2 g/L in these sectors, with a maximum of 4.5 g/L locally. This salinity increase in those places is due to the seawater intrusion resulting from overpumping and conjugated to reduced recharge of the aquifer.

3. MODELING SEAWATER INTRUSION IN THE CHAOUIA AQUIFER

3.1 Simulation code SEAWAT

The original SEAWAT code was written by Guo and Bennett (1998) to simulate ground water flow and salt water intrusion in coastal environments. SEAWAT uses a modified version of MODFLOW (McDonald and Harbaugh, 1988) to solve the variable density, ground water flow equation and MT3D (Zheng, 1998) to solve the solute-transport equation. The SEAWAT code uses a one-step lag between solutions of flow and transport. This means that MT3D runs for a time step, and then MODFLOW runs for the same time step using the last concentrations from MT3D to calculate the density terms in the flow equation. For the next time step, velocities from the current MODFLOW solution are used by MT3D to solve the transport

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equation. For most simulations, the one-step lag does not introduce significant error, and it can be reduced or evaluated by decreasing the length of the time step.

3.2 Model Set-up

Mesh discretization and boundary conditions

The developed model simulates transient variable density groundwater flow coupled to solute transport for a period from 1960 to 2002. This model is based on the hydrogeological description presented in the previous sections. Regular spaced finite-difference cells of 1 km × 1 km on the horizontal plane are constructed (Figure 3). Hence, the grid consists of 24 rows, 71 columns and 8 regular spaced layers on the vertical direction.

In a first step, the model developed is expected to simulate the global aquifer behavior. However, for studying local variations of the seawater intrusion, a refined grid is necessary, as will be developed in the further sections.

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Figure 3. Finite-difference grid and boundary conditions for the Chaouia regional model.

Boundary conditions and aquifer parameters

The hydraulic aquifer parameters assigned to the model are based on those produced from the pumping tests performed in the aquifer system. The hydraulic conductivity distribution, for tests carried out in the area, shows that values vary from 7×10-4 m/s to 2×10-6 m/s on the western area, with a mean value in general of 2×10-4

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m/s. At the Eastern part the average value of the hydraulic conductivity is around 6×10-4 m/s.

Only six values of the storage parameter are available for the Chaouia coastal aquifer. Measurements that carried-out, were concentrated only along the aquifer coastal part, and the measured values range from 0.13% to 7%. In the rest of the aquifer, parameters were obtained from Moroccan hydrogeological literature for similar type of soils (DRPE, 1995).

Constant head and concentrations are specified to the cells along the Atlantic coast, respectively 0.0 m and 35 kg/m3. The head for each cell is converted to freshwater head using the specified salt concentration of 35 kg/m3 at the center elevation of the cell. Constant head along the Oum Er Rabia river presents at layer 3, expressed by a slope of 0.1% from the sea. The concentration is assumed to be 0.0. The lateral flow of groundwater upstream is represented by a general head boundary based on the groundwater level record. The aquifer bottom is a no flux boundary, while on the top of the aquifer a recharge influx is assigned. The northern and southern west boundaries are assumed to be no flow boundaries, due to the existence of impervious layers of schist.

3.3 Calibration and model results

First the numerical model is established and tested against steady state groundwater flow on 1949. During calculation, measured and calculated groundwater heads are compared, and the difference referred to as the residual. Figure 4 shows the calibrated, simulated groundwater heads in the Chaouia aquifer for year 1949 conditions. The results show good agreement between calculated and measured heads, and the established freshwater-saltwater interface was far away from inland. Table 1 shows also the resulting water balance with different components, including the discharge to the sea, which is estimated to 43.85 Mm3/yr.

It is clear from Table 1 that the main component of the input corresponds to the recharge from precipitation and that the ocean boundary constitutes the main outlet of the aquifer.

Transient simulations are conducted starting from the initial heads and concentration with reference to 1960 steady state conditions assuming that there was no change in groundwater abstraction between 1949 and 1960. The initial concentrations in the aquifer were assumed to be 0 kg/m3. Transient calculation was performed for the 1960-2001 target period (41 years).

The major pumping started from 1960 and was changed with lateral flows for the specified stress periods, based on the field investigations and the recorded data. The results of the transient calculations are prescribed in Figure 5 which shows satisfactory agreement between measured and calculated heads in different target observation wells for the considered simulation period. These transient simulations have led to answer some questions raised by the decision makers and local managers such as:

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- To determine the date of the beginning of the seawater intrusion in the aquifer and to follow up its evolution;

- To identify invaded zones by this intrusion, as well as their degree of contamination; and

- To quantify the seawater intrusion volume, as well as the other components of the hydraulic mass balance.

Table 1. Calculated mass balance in 1960 under the steady state conditions.

Water balance component Volumes in Mm3 /yr

Recharge from precipitation 52.67

Lateral flow from upstream aquifer. 6.24Inflow

Total 58.91

Discharge to Atlantic Ocean. 43.85

Discharge to Oum Er Rbia river 1.20

Evaporation 13.34

Agricultural and domestic abstraction 0.53

Outflow

Total 58.91

Error % 7 E-6

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Figure 4. Calibrated and observed water level for the year 1949.

Figure 5. Calibrated and observed water level for observation wells.

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Table 2 illustrates the results of different water balance terms between 1965 and 2001, as obtained from the transient simulations. This table shows that the intensive groundwater pumping, especially between 1965 and 1985, resulted in less storage in the aquifer which reached 24 Mm3 in 1985. As a consequence, seawater intrusion advances inland and continues to progress until 1995. The concentration values calculated by the model give the following invaded zones by seawater intrusion and the extent:

- The first zones invaded by the seawater intrusion in the beginning of the 1980’s are located north of Azemmour south west. Afterward, this invaded zone extended along the inshore line, on about 20 km of length and 2 km of width as shown by Figures 6 and 7.- The aquifer is strongly contaminated on the S-W coastal zone, in which the toe interface does not exceed the strip of 1km of large. Beyond this zone the contamination of the aquifer is limited; - Generally, the calculated concentration values exceed 10 g/L in the aquifer bottom as demonstrated by several profiles of Figure 8.

The relative vulnerability of the south-west part of the aquifer to the seawater intrusion in comparison with the rest of the coast is explained by the following:

the relatively high hydraulic conductivity of the fissured limestone located in this sector of the coastal aquifers; the structure of the aquifer bottom that is deeper in this zone and reaches its maximum depth (between -60 m to -90 m) in this sector; and the high concentrations of the pumping wells in this sector.

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Figure 6. Simulated salinity distribution in 1985.

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Table 2. Calculated mass balance for the transient simulations.

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Figure 7. Simulated salinity distribution in 1995.

Figure 8. Salinity evolution in profiles of the model layers. In order to study the seawater intrusion effect on the south-western part, a model

of transverse section with a refined mesh has been developed. The objective of this local model is to simulate the extent of the saltwater intrusion with a finer spatial

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resolution in two dimensions. The model results will be of great importance for the management policy at the local level.

The mesh cells dimension is reduced to 100 m along the x-axis for the same simulation period adopted for the global model.

Figure 9 illustrates the results of this transverse model in transient conditions. These show that seawater intrusion started in 1980-85 by contaminating the aquifer on 100 m from the coast line; and the estimated extent of the seawater intrusion wedge reaches 1300 m in 2001.

Figure 9. Result of the cross sectional simulation model in 2001.

4. PREDICTED RESULTS FOR RATIONAL WATER MANAGEMENT

The calibrated model can be exploited as a management tool for simulating the response of the aquifer, including the quantitative and qualitative aspects (variation of concentration in the Chaouia aquifer) based on proposed planning schemes by the local managers.

As mentioned in the previous sections, measured withdrawal of groundwater for irrigation supply, especially at the south-west sector, is the main cause of seawater intrusion into the aquifer, conjunctive decrease in groundwater withdrawal with artificial recharge is an important alternative to be considered in the future to prevent any further intrusion of saltwater inland. For this purpose, planning scenarios schemes were designed to use the calibrated model for simulating the future changes in drawdown and salinity concentrations in a period of 40 years:

- The first scenario consists on the design of an irrigation project from surface water pumped from the Oum Er Rbia river. Pumping from the aquifer has to be stopped on the inshore strip to the East of Azemmour along an area of 3 km of width and of about 20 km of length. A brutal reduction of withdrawals occurs when the irrigation project starts in 2005; this leads to a quasi-stabilization withdrawal of about 22 Mm3/y until the end of the simulation period.

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- The second scenario (the worst) assumes that the irrigation project would not be achieved and the pumping from the aquifer of about 30.5 Mm3/y continues to occur. The evolution of the groundwater quality of the aquifer is analyzed in order to define the affected area by seawater intrusion.

- The third scenario assumes maintaining the present exploitation and improves the overexploitation by artificial recharge of the aquifer through a series of injection wells upstream of the highly contaminated zone (300 l/s).

In all cases, conditions of average annual rainfall infiltration rate (normal conditions denoted as case 1, 3, and 5) and minimal annual rainfall infiltration rate about 70% of the normal infiltration rate (drought conditions, denoted as case 2, 3 and 6), are considered.

Results of scenario simulations

The corresponding scenario simulations are performed for a period of 40 years (2001-2040). For scenario 1 (case 1) (Table 3), although the seawater intrusion volume decreases and the equilibrium is established after the year 2010 in response to the irrigation project from surface water, the groundwater quality in the south-western sector would remain of high salinity concentrations. The improvement of the water quality is a slow process, and would only be felt after the year 2020, when the width of the invaded zone by seawater intrusion would be reduced 1 km and the salinity would decrease significantly, as it is shown in Figure 10 (1) and 11.

Table 3. Calculated volumes of seawater intrusion for scenario 1 (in Mm3/yr)

Year 2005 2010 2015 2020 2030 2040

case 1 1.45 0.2 0 0 0 0

case 2 2 0.42 0.27 0.26 0.31 0.5

The results given also in Table 3 for scenario 1 (case 2) shows that the seawater intrusion volume decreased progressively until the year 2020; afterwards it continues to increase. The expected groundwater quality is not improving as shown by Figure 10(2) 12. The north-eastern sector of the coastal part would be also concerned by this intrusion. The improvement of the groundwater quality would remain very slow by 2020, due to the reduced discharge toward the sea, which is 30% less (drought conditions) compared to the discharge in case 1 (normal conditions).

The results of the first worst scenario 2 (case 3) (Table 4), show that the groundwater quality in the coastal zone is alarming, because the salinity concentration increases much as shown by Figure 10(3) and 13. Indeed scenario 2 (case 3), with the average annual rainfall infiltration rate, leads to reach after the year 2005 a new equilibrium with a practically a steady state volume of seawater intrusion (1.4 Mm3/yr). However, the salinity in groundwater would continue to increase and would exceed 15g/L by 2020 in the deep aquifer levels.

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Table 4. Calculated volumes of seawater intrusion for scenario 2 (in Mm3/yr)

Year 2005 2010 2015 2020 2030 2040

Case 3 1.45 1.42 1.40 1.41 1.43 1.47

Case 4 2 2.3 2.6 3 3.4 4

For the second worst scenario 2 (case 4) (Table 4), the salinity and the seawater intrusion volume would increase continuously and no stabilization would be established. The seawater intrusion volume would reach 3 Mm3/yr by 2020; the salinity would exceed 20 g/L in the deepest aquifer levels. In addition, the invaded zone by the seawater intrusion would spread over 3 km and the northern sector of the coastal part would be also affected significantly by this intrusion (Figure 10(4) and 14).

For scenario 3 (case 5) the seawater intrusion volume decreases and the equilibrium is restablished after the year 2005 in response to the artificial recharge project from surface water (Table 5), the groundwater quality in the south-western sector improves significantly only after 2010, due to the slow process, Figure 10 (5) and 15 illustrate this situation.

Table 5. Calculated volumes of seawater intrusion for scenario 3(in Mm3/yr)

Année 2005 2010 2020 2030 2040

Case 5 1.45 0 0 0 0

Case 6 2 0.06 0.26 0.31 0.5

The results given also in Table 5 for scenario 3 (case 6) shows that the seawater intrusion volume increases progressively after 2005; the expected groundwater quality is improving slowly as shown by Figure 10(6) and 16. The north-eastern sector of the coastal part would be concerned by this intrusion.

In conclusion, we can say that the improvement of the groundwater quality in the aquifer is conditioned by the withdrawal decrease, and by increasing the discharge toward the ocean which mainly depend on the aquifer recharge/discharge conditions. Hence, the predicted results from planning scenarios show that a sustainable management of the aquifer, especially in its coastal part, requires the use of surface water which will contribute first to the economic development of the irrigated area by restoring the abandoned exploitation; and secondly to improve significantly the groundwater quality in the invaded sectors of the aquifer by seawater intrusion.

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Figure 10. Salinity variation in profiles for the model layers (Scenario 1, 2, and 3)

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20 - 30

> 30

LEGEND

Figure 11. Results of salinity distribution for scenario 1(1) in 2020.

214

#

#

#

#

#

#

#

# #

#

#

#

ATLANTIC OCEAN

Casablanca

Berrechid

Haouzia

S.SaîdM'achou

O.Oum Er Rbia

O.H

ouara

O.M

erzeg

O. B

ou Skoura

Rhnimiyne

Bouskoura

Nouacer

Médiouna

Derroura

Soualem Trifia

Bir Jdid

ChtoukaAzemmour

Od. Harriz

Dar Bouazza

N

EW

S

220000

220000

240000

240000

260000

260000

280000

280000

300000

300000

280000 280000

300000 300000

320000 320000

340000 340000

Salinity in g/l <5

5 - 10

10 - 20

20 - 30

> 30

LEGEND


#

#

#

#

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#

# #

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#

#

ATLANTICOCEAN

Casablanca

Berrechid

Haouzia

S.SaîdM'achou

O.Oum Er Rbia

O.H

ouara

O.M

erzeg

O. B

ou Skoura

Rhnimiyne

Bouskoura

Nouacer

Médiouna

Derroura

Soualem Trifia

Bir Jdid

ChtoukaAzemmour

Od. Harriz

Dar Bouazza

N

EW

S

220000

220000

240000

240000

260000

260000

280000

280000

300000

300000

280000 280000

300000 300000

320000 320000

340000 340000

Salinity in g/l< 5

5 - 10

10 - 20

20 - 30

> 30

LEGEND


215

#

#

#

#

#

#

#

# #

#

#

#

ATLANTIC OCEAN

Casablanca

Berrechid

Haouzia

S.SaîdM'achou

O.Oum Er Rbia

O.H

ouara

O.M

erzeg

O. B

ou Skoura

Rhnimiyne

Bouskoura

Nouacer

Médiouna

Derroura

Soualem Trifia

Bir Jdid

ChtoukaAzemmour

Od. Harriz

Dar Bouazza

N

EW

S

220000

220000

240000

240000

260000

260000

280000

280000

300000

300000

280000 280000

300000 300000

320000 320000

340000 340000

Salinity in g/l < 5

5 - 10

10 - 20

20 - 30

> 30

LEGEND


Figure 15. Results of salinity distribution for scenario 3 (5) in 2020

216

Figure 16. Results of salinity distribution for scenario 3(6) in 2020

5. CONCLUDING REMARKS

This study constitutes a methodological approach to generalize modeling of seawater intrusion to coastal aquifers in Morocco. In fact, this phenomenon remained until now far from being well known for these aquifers. The present case study of the Chaouia coastal aquifer is an example of a pilot study to develop also for several coastal aquifers threatened by this phenomenon, as Morocco is extended over more than 3500 km of coasts.

All data used in this application have been analyzed, structured and organized in a database built-in via the GIS “Arc View " code and integrated into the conceptual model.

The coupled flow and transport code (SEAWAT) was applied to analyze seawater intrusion in the Chaouai coastal aquifer in both qualitative and quantitative issues. The model results showed that seawater intrusion started by 1980-1985 in the south-western sector of the coastal part and developed within the time, due to intensive pumping from the wells and drought conditions.

The numerical model is also applied to test the response of the aquifer of three planning scenarios for a period of 40 years. The first one is the design of an irrigation project from surface water and stopping groundwater withdrawal in the south-west sector; the second scenario is a continuous pumping from the aquifer, and the third one is the design of a series of artificial recharge wells along the south western coast. It is predicted that the first scenario will reduce considerably the quantity of seawater

217

intrusion and improve the groundwater quality, although in a slow manner in the south-western sector. However, the results from the second scenario will induce a considerable quantity of seawater intrusion, and its extent progressed more into the inland aquifer and will reach the north sector of the coastal part. The third scenario will also reduce considerably the seawater intrusion volume and improve the water quality in a similar way as scenario 1.

Given missing data with respect to the aquifer, more recommendations are necessary to improve the results of this model:

- Measurements of salinity on the vertical profiles are necessary in the invaded zone by the seawater intrusion, the measurement values will contribute to improve the calibrated model.

- Some tracing element tests in the coastal part have to be carried out in order to estimate the dispersivity parameters of the aquifer;

- Refinements of the model cells have to be performed, especially near the shore where the area is more threatened by the seawater intrusion.

Acknowledgments :

The authors are very thankful to the hydraulic department (Direction Générale de l’Hydraulique in Morocco) for the fruitful and successful collaborations between the University and the Administration in conducting this study.

References:

Bentayeb, A, (1972). Etude hydrogéologique de la Chaouia côtière avec essai de simulation mathématique en régime permanent. Thèse Univ. Scien. Tech. du Languedoc, Montpellier.

Direction de la Recherche et de la Planification des Eaux (DRPE) (1995). Etude hydrogéologique et modélisation mathématique de la nappe de la Chaouia côtière, Direction Générale de l’Hydraulique, Rabat (Internal report).

DRPE, (1994). Reconnaissance par prospection électrique dans la Chaouia côtière (project carried out by LPEE) Rabat (Internal report).

Guo, W., Bennett. G.D. (1998). Simulation of saline/fresh water flows using MODFLOW In: Proc MODFLOW ‘98 Conference. International Ground Water Modeling Center, Golden, Colorado, 1: 267–274.

Langevin, C.D. and Guo W. (2002). User’s Guide to SEAWAT, A computer program for simulation of three-dimensional variable density groundwater flow. U.S Geological Survey, Open-File Report 01-434, Tallahassee, Florida.

218

McDonald M.G., and Harbaugh. A.W. (1988) A modular three dimensional finite-difference ground-water flow model. U.S. Geological Survey Techniques of Water Resources Investigations, Book 6.

Zheng, C., Wang. P.P (1998) MT3DMS, A modular three dimensional multi-species transport model for simulation of advection, dispersion and chemical reactions of contaminants in groundwater systems: Vicksburg, Mississippi, Waterways Experiment Station, U.S. Army Corps of Engineers.

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Performing Unbiased Groundwater Modeling: Application of The Theory of Regionalized Variables

Shakeel Ahmed, Aadil Nabi Bhat and Shazrah Owais

Indo-French Centre for Groundwater Research National Geophysical Research Institute

Hyderabad-500007, India

Abstract:Importance of groundwater is extremely obvious in arid and semi-arid regions.

However, groundwater, an important constituent of hydrological cycle and major source of water supply in various sectors, is getting depleted due to over-exploitation, because of high population growth as well as extensive agricultural uses and urgently requires a reliable management option. It is essential to have a thorough understanding of complex processes viz., physical, chemical and/or biological occurring in the system for groundwater assessment and management. To understand these complexities groundwater model plays an important role. Groundwater models are simplified, conceptual representations of a part of the hydrologic cycle. They are primarily used for hydrologic prediction (hydraulic head, flow rates and solute concentration) and for understanding hydrologic processes (contaminant migration, solute transport etc.). However, in most regional studies the modeler has only a few measurements of head, the results of some pumping tests, and a vague idea of the boundary conditions, leakage, or rate of recharge. Because of this models are calibrated with fewer available data, which would significantly lead to erroneous results. Geostatistics, based on theory of regionalized variables has now-a-days found application in almost all the domains of hydrogeology from parameter estimation to predictive modeling for groundwater management. It could be applied at each step of hydrogeological modeling studies viz., from data collection network designing, parameter estimation for fabrication and calibration of aquifer modeling. Kriging, best linear unbiased estimator of the regionalized variables, is a potential tool in increasing the accuracy of model calibration as it provides the best estimates of the system parameters with a measure of confidence (estimation error) that can be used in performing simulations with an ease for the modeler to investigate the model uncertainties. Thus with the help of kriging, equally probable possibilities can be generated that conforms the available data. The proposed methodology allows the users to attain a better picture of the aquifer system and would assess the accuracy and reliability of model predictions.

Keywords: Geostatistics; Kriging; Simulations; Estimation error.

221

INTRODUCTION

Models are nowadays widely used in hydrogeologic studies mainly for understanding and the interpretation of the issues having complex interaction of many variables in the system. These are used to calculate the rate and direction of movement of groundwater through aquifers and confining units in the subsurface and for the predictions about the hydraulic heads, flow rates and solute concentrations. These calculations are referred to as simulations. The simulation of groundwater flow requires a thorough understanding of the hydrogeologic characteristics of the site. The hydrogeologic investigation should include a complete characterization of the subsurface extent and thickness of aquifers and confining units (hydrogeologic framework), hydrologic boundaries (also referred to as boundary conditions), which control the rate and direction of movement of groundwater, hydraulic properties of the aquifers and confining units, and the description of the horizontal and vertical distribution of hydraulic head throughout the modeled area for both beginning (initial conditions), equilibrium (steady-state conditions) and transitional conditions when hydraulic head may vary with time (transient conditions), and distribution and magnitude of groundwater recharge. These models are computer based numerical solutions to the boundary value problems and thus can be summarized in few steps as ‘Specified hydrogeological parameters or inputs, given initial and boundary conditions and lastly the solution of the differential equations’. However, when the predictions have to be made for a particular area the available data is certainly not sufficient or the modeler has only a few measurements of head, the results of some pumping tests, and a vague idea of the boundary conditions, leakage, or rates of recharge. Because of this models are calibrated with fewer available data, which would significantly lead to erroneous results.

There is no doubt that the mathematical and computational aspect of groundwater modeling has reached a satisfactory level of development. The focus of research in recent years has shifted to the problems of parameter identification and uncertainty quantification. Hydrogeological parameters display a large spatial heterogeneity, with possible variations of several orders of magnitude within a short distance. This spatial variability is difficult to characterize in a deterministic way. However, a statistical analysis shows that hydrogeological parameters do not vary in space in a purely random fashion. There is some structure to this spatial variability that can be characterized only in a statistical way.

During the last few decades numerous mathematical approaches have been used to estimate hydrogeological parameters from scattered or scant data. The study of regionalized variables starts from the ability to interpolate a given field starting from a limited number of observations, but preserving the theoretical spatial correlation. This is accomplished by means of a geostatistical technique called kriging. The advantage of the geostatistical estimation technique is that the variance of estimation error can be

222

calculated at any point without knowing the actual measurement on that point. Compared with other interpolation techniques, kriging is advantageous because it considers the number and spatial configuration of the observation points, the position of the data point within the region of interest, the distances between the data points with respect to the area of interest and the spatial continuity of the interpolated variable.

THE THEORY OF REGIONALIZED VARIABLES: GEOSTATISTICS

Geostatistics, a statistical technique based on the theory of regionalized variables, is used to describe spatial relationship among the data sets. It was initially used by mining industry as high cost of drilling made the analysis of the data extremely important. The most important problem faced in the mining is the prediction of the ore grade in the mining block from observed samples that are irregularly spaced. The basic tool in geostatistics, the variogram, is used to quantify spatial correlations between observations. The estimation procedure Kriging is named after D. Krige, who and his colleague’s applied statistical techniques to ore reserves in 1950’s. Due to lot of advantages of kriging, the application of kriging can be found in very different disciplines ranging from mining and geology to soil science, hydrology, meteorology, environmental sciences, agriculture etc. Kriging can be applied wherever a continuous measure is made on a sample at a particular location in space and/or time (Cressie, 1991; Armstrong, 1998). Kriging is best to adopt for spatially dependent variables, where it is necessary to use the limited data in order to infer the real nature as precisely as possible.

The application of kriging to groundwater hydrology was initiated by Delhomme (1976,1978,1979), followed by a number of authors, viz., Delfiner and Delhomme (1983), Marsily et al (1984), Marsily (1986), Aboufirassi and Marino (1983, 1984), Gambolti and Volpi (1979) and so many else. In geostatistics, the geological phenomenon is considered as a function of space and or time. If z(x) represents any random function with measured values at “N” locations in space z(xi) i = 1,2,3,…N and if the value of the function z has to be estimated at the point xo. Each of z1, z2, z3,….zN contributes a part of it to zo and taking this in to account, therefore, the expected value of zo is zo*, which is equal to

z*(xo) = 1z1 + 2z2 + 3z3+ 4z4+………. NzN

The kriging estimate can be defined as:

z*(xo) = z(xi) (1) N

ii

1

Where z*(xo) is the estimation of function z(x) at the point xo and i are the weighting factors.

For the purpose of making this predictor to be unbiased the following conditions need to be satisfied.

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On an average the difference between the estimated value and the true (unknown) value that is the expected value of estimation error should be zero. This un-biased condition is also sometimes called as universality condition.

E [z*(xo)- z(xo) ] = 0 (2)

The condition of optimality means the variance of the estimation error should be minimum.

K2 (xo)= var [(z*(xo)- z(xo)] is minimum. (3)

Using Equation 1 and 2, we get N

ii

1

=1 (4)

expanding Equation 3, we obtain

K2(xo)= E[z (xi) z (xj) + E [z*(xo)2 – 2 E [z (xi) z*(xo)] (5)

N

ii

1

N

jj

1

N

ii

1

The best unbiased linear estimator is the one which minimizes K2(xo) under the

constraint of Equation 4. Introducing the Langrange multipliers and adding the term -2

( -1), we obtain N

ii

1

Q = K2(xo) - 2 ( -1)

N

ii

1

= C (xi , xj) + C (O) –N

ii

1

N

jj

1

2 C (xi , xo) - 2 + 2 (6) N

ii

1

N

ii

1

to minimize the above equation, making partial differentiation of Q w.r.t. i and μ and equal to zero, we obtain the following kriging equations:

N

jj

1C (xi , xj) - = C (xi , xo) (7)

i = 1,2,3,…..N

11

N

jj (8)

where C (xi , xj) is the covariance between points xi and xj. Substituting Equation 7 into Equation 5, we obtain the variance of estimation error.

K2(xo) = C (O) - C (xi , xo) + (10)

N

ii

1

224

The square root of this equation gives us standard deviation K(xo), which means that with the 95% confidence, the true value will be within z*(xo) 2 K(xo) range.

In case the covariance cannot be defined, we can derive the following kriging equations:

N

jj

1 (xi , xj) + = (xi , xo) (11)

i =1,2,3,…..N

11

N

jj (12)

and the variance of the estimation error becomes as:

k2 = (xi , xo) + (13)

N

ii

1

Equation 12 and 13 are a set of (N+1) linear equations with (N+1) unknowns and on solving them we obtain the value of i, which are used to calculate the Equation 1 and 10 or 13.

Kriging is the best linear unbaised estimator (BLUE) as this estimator is a linear function of the data with weights calculated according to the specifications of unbaisedness and minimum variance. The weights are determined by solving a system of linear equations with coefficients that depends only on the variogram which describes the structure of the family of the functions. The weights are not selected on the basis of some arbitrary rule, but infact depends on how the function varies in space. A major advantage of kriging is that it provides the way to estimate the magnitude of the estimation error, which is the rational measure of the reliabilty of the estimate. Since the variance of estimation error depends on the variogram and the location of measurements, therefore, before deciding a new location for measurement or deleting an existing measuremnet point, a variance can be calculated and a better network of data can be designed based on the minimum varaince even proir to any measurement.

The Variogram It has been observed that all the important hydrogeological properties and

parameters such as piezometric head, transmissivity or hydraulic conductivity, storage coefficient, yield, thickness of aquifer, hydrochemical parameters, etc. are all functions of space. According to De Marsily (1986) these variables (known as the regionalized variables) are not purely random, and there is some kind of correlation in the spatial distribution of their magnitudes. The spatial correlation of such variables is called the structure, and is normally defined by the variogram. The experimental variogram measures the average dissimilarity between data separated by a vector h (Goovaerts, 1997). It is calculated according to the following formula.

225

2

121 hN

ixzhxz

hNh

h = separation distance between two points, also called the lag distance.

A generalized formula to calculate the experimental variogram from a set of scattered data can be written as follows (Ahmed, 1995):

]) ,xz(-) ,d+x([z N21 = ) ,d(

dN

i

2ii

d 1

ˆˆˆ (14)

where,

+ - ,d+d d d-d ˆˆ (15)

with (16) N1 = ,d

N1 = d

dd N

ii

d

N

ii

d 11

ˆˆ

Where d and are the initially chosen lag and direction of the variogram with d and as tolerance on lag and direction respectively. d and are actual lag and direction for the corresponding calculated variogram. Nd is the number of pairs for a particular lag and direction. The additional Equation 16 avoids the rounding-off error of pre-decided lags and directions (only multiples of the initial lag and fixed values of only are taken in conventional cases). It is very important to account for every term carefully while calculating variograms. If the data are collected on a regular grid, d and can be taken as zero and d and becomes d and respectively. Often, hydrogeological parameters exhibit anisotropy and hence variograms should be calculated at least in 2 to 4 directions to ensure existence or absence of anisotropy. Of course, a large number of samples are required in that case.

The variogram model is the principal input for both interpolation and simulation schemes. However, modeling the variogram is not a unique process. Various studies (Dagan and Neuman, 1997, Cushman, 1990) tried to correlate the mathematical expressions normally used to describe variogram models to the physical characteristics of the parameters.

Variograms obtained by modeling an experimental variogram are often not unique. It is therefore, necessary to validate it. Cross-validation is performed by estimating the random function at the points where realizations are available (i.e. at data points) and comparing the estimate with data (Ahmed and Gupta, 1989). In this exercise, measured values are removed from the data set one by one and the same is repeated for the entire data set. Thus, all the measurements points, the measured value (z), the estimated value (z*) and the variance of the estimation error ( 2) become available. This leads to computing following statistics:

226

N1 N

iii zz

1

* 0.0)( (17)

N1 N

iii zz

1

2* min)( (18)

N1 N

iiii zz

1

22* 1/)( (19)

0.2*

i

ii zzi (20)

Various parameters of the variogram model are gradually modified to obtain satisfactory values of Equation 17 to 20. During the cross-validation many important tasks are accomplished, such as:

Testing the validity of the structural model.

Deciding the optimum neighbourhood for estimation.

Selecting suitable combinations of additional information particularly in case of multivariate estimation.

Sorting out unreliable data.

APPLICATION OF GEOSTATISTICS IN GROUNDWATER MODELING Theoretical Aspects

The formulation of the general groundwater flow equation in porous media is based on mass conservation and Darcy’s law, and can be represented as follows in three-dimensions (see e.g. Bear, 1979; Mercer and Faust, 1981; Marsily, 1986):

)21(..........'thR

zh

zyh

yxh

x Sszyx

where Kx, Ky and Kz [LT-1] are the hydraulic conductivities (or permeability) along the x,y and z coordinates, respectively, assumed to be the principal directions of anisotropy, h[L] is the hydraulic head given by h=p/ g+z (p is the pressure, the density of water, gthe gravity constant, and z the elevation); h is also the water level measured in piezometers; R [T-1] is an external volumetric flux per unit volume entering or exiting the system and sS [L-1] is the specific storage coefficient.

227

Equation 21 can be integrated over the vertical (two-dimensional flow) and simplified when the flow takes place in confined conditions in the aquifer, and the head is assumed constant over the vertical:

(22).............thSR

yhT

yh

xhT

xh

yx

where Tx and Ty [L²T-1] are the transmissivities (product of the hydraulic conductivity and saturated thickness of the aquifer) in the x and y directions, assumed to be the principal directions of anisotropy in the plane, respectively; R [LT-1] is an external volumetric flux per horizontal unit area entering or exiting the aquifer, in practice, it represents recharge, or sometimes evapotranspiration from a shallow aquifer, or else leakage fluxes between aquifers; [-] is the storage coefficient (product of S sS , the specific storage coefficient, by the thickness of the aquifer). In case of unconfined conditions, Equation 21 can be integrated over the vertical and simplified as follows:

)23........(..........thSR

yhhK

yh

xhhK

xh

yyx

where yS [-] indicates the specific yield of the water-table aquifer. In this case, the hydraulic head is expressed taking the bottom of the aquifer (assumed horizontal) as the z=0 reference plane, so that the head h is also the saturated thickness of the aquifer. Since this equation is non linear, and difficult to deal with, in many instances, Equation 22 is used as an approximation for unconfined aquifers as well, making the simplifying assumption that the product K*h of the saturated thickness h of the aquifer by its permeability K is equal to a constant, the transmissivity T, even if the saturated thickness varies with time. In that case, the storage coefficient S in Equation 22 is replaced by the specific yield yS .

These equations with the appropriate boundary and initial conditions can be solved by standard numerical methods such as Finite Differences or Finite Elements. A large body of literature is available on the analytical or numerical formulation, development and solution of the groundwater flow equations. Some useful references are: Freeze (1971); Remson et al (1971); Prickett (1975); Trescott et al (1976); Pinder and Gray (1977); Cooley (1977, 1979, 1982); Brebbia (1978); Mercer and Faust (1981); Wang and Anderson (1982); Townley and Wilson (1985); Marsily (1986).

The major steps involved in performing aquifer modeling of an aquifer are as follows :

1. Conceptualization of the aquifer system and establishment of various flows.

228

2. Preparing a database of the essential and available parameters with their georeference.

3. Analyzing the variability of various paramters in the area. 4. Discretization of the aquifer system into meshes with theoretical and practical

constraints.5. Fabrication of the aquifer model by assigning various parameters to each mesh. 6. Execution of model under steady and transient conditions and comparing the

output with filed values. 7. Calibration of aquifer model to eleminate the mismatches. 8. Performing sensitivity analyses to determine the sensitivity and priority of various

parameters 9. Finalization of the calibrated model simulating the flow in the aquifer system. 10. Building the futuristic scenarios and predcting the waterlevels using the

calibrated model.

The entire procedure involves a large number of decisions and hence the risk of high biasness. The present work analyses the pratcial problems and constraints faced at all levels and to solve them in an unbiased way using the theory of regionalized variables.

Discretization

This is the first step in the aquifer modeling. Here the modelers have to descritize the area into large number of uniform grids or meshes over which it is supposed that there is no variation in aquifer properties. Although with the invent of advanced computers, the computation of large number of grids/meshes is not a difficult task but the preparation of data to be assigned to each grid is time consuming. The main question arises here is how to decide the number of grid and their sizes? Here the use of geostatistics lies in the modeling. Initially the area is divided in to very finer grids and with the help of geostatistical estimation procedure, the variance of estimation error over each grid is calculated. As we know that the closer the spatial difference, the closer will be the values and thus final grids sizes can be decided by uniting the grids having same or close values of the variance of estimation error. Thus the grid size is decided on the basis of geostatistical estimation which provide unbiased, minimum variance and with a unique value over the entire area of the mesh as block kriging is used.

Model Fabrication After discretization of the area in to optimal grid sizes, the hydrogeological

parameters are required over each grid. It is often very difficult to determine these parameters accurately in the field over such a large number of points either due to economic reason or because of access to the area. The data is always available only at few places in the area. Now it is necessary to link the two situations, and to supply the parameter values for all the discretized volumes from the available observed data.

229

It has been observed that the deterministic approaches are not much applicable due to great spatio-temporal variability of the hydrogeological parameters. The field heterogeneity of the groundwater basins is often inextricable and very difficult to analyze with deterministic methods (e.g., Bakr et al., 1978; Delhomme, 1979; Ganoulis and Morel-Seytoux, 1985). In-situ measurements at the basin scale have shown that physical properties of the hydrogeological variables are highly irregular. However, this spatial variability is not, in general, purely random and the variables exhibit some kind of correlation in their spatial distribution.

It is more convenient and common to introduce geostatistics in a probabilistic framework, for performing estimation considering the spatio-temporal variability of the parameters. By using geostatistical estimation techniques, it is not only possible to evaluate the values of parameters at each and every grid but the level of accuracy in terms of variance of estimation error can also be obtained prior to estimation.

Calibration

Calibration is the process of modifying the input parameters to a groundwater model until the output from the model matches an observed set of data with in some acceptable criteria. This criterion is decided with the help of geostatistics. A confidence interval is given by the standard deviation of the estimation error that provides a useful guide to parameter modification at each mesh and to check that the calculated value falls inside the confidence interval of the observed values.

The calibration process typically involves calibrating the steady state and transient state conditions. In steady state condition unbiased calibration of aquifer model is obtained by using kriging estimation variance and the result lies within the 95% of confidence interval. Mathematically it can be stated as,

i

imi hh

< 2 (24)

Where,hi

m = modeled water level. hi

* = estimated water level.

i = standard deviation of the estimation error.

Similarly in case of transient condition, the calibration of the aquifer model in an unbiased way is performed using hydraulic conductivity within 95% confidence interval of estimation variance. Mathematically it can be stated as,

kT

k KKK 22 (25)

where,K* = estimated hydraulic conductivity KT = observed hydraulic conductivity

230

k = standard deviation of the estimation error.

Prediction

After the model has been succesfully calibrated it is now ready for predictive simulation. Thus the model now can be used for predicting some future groundwater flow or contaminant transport conditions. It can also be used for evaluating different remediation alternatives. However errors and uncertainities in a groundwater flow analysis and solute transport analysis make any model prediction no better than approximation unless all the model predictions should be expressed as a range of possible outcomes that reflects the assumptions involved and uncertainity in the model input data and parameter values. In spite of all unbiased and best efforts, still a large number of calibrated models could be avialable for prediction. Thus to make them unique, a double sum is calculated using the statnadard deviation of the water level estimation error also for all the time step and for all the meshes. Thus a calibrated model providing minimum value of this sum could be used for prediction.

A CASE STUDY FROM THE MAHESHWARAM WATERSHED, ANDHRA PRADESH, INDIA

The Maheshwaram watershed of about 53 km² in the Ranga Reddy district (Figure 1) of, Andhra Pradesh, India, is underlain by granitic rocks. This watershed is a representative Southern India catchment in terms of overexploitation of its weathered hard-rock aquifer (more than 700 borewells in use), its cropping pattern, rural socio-economy (based mainly on traditional agriculture), agricultural practices and semi-arid climate. The granite outcrops in and around Maheshwaram form part of the largest of all granite bodies recorded in Peninsular India. Alcaline intrusions, aplite, pegmatite, epidote, quartz veins and dolerite dykes traverse the granite. There are three types of fracture patterns in the area, viz. (i) mineralised fractures, (ii) fractures traversed by dykes, and (iii) late-stage fractures represented by joints. The vertical fracture pattern is partly responsible for the development of the weathered zone and the horizontal fractures are the result of the weathering. Hydrogeologically, the aquifer occurs both in the weathered zone and in the underlying weathered-fractured zone. However, due to deep drilling and heavy groundwater withdrawal, the weathered zone has now become dry. About 150 dug wells were examined and the nature of the weathering was studied. The weathered-zone profiles range in thickness from 1 to 5 m below ground level (bgl). They are followed by semi-weathered and fractured zones that reach down to 20 m bgl. The weathering of the granite has occurred in different phases and the granitic batholith appears to be a composite body that has emerged in different places and not as a single body. One set of pegmatite veins displacing another set of pegmatitic veins has been observed in some well sections. Joints are well developed in the main directions – N 0° – 15°E, NE-SW, and NW-SE that vary slightly from place to place.

231

Figure 1. Geographical location of the study area and the watershed

The groundwater flow system is local, i.e. with its recharge area at a topographic high and its discharge area at a topographic low adjacent to each other. Intermediate and regional groundwater flow systems also exist since there is significant hydraulic conductivity at depth. Aquifers occur in the permeable saprolite (weathered) layer, as well as in the weathered-fractured zone of the bedrock and the quartz pegmatite intrusive veins when they are jointed and fractured. Thus only the development of the saprolite zone and the fracturing and interconnectivity between the various fractures allow a potential aquifer to develop, provided that a recharge zone is connected to the groundwater system.

Mean annual precipitation (P) is about 750 mm, of which more than 90% falls during the monsoon season. The mean annual temperature is about 26 °C, although in summer ("Rabi" season from March to May), the maximum temperature can reach 45 °C. The resulting potential evaporation from the soil plus transpiration by plants (PET) is 1,800 mm/year. Therefore, the aridity index (AI = P/PET = 0.42) is 0.2 < AI < 0.5, typical of semiarid areas (UNEP 1992). Although the annual rainfall is around 750 mm and the recharge around 10-15%, the water levels have been lowered by about 10 m during the last two decades due to intensive exploitation. The transmissivity of the fractured aquifer was measured by seven pumping tests in wells and it varies considerably from about 1.7×10-5 m²/s to about 1.7×10-3 m²/s, and a low storage

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coefficient (0.6%) indicates a weak storage potential of the aquifer (Maréchal et al., 2004). The withdrawal which increases year by year has to be controlled to allow recharge by rainfall to maintain or restore the productive capacity of the depleted aquifer.

Our data show that the weathered-fractured layer is conductive mainly from the surface down to a depth of 35 m, the range within which conductive fracture zones with transmissivities greater than 5×10-6 m2/s are observed (Maréchal et al. 2004). The lower limit corresponds to the top of the unweathered basement, which contains few or poorly conductive fractures (T < 5×10-6 m2/s) and where only local deep tectonic fractures are assumed to be significantly conductive at great depths, as observed by studies in Sweden (Talbot and Sirat, 2001), Finland (Elo, 1992) and the United States (Stuckless and Dudley, 2002). An unpublished study of the same area has statistically confirmed this result by the analyses of airlift flow rates in 288 boreholes, 10 to 90 m deep, where the cumulative airlift flow rate ranges from 1.5 m3/h to 45 m3/h, and the mean value increases drastically in the weathered-fractured layer at depths between 20 and 30 m. Below 30 m, the flow rate is constant and does not increase with depth. In practice, drilling deeper than the bottom of the weathered-fractured layer (30 – 35 m) does not increase the probability of improving the well discharge. The data confirm that the weathered-fractured layer is the most productive part of the hard-rock aquifer, as already shown elsewhere by other authors (e.g. Houston and Lewis 1988, Taylor and Howard 2000).

Two different scales of fracture networks are identified and characterized by hydraulic tests (Maréchal et al., 2004): a primary fracture network (PFN) which affects the matrix at the decimetre scale, and a secondary fracture network (SFN) affecting the blocks at the borehole scale. The latter is the first one described below.

The secondary network is composed of two conductive fractures sets - a horizontal one (HSFN) and a sub-vertical one (VSFN) as observed on outcrops. They are the main contributors to the permeability of the weathered-fractured layer. The average vertical density of the horizontal conductive set ranges from 0.15 m-1 to 0.24 m-1, with a fracture length of a few tens of meters (10 - 30 m in diameter for the only available data). This corresponds to a mean vertical thickness of the blocks ranging from 4 m to 7 m. The strong dependence of the permeability on the density of the conductive fractures indicates that individual fractures contribute more or less equally to the bulk horizontal conductivity (Kr = 10-5 m/s) of the aquifer. No strong heterogeneity is detected in the distribution of the hydraulic conductivities of the fractures, and therefore no scale effect was inferred at the borehole scale. The sub-vertical conductive fracture set connects the horizontal network, ensuring a vertical permeability (Kz = 10-6 m/s) and a good connectivity in the aquifer. Nevertheless, the sub-vertical set of fractures is less permeable than the horizontal one, introducing a horizontal-to-vertical anisotropy ratio for the permeability close to 10 due to the preponderance of horizontal fractures.

As discussed earlier, the horizontal fracture set is due to the weathering processes, through the expansion of the micaceous minerals, which induces cracks in the rock. These fractures are mostly sub-parallel to the contemporaneous weathering surface, as in

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the flat Maheshwaram watershed where they are mostly horizontal (Maréchal et al., 2003). In the field, the bore wells drilled with a fairly homogeneous spacing throughout the watershed confirm this conclusion: of 288 wells, 257 (89%) were drilled deeper than 20 meters and 98% of these are productive. In any case, the probability of a vertical well crossing a horizontal fracture induced by such wide-scale weathering is very high.

The primary fracture network (PFN), operating at the block scale, increases the original matrix permeability of Km = 10-14 – 10-9 m/s to Kb = 4×10-8 m/s. Regarding the matrix storage, it should also contribute to the storage coefficient in the blocks of Sb = 5.7 x 10-3. The storage in the blocks represents 91% of the total specific yield (Sy = 6.3×10-3)of the aquifer; storage in the secondary fracture network accounts for the rest. This high storage in the blocks (in both the matrix and the PFN) and the generation of the PFN would result from the first stage of the weathering process itself. The development of the secondary fracture network (SFN) is the second stage in the weathering process: that is why the words “primary” and “secondary” have been chosen to qualify the different levels of fracture networks. The obtained storage values are compatible with those of typical unconfined aquifers in low-permeability sedimentary layers.

The universal character of granite weathering and its worldwide distribution underline the importance of understanding its impact on the hydrodynamic properties of the aquifers in these environments.

Several input parameters were analysed for their variability to assess the areas of high, low and medium variability. Finally the water level gradient have been kriged all over the area by taking initially a reasonable fine grids of 200m by 200m size. The estimation error of the estimates of the water level gradient in a pre-monson season when the pumping has been least and water levels were not disturbed by the anthropogenic activity, were taken as guiding parameter and by joining the grids with uniform values the meshes were made courser and courser. Figure 2 shows the experimental and theoretical variogram of the water level gradient.

Figure 3 depicts a reasonable discretization of the aquifer system with variable meshe sizes in a nested square fashion. The discretization thus are made based on the criteria of parameters varibility and distribution of the estimation error. However, this has to be refined as the software MODFLOW does not permit a nested square grids and due to this constraints, the meshes were further modified but this change was made only to suit the software. There are software that may permit the use of originally obtained mesh discretization. The meshe sizes thus used are 200m by 200m, 200m by 400m, 400m by 200m and 400m by 400m with a total of 764 meshes in the watershed (Figure 4).

The model discreized thus has reasonable logics and have been obtained following criteria laid down mainly on the variability of the paramters. The system parameter mainly K values obtained from the hydraulic tests on 25 wells fairly distributed in the area, were kriged using a suitable variogram and obtained the values over all the 764 meshes of the model. Of course, the field values were log-transformed and then the estimated values were backword transformed by taking anti-logs. The Specific Yield

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values were not available so two zones were assigend. The eastern and western boundaries are taken as no flow boundaries as the major flow direction is from south to north. The remaining nodes at the boundaries are taken as fixed head boundaries as the water levels are well known, Other input paramters such as rainfall recharge, groundwater withdrawal, groundwater evaporation, Irrigation return flow from irrigated fields etc. are treated as regionalized variables and they have all beenestimated using theory of regionalized variables. Figures 5 and 6 show the variogram of recharge percentage and the map of the estimated recharge percentage over the area

Figures 7 and 8 show the variogram and the estimates of groundwater withdrawals.

In this case the number of meshes and number of pumping wells are more or less conciding and then values were estimated at each grids using block estimation. The model with these paramter values was run for the steady and transient conditions. Figures 9 show the estimated values of initial water level that was used for steady state simulation.

Figure 2. Experimental and Theoretical Variogram of the waterlevel gradient

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Figure 3. The discretization of the aquifer into Nested Square meshes

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Figure 4. Variable size mesh discretization suitable to MODFLOW

Figure 5. Variogram of the percentage of rainfall recharge

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Figure 6. The map showing variation of recharge percentage to rainfall

Figure 7. Variogram of the groundwater withdrawal with a high nugget effect.

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Figure 8. Estimated values of the groundwater withdrawal for a typical period

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Figure 9. Piezometric map in the area for the initial conditions (June 2000)

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The most important aspect in deaaling with and removing biasness from the aquifer modeling is during calibration and acceptance of the model. The Equation 24 has been used for testing the acceptability and zones of calibration if required and the changes during the calibration was based on the Equation 25. The values of Equation 24 is thus plotted in a map at Figure 10 that very intrestingly shows the meshes needed caibration chnages and these chnages were carried out using Equation 25.

The comparison during the transient state was also made using the geostatistics.

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Figure 10. Map showing the areas that need calibration and the areas where the comparison of water-levels are satisfactory.

CONCLUSIONS It is often very difficult to analyse field hetrogeneties of groundwater basins with

deterministic methods. Thus stochastic methods has been used especially for the estimation of the hydrogeological parameters. The theory of regionalized variables provides a sound stochastic model for the study of spatial phenomena and its applicaion in the field of hydrogeology has been found to be extremely fruitful. The advantage of

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geostatistical methods is that it provide the variance of estimated error together with the estimated value. Of course there are tremendous applications of this method particularly in groundwater modelling:

The closer the values of aquifer parameters to reality the faster will be the model calibration. Better estimated values with lower estimation variance are initially assigned to the nodes of aquifer model using geostatistical estimation.

An assumption is made in aquifer modeling that a single value of system parameter represents the entire grid(although, very small). Averaging over a block in two or three dimension can be obtained through block estimation.

An optimal grid size and number of nodes in discretizing aquifer system, can be obtained and best locations for the new control points can be predicted.

A confidence interval given by the standard estimation of the estimation error provides a useful guide to K/T modification over each grid and to check that the calculated heads fall inside the confidence interval of the observed heads.

A performance analysis of the calibrated model can be achieved to decide the best calibrated model using variance of the estimated error which can be used for prediction.

The main handicap in applying geostatistical techniques to the hydrogeological problems is the scarcity of the observed data. Careful estimation of the variogram and cross-validation of the variogram models are therefore essential. Thus, a few modifications and improvement to the existing techniques permits to utilize hydrogeological data successfully in prediction of aquifer parameters. One such modification which has found much use is ˝kriging with external drift˝.

In geostatistics assumptions are made on the nature of the underlying random function (such as the intrinsic hypothesis or second order stationarity). These assumptions are critical to the success of the application of these techniques.

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Morocco, J. Math. Geol., 15(4), 537-551. Aboufirassi, M., and Marino, M.A., (1984). Cokriging of aquifer transmissivities from

field measurements of transmissivity and specific capacity, J. Math. Geol., 16(1), 19-35.

Ahmed, S., (2001). Rationalization of Aquifer Parameters for Aquifer Modeling Including Monitoring Network Design, Modeling in Hydrogeology, UNESCO International Hydrological Programme, Allied Publishers Limited, 39-57.

Ahmed, S., and Marsily, G. de, (1988). Some application of multivariate kriging in groundwater hydrology. Science de la Terre, Serie Informatique, 28: 1-25.

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Ahmed, S., Sankaran, S., and Gupta, C.P., (1995). Variographic analysis of some hydrogeological parameters: Use of geological soft data, Jour. of Environ. Hydrology, 3 (2), 28-35.

Bakr, A. A., Gelhar, L. W., Gutjahr, A. L., and MacMillan, R. R., (1978), Stochastic analysis of spatial variability in subsurface flows 1. Comparison of one- and three-dimensional flows, Water Resour. Res. 14(2), 263–271.

Bear, J. (1979) Hydraulics of Groundwater, McGraw-Hill, New York. Brebbia, C.A. (1978) The Boundary Element Method for Engineers. Pentech Press,

London.Cooley, R.L. (1977) A method of estimating parameters and assessing reliability for

models of steady state groundwater flow, 1, Theory and numerical property. Water Resour.Res. 13 (2): 318-324.

Cooley, R.L. (1979) A method of estimating parameters and assessing reliability for models of steady state groundwater flow, 2, Application of statistical analysis. Water Resour. Res. 15 (3): 603-617.

Cooley, R.L. (1982) Incorporation of prior information into non-linear regression groundwater flow models. 1. Theory. Water Resour.Res. 18 (4): 965-976.

Cressie, N. A., (1991). Statistics for spatial data. Wiley Series in Probability and Mathematical Statistics. John Wiley, New York.

Dagan, G., (1985). Stochastic modeling of groundwater flow by unconditional and conditional probabilities. Water Resour. Res., Vol. 21, no.1, 65-72.

Delfiner, P. and Delhomme, J.P., (1973). Optimum Interpolation by kriging, in “Display and Analysis of Spatial Data”, J.C. Davis and M.J. McCullough (eds.), 96-114, Wiley and Sons, London.

Delhomme, J.P., (1976). Application de la theorie des variables regionalisees dans les sciences de l’eau, Doctoral Thesis, Ecole Des Mines De Paris Fontainebleau, France.

Delhomme, J.P., (1978). Kriging in the hydrosciences, Advances in Water Resources, 1 no. 5: 252-266.

Delhomme, J.P., (1979). Spatial variability and uncertainty in groundwater flow parameters: a geostatistical approach, Water Resour. Res. 15(2), 269-280.

Elo, S. (1992) Geophysical indications of deep fractures in the Narankavaara-Syote and Kandalaksha-Puolanka zones, Geological Survey of Finland, 13: 43-50.

Freeze, R.A. (1971) Three-dimensional, transient, saturated-unsaturated flow in a groundwater basin. Water Resour. Res. 7(2): 347-366.

Gambolati, G. and Volpi, G., 1979(a). Groundwater contour mapping in Venice by Stochastic Interpolators, 1. Theory, Water Resour. Res., 15(2), 281-290.

Gambolati, G., and Volpi, G., (1979). A conceptual deterministic analysis of the kriging technique in hydrology. Water Resour. Res., Vol.15, no.3, 625-629.

Ganoulis, J., and Morel-Seytoux, H (1985). Application of stochastic methods to the study of aquifer systems Project IHP-II A.1.9.2. UNESCO.

Goovaerts, P., (1997). Geostatistics for Natural Resources Evaluation. New York: Oxford University Press.

Houston, J.F.T. and Lewis, R.T. (1988) The Victoria Province drought relief project, II. Borehole yield relationships, Groundwater, 26(4): 418-426.

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Kitanidis, P.K. (1997). Introduction to Geostatistics: Applications in Hydrogeology. Cambridge University Press, New York.

Maréchal, J.C., Dewandel, B., Subrahmanyam, K. (2004), Use of hydraulic tests at different scales to characterize fracture network properties in the weathered-fractured layer of a hard rock aquifer, Water Resour. Res., 40, W11508, doi:10.1029/2004WR003137

Maréchal, J.C., Wyns, R., Lachassagne, P., Subrahmanyam, K. and Touchard, F. (2003) Anisotropie verticale de la perméabilité de l’horizon fissuré des aquifères de socle : concordance avec la structure géologique des profils d’altération, C.R. Geoscience, 335 : 451-460.

Marsily, G de., 1986. Quantitative Hydrogeology: Groundwater hydrology for Engineers: Academic Press: 203-206.

Marsily, G de., Lavedan, G., Boucher, M., and Fasanini, G., (1984). Interpretation of interference tests in a well field using geostatistical techniques to fit the permeability distribution in a reservoir model, in Geostatistics for Natural Resources Characterization, part 2, edited by G. Verly et al., 831-849, D. Reidel, Hingham, Mass.

Marsily, G. de and Ahmed, S., (1987). Application of kriging techniques in groundwater hydrology, Jr. Geol. Soc. of India, Vol. 29, No.1, 47-69.

Mercer, J.W., and Faust, Ch. R. (1981) Groundwater modelling. National Water Well Assoc., Columbus, Ohio.

Pinder, G. F., and Gray, W.G. (1977) Finite Element Simulation in Surface and Subsurface Hydrology. Academic Press, London.

Remson, I., Hornberger, G.M., and Molz, F.J. (1971) Numerical Methods in Subsurface Hydrology with an Introduction to the Finite Element Method. Wiley (Intersciences), New York.

Stuckless, J.S. and Dudley, W.W. (2002) The geohydrologic setting of Yucca Mountain, Nevada, Applied Geochemistry, 17(6): 659-682.

Talbot, C.J. and Sirat, M. (2001) Stress control of hydraulic conductivity in fracture-saturated Swedish bedrock, Engineering Geology, 61(2-3): 145-153.

Taylor, R. and Howard, K. (2000) A tectono-geomorphic model of the hydrogeology of deeply weathered crystalline rock: evidence from Uganda, Hydrogeology J., 8(3): 279-294.

Townley, L.R. and J.L. Wilson (1985) Computationally efficient algorithms for parameter estimation and uncertainty propagation in numerical models of groundwater flow, Water Resour. Res., 21(12):1851-1860.

Trescott, P. C., Pinder, G.F., and Carson, S.P. (1976) Finite difference model for aquifer simulation in two dimensions with results of numerical experiments. In “ Technique of Water Resources Investigations of the USGS,”

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Wang, H. F., and Anderson, M. P. (1982) Introduction to Groundwater Modeling. Finite Difference and Finite Element Methods. Freeman, San Francisco, Calif.

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An evaluation of Jacob’s method for the interpretationof pumping tests in heterogeneous formations

Peter M. Meier, Jesus Carrera, and Xavier Sanchez-VilaDepartament Enginyeria del Terreny i Cartografica, ETS Enginyers de Camins, Canals i Ports de BarcelonaUniversitat Politecnica de Catalunya, Barcelona, Spain

Abstract. Most pumping tests are interpreted using the classical Theis assumption oflarge-scale homogeneity with various corrections to account for early time behavior ofdrawdown curves. When drawdowns are plotted versus log time, late time data oftendelineate a straight line, which is consistent with Jacob’s approximation of Theis’ solutionbut may seem surprising in view of the heterogeneity of natural media. The aim of ourwork is to show that Jacob’s method leads to a good approximation of the effectivetransmissivity of heterogeneous media when constrained to late time data. A review ofseveral multiwell pumping tests demonstrates that when drawdown curves from eachobservation well are interpreted separately, they produce very similar transmissivity Testimates. However, the corresponding estimates for storativity span a broad range. Thisbehavior is verified numerically for several models of formation heterogeneity. A verysignificant finding of the numerical investigation is that T values estimated using simulateddrawdown from individual observation wells are all very close to the effective T value forparallel flow. This was observed even in nonmultiGaussian T fields, where high T zonesare well connected and where the effective T is larger than the geometric average of pointvalues. This implies that Jacob’s method can be used for estimating effective T values inmany, if not most, formations.

1. Introduction1.1. Basic Concepts and Previous Work

Formation transmissivity is the most important parameter tobe determined in many hydrogeological problems. Transmis-sivity is often determined in the field using pumping tests.Hydrogeologists have to rely on an interpretation model forthe evaluation of pumping test data. Constant rate pumpingtests are most often interpreted using the Theis [1935] methodor Jacob’s semilogarithmic approximation [Cooper and Jacob,1946]. Both of these techniques use the temporal evolution ofpumping-induced drawdown to obtain estimates of transmis-sivity and storativity under the assumptions of homogeneity, atwo-dimensional domain, and confined conditions. Because ofthe importance of Jacob’s method for this work, we review itbriefly here (further details can be found in most hydrogeologytext books; see, e.g., Freeze and Cherry [1979]). Jacob’s methodis based on the fact that the Theis well function plots as astraight line on semilogarithmic paper at large dimensionlesstimes. Hence the method consists of drawing a straight linethrough the late time data points and extending it backward tothe point of zero drawdown (time axis intercept), which isdesignated t0. The drawdown per log cycle is obtained fromthe slope m of the line. Values for transmissivity T and stor-ativity S can then be found from

T �2.3Q4�m (1)

S �2.25Tt0

r2 (2)

where Q is the constant pumping rate and r is the radialdistance to the observation well; if drawdowns are measured atthe pumping well, r is equal to the effective radius of the well.The method is considered a valid approximation of the Theissolution if the slope can be derived from data points with u �r2S/(4Tt) smaller than 0.03 for which the approximation erroris �1%.

The analytical solution underlying the Theis and Jacob tech-niques is based on the assumption of aquifer homogeneity.Other analytical solutions assume that the aquifer can be sub-divided into, at most, two or three regions with uniform pa-rameters [e.g., Streltsova, 1988; Butler, 1988, 1990; Butler andLiu, 1991, 1993]. Although such configurations are simplifica-tions of the complex structure of real heterogeneous forma-tions, they can provide insight into the behavior of drawdownin heterogeneous formations. For example, Butler [1990] ex-plains how the Theis and Jacob methods lead to different Tvalues in heterogeneous aquifers because they apply differentweightings to different portions of the drawdown curve. Asshown in (1), pumping test analysis using Jacob’s method isbased on the rate of drawdown change. When T only dependson radial distance (axial symmetry), such change reflects aqui-fer properties only within a ring of influence through which thefront of the pressure depression passes within the consideredtime interval. Therefore estimated transmissivity is indepen-dent of the aquifer properties between the inner radius of thering of influence and the pumping well. However, storage es-timates depend on the variations in T between the pumpingwell and the front of the cone of depression [Butler, 1988].Butler and Liu [1993] conceptualized the nonuniform aquiferas a uniform matrix into which a disk of anomalous propertieshas been placed. They found, among other things, that Jacob’smethod can be used in any laterally nonuniform system toestimate matrix transmissivity if the flow to the pumping well is

Copyright 1998 by the American Geophysical Union.

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approximately radial during the period of analysis. Oliver[1990] used a perturbation approach to determine the weight-ing function according to which formation permeabilities areaveraged in the determination of apparent values from Jacob’smethod for pumping test analysis. He found that the averagingarea is an annular region of the formation and that both theaveraging area and the radius of investigation of a pumpingtest increase with time. Oliver [1993] used a perturbation ap-proach to study the effect of two-dimensional spatial variationsin transmissivity and storativity. He concluded that the area ofinvestigation of a pumping test is bounded by an ellipse thatencloses the pumping well and the observation well and thatobservation well drawdown is relatively sensitive to near-welltransmissivity, whereas the observation well drawdown deriv-ative is influenced by the near-well properties for a finite pe-riod only.

Previous numerical investigations of transient radial flow inheterogeneous systems resembling real formations by Warrenand Price [1961], Lachassagne et al. [1989], and Butler [1991]have been concerned with estimates of transmissivity frompumping tests and their relationship to statistical parametersof the heterogeneous transmissivity field. These authors con-clude that transmissivity values obtained from late time datausing standard analysis techniques and especially Jacob’smethod are close to the geometric mean of the transmissivityfields and are a reasonable representation of transmissivity onthe regional scale (assuming boundary effects are negligible).Gomez-Hernandez et al. [1995] simulate pumping tests in non-stationary anisotropic T fields and conclude that the simulateddrawdown curves can be misinterpreted as typical responsesfrom a variety of homogeneous aquifer conceptual models(e.g., leaky aquifers, double porosity systems, Theis withboundary effects, and others). Herweijer [1996] simulatespumping and tracer tests in T fields corresponding to both adeterministic sedimentological facies model and a Gaussiangeostatistical model. He concludes that early time portions ofpumping test data reveals high conductivity interwell pathways,which dominate solute transport.

1.2. Working Hypothesis and Scope of the Work

We have frequently observed that when applying Jacob’smethod to late time drawdown data at several observationwells, transmissivity estimates Test tend to be fairly constant,while the storativity estimates Sest display a great spatial vari-ability. Similar observations have previously been reported bySchad and Teutsch [1994] and Herweijer and Young [1991] instudies in which the Theis method is used to analyze drawdowndata from observation wells in heterogeneous alluvial aquifers.

This behavior is surprising in most cases because indepen-dent information (such as short term hydraulic tests, geologicalinformation, etc.) often suggests that the actual formationtransmissivity T is strongly heterogeneous. Conversely, the ac-tual aquifer storativity S is not thought to be highly variable.The variability of the storativity estimates is sometimes attrib-uted to radial flow in homogeneous anisotropic formations[e.g., Streltsova, 1988]. However, methods accounting for ho-mogeneous anisotropic transmissivity [e.g., Neuman et al.,1984; Hantush, 1966; Papadopulos, 1965] did not provide con-sistent results when applied to our field cases. Storativity de-pends on porosity and the rock and water compressibility. Thelatter are believed to be fairly constant at the local scale.Porosity usually varies within a small range compared withhydraulic conductivity, as reported by Guimera and Carrera

[1997] for fractured rocks, Ptak and Teutsch [1994] for a porousaquifer, Bachu and Underschulz [1992] for sandstone cores,and Neuzil [1994] for clays and shales. Because of this, we haveassumed storativity to be homogeneous for our numericalwork.

The above paragraphs appear paradoxical: Long-term hy-draulic tests lead to small variability of Test and large variabilityof Sest, while the opposite should be expected for point valuesof T and S. We conjecture that this paradox can be attributedto the fact that methods developed for homogeneous mediaare being used for interpretation of tests performed in heter-ogeneous formations.

Our work is centered around the above conjecture. Hencewe first provide some field observations to show that this sit-uation indeed exists. Second, we use a series of numericalsimulations to show that Jacob’s method yields nearly constantTest and spatially variable Sest in aquifers with variable T andconstant S. Third, we investigate how the spatial correlationstructure of T and the variance of the input log T field affectthe results. Finally, we compare the value of transmissivityobtained using Jacob’s method with the effective transmissivityderived for parallel flow.

2. Field ObservationsWe have obtained similar Test and strongly varying Sest val-

ues when using Jacob’s method at several field studies in verydifferent types of heterogeneous media, including a single-shear zone in granitic rock, a fractured gneiss formation, andalluvial aquifers. In what follows, we first present field datafrom the migration experiment of the joint project of Nagra(Swiss National Cooperative for the Disposal of RadioactiveWaste) and PNC (Power Reactor and Nuclear Fuel Develop-ment Corporation, Japan) at the Grimsel underground rocklaboratory, located in the Swiss Aar Massif within graniticrocks. Second, we show the results from the interpretation of along-term pumping test in a fractured gneiss formation at ElCabril, which is the site for storage of low- and intermediate-level radioactive waste in southern Spain. Third, we presenttwo other cases from the literature in which different authorshave reported a similar behavior of Test and Sest values inalluvial aquifers.

2.1. Migration Experiment at the Grimsel Test Site

The Grimsel rock laboratory consists of galleries lying 450 mbelow the surface within granitic rock. A migration experiment[Frick et al., 1992] was conducted within an almost verticalshear zone that dips parallel to the foliation of the host rock.This shear zone, henceforth designated as the migration shearzone, is more or less isolated from other shear zones andconsists of a series of subparallel fractures. Mylonite, cataclas-tite, and other gouge materials are observed within the sub-fractures, influencing the permeability within the shear zone.The migration shear zone can be conceptualized as a verticalzone of intersecting fractures with a thickness of �0.5 m andcan be considered to be a two-dimensional feature at the scaleof the tests described here. Hydrological exploration involveddrilling eight boreholes into the shear zone. Locations of theboreholes and the laboratory gallery within the shear zoneplane are shown in Figure 1. The intervals in the boreholeswere isolated with packers for the performance of hydraulicand tracer tests.

Single-hole short-term pumping tests of 5–10 s in duration

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(rates between 10 and 200 mL/min) were conducted to obtainpoint estimates for the transmissivity. These tests were ana-lyzed with the MARIAJ code [Carbonell and Carrera, 1992],which uses an automatic optimization procedure coupled withthe general analytical solution of Barker [1988]. Transmissivityestimates range between 10�8 and 5 � 10�6 m2/s, indicatingstrong local heterogeneity, while storativity estimates vary from5 � 10�10 to 10�5 m2/s [Meier, 1997]. The large variation in theS estimates is believed to reflect the uncertainties in estimatingS from single-well tests but may also be a product of an analysismethod based on an analytical solution for homogeneous aqui-fers. Cross-hole constant-rate pumping tests of �2 hours induration were performed in boreholes 4, 6, 8, 9, and 10 usingpumping rates of 200 mL/min. Analysis of the late time data atboth the pumping and the observation wells with Jacob’smethod resulted in a virtually constant Test value of �10�6

m2/s for all tests and in a broad range of Sest values extendingfrom 4 � 10�7 to 5 � 10�2 m2/s [Meier, 1997]. Transmissivitiesand storativities are displayed in Figures 2a and 2b for thesingle-hole tests and for the cross-hole test performed bypumping in borehole 9. A semilog plot of the drawdown curvesfor the cross-hole pumping test in borehole 9 is shown inFigure 3 as an example of the drawdown behavior seen in allthe cross-hole tests. Straight lines with similar slopes startdeveloping after about 3500 s for all drawdown curves. Bound-ary effects due to the influence of the gallery are not observedduring these tests. Note that inverse geostatistical modeling ofthe steady state hydraulic head field and the transient draw-down data reveals a variation in transmissivity of several ordersof magnitude and the existence of highly transmissive channelswithin the shear zone [Meier et al., 1997]. Such channels may beresponsible for the relatively large drawdown at borehole 5shown in Figure 3.

2.2. El Cabril

The El Cabril site is located in a very sparsely populated areawest of Cordoba in southern Spain. The underlying formationsare composed of biotitic gneiss and metaarkoses. These for-mations were affected by the compressive Hercynian deforma-tion, so the formations follow a predominantly WNW-ESE andNW-SE trend and are characterized by primarily vertical dips.Several sets of transverse fractures (directions 60�N, 90�N, and

N-S) are clearly indicated on the surface [Carrera et al., 1993].A large number of slug and short-term pumping tests werecarried out at this site. The transmissivities obtained fromthese tests vary over 5 orders of magnitude. A long-term pump-ing test was also performed [Bureau de Recherches Geologiqueset Minieres, 1990]. The locations of the pumping (S33) andobservation wells are shown in Figure 4. The drawdown curvesfrom individual observation wells were interpreted separatelyusing the MARIAJ code [Carbonell and Carrera, 1992] underthe assumption of a homogeneous and isotropic medium. Theestimated parameters obtained for each well are shown inFigure 5. The Test values range from 0.38 to 0.52 m2/d, whilethe Sest values range from 10�4 to 3 � 10�2 m2/d. Thus the Test

values are almost constant, while the Sest values range over

Figure 1. Locations of the boreholes and the laboratory gal-lery in the migration shear zone at the Grimsel test site.

Figure 2. Results from single-hole and cross-hole tests con-ducted in the migration shear zone for (a) transmissivity esti-mates and (b) storativity estimates. Note that transmissivitiesderived from short-term single-hole tests display a large vari-ability in contrast to transmissivities from cross-hole tests,which remain virtually constant. Storativity estimates span awide range of values for single-hole and cross-hole tests.

Figure 3. Semilog plot of the drawdown observed at thepumping well and the observation points during a cross-holetest in the migration shear zone.

1013MEIER ET AL.: JACOB’S METHOD AND HETEROGENEOUS FORMATIONS

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more than 2 orders of magnitude. This test was later inter-preted by means of a finite element model automatically cali-brated with the approach of Carrera and Neuman [1986].Drawdowns could only be closely matched by assuming thatS227 (the observation well at which the smallest Sest was ob-tained) was connected to S33 through a fracture, a likely situ-ation given the geology at the site. Note that a convergent flowtracer test was performed at this site by injecting a differenttracer into each of the observation wells displayed in Figure 4while well S33 was pumped. Porosities were estimated underthe assumption of homogeneity using the TRAZADOR code[Benet and Carrera, 1992]. The pattern of estimated porositiesis very similar to that of Sest in Figure 5 [Sanchez-Vila et al.,1992; Carrera, 1993]. This suggests that relatively high trans-missivity pathways are reflected not only in Sest but also inestimated porosity when homogeneous media solutions areused.

2.3. Field Observations Reported in the Literature

Schad and Teutsch [1994] provide an extensive data set froma series of pumping tests performed in a highly heterogeneousaquifer at the Horkheimer Insel field site in the Neckar Valleyof Germany. Because of its relevance to our work, we reviewbriefly some parts of their study. The unconfined aquifer inwhich the tests were performed consists of �4 m of poorlysorted sand and gravel deposits of a braided river origin. Thisaquifer is overlain by 5–6 m of mostly clayey flood plain de-posits and underlain by a hydraulically tight clay. Hydraulicconductivity data derived from grain-size distributions had anoverall variance of log conductivity equal to 2.35, indicatingsignificant heterogeneity. Schad and Teutsch performed 26pumping tests (duration of 2 hours) at different wells at thesite. During each of these tests, up to four observation wellswere monitored at radial distances of 2 to 36 m from thepumping well. The analysis of the early time data, using theTheis method, resulted in a broad range of Test values (1.6 �10�2 to 3.2 � 10�1 m2/s) and in a very broad range of Sest

values (5.8 � 10�4 to 1.0 � 10�1 m2/s). The analysis of the latetime data, using the Theis method, resulted in a small range ofTest values (2.4 � 10�2 to 5.0 � 10�2 m2/s) and in a broadrange of Sest values (1.7 � 10�2 to 1.3 � 10�1 m2/s). They alsoconducted a long-term (96 hour) pumping test in which draw-down was monitored at 23 observation wells. The analysis ofthis test with the Theis method resulted in a very small rangeof Test values (2.9 � 10�2 to 3.5 � 10�2 m2/s) and in a broadrange of Sest values (2.6 � 10�2 to 1.1 � 10�1 m2/s). Schad andTeutsch also mention that Sest values show a large variability atshort observation distances and a progressively smaller vari-ability for larger distances. They point out that the variations inSest should be a function of the relative transmissivity of theflow path between the pumping and the observation well. Ifthis path has a high relative transmissivity, Sest should be smalland vice versa.

Herweijer and Young [1991] present a qualitative model ofaquifer heterogeneity for the interpretation of spatial and tem-poral variability of hydraulic parameters derived from pump-ing tests in a heterogeneous fluvial aquifer at Columbus AirForce Base, Mississippi. Their results from pumping tests an-alyzed with the Theis method cover a wide range for storativityestimates and a relatively small range for transmissivity esti-mates. They conclude that the storage estimates show a strongdependence on the flow pattern induced by pumping and thedegree of interconnection of high hydraulic conductivity lensesbetween the pumping and observation wells.

3. Numerical Simulation MethodologyWe performed a series of numerical simulations to corrob-

orate our field observations and to further develop those ofSchad and Teutsch [1994]. Our methodology, which is similarto that used by Lachassagne et al. [1989], consists of the fol-lowing steps: (1) generation of two-dimensional (2-D) hetero-geneous T fields, (2) simulation of transient radial flow towarda pumping well in the generated T fields using a finite elementcode, (3) analysis of the drawdown curves simulated at allnodes of the grid using Jacob’s method, (4) generation of mapsshowing the spatial distribution of the Test and Sest values anddrawdown, and (5) analysis of the results. Regarding the anal-ysis of the results, we first compare the spatial Sest distributionwith the spatial structure of the input T fields. Second, wecheck whether the average of the Test values (designated asTavg) is somewhat representative of the full domain by com-paring it with the effective transmissivity obtained from theo-retical considerations and from numerical analysis of parallelflow in the same T fields.

Figure 4. Location of pumping and observation wells at theEl Cabril site.

Figure 5. Transmissivity and storativity estimates obtainedfrom the interpretation of drawdown curves at each observa-tion well for the long-term pumping test in borehole S33 at theEl Cabril site. Again, T estimates remain virtually constant,while S estimates span 3 orders of magnitude.


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We first describe in detail the methodology and then exam-ine the influence of the variance and spatial correlation struc-ture of T on the results. Our investigations include multilog-normal, nonmultilognormal, and fractal T fields.

The simulated random transmissivity fields are square inshape and are discretized into 500 � 500 square elements ofunit area. We assign T values, which are generated using thecomputer code GCOSIM3D [Gomez-Hernandez and Journel,1993], to each of the elements and consider these as pointvalues.

To minimize boundary effects, the heterogeneous domain of500 � 500 square elements of unit area is bounded on all sidesby 20 rectangular elements of increasing size as shown sche-matically in Figure 6. The T values of these elements are setequal to the geometric mean transmissivity of the random Tfield (Tg � 1.0). Constant head conditions are applied to theexternal boundaries of the full domain. This full domain issquare in shape and consists of 291,600 elements. A homoge-neous input storativity of 1.0 is used for all the simulations.Although 1.0 is a physically implausible value for storativity,this assignment has no effect on the results (S � 1.0 waschosen so that the average diffusivity (T/S) of the heteroge-neous domain was 1.0). The pumping well is located at thecenter of the domain. The transient flow equation is solvedwith the finite element computer code FAITH [Sanchez-Vila etal., 1992], which is especially designed to work with a largenumber of elements. Drawdown values are calculated every 10time units and are written to a file every 100 time units for allnodes of the finite element mesh. The Test and Sest values arecalculated with Jacob’s method for all nodes by fitting astraight line through the last three drawdown points. Specialcare is taken to make sure that drawdown curves are notsensitive to the homogeneous outer zone surrounding the het-

erogeneous domain. Several preliminary simulations showedthat this influence starts at �30,000 time units. Our simulationsare therefore terminated at 20,000 time units, and our meth-odology is considered valid in a square domain of 100 units ona side centered on the well. This small heterogeneous domainis schematically indicated in Figure 6.

Note that this methodology involves applying Jacob’smethod over the same time range for all observation points.This results in Jacob’s method being applied at different di-mensionless times (1/u � 4Tt/(r2S)) at different points in thedomain. Since the appropriateness of the approximation un-derlying Jacob’s method is a function of the dimensionlesstime, the appropriateness of this approximation varies as afunction of location within the domain. For example, the di-mensionless time is 15.79 for a radius of 71 units, correspond-ing to the distance between the pumping well and the cornersof the domain of valid methodology, for t � 19,900, which isthe average time of the small time period from 19,800 to 20,000time units for which Jacob’s method is applied, for an averageT � 1.0 and for S � 1.0. For 1/u � 15.79 the error in usingJacob’s method instead of the Theis method is about 3.0%,assuming that these criteria are valid for heterogeneous con-ditions. The errors are considerably smaller at the center of thedomain (about 0.2% for a radius of 30 units and the sameparameters used above). Therefore we expect to obtain a slightradial trend with smaller T values in the center and larger Tvalues at further distances from the pumping well as a conse-quence of applying Jacob’s method over the same time rangefor all observation points. However, we want to stress that theabove considerations are strictly valid only for homogeneousconditions and may not be appropriate for highly heteroge-neous T fields.

The arithmetic mean of the 10,000 Test values within the

Figure 6. Schematic drawing of the model setup indicating the discretization scheme in the top left quad-rant, the pumping well location, the full model domain, the heterogeneous domain, and the heterogeneoussubdomain for which our methodology is considered valid.


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small domain, which are obtained using Jacob’s method, willbe designated as Tavg throughout the remainder of this work.Tavg is compared (1) with the geometric sample mean (Tg) ofthe input T fields and (2) with the effective transmissivityobtained from parallel flow simulations (Teffp) using the sameT fields as in the radial flow simulations. The motivation forthis twofold comparison is based on the work of Sanchez-Vilaet al. [1996]. This work suggests that the assumption of multi-lognormality may not be valid in many real field cases, even ifpoint T values display a lognormal distribution and, further-more, that the appearance of scale effects in T may be attrib-uted to departures from multilognormality. Here the term“scale effect” is used to describe the apparent increase of Twith the size of the definition domain and, more specifically,the increase of large-scale effective T with respect to the geo-metric average of point values. They performed 2-D flow sim-ulations with a mean parallel gradient in nonmultilognormal Tfields and showed that the effective transmissivity Teffp doesnot necessarily agree with Tg, contrary to Teff derived formultilognormal distributions. Following their approach, we ob-tain Teffp for T fields by solving the steady state flow equationunder very simple boundary conditions as shown in Figure 7.When applied to our heterogeneous T fields (large domain of500 � 500 cells), we obtain a mean gradient that is constantand parallel to the x coordinate axis. Under such conditions,the effective transmissivity in the x direction is given byTeffpx � QT � L/�H, where QT is the total flux per unit widthinto (and out of) the domain and L is its length. A change ofboundary conditions is used to find the effective transmissivityin the y direction Teffpy. The geometric mean Teffp �TeffpxTeffpy is considered an appropriate quantity for thecomparison with our simulation results (Tavg). The Tg, Teffp,and Tavg values for all our test cases are given in Table 1.

It should be stressed that a single effective transmissivity,defined as the tensor that relates the expected values of flowand head gradient, can only be defined for parallel flow con-ditions, for which it is equal to Tg under relatively broadconditions [Matheron, 1967]. De Marsily [1986, p. 82] states, “Ifthe flow is not uniform (converging radial, for example), thereis no law of composition, constant in time, that makes it pos-sible to define a mean darcian permeability. This problem isquite worrying from the conceptual viewpoint in so far as it is

precisely through pumping tests in wells that the transmissivityof an aquifer is measured in situ.” It should be stressed that theproblem of an effective transmissivity for radial flow conditionsis not examined here, because we are primarily interested inthe relationship between the transmissivity estimates providedby an interpretation model for pumping test analysis (in ourcase Jacob’s method) and the effective transmissivity for par-allel flow conditions (Teffp), which is a representative param-eter for most problems in hydrogeology.

In section 4 we examine the relation between Teffp and Tavg

for three conceptual models for aquifer heterogeneity: (1) amultilognormal T field, (2) a nonmultilognormal T field withconnected high T zones without a large-scale trend, and (3) afractal field with a large-scale trend.

4. Test Cases4.1. Test Case 1: Multilognormal Random Field,Effect of log T Variance �Y

2

The objective of the first test case is to corroborate the fieldobservations and to investigate the influence of �Y

2 on Test andSest. We simulate three T fields, assuming Y � ln T is astationary multinormal random function characterized by themean Y� (�0.0 for the three T fields), the variances (�Y

2 �0.25, 1.0, and 4.0), and the variogram (a spherical isotropicmodel with a range of 26 unit lengths, which corresponds to anintegral scale of �10 unit lengths). The T fields are generatedby conditioning on the four square elements surrounding thenode located at the center, which serves as the pumping well,fixing their values to Tg � 1.0, Tg being the geometric meanof the point T values. The effect of varying the transmissivity atthe well (Tw) will be investigated in test case 3. The T fieldwith �Y

2 � 1.0 is shown in Figure 8a, where the black areasrepresent the most transmissive quartile and the white areasrepresent the least transmissive one. Table 1 indicates thatTeffp agrees well with the sample Tg of the input T fields, asexpected for multilognormal fields.

Figures 8c and 8d show the simulated drawdown curves atseveral observation points located in different directions (1 to8) but at the same distance R from the pumping well (see

Figure 7. Boundary conditions used to compute effectivetransmissivity for parallel flow conditions in a heterogeneous Tfield.

Table 1. Comparison Between Effective Transmissivity forParallel Flow Conditions and Average TransmissivityEstimates Obtained From Jacob’s Method for DifferentIntegral Scales, Pumping Well Transmissivity, log TVariances, and Geometric Mean Transmissivity

TestCase Tw I �Y

2 Tg Teffp/Tg

Tavg/Tg, Range

1 1.00 10 0.25 1.00 1.00 1.01 2.1E-41 1.00 10 1.01 1.03 1.00 0.97 2.6E-41 1.00 10 4.04 1.00 0.99 0.96 8.5E-43 1.65 10 1.01 1.03 1.00 0.97 2.5E-43 0.61 10 1.01 1.03 1.00 0.97 2.7E-44 1.00 25 1.03 1.00 1.10 1.13 4.2E-44 1.00 25 2.06 1.02 1.24 1.23 6.3E-44 1.00 25 4.12 1.00 1.58 1.48 7.6E-45 1.00 � � � 0.74 0.97 0.97 1.07 1.2E-3

Tw, pumping well transmissivity; I, integral scale; �Y2 , log T variance;

Tg, geometric mean transmissivity; Teffp, effective transmissivity forparallel flow conditions; Tavg, average transmissivity estimate. Noticethat Teffp and Tavg are very close except when T is nonstationary (testcase 5). Notice also the small range of Tavg. Read 1.01 2.1E-4 as1.01 2.1 � 10�4.


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Figure 8b), for the T fields with �Y2 � 0.25 and 4.0. Results are

presented for R � 10 and R � 30. The corresponding draw-down curves for a homogeneous T field with T � Tg � 1.0and S � 1.0 are shown for comparison. Note that only a small

number of drawdown points are shown on each curve to keepthe figures clear. All drawdown curves exhibit late time slopeswhich are similar to that corresponding to the homogeneouscase. In Figure 8d the drawdown curves at observation points

Figure 8. Test case 1. (a) Multilognormal random T field of 500 � 500 units with �Y2 � 1.0 and (b)

subdomain of 100 � 100 units for which the methodology is considered valid are shown here. “W” indicatesthe well location. The numbers indicate the observation points at a radial distance R � 30 units. The solidtriangles indicate the observation points at R � 10 units. (c) and (d) Semilog plots of the simulated drawdownat the observation points shown in Figure 8b for �Y

2 � 0.25 and 4.0 are shown. Note that late time slopes areapproximately equal for all curves.


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2 and 5 at radial distances of 30 unit lengths from the wellillustrate the variability in early time pressure response in het-erogeneous media. Observation point 2 and the well are con-nected through a zone of low T, which can easily be recognizedin Figure 8b. The drawdown response is therefore delayed withrespect to the homogeneous case. Jacob’s method yields a Sest

greater than the formation S because t0, the intercept with thetime axis of the straight line fitted through the last data points,is greater than that of the homogeneous case. On the contrary,observation point 5 and the well are connected through a zoneof high T. The drawdown response is therefore earlier in timethan the homogeneous case, resulting in a Sest smaller than theformation S. A high �Y

2 results in greater differences betweenthe drawdown curves at the same radial distance to the well,which in turn results in a higher variability of Sest and viceversa. The differences between the drawdown curves decreasewith increasing radial distance to the well, causing a decreaseof the Sest variance with the radial distance from the well, asthe flow process samples a larger number of heterogeneities.This is consistent with the field observations of Schad andTeutsch [1994] discussed in section 2. Note that for R � 10 thesolid line (homogeneous case) is below all responses in Figure8c and all but one in Figure 8d for times later than 200 timeunits at which the pressure front has already passed the obser-vation points at R � 10. This is due to the increased slopes ofthe drawdown curves (with respect to the homogeneous case)between �100 and 500 time units. The increased slopes reflectthe relatively low geometric average of �0.8 of the point Tvalues within the annular ring between radial distances of �10and 30 units from the well. The increased slopes of the draw-down curves between 100 and 500 time units result in a generaldecrease of the Sest values for the observation points at R �10. In summary, Sest values seem to depend not only on thetransmissivity connecting the well and the observation pointsbut also on transmissivities within a certain catchment areasurrounding both the well and the observation points.

The spatial distribution of Sest and Test and the drawdownfor the T fields with �Y

2 � 0.25 and 4.0 are shown in Figures9a–9f. The following can be concluded from these figures: (1)The Test values obtained from Jacob’s method are very close toTg � 1.0, which is equal to Teffp in multilognormal T fields (asdiscussed in section 3, the radial trend of the Test distributionsmay be attributed to applying Jacob’s method over the sametime range to all points, regardless of the distance to thepumping well); (2) the Sest values vary strongly in space despitethe fact that the formation S was homogeneous; (3) the spatialstructures of Sest resemble each other for different values of�Y

2 (see Figure 10c for �Y2 � 1.0), and the variance of Sest

depends on �Y2 ; and (4) a dependence between the heteroge-

neous input T field (Figure 8b) and the spatial distribution ofSest does exist, although the Sest field is smoother. Note that wehave also found that results are independent of the integralscale (I) by performing the same test with different values of I.

4.2. Test Case 2: Multilognormal Random Field,Effect of Support Scale

The idea behind test case 2 is to examine what happens whenwe sample the field more coarsely and assign the data as blockvalues. A new multilognormal random field with increasedsupport scale is generated using the T field with �Y

2 � 1.0corresponding to the first test case, while maintaining Y� andchanging only slightly �Y

2 and the variogram. The generationprocedure is the following: The T value of one element of the

original T field is assigned to its surrounding elements, formingblocks of 3 � 3 unit lengths. This procedure is repeated forevery third element in the x and y directions for the T fieldshown in Figure 8a. The central domain of the resulting T fieldwith the increased support scale is shown in Figure 10a. Com-parison with Figure 8b indicates that the main features of theoriginal T field are maintained. The spatial distributions of Sest

obtained for the T field with increased support scale and forthe original T field are shown in Figures 10b and 10c, respec-tively. The change of support scale results only in minorchanges of the spatial Sest distribution. The increase of supportscale has virtually no effect on the spatial distribution of Test,which is therefore not shown.

4.3. Test Case 3: Multilognormal Random Field,Effect of Transmissivity at the Well

In the T fields presented in the first two cases the transmis-sivity at the well (Tw) was selected to be equal to Tg. In testcase 3 we are interested in examining the effects that changingthe Tw value would have on the Sest and Test values. For thispurpose, two new fields were generated by conditioning thesimulations on Tw values different from Tg � 1.0. The twonew values were Tw � 0.61 and Tw � 1.65. The statisticalparameters and the random number seed corresponding to thefirst test case were maintained. Comparison of the subdomainsof the fields generated with Tw � 1.0, Tw � 0.61, and Tw �1.65 (Figures 8b, 11a, and 11b, respectively) indicates that onlythe T values at the central domain of �20 � 20 elements aresignificantly changed.

The spatial distributions of Sest and Test are shown in Fig-ures 11c–11f. The Tavg values for the two fields remain prac-tically unchanged with respect to the first test case (see Table1). This is in conceptual agreement with the results of Butler[1988], Butler and Liu [1993], and Oliver [1993], who state thatmethods of transmissivity estimation that use the late time rateof change of drawdown, instead of the total drawdown, are lesslikely to be influenced by near-well nonuniformity. Note thatthe radial trend of the Test distributions, which is similar to theone observed for test case 1, may be attributed to applyingJacob’s method over the same time range to all points, regard-less of the distance to the pumping well.

Noticeable differences between the two simulations are re-stricted to the Sest in the central domain where the input Tfields are different. High Tw results in high Sest next to the welland vice versa. Note that the results within a radial distance of�5 unit lengths around the well are considered to be affectedby numerical errors due to a poor spatial discretization in thevicinity of the well. However, our qualitative interpretation ofthe simulations seems to be in agreement with the commonknowledge that Sest obtained from single-hole hydraulic testscan disagree considerably with formation S (a detailed discus-sion about this topic is given by Butler [1988, 1990]). For thecase of Sest that is considerably greater than formation S weexpect that the pumping well is situated within a zone of T �Test or vice versa. The formation S has then to be calculatedfrom rock and fluid properties, which is a common practice inthe field of reservoir engineering. Note that differences be-tween Sest and real aquifer S are generally attributed to skineffects [e.g., Earlougher, 1977], which are conceptually equiv-alent to heterogeneities near the well. The fact that the Sest

values at the well are highly sensitive to near-well nonunifor-mity may potentially be utilized to characterize small-scaleheterogeneity using Sest values from single-hole tests at differ-


252

ent locations. However, the applicability of these findings toreal field data will strongly depend on several factors fre-quently encountered in field situations. These factors include,among others, skin effects due to well drilling and develop-ment, well losses, well bore storage effects, and delayed drainage.

4.4. Test Case 4: Nonmultilognormal Field,Effect of �Y

2

Test cases 1–3 show that Jacob’s method yields values veryclose to Tg and Teffp when applied to pumping test data frommultilognormal T fields. The objective of this test case is to

Figure 9. Test case 1: (a) and (b) spatial distribution of Sest for �Y2 � 0.25 and 4.0, (c) and (d) spatial

distribution of Test for �Y2 � 0.25 and 4.0, and (e) and (f) drawdown distribution for �Y

2 � 0.25 and 4.0. Anoteworthy feature of Figures 9c and 9d is the very small range of Test values, suggesting that T estimates inthis field are independent of the location of the observation points.


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extend our study to nonmultilognormal T fields. For this pur-pose, we use field number 4 of Sanchez-Vila et al. [1996]. Thisfield is shown in Figure 12a. It was generated by connectinghigh T values of an originally multilognormal field while main-taining the histogram for the In T field and the normality of the

univariate distribution of T values. The variogram and themultilognormality of the original field were not retained. Thegeneration procedure is explained by Sanchez-Vila et al. [1996].We believe that nonmultilognormal T fields like the one shownin Figure 12a, having well-connected high T zones, are betterconceptual models for fractured rocks or alluvial aquifers thanmultilognormal T fields. Sanchez-Vila et al. [1996] have shownthat Teffp may depend on �Y

2 in nonmultilognormal T fields.Here we examine whether the good agreement between Teffp

and Tavg obtained for multilognormal T fields holds also fornonmultilognormal T fields.

Radial flow was simulated in three T fields with Y� � 0.0and �Y

2 values of 1.0, 2.1, and 4.1. The spatial Test distributionsare shown in Figures 12c, 12e, and 12f. Note that the radialtrend of the Test distributions, which is similar to the onesobserved for test cases 1–3, may be attributed to applyingJacob’s method over the same time range for all observationpoints, regardless of the distance to the pumping well.

Table 1 shows that Tavg agrees well with Teffp but is quitedifferent from Tg at large �Y

2 . These results indicate thatperforming a pumping test and using the slope of the draw-down curve in the analysis allow a value of transmissivity to beobtained that is very close to the effective transmissivity of thearea influenced by the test, regardless of whether the T field ismultilognormal. We have observed this type of behavior onseveral occasions. The most dramatic is the one reported byCarrera et al. [1990], who found T derived from a long-termpumping test to be 20 times larger than the geometric averageof short-term single-hole tests performed in a fractured me-dium at nine wells at the Chalk River site in Canada. Note thathighly transmissive channels as conceptualized in Figure 12aare frequently observed within fractured zones. Hence we arelead to the conclusion that transmissivities derived from long-term pumping tests are frequently larger than those derivedfrom the geometric mean of short-term single-hole tests (seeSanchez-Vila et al. [1996] for a further discussion of this issue).The main finding from this section is that an effective trans-missivity for parallel flow is that obtained from long-termpumping tests, rather than the geometric average of local val-ues.

The spatial Sest distribution is shown for �Y2 � 1.0 in Figure

12d. Unlike the other test cases, it is difficult to recognize herea correlation between the input T field and the spatial Sest

distribution. The fine structure of the input T field is notreflected by the Sest distribution, which seems to be the resultof a spatial integration. Therefore we conjecture that Sest de-pends not only on the nature of the connection between thewell and the observation points but also on all the Y values ina certain catchment area surrounding both the well and theobservation points. This aspect is further explored by X.Sanchez-Vila et al. (Pumping tests in heterogeneous auquifers:An analytical study of what can be really obtained from theirinterpretation using Jacob’s method, submitted to Water Re-sources Research, 1998). Note that the spatial Sest distributionsfor the other �Y

2 qualitatively resemble the one shown in Fig-ure 12d and are therefore not presented.

4.5. Test Case 5: Fractal Field

A log transmissivity field was generated having a power lawsemivariogram �(h) � c h2� with � � 0.25 and c � 0.027,as proposed by Neuman [1990, 1994]. This T field, shown inFigure 13a, is generated by conditioning on the transmissivity

Figure 10. Test case 2: (a) subdomain (100 � 100 units) withincreased support scale (T blocks of 3 � 3 units) and with thegeostatistical parameters of test case 1, (b) spatial distributionof Sest for test case 2, and (c) spatial distribution of Sest for testcase 1 with original support scale (T blocks of 1 � 1 units) and�Y

2 � 1.0.


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value (Tw � 1.0) at the well. The field has a sample meanY� � �0.03 and a sample variance �Y

2 � 0.74.The spatial Test distribution, shown in Figure 13c, may be

affected by the spatial distribution of the extreme transmissiv-ity values which are located at the corners of the large domain,as the radial pattern observed in test cases 1–4 is distorted. Inthis case, the original T field is markedly nonstationary. Hence

it is not surprising that some trend can be observed in Test.However, the range of Test is relatively small. The fact thatTavg � Tg � Teffp (see Table 1) may indicate a problem for thedefinition of an effective T value for our fractal field.According to the relation given by Neuman [1994], Teff(L) �Tg exp (cL2�[0.5 � �/E]), where L is the characteristiclength of a rock volume which acts as the support for a per-

Figure 11. Test case 3: (a) and (b) Subdomains for T fields with Tw � 0.61 and 1.65 (�Y2 � 1.0) (the

pumping well locations are in the center of the subdomain), (c) and (d) spatial distributions of Sest for Tw �0.61 and 1.65, and (e) and (f) spatial distributions of Test for Tw � 0.61 and 1.65.


255

Figure 12. Test case 4: (a) nonmultilognormal random T field with �Y2 � 1.0 (500 � 500 units), (b)

subdomain (100 � 100 units) for which our methodology is considered valid, (c) spatial distribution of Test for�Y

2 � 1.0, (d) spatial distribution of Sest for �Y2 � 1.0, and (e) and (f) spatial distribution of Test for �Y

2 �2.1 and 4.1.


256

meability test, E is the dimension of the flow condition(1, 2, or 3), and � � 1 for infinite isotropic media, 0 � � � 1for bounded isotropic media, and 1 � � � E for infiniteanisotropic media. Using the above relation with our Tavg,Teffp, Tg, and L � 700 unit lengths, we obtain � � 0.82 forTavg, indicating a bounded isotropic medium, and � � 1.0 forTeffp, indicating an anisotropic medium. Inspection of Figure13a reveals the somewhat anisotropic nature of this field. How-ever, because the well is located at the center of the field andthe extreme T values are concentrated at the corners, thedrawdown behavior is similar to the one for a bounded do-main. The high T zones at the bottom left and the top rightcorners act similar to constant head boundaries, whereas thelow T zones at the other corners act similar to no-flow bound-aries.

The spatial distribution of Sest (Figure 13d) seems to bedominated by the heterogeneities of the small domain pre-

sented in Figure 13b. High Sest correspond to zones with a lowconnectivity to the well, for example, the bottom right cornersof Figures 13b and 13d, and vice versa.

5. Summary and Conclusions1. Our numerical simulations of pumping tests in aquifers

with heterogeneous transmissivity T and homogeneous storat-ivity S show that applying Jacob’s method to late time draw-down data at observation points results in a strong spatialvariability of storativity estimates Sest and in an almost homo-geneous spatial distribution of transmissivity estimates Test.This is in agreement with field observations.

2. Our simulations indicate that Sest depends mainly on theconnectivity between the well and the observation points. Thiswas suggested by Schad and Teutsch [1994] on the basis of fieldobservations and is consistent with the discussions of Butler

Figure 13. Test case 5: (a) fractal T field of 500 � 500 units, (b) subdomain (100 � 100 units) for which ourmethodology is considered valid, (c) spatial distribution of Test, and (d) spatial distribution of Sest.


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[1990] and Butler and Liu [1993]. Furthermore, Sest valuesseem to depend also on the transmissivities within a certaincatchment area surrounding both the well and the observationpoints.

3. As has been shown in previous investigations, nonuni-formities near the well have no effect on Test. However, Sest

values vary considerably in the near-well region. A well locatedwithin a zone of T � Test results in Sest values considerablygreater than the formation S and vice versa. This result hasbeen documented in the petroleum engineering and ground-water literature [e.g., Barker and Herbert, 1982; Butler, 1988,1990]. However, these results were mostly considered withinthe context of artificially induced heterogeneities around a well(e.g., skin effects due to drilling and development). We con-jecture that these results could potentially be used to charac-terize natural small-scale heterogeneity (still much larger thanthe well radius), using Sest values from single-hole pumpingtests at different locations, provided that artificially inducedheterogeneities at the well are either measured by other meansor are negligible in comparison with natural small-scale heter-ogeneity.

4. Test values show an almost homogeneous spatial distri-bution for stationary input T fields. The smooth radial trend ofthe Test distributions obtained for all test cases may largely beattributed to applying Jacob’s method over the same timerange for all observation points, regardless of the distance tothe pumping well. However, in the case of the fractal T fieldthe spatial distribution of Test may also be affected by thespatial trend of the input T field, although the enormous trans-missivity contrasts of the input field are smoothed out dramat-ically. Furthermore, the arithmetic average of the Test valuesdoes not coincide with the effective transmissivity for parallelflow conditions in the fractal field (Teffp/Tavg � 0.9), althoughit should be noted that the concept of effective transmissivity isonly valid for stationary fields.

5. Transmissivity estimates obtained from Jacob’s methodunder radial flow conditions are close to the effective trans-missivity for parallel flow conditions for the area of influenceof the pumping test in multilognormal and nonmultilognormalT fields. We consider nonmultilognormal T fields, with well-connected zones of relatively high T, to be realistic conceptualmodels for many, if not most, heterogeneous formations.Therefore the above conclusion bears two significant implica-tions. First, Jacob’s method does indeed work very well in mostheterogeneous formations. Second, the problem that a singleeffective transmissivity value depending only on the statisticalproperties of a T field cannot be derived for radial flow con-ditions does not affect the pumping test interpretation. Jacob’smethod directly provides a representative transmissivity whichcan be used for most hydrogeological problems dealing withparallel flow.

6. Our numerical work is based on the assumption of two-dimensional and confined flow conditions and might thereforeidealize complexities associated with field tests, especially inunconfined aquifers. Three-dimensional flow conditions mightalso be of concern in cases where the aquifer thickness is of asimilar magnitude to the distance between the pumping welland the observation wells. The importance of such effectsshould be investigated in further studies.

Acknowledgments. Comments by G. Teutsch of the University ofTubingen, S. Young of Hydrogeologic Inc., and J. J. Butler of the

Kansas Geological Survey have helped improve the final text and aregratefully acknowledged. The first author acknowledges financial sup-port from SATW (Swiss Academy of Engineering Sciences) and Nagra(Swiss National Cooperative for the Disposal of Radioactive Waste).The work of the remaining authors was funded by ENRESA (SpanishNuclear Waste Management Company), 1995SGR00405 (CIRIT), andAMB95-0407 (CICYT). The data from the Grimsel test site werekindly supplied by Nagra and PNC (Power Reactor and Nuclear FuelDevelopment Corporation, Japan).

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J. Carrera, P. M. Meier, and X. Sanchez-Vila, Department Enginy-eria del Terreny i Cartografica, E. T. S. Enginyers de Camins, Canalsi Ports de Barcelona, Universitat Politecnica de Catalunya, Modul D-2,08034 Barcelona, Spain. (e-mail: [email protected];[email protected]; [email protected])

(Received December 31, 1996; revised December 23, 1997;accepted December 29, 1997.)


259

A procedure for the solution of multicomponent

reactive transport problems

M. De Simoni

Dipartimento di Ingegneria Idraulica, Ambientale, Infrastrutture Viarie, Rilevamento, Politecnico di Milano, Milan, Italy

J. Carrera and X. Sanchez-Vila

Department of Geotechnical Engineering and Geosciences, Technical University of Catalonia, Barcelona, Spain

A. Guadagnini

Dipartimento di Ingegneria Idraulica, Ambientale, Infrastrutture Viarie, Rilevamento, Politecnico di Milano, Milan, Italy

Received 23 February 2005; revised 16 July 2005; accepted 1 August 2005; published 8 November 2005.

[1] Modeling transport of reactive solutes is a challenging problem, necessary forunderstanding the fate of pollutants and geochemical processes occurring in aquifers,rivers, estuaries, and oceans. Geochemical processes involving multiple reactive speciesare generally analyzed using advanced numerical codes. The resulting complexity hasinhibited the development of analytical solutions for multicomponent heterogeneousreactions such as precipitation/dissolution. We present a procedure to solve groundwaterreactive transport in the case of homogeneous and classical heterogeneous equilibriumreactions induced by mixing different waters. The methodology consists of four steps:(1) defining conservative components to decouple the solution of chemical equilibriumequations from species mass balances, (2) solving the transport equations for theconservative components, (3) performing speciation calculations to obtain concentrationsof aqueous species, and (4) substituting the latter into the transport equations to evaluatereaction rates. We then obtain the space-time distribution of concentrations andreaction rates. The key result is that when the equilibrium constant does not vary in spaceor time, the reaction rate is proportional to the rate of mixing, rT u D ru, where u is thevector of conservative components concentrations and D is the dispersion tensor. Themethodology can be used to test numerical codes by setting benchmark problems butalso to derive closed-form analytical solutions whenever steps 2 and 3 are simple, asillustrated by the application to a binary system. This application clearly elucidates that ina three-dimensional problem both chemical and transport parameters are equally importantin controlling the process.

Citation: De Simoni, M., J. Carrera, X. Sanchez-Vila, and A. Guadagnini (2005), A procedure for the solution of multicomponent

reactive transport problems, Water Resour. Res., 41, W11410, doi:10.1029/2005WR004056.

1. Introduction

[2] Reactive transport modeling refers to the transport ofa (possibly large) number of aqueous species that reactamong themselves and with the solid or gaseous phases. It isrelevant because it helps in understanding the fate ofpollutants in surface and groundwater bodies, the hydro-chemistry of aquifers, and many geological processes [e.g.,Gabrovsek and Dreybrodt, 2000; Freedman et al., 2003;Salas and Ayora, 2004; Emmanuel and Berkowitz, 2005].Unfortunately, it is complex, both conceptually and math-ematically. Modeling reactive transport involves two cou-pled ingredients: (1) the mass balance of all participatingspecies, which is expressed by the solute transport equationfor mobile species; and (2) a set of equations describing thereactions among species. The nature of participating speciesand reactions leads to a broad range of possible behaviors of

the system [Rubin, 1983; Molins et al., 2004]. Moreover,reactions such as dissolution can also modify the aquiferproperties (i.e., porosity and hydraulic conductivity) [e.g.,Wood and Hewett, 1982; Philips, 1991; Kang et al., 2003;Singurindy and Berkowitz, 2003; Singurindy et al., 2004].Therefore it is not surprising that numerous general math-ematical formulations to solve reactive transport problemsare available [e.g., Rubin, 1990, 1992; Yeh and Tripathi,1991; Friedly and Rubin, 1992; Lichtner, 1996; Steefel andMacQuarrie, 1996; Clement et al., 1998; Saaltink et al.,1998, 2001; Tebes-Stevens et al., 1998; Robinson et al.,2000; Molins et al., 2004]. The resulting sets of governingequations have been included in a large number of reactivetransport codes that can handle several species with differ-ent types of reactions (see Saaltink et al. [2004] for a list ofcodes).[3] As opposed to numerical solutions, very few analyt-

ical results are available. These are of general interest, asthey provide insights on the nature of the solution and allowevaluating the relative importance of the involved parame-

Copyright 2005 by the American Geophysical Union.0043-1397/05/2005WR004056$09.00

WATER RESOURCES RESEARCH, VOL. 41, W11410, doi:10.1029/2005WR004056, 2005

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ters and processes. Analytical and semianalytical solutionsare available in the case of linear and nonlinear equilibriumsorption reactions [e.g., Serrano, 2003, and referencestherein]. Analytical solutions are also available to solvereactive transport problems in case of (networks of) first-order kinetic reactions [Sun et al., 1999, 2004; Clement,2001; Serrano, 2003; Quezada et al., 2004]. These meth-odologies are appropriate for radioactive decay chains andfor many biochemical processes. However, they are difficultto extend to geochemical processes whose rates are nonlin-ear functions of the concentrations of the dissolved species,or to cases in which equilibrium conditions can be assumedand the reaction rates cannot be written explicitly.[4] Because transport times are generally large in aqui-

fers, most aqueous reactions and many dissolution/precip-itation reactions can be modeled with the assumption ofchemical equilibrium conditions at each point of thedomain. As equilibrium is reached instantaneously, theamount of reactants evolving into products depends onthe rate at which they mix, which is, in turn, governed bytransport. To illustrate the concept, Figure 1a depicts areaction of pure dissolution/precipitation in the mixing oftwo waters. The concentrations of the dissolved species, c1and c2, in both mixing waters satisfy the equilibriumcondition c1c2 = K, where K is the equilibrium constant.For conservative species, concentrations of the mixturewould be a linear function of the mixing waters concen-trations, while equilibrium concentrations for reactivespecies are governed by a number of factors, includingpH, salinity and temperature [e.g., Wigley and Plummer,1976]. As a result, it is generally difficult to predict whetherthe mixture will precipitate or dissolve, and to estimate theamount of precipitated/dissolved mass.[5] In this paper we propose a concise methodology to

analyze groundwater reactive transport in the presence ofgeneral homogeneous and classical heterogeneous equilib-rium reactions (according to the classification of Rubin[1983]) caused by mixing of different waters. We analyzesituations where all reactions in the system can be assumedat equilibrium conditions at all times. Emphasis is placed onevaluating not only concentrations, but also reaction rates.The method is first applied to a binary system, providing anoriginal analytical expression for the reaction rate in thecase of mixing-driven precipitation. The methodology isthen extended to deal with all kinds of classical heteroge-neous and homogeneous equilibrium reactions. The com-plete methodology allows the mathematical decoupling ofthe problem into four steps: (1) definition of mobileconservative components of the system, (2) transport ofthese components, (3) speciation, and (4) evaluation ofreaction rates and mass change of constant activity species.The method is proposed as a useful tool in deriving furtheranalytical solutions describing reactive transport processes.It can also be of practical use in developing, adapting, ortesting numerical codes to solve solute transport problemsinvolving multiple reactions.[6] The outline of the paper is as follows. In section 2 we

present the mathematical formulation of the problem.Section 3 is focused on the reactive transport problem inthe presence of a pure dissolution/precipitation reaction.Specifically, we analyze in detail a three-dimensionalporous medium with uniform flow carrying two dissolved

constituents at equilibrium with a solid mineral. We derive aclosed-form solution for the mixing-driven precipitationprocess induced by an instantaneous point-like injectionof water containing the same constituents of the residentwater, but at a different equilibrium state. Finally, in

Figure 1. Schematic representation of a pure dissolution/precipitation process during the mixing of two waters. (a)Concentrations of the dissolved species, c1 and c2, in themixing waters satisfy the equilibrium condition c1c2 = K, Kbeing the equilibrium constant. From the conservativemixing point (at which the concentrations are weightedarithmetic averages of the input c values), the mixtureevolves toward the equilibrium curve following a straightline (whose slope is given by the ratio of the stoichiometriccoefficients and is equal to 1 in the simple binary casepresented in the text). (b) Example of the evaluation of(22a). Function u (arbitrarily chosen, for illustrativepurposes) and the two factors in (22a) are displayed versusthe distance from the location at which u reaches itsmaximum value. Note the different scales: @2c2/@u

2 isnearly constant, while the term accounting for mixingdisplays the largest variations.

W11410 DE SIMONI ET AL.: MULTICOMPONENT REACTIVE TRANSPORT PROBLEMS W11410

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section 4 we generalize the formulation to solute transportproblems involving multiple reactions.

2. Preliminary Concepts

2.1. Chemical Equilibrium

[7] Equilibrium reactions are described by mass actionlaws, which in the case of multiple species can be written ina compact form as [Saaltink et al., 1998]

Se log a ¼ logK0; ð1Þwhere Se is the stoichiometric matrix, that is a Nr � Ns

matrix (Nr and Ns being the number of reactions and thetotal number of chemical species, respectively) containingthe stoichiometric coefficients of the reactions, a is thevector of activities of all species and K0 is the vector ofequilibrium constants. In general, the activity of aqueousspecies, aa, is a mildly nonlinear function of aqueousconcentrations, ca, which can be written as

log aa ¼ log ca þ logg cað Þ; ð2Þ

where the activity coefficient, g, may be a function of allaqueous concentrations and can be computed in severalways, the Debye-Huckel equation being a frequent choicefor moderate salinities [Helgeson and Kirkham, 1974]. Fordilute solutions, which are frequently observed in many realgroundwater problems, the activity coefficients can beassumed as unitary. If some reactions involve pure phasesor, in general, mixtures with fixed proportions of constitu-ents or minerals or gas phases, the activities of thecorresponding species can be assumed to be fixed andconstant (e.g., activity is unity for pure minerals and water).[8] Let Nc be the number of constant activity species. We

split vector a (equation (1)) into two parts, ac and aa,containing the constant activity and the remaining aqueousspecies, respectively. Here and in the following, subscripts cand a are used to identify the constant activity and theremaining aqueous species, respectively. Matrix Se is alsodivided into two parts, that is, Se = (SeajSec), where Sec andSea contain the stoichiometric coefficients of constantactivity species and of the aqueous species, respectively.Since Nc activities are fixed and (1) can be used to eliminateNr activities, we can now pose the problem in terms of Ns �Nc � Nr independent activities (conforming with Gibbsphase rule). Once these are calculated, we would obtain theremaining Nr activities on the basis of the chemical equi-librium relations (equation (1)). We call as primary thosespecies corresponding to the independent Ns � Nc � Nr

activities; the remaining Nr species are called secondary. Wethen split Sea into two parts, that is, Sea = (S0eajS00ea), whereS0ea and S00ea contain the stoichiometric coefficients of theprimary and secondary species, respectively.[9] It is convenient to redefine the chemical system so that

the matrix of the stoichiometric coefficients of the secondaryspecies coincides with the opposite of the identity matrix, I.Mathematically, this is equivalent to multiplying equation (1)by (S00ea)

�1 (a proper choice of primary and secondary speciesleads to S00ea invertible). As activities ac are fixed and known,we can rewrite the mass action law (1) as

log a00a ¼ S0ea log a0a � logK; ð3Þ

where a0a and a00a are the activities of primary and secondaryspecies, respectively; S00ea = �I and S0ea are the redefinedmatrices of stoichiometric coefficients of secondary andprimary species, respectively; K is the redefined vector ofequilibrium constants, log K = (S00ea)

�1 (log K0 � Sec log ac).With these definitions and upon assuming unitary activitycoefficients, the mass action law (1) reduces to

Sea log ca ¼ logK: ð4Þ

When the assumption of unitary activity coefficients is notjustified, one needs (2) to relate activities and concentrations.However, it is still possible to write the mass action law interms of concentrations. This is accomplished by substitut-ing K with an equivalent equilibrium constant, K*, definedas

logK* ¼ logK � Sea logg cað Þ; ð5Þ

where K* is a function of ca, besides other possible inde-pendent state variables, such as temperature.[10] In the following we present all chemical equations in

terms of concentrations, using (4). Appendix A is devotedto the discussion of an example to facilitate understandingof the meaning of the vectors and matrices appearing in themass action law.

2.2. Mass Balances

[11] Mass balance of each species can be written in aconcise vector notation as

@ mð Þ@t¼MLt cð Þ þ f : ð6Þ

Here, vector m contains the mass of species per unit volumeof porous medium; it can be split into two parts, mc and ma,respectively related to the constant activity species and tothe remaining species. Vector c contains the concentrationsof species (mi = nci for mobile species, where n is porosity).Matrix M is diagonal and its diagonal terms are unity whena given species is mobile and zero otherwise; f is a generalsource/sink term. The linear operator Lt(ci) appearing in (6)is defined as

Lt cið Þ ¼ �r � q cið Þ þ r � nDrcið Þ; ð7Þ

where D is the dispersion tensor and q is Darcy’s flux.[12] Assuming that the source/sink terms are only due to

chemical reactions and that the system is always at chemicalequilibrium, we can express f as

f ¼ STe r; ð8Þ

where r is the vector of reaction rates (expressed per unitvolume of medium) and Se is defined after (1). We noticethat reactive transport processes also affect immobile spe-cies. For instance, while a mineral at equilibrium with thesolution is not transported, its mass is changed in order toallow the system to attain equilibrium.[13] A reactive transport process is completely described

by the concentrations of the Ns species, together with thedefinition of the Nr (unknown) reactions rates, including theNc rates of mass change of the constant activity species. To


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this end, one has to solve the Ns mass balance equations (6)together with the Nr equilibrium equations (4). While theproblem is very complex because of the nonlinearity of thegoverning system, it can be significantly simplified uponintroducing the concept of components [e.g., Rubin, 1990,1992; Friedly and Rubin, 1992; Steefel and MacQuarrie,1996; Saaltink et al., 1998, 2001; Molins et al., 2004], asdescribed below.

2.3. Components

[14] Components are linear combinations of specieswhose mass is not affected by equilibrium reactions. Theirintroduction is convenient since it allows eliminating thechemical reactions source term in the transport equations.Most of the solution methods for reactive transport prob-lems mentioned in the introduction are based on thisconcept. We introduce the components matrix, U, definedsuch that

USTea ¼ 0: ð9ÞRecalling that the system was defined so that Sea = (S0eaj�I),a widely used expression for U is

U ¼ INuj S0Tea

� �; ð10Þ

where Nu = Ns � Nc � Nr. The expression for thecomponents matrix is not unique. The motivation ofexpressing U by (9) is that multiplying (6) by U, yieldsthe equation

@ nuð Þ@t¼ Lt uð Þ; ð11Þ

where u = Uc is a vector of Nu components. Equation (11)represents Nu transport equations where no reaction sink/source terms appear. That is, components are indeed con-servative quantities in systems at equilibrium. A generalformulation for decoupling transport equations in the pres-ence of kinetic reactions is given by Molins et al. [2004].[15] In the following we first solve a simple binary

system and then generalize the methodology to deal withclassical heterogeneous and homogeneous chemical reac-tions at equilibrium.

3. Binary System

3.1. Problem Statement

[16] We consider a reaction of pure dissolution/precipita-tion [Philips, 1991] at equilibrium, where an immobile solidmineral S3s dissolves reversibly to yield ions B1 and B2:

B1 þ B2�! � S3s: ð12Þ

We further assume that the mineral, S3s, is a pure phase, sothat its activity equals 1. With the notation of section 2,vector ma contains the mass per unit volume of medium,m1 = nc1 and m2 = nc2, of the aqueous species B1 and B2

respectively, while mc contains the mass per unit volume ofmedium, m3, of the solid mineral, S3s. The stoichiometricmatrix of the system described by (12) is

Se ¼ �1 � 1 1ð Þ: ð13Þ

For this system, Se can be split as

Sec ¼ 1ð Þ ð14aÞ

Sea ¼ S0ea j S00ea� � ¼ �1 j �1ð Þ: ð14bÞ

Matrices Sec and Sea contain the stoichiometric coefficientsrelated to the constant activity species and to the remainingaqueous species, respectively. Equation (14b) implicitlyidentifies B1 as primary species and B2 as secondary.[17] Thus the mass action law (equation (4)) for the

considered system is expressed as

log c1 þ log c2 ¼ logK: ð15Þ

The equilibrium constant, K, is strictly related to solubilityof the solid phase, S3s, and usually depends on temperature,pressure and chemical composition of the solution [e.g.,Philips, 1991; Berkowitz et al., 2003].[18] Recalling (8), the mass balance equations for the

three species are

@ nc1ð Þ@t� Lt c1ð Þ ¼ �r ð16aÞ

@ nc2ð Þ@t� Lt c2ð Þ ¼ �r ð16bÞ

@ m3ð Þ@t¼ r; ð16cÞ

where r is the rate of the reaction (12). The reaction rateexpresses the moles of B1 (and B2) that precipitate in orderto maintain equilibrium conditions at all points in themoving fluid (as seen by (16a) and (16b)) and coincideswith the rate of change of m3 (as seen by (16c)).[19] In order to obtain the space-time distribution of the

concentrations of the two aqueous species and thereaction rate, we need to solve the nonlinear problemdescribed by the mass balances of the aqueous species(equations (16a) and (16b)) and the local equilibriumcondition (equation (15)). The rate of change in the solidmineral mass is then provided by (16c).

3.2. Methodology of Solution

[20] The general procedure to solve a multispecies trans-port process will be detailed in section 4. Here, we start byillustrating the application of the method to the binarysystem described in section 3.1 and provide a closed-formanalytical solution.[21] In essence, the solution procedure develops accord-

ing to the following steps:[22] 1. Definition of mobile conservative components of

the system: Using (10) for U and (14b) for Sea, yields

U ¼ 1 � 1ð Þ; ð17Þ

this implies that one needs to solve the transport problem foronly one conservative component u = Uca = c1 � c2. Noticethat simply subtracting (16b) from (16a) leads to the


264

equation which governs the transport of the conservativecomponent u = (c1 � c2). In other words, dissolution orprecipitation of the mineral S3s equally affects c1 and c2 sothat the difference (c1 � c2) is not altered.[23] 2. Transport of the conservative components: Since

only one component is evidenced in this case, one needs tosolve (11) for u.[24] 3. Speciation: Here one needs to compute the con-

centrations of (Ns � Nc) mobile species from the concen-trations of the components. In our binary system, thisimplies solving

c1 � c2 ¼ u ð18aÞ

log c1 þ log c2 ¼ logK: ð18bÞ

[25] Assuming that K is independent of c1 and c2, thesolution of (18a) and (18b) is

c1 ¼ uþ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2 þ 4Kp

2ð19aÞ

c2 ¼ �uþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiu2 þ 4Kp

2: ð19bÞ

[26] 4. Evaluation of the reaction rate: Substitution of theconcentration of the secondary species, B2, into its transportequation (16b) leads to (see Appendix B for details)

r

n¼ @c2

@K� @K

@tþ 1

nLt Kð Þ

� �þ @2c2

@u2rTuDruþ 2

@2c2

@u@KrTuDrK

þ @2c2

@K2rTKDrK: ð20Þ

When K is a function of conservative quantities such assalinity, s, equation (20) can be rewritten (see Appendix Cfor details) as

r

n¼ @2c2

@s2rT sDrsþ @2c2

@u2rTuDruþ 2

@2c2

@u@srTuDrs: ð21Þ

Moreover, in the case of constant K, equation (20) reducesto

r

n¼ @2c2

@u2rTuDru� � ð22aÞ

@2c2

@u2¼ 2K

u2 þ 4Kð Þ3=2: ð22bÞ

In general, the reaction rates should be used to compute themass change of solid mineral. In turn, this would cause amodification in the medium properties. Here, however, wewill neglect such changes assuming that modification inthe solid mass due to transport involves very thin layers ofthe matrix [Rubin, 1983] and no significant variations of thepore system occur.[27] The results encapsulated in (20), (21) and (22)

deserve some discussion. First of all, under chemical

equilibrium conditions, they provide a way to computedirectly the rate of dissolution/precipitation as a functionof quantities such as the concentrations of components, theequilibrium constants, and the dispersion coefficients, with-out the need to actually evaluate the concentrations of thedissolved species. We note that (21) includes the model ofPhilips [1991] as a particular case.[28] Furthermore, equation (22) shows that the reaction

rate is always positive (i.e., precipitation occurs) in systemswhere K is constant. This is consistent with the comments ofRubin [1983], who points out that, in the case of the reactiondescribed by (12), reactive transport processes cannot resultin dissolution of the solid mineral. This is also evident fromFigure 1a. Figure 1a displays another interesting feature.The equilibrium point can be obtained by drawing a linefrom the conservative mixing point toward the equilibriumline. The slope of this line is equal to the ratio of thestoichiometric coefficients and is equal to 1 in our example.[29] Equations (20) and (22) clearly demonstrate that the

reaction rate depends on chemistry, which controls @2c2/@u2

(equation (22b)), but also on transport processes, controllingthe gradient of u. Figure 1b qualitatively shows a syntheticexample which allows evaluating the relative importance ofthe two factors (chemistry and transport) in the reaction rate.We note that it would be hard to make general statementsabout which one is more important, which is consistent withthe sensitivity analysis of Tebes-Stevens et al. [2001]. Oneof the most paradoxical features of Figure 1b is that reactiondoes not necessarily take place where concentrations attaintheir maximum values. In fact, the reaction rate equals zerowhen u is maximum or minimum (i.e., when c1 or c2 reachtheir maximum value, respectively). In general (see equa-tion (20)), the reaction rate depends on both the gradients ofu and K. A nonzero gradient of K, for example, can occurwhen the mixing of different waters induces spatial vari-ability in temperature or salinity [Berkowitz et al., 2003;Rezaei et al., 2005].[30] It is interesting to notice that, in the absence of the

first contribution on the right hand side of (20), all terms areproportional to the dispersion tensor, D, thus strengtheningthe relevance of mixing processes to the development ofsuch reactions. In particular, the term rT u D ru can beused as a measure of the mixing rate, which is consistentwith the concept of dilution index, as defined by Kitanidis[1994] on the basis of entropy arguments. This result alsosuggests that evaluating mixing rates may help to properlyidentify not only the sources of water [Carrera et al., 2004],but also the geochemical processes occurring in the system.[31] In general, the amount of S3s that can precipitate is

controlled by the less abundant species. Thus we expectprecipitation to be highest when mixing induces similarvalues of c1 and c2. Mathematically, this is evidenced by thedependence of the reaction rate on @2c2/@u

2, that reaches amaximum when u = 0 (see equation (22b)). We also noticethat the rate (22a) increases with decreasing K, sincesolubility also decreases (in a system described by (12),solubility is the square root of K).[32] The methodology we have presented can easily be

extended to deal with solute transport in the presence ofmultiple reactions. This is shown in section 4. It is thenrelevant to stress the point that analytical solutions for thesetypes of systems are possible whenever an analytical solu-


265

tion is available for the conservative transport problem (11).One such case is discussed below.

3.3. Analytical Solution: Pulse Injection in aBinary System

[33] In this section we apply the general methodology toderive a closed-form solution for a mixing-driven precipi-tation reaction. Closed-form solutions provide basic meansto investigate the physical underlying processes and toanalyze the relative importance of the parameters involved.Moreover, analytical solutions are potentially useful asbenchmark for numerical codes and can be of assistancein developing methodologies for the setup of laboratoryexperiments and procedures for data analysis and/orinterpretation.[34] We consider a three-dimensional homogeneous

porous formation, of constant porosity, n, under uniformflow conditions. The system is affected by an instantaneouspoint-like injection of water containing the same constitu-ents as the initial resident water. This problem can be usedas a kernel for other injection functions. We consider thecase of constant K. We assume that the velocity is alignedwith the x coordinate, V = q/n = Vix (ix being the unit vectorparallel to the x axes), and that the dispersion tensor isdiagonal, DL and DT respectively being its longitudinal andtransverse components. We further assume that the time ofthe reaction is small as compared to the typical time oftransport (i.e., equilibrium condition).[35] The reactive transport system is governed by (12),

(15), and (16). The boundary conditions are

ci1 ¼ ci x!1; tð Þ ¼ ci0; i ¼ 1; 2: ð23Þ

Initially, we displace resident water by injecting a volume Veof solution with concentration ciext = ci0 + cie. In order tofind an analytical solution for small Ve, it is mathematicallyconvenient to write the initial concentration condition, afterequilibrium is reached at the injection point, as

ci x; t ¼ 0ð Þ ¼ cieVed xð Þ=nþ ci 0; i ¼ 1; 2; ð24Þ

where equilibrium must be satisfied at all points in theaquifer. This implies that (1) c10c20 = K, and (2) (c10 + c1e)(c20 + c2e) = K. From these two equilibrium conditions, itshould be clear that c1e and c2e would have different signs.[36] With this in mind, we then follow the steps detailed

in section 3.2. The (conservative) component u = c1 � c2satisfies (11) with boundary and initial conditions

u1 ¼ u x!1; tð Þ ¼ c1 x!1; tð Þ � c2 x!1; tð Þ ¼ u0 ð25aÞ

u x; t ¼ 0ð Þ ¼ ueVed xð Þ=nþ u0: ð25bÞ

With this definition, ue is then the excess of the injectedcomponent u that remains in the aquifer immediately afterinjection. The solution of (11), subject to boundary andinitial conditions (25a) and (25b) is [Domenico andSchwartz, 1997, p. 380]

u x; tð Þ ¼ u0 þ 18

2pð Þ1=2ueeVd

exp � 1

2r2

� �; ð26Þ

where r is the normalized radial distance from the center ofthe plume, defined as

r ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix� Vtð Þ22tDL

þ y2 þ z2

2 tDT

s: ð27Þ

The dimensionless quantity eVd is the ratio between thevolume containing about the 99, 7% of the excess ofinjected mass [Domenico and Schwartz, 1997] and theinjection volume

eVd ¼ 72pnffiffiffi2p

t3=2ffiffiffiffiffiffiDL

pDT

Ve

ð28Þ

and is a measure of the temporal evolution of the dispersiveeffect.[37] Substituting (26) into (19a) and (19b), we obtain the

concentrations, c1 and c2, of the dissolved species. Finallythe expression of the local mineral mass precipitation rate, r,per unit volume of medium is derived from (22) as

r x; tð Þ ¼ nK

t

18

2pð Þ1=2ueeVd

� �2

r2 exp �r2½ �

u0 þ 18

2pð Þ1=2ueeVd

exp � 12r2

� �2

þ4K" #3=2 : ð29Þ

We notice that the rate, r, vanishes in the trivial case of ue =0, since the concentrations of the two injected species arethe same as the resident ones and there is no excess ofinjected mass in the solution right after the injection. Therate vanishes also when eVd ! 1. The latter situationdescribes a scenario where gradients of the component u arenegligible because of large dispersion effects or because alarge amount of time has elapsed since injection.[38] To illustrate the features of this solution we consider

the following problem: we start with a resident watercharacterized by c10/

ffiffiffiffiKp

= 0.25 and c20/ffiffiffiffiKp

= 4.0 (u0/ffiffiffiffiKp

= �3.75); we then inject water from an external source,characterized by c1ext/

ffiffiffiffiKp

= 0.184, c2ext/ffiffiffiffiKp

= 5.434(uext/

ffiffiffiffiKp

= � 5.25). Equilibrium condition is satisfied whenc1e/

ffiffiffiffiKp

= �0.066, c2e/ffiffiffiffiKp

= 1.434 (ue/ffiffiffiffiKp

= �1.5).Figure 2a depicts the dependence of the dimensionlessconcentrations ~c1 = c1/

ffiffiffiffiKp

and ~c2 = c2/ffiffiffiffiKp

on the normalizeddistance from the center of the (moving) plume, (x � Vt)/ffiffiffiffiffiffiffiffiffiffi2tDL

p(while z = y = 0), and eVd = 3.5. Dimensionless

concentrations (~c1NR, ~c2NR) for the corresponding nonreac-tive system are also shown for comparison. Notice that whileboth concentrations are higher than the initial ones in thenonreactive case, in the reactive case ~c1 decreases while ~c2increases (in agreement with the fact that c1e < 0 and c2e > 0).[39] The spatial distribution of the (local) dimensionless

reaction rate, ~r = rt/(nffiffiffiffiKp

), for the same conditions ofFigure 2a, is depicted in Figure 2b. A comparison ofFigures 2a and 2b clearly elucidates that the system is chem-ically active (i.e., the reaction rate is significant) at locationswhere concentration gradients of both species are relevant.This implies, in turn, that strong gradients of the component ugive rise to significant reaction rates, in agreement with (22a).In general, no reactions occurwithin the systemwhen concen-tration gradients vanish.As a consequence, no reaction occurs


266

at the (moving) center of the plume, that is, at the points ofhighest (or lowest) concentration values.[40] Figures 3a and 3b display the same quantities of

Figures 2a and 2b but for eVd = 16. A comparison betweenFigures 2a and 3a reveals that the concentration profilesdisplay a lower amplitude when eVd increases. A larger valuefor eVd (equation (28)) can be seen as an increase either inelapsed time or in the effects of dispersion for a given time.Thus it is expected that more dispersed plumes are associ-ated to weaker gradients of u and thus to reduced precip-itation rates. This behavior is not observed in the range ofsmall dispersion effects (eVd � 1), where an increase ofdispersion phenomena enhances the mixing process and therelated reaction. This feature can be observed mathemati-cally by taking the limit of (29) when eVd tends to zero.

From (29) we also observe that a change in DT has strongereffects rather than a modification in DL. This is so since DT

is not squared in the definition of eVd and is also clear whenconsidering that dispersion is enhanced along two directionsby increasing DT, while an increase in DL affects only onespatial direction. We will comment further on this topic inthe following.[41] The sensitivity of the reaction rate to ue for a given

u0 is presented in Figure 4, with reference to a residentwater with u0/

ffiffiffiffiKp

= �20 (c10/ffiffiffiffiKp

= 0.05; c20/ffiffiffiffiKp

= 20.05).We start by considering the case with a negative ue (ue/

ffiffiffiffiKp

=�30). The dimensionless mixing volume is set as eVd = 3.5.The shape of the reaction rate function is displayed inFigure 4a at the plane z = 0. From the plot it is clear that, atany given time, precipitation would concentrate in a (three-

Figure 2. Dependence of (a) dimensionless concentrations~c1, ~c2 and (b) dimensionless reaction rate, ~r, on thenormalized distance from the center of the (moving) plume,(x � Vt)/

ffiffiffiffiffiffiffiffiffiffi2tDL

p, for z = y = 0, eVd = 3.5, ue/

ffiffiffiffiKp

= �1.5, andu0/

ffiffiffiffiKp

= �3.75. In Figure 2a, the dimensionless concentra-tions ~c1NR, ~c2NR are those of a nonreactive system; thedashed line indicates the initial concentrations of the twospecies within the system.

Figure 3. Dependence of (a) dimensionless concentrations~c1, ~c2 and (b) dimensionless reaction rate, ~r, on thenormalized distance from the center of the (moving) plume,(x � Vt)/


p, for z = y = 0, eVd = 16, ue/

ffiffiffiffiKp

= �1.5, andu0/

ffiffiffiffiKp

= �3.75. In Figure 3a, the dimensionless concentra-tions ~c1NR, ~c2NR are those of a nonreactive system; thedashed line indicates the initial concentrations of the twospecies within the system.


267

dimensional) aureole around the moving center of the plume.The actual location and shape of this aureole is governedby (29), and would depend on DL, DT and ue. One shouldnote that Figure 4a displays an artificial symmetry, ascoordinate x is normalized by


p, while the y direction

is normalized byffiffiffiffiffiffiffiffiffiffi2tDT

p. Figure 4b depicts radial profiles

of the dimensionless reaction rate, ~r, for different ue.Because of the symmetry of the solution with respect tothe normalized coordinates, we display only radial profilesstarting from the plume center. Figure 4b is organized insuch a way that while the curves resulting from positivevalues of ue are displayed on the larger scale, those arisingby negative values of ue are displayed within the insert.Figure 4b reveals that the reaction rate is larger when ue/ffiffiffiffiKp

and u0/ffiffiffiffiKp

have opposite signs. An explanation ofthis is provided with the aid of Figure 5 where, for thesake of discussion, we consider the effect of jue/

ffiffiffiffiKp j =

30. From the points reached on the equilibrium curveright after injection, characterized by (cie + ci0)/

ffiffiffiffiKp

,concentrations would evolve until they reach the asymp-totic condition characterized by the same concentrationsas the resident water. In the case of conservative solutes,the paths would be the straight lines depicted in Figure 5as dashed lines. For reactive solutes the path would bealong the hyperbola (equilibrium line: ~c1~c2 = 1). FromFigure 5 we see that there is no symmetry: in the case ofue < 0 reactive and conservative solutes paths are quite

Figure 4. Dependence of the dimensionless reaction rate, ~r, on the dimensionless distance from thecenter of mass of the plume for u0/

ffiffiffiffiKp

= �20 and eVd = 3.5: (a) three-dimensional view for ue/ffiffiffiffiKp

= �30and (b) radial sections for ue/

ffiffiffiffiKp

= �20, �30, �40, 20, 30, and 40.

Figure 5. Scheme of a pure dissolution/precipitationreaction induced by a point-like injection. The equilibriumcondition at the injection point is characterized by jue/

ffiffiffiffiKp j =

30, while the initial water is identified by u0/ffiffiffiffiKp

= �20 (theinitial point is also the final equilibrium point). Dashed linesoutline the path of the reaction in the case of conservativesolutes. For reactive solutes, the path would develop alongthe hyperbola (equilibrium line).


268

similar, which is not the case when ue > 0. As aconsequence, in the latter case a larger amount of precip-itation is needed in order to reach equilibrium at alllocations and times. Going back to Figure 4b, we observethat the largest absolute value of ue/

ffiffiffiffiKp

produces thelargest reaction rate, as a consequence of the mixing ofincreasingly different waters, thus inducing significantgradients of u.[42] Knowledge of the local rate is essential to evaluate

the global reaction rate

rW tð Þ ¼ZW

r x; tð Þ dW; ð30aÞ

which is an integral measure of the rate of precipitation ofthe mineral mass of the system at any given time. Here, W isthe entire volume of the medium. This information is ofpractical interest for experimental applications devoted toanalyze precipitation/dissolution processes, such as thosereported by Berkowitz et al. [2003] and Singurindy et al.[2004].[43] In general a closed form solution for the integral

(30a) does not exist. Upon using spherical coordinates, thethree-dimensional integral (30a) can be written as thefollowing one-dimensional integral,

rW ¼ nK

t

Ve=n

36peVd

Z10

18

2pð Þ1=2ueeVd

!2

r2 exp �r2

� u0 þ 18

2pð Þ1=2ueeVd

exp � 1

2r2

� � !2

þ4K24 35�3=24p r2dr:

ð30bÞ

As evidenced by (30b) all information related to dispersion isconcentrated in eVd. Thus the sensitivity of rW on dispersioncan be analyzed from the following expression,

@rW@DL

¼ @rW

@eVd

@eVd

@DL

¼ @rW

@eVd

eVd

2DL

¼ DT

2DL

@rW@DT

: ð31Þ

Equation (31) shows that the overall reaction rate, rW, is muchmore sensitive to DT than to DL. Considering that long-itudinal dispersion is often taken to be about five to ten timesof transverse dispersion, equation (31) implies that theoverall reaction rate would be ten to twenty times moresensitive to DT than to DL. Furthermore, the input ofsolutes is frequently continuous in time and reactants donot enter directly into the flow domain, as they are laterallydriven into it. Both factors would tend to enhance the role oftransverse dispersion in reactive transport problems. This isparticularly relevant in view of the uncertainties surroundingthe actual values of transverse dispersion. While some arguethat transverse dispersion tends to zero in three-dimensionaldomains for large travel times [Dagan, 1989], others[Neuman and Zhang, 1990; Zhang and Neuman, 1990;Attinger et al., 2004] show that this is not the case, thuscorroborating the idea that the actual transverse dispersion islinked to the interplay between spatial heterogeneity andtime fluctuations of velocity [Cirpka and Attinger, 2003;Dentz and Carrera, 2003].

[44] An asymptotic expression for (30b) (see Appendix Dfor details) can be derived analytically in the presence oflarge dispersion effects (i.e., large eVd) as

rW tð Þ ¼ VeK

t

u2eeVd

27

4p1=2 c10 þ c20ð Þ3 : ð32Þ

Recalling the definition of eVd (equation (28)), expression(32) reveals that, for large eVd, the global reaction rate, rW,decreases with time proportionally to t�5/2. Figure 6 depictsthe dependence of the dimensionless global reaction rate,~rW = rW t/(Ve

ffiffiffiffiKp

), on eVd for different ue/ffiffiffiffiKp


=�20; bold lines are the asymptotic solution (32), while thinlines with symbols correspond to ~rW as obtained bynumerical solution of (30b). We observe that ~rW increaseswith eVd in the range of small dispersion effects (eVd < 9),since an increase of dispersion effects enhances the mixingprocess and the related reaction is enhanced. On thecontrary, ~rW is a decreasing function of eVd for largerdispersion effects (eVd > 9), as the prevailing effect of theenhanced dispersion effects is now a reduction ofconcentration gradients and thus of precipitation. Consis-tently with our previous comments, for small dispersioneffects ~rW is larger when ue/

ffiffiffiffiKp


have oppositesigns, while it is independent of the sign of ue/

ffiffiffiffiKp

andu0/

ffiffiffiffiKp

when dispersion effects are large.[45] We note that the asymptotic solution (32) coincides

with the exact results of the numerical integration of (30b)when eVd � 100. Considering, as an example, a mediumcharacterized by DT = 0.21 m2/day and DL = 2.1 m2/day andsetting Ve/n = 1 m3, this means that the precipitationphenomenon starts decreasing according to a factor t�5/2

after 1 day from the injection. This suggests that the bulk ofthe reactive process develops at early time and is confinedwithin a region of the domain which is relatively close tothe injection point.[46] As a final result, Figures 7a and 7b depict the spatial

distribution of the dimensionless total rate of precipitation

ert xð Þ ¼Z10

r x; tð Þ dt

nffiffiffiffiKp ; ð33Þ

obtained by numerical integration of (29) on the horizontalplane z = 0 for the entire duration of the process. Figures 7aand 7b have been obtained on the basis of the initialcondition u0/

ffiffiffiffiKp

= �20, while the boundary conditions areue/

ffiffiffiffiKp

= ±20.[47] Figures 7a and 7b corroborates the findings that the

system is active within a region close to the injection point.We note that the reaction can be considered exhausted whenVx/DL > 0.5 and Vy/DT > 2. Considering, as an example, amedium characterized by DT = 0.21 m2/day and DL =2.1 m2/day, setting Ve/n = 1 m3 and V = 0.5 m/day, thismeans that the precipitation process is negligible for x >2.1 m and y > 0.85 m.

4. Generalization of the Methodology of Solution

[48] We now extend the methodology illustrated insection 3 to describe multispecies transport processes inthe presence of generic homogeneous and classical hetero-


269

geneous reactions. We assume that all the variable activityspecies are mobile, so that mass balances (equation (6)) fordissolved and constant activity species are expressed as

@ ncað Þ@t

¼ Lt cað Þ þ STear ð34aÞ@ mcð Þ@t¼McLt mcð Þ þ STecr; ð34bÞ

where matrix Mc is diagonal and its diagonal terms equal 1when a given constant activity species is mobile or zerootherwise. Our aim is to evaluate the concentrations ofdissolved species, ca (Ns � Nc unknowns) and the rates ofthe reactions, r (Nr unknowns). Once r is known, (34b)provides the rates of mass change of the constant activityspecies. In order to calculate ca and r, we need to solve thealgebraic-differential system given by the Ns � Nc partialdifferential equations of transport for ca (equation (34a))and the Nr nonlinear algebraic equilibrium conditions(equation (4)). The general methodology of solution of thissystem of equations is detailed in the following.

4.1. Step 1: Definition of MobileConservative Components

[49] The first step of the methodology consists of defin-ing the chemical system (species and reactions, that is, thestoichiometric matrix of (1)) and reducing the number ofequations describing transport of dissolved species byeliminating the source term in (34a). To this end, we usethe components matrix, U, defined by (9).

4.2. Step 2: Transport of Components

[50] Since components are conservative, equation (11)can be used to describe their transport. The initial and/or

Figure 6. Dependence of the dimensionless global reac-tion rate, ~rW, on the dimensionless dispersive volume, eVd,for different ue/

ffiffiffiffiKp


= �20. Thin lines withsymbols correspond to ~rW as obtained by numericalintegration of (30b); thick lines correspond to theasymptotic solution (32).

Figure 7. Spatial distribution of the dimensionless total rate of precipitation, ~rt, as obtained bynumerical integration of (29) on the horizontal plane z = 0 during the entire process, for u0/

ffiffiffiffiKp

= �20,(a) ue/

ffiffiffiffiKp

= 20 and (b) ue/ffiffiffiffiKp

= �20.


270

boundary conditions for (11) are given by u0 = Uc0, c0being the vector of initial and/or boundary species concen-trations. Since the reaction sink/source terms have beeneliminated, all components are independent. Moreover, theresulting transport equation is identical for all components,the only difference being in the boundary conditions. Onlyin some particular cases (such as in the example ofsection 3.3) a closed-form analytical solution is possible.Otherwise, any conventional transport simulator can beused. The underlying hypotheses are that (1) the mobilespecies are subject to the same advective flow field and(2) the same dispersion processes apply to all species, withequal dispersion coefficients. Other than that, the formula-tion does not impose any additional restrictions on thenature of flow. That is, flow can be steady or transient,saturated or unsaturated, single phase or multiphase, andtemperature may be constant or variable.

4.3. Step 3: Speciation

[51] This step is devoted to evaluating the concentrationsof the Ns � Nc aqueous species, ca. The components,computed in step 2, provide N u = Ns � Nc � Nr equations.The mass action law for each reaction provides the remain-ing Nr equations. Thus one needs to solve the nonlinearalgebraic system

u ¼ Uca ð35aÞ

log c00a ¼ S0ea log c0a � logK: ð35bÞ

It is clear that the system is nonlinear, both explicitly(simultaneous linear dependence on ca and log ca) andimplicitly (in general, K may depend on activity coefficientsand thus be a nonlinear function of ca, as discussed insection 2.1). The solution of this system can be verycomplex. However, in some geochemical problems (such asthe binary system of section 3) it is possible to derive ananalytical solution. Formally, one can substitute (35b) into(35a), obtaining Nu nonlinear algebraic equations for c0a.Their solution renders c0a, as a function of u and K, which isthen substituted in (35b) to obtain c00a. In summary, we canwrite

c0a ¼ c0a u;Kð Þ ð36aÞ

c00a ¼ c00a u;Kð Þ: ð36bÞ

4.4. Step 4: Evaluation of Reaction Rates andMass Change of Constant Activity Species

[52] Here, we substitute the concentrations computed instep 3 into the mass balance equations to obtain the Nr

reaction rates. We recall (section 2) that each equilibriumreaction yields a secondary species and that thecorresponding transport equation solely depends on itsreaction rate, as the redefined matrix of the stoichiometriccoefficients of the secondary aqueous species coincides withthe opposite of the identity matrix, I. Therefore, consideringthe Nr transport equations of the secondary species leads to

r ¼ Lt c00a� �� @ n c00a

� �@t

: ð37Þ

Substituting the functional dependences of (36b) into (37),it is possible to evaluate the rates of the reactions (seeAppendix E for details). The rate of the mth reaction (weswitch temporarily to index notation to avoid ambiguities) isgiven by

rm

n¼XNr

p¼1

@c00am@Kp

� @Kp

@t� V � rKp þr � DrKp

� �� þXNu

i¼1

XNu

j¼1

@2c00am@ui@uj

rTuiDruj þ 2XNu

i¼1

XNr

q¼1

@2c00am@ui@Kq

rTuiDrKq

þXNr

p¼1

XNr

q¼1

@2c00am@Kp@Kq

rTKpDrKq: ð38Þ

This equation can be simplified when K is a function ofstate variables satisfying nonreactive transport equation(recall equation (21)) and, especially, when K is constant.The latter condition leads to

rm

n¼XNu

i¼1

XNu

j¼1

@2c00am@ui@uj

rTuiDruj: ð39Þ

Reverting to vector notation, (39) can then be expressed as

r ¼ nHrTuDru; ð40Þ

where H is the vector of Hessian matrices (a third-ordertensor) of the reactions (as represented by the correspondingsecondary species) with respect to the components.[53] The rate of mass change of constant activity species

is then obtained by substituting the reaction rates into thecorresponding mass balance equations (34b). In the case ofimmobile species (minerals subject to precipitation or dis-solution), the rate of mass change is given directly by thereaction rate. Knowledge of these rates of solid mass changeis of particular interest, since it is a prerequisite to estimatethe characteristic time of changes in medium properties[Wood and Hewett, 1982; Philips, 1991].[54] When the constant activity species are mobile (e.g.,

dissolved gases or some colloids), the formulation is stillvalid but more complex, as one would need to use thecomplete equation (34b) to obtain the reaction rate. If thespecies is water, then one is rarely interested in finding itsconcentration, other than in special cases, such as in thepresence of osmotic effects. In such cases our formulation isstill valid, but water would not be a constant activity speciesanymore.

4.5. Additional Considerations

[55] As previously pointed out, a key feature of ourmethodology is that it allows obtaining concentrations ofsolutes and reaction rates independently of constant activityspecies.[56] As compared to existing formulations proposed in

the literature to solve multispecies transport processes in thepresence of a generic number of homogeneous and classicalheterogeneous reactions [Rubin, 1990, 1992; Friedly andRubin, 1992; Steefel and MacQuarrie, 1996; Saaltink et al.,1998, 2001; Molins et al., 2004], our method is simpler andmore concise. It allows eliminating constant concentrationspecies from the beginning, thus simplifying the formula-tion from the outset.


271

[57] The method leads to general expressions for reactionrates. The latter should then be used to compute the masschange of solid mineral, which may induce modifications inthe medium properties. Here we neglect such changes andassume that modifications in solid mass due to transportinvolve very thin layers of the matrix [Rubin, 1983] and nosignificant variations of the pore system occur. In manypractical problems minerals may appear (in response to thechanges caused by mixing and reactions) or disappear (bybeing totally dissolved) in zones of the flow domain. Thiscan be handled by defining different chemical systems ineach zone. Unfortunately, this would lead to severe math-ematical complications.[58] The first three steps of the proposed method are

similar to those implemented in many computer codes, suchas HYDROGEOCHEM [Yeh and Tripathi , 1991],PHREEQC [Parkhurst, 1995] and RETRASO [Saaltink etal., 2004] (which also includes a detailed list of codes). Theevaluation of reaction rates using (38) or (40) is no lesscumbersome than simply substituting the concentrations ofsecondary species into the transport simulator to solve for r,as implied by (37). In fact, conventional speciation codes donot yield the second derivatives of secondary species. Ourequations yield valuable insights into the basic processescontrolling the rate of equilibrium reactions and allow forexplicit evaluations of these rates in cases when an analyt-ical solution is feasible.

5. Conclusions

[59] Our work leads to the following major conclusions:[60] 1. The methodology we propose allows obtaining the

local concentrations of dissolved species and the rates, r, ofthe reactions occurring in the system. It also allows evaluatingthe rate of change in themass of the solidminerals involved inthe phenomena as a function of r; this is a prerequisite toestimate the characteristic timescale of changes in mediumproperties [Wood and Hewett, 1982; Philips, 1991].[61] 2. An appealing feature of our method is that it

allows separating the solution of chemical equations to themass balance equations of dissolved species. This leads to areduced number of transport equations to be solved, and,more importantly, deconstructs the problem to the solutionof a set of independent nonreactive advection-dispersionequations.[62] 3. From a practical standpoint, our methodology is a

very powerful tool in deriving analytical solutions formultispecies system where homogeneous or classical het-erogeneous reactions occur [Rubin, 1983] on the basis ofknown classical solutions describing nonreactive processes.It can also be used to develop and/or simplify numericalcodes for solving reactive transport problems.[63] 4. As compared to formulations previously proposed

in the literature to solve multispecies transport phenomenain the presence of a generic number of homogeneous andclassical heterogeneous reactions [Rubin, 1990, 1992;Friedly and Rubin, 1992; Steefel and MacQuarrie, 1996;Saaltink et al., 1998, 2001; Molins et al., 2004], our methodis simpler and more concise. Moreover, it permits not onlyto evaluate concentrations of dissolved species, but alsoprovides general expressions for the rates of the systemreactions and for the rate of change in solid mass involvedin the phenomena.

[64] 5. The general expression provided for the reactionrates highlights that mixing processes control equilibriumreaction rates and evidences the possibility of inducingreaction by simply mixing waters at different equilibriumconditions. This type of mixing-driven chemical reactionscan help in explaining the enhancement of reaction pro-cesses observed when different solutions mix in carbonatesystems [Gabrovsek and Dreybrodt, 2000; Corbella et al.,2003; Rezaei et al., 2005].[65] 6. We applied the general methodology to a binary

system and derived an original closed-form analyticalsolution of a mixing-driven precipitation reaction inducedby a pulse injection in a three-dimensional homogeneousporous medium in the presence of uniform flow. Our resultsprove that in binary systems the mixing process of twowaters, both of which are at equilibrium, induces precipita-tion processes throughout the system. The solution alsodemonstrates that (1) the presence of precipitation signifi-cantly modifies the concentrations of aqueous species, whencompared to a nonreactive situation; (2) the features of thereactive process are strongly dependent on flow character-istics and on the difference in the concentrations of both theinitial and the injected water; (3) the bulk of the reactiveprocess develops at early time, thus remaining confinedwithin a region of the domain which is relatively close tothe injection point; and (4) the overall reaction rates aremore sensitive to transverse than to longitudinal dispersion.

Appendix A: Illustrative Example of theChemical Equilibrium Formulation

[66] As an example to help understanding the meaning ofvectors and matrices employed within the mass action law(1), we analyze the system of reactions leading to dedolo-mitization [Ayora et al., 1998]. The system is governed byfour equilibrium reactions (Nr = 4)

CO2�3 ¼ HCO�3 � Hþ ðA1aÞ

CO2 ¼ HCO�3 þ Hþ � H2O ðA1bÞ

CaMg CO3ð Þ2s¼ Ca2þ þMg2þ þ 2CO2�3 ðA1cÞ

CaCO3s ¼ Ca2þ þ CO2�3 : ðA1dÞ

We identify H2O, CaCO3(s) and CaMg(CO3)2s as constantactivities species (Nc = 3; one aqueous and two solidspecies). As the total number of species involved is Ns = 9,we need to define Ns � Nr � Ns = 2 primary species. Wethen define H+ and HCO3

� as the primary species and CO32�,

CO2, Mg2+ and Ca2+ as the secondary species.[67] We write the vector of activities, a, as

a ¼ Hþ;HCO�3 ;CO2�3 ;CO2;Mg2þ;

�Ca2þ;CaMg CO3ð Þ2s;CaCO3s;H2OÞT ; ðA2Þ

which is written respecting the order presented in section 2,first the primary species, then the secondary, and last theconstant activity species. The stoichiometric matrix is


272

Se ¼

Hþ HCO�3 CO2�3 CO2 Mg2þ Ca2þ CaMg CO3ð Þ2s CaCO3s H2O

�1 1 �1 0 0 0 0 0 0

1 1 0 �1 0 0 0 0 �10 0 2 0 1 1 �1 0 0

0 0 1 0 0 1 0 �1 0

0BBBB@1CCCCA

¼ Sea j Secð Þ ¼ S0ea S

00ea Sec

� �:

Note that negative and positive coefficients in Se are relatedto reactants and products of the reactions, respectively.Multiplying S0ea and S00ea by �(S00ea)�1 we rewrite thestoichiometric matrix so that the matrix of the stoichio-metric coefficients of the secondary species coincides withthe opposite of the identity matrix, I, and redefine thematrices Sea as

S0ea j �I� � ¼

Hþ HCO�3 CO2�3 CO2 Mg2þ Ca2þ

�1 1 �1 0 0 0

1 1 0 �1 0 0

1 �1 0 0 �1 0

1 �1 0 0 0 �1

0BBBB@1CCCCA:

ðA4Þ

This last operation has simply implied rewriting the last tworeactions (A1c) and (A1d) in terms of primary species (i.e.,eliminating CO3

2�), so that the last two reactions now read

Mg2þ ¼ CaMg CO3ð Þ2s�CaCO3s þ Hþ � HCO�3 ðA5aÞ

Ca2þ ¼ CaCO3s þ Hþ � HCO�3 : ðA5bÞ

Appendix B: Reaction Rate for a Binary System

[68] Here we detail the steps leading to a general expres-sion for the rate of the reaction of pure precipitation/dissolution processes in a binary system (given byequation (20)). We start from the transport equation of thespecies B2 (equation (16b)) and assume n to be constant.Recalling (19b), the time derivative of c2 can be expressedas

@ n c2 u;Kð Þ½ �@t

¼ n@c2@u

@u

@tþ @c2

@K

@K

@t

� �: ðB1Þ

We introduce the advective and diffusive linear operators,Lt_adv and Lt_d, defined as

Lt cð Þ ¼ n Lt adv cð Þ þ Lt d cð Þ½ �; ðB2Þwhere

Lt adv cð Þ ¼ �V � rc ðB3aÞ

Lt d cð Þ ¼ r � Drcð Þ ðB3bÞ

and V = q/n.

[69] Applying the advective operator to c2 leads to

Lt adv c2ð Þ ¼ @c2@u�V � ruð Þ þ @c2

@K�V � rKð Þ; ðB4Þ

while applying the diffusive operator to c2 leads to

Lt d c2ð Þ ¼ r � @c2@u

Druð Þ� �

þr � @c2@K

DrKð Þ� �

¼ @c2@ur � Druð Þ þ @c2

@Kr � DrKð Þ þ @2c2

@u2rTuDru

þ 2@2c2

@u@KrTuDrK þ @2c2

@K2rTKDrK: ðB5Þ

Substituting (B4) and (B5) into the transport operator (B2),applied to c2, leads to

Lt c2ð Þ ¼ n@c2@u�V � ruþr � Druð Þ½ �

þ @c2

@K�V � rK þr � DrKð Þ½ �

þ @2c2

@u2rTuDruþ 2

@2c2

@u@KrTuDrK

þ @2c2

@K2rTKDrK

�: ðB6Þ

Finally, substituting (B1) and (B6) into (16b) and recalling(11), yields

r

n¼ @c2

@K� @K

@tþ 1

nLt Kð Þ

� �þ @2c2

@u2rTuDruþ 2

@2c2

@u@KrTuDrK

þ @2c2

@K2rTKDrK: ðB7Þ

Appendix C: Reaction Rate for a BinarySystem When the EquilibriumConstant Depends on Conservative Quantities

[70] This Appendix is devoted to finding the expression forthe rate of the reaction of pure precipitation/dissolution, whenthe equilibrium constant, K, is a function of a given quantity,s, satisfying a nonreactive form of the advection-dispersionequation. This could be, for example, the case of salinity.[71] We start from the general expression of the reaction

rate (20)

r

n¼ @c2

@K� @K

@tþ 1

nLt Kð Þ

� �þ @2c2

@u2rTuDruþ 2

@2c2

@u@KrTuDrK

þ @2c2

@K2rTKDrK: ðC1Þ

(A3)


273

As K is function of s, the time derivative of K can beexpressed as

@ K sð Þ½ �@t

¼ @K

@s

@s

@t: ðC2Þ

Recalling the definition of advective and diffusive linearoperators, Lt_adv and Lt_d, (equations (B3a) and (B3b)), wesplit the operator Lt (K) as

1

nLt Kð Þ ¼ Lt adv Kð Þ þ Lt d Kð Þ: ðC3Þ

Applying the advective operator to K leads to

Lt adv Kð Þ ¼ @K

@s�V � rsð Þ: ðC4Þ

The diffusive operator acting on K provides

Lt d Kð Þ ¼ r � @K

@sDrsð Þ

� �¼ @K

@sr � Drsð Þ þ @2K

@s2rT sDrs:

ðC5Þ

By virtue of (C4) and (C5), (C3) results

1

nLt Kð Þ ¼ @K

@s�V � rsþr � Drsð Þ½ �

þ @2K

@s2rT sDrs

�: ðC6Þ

Substituting (C2) and (C6) into (C1), and recalling that ssatisfies a nonreactive format of the advection-dispersionequation, leads to

r

n¼ @c2

@K

@2K

@s2rT sDrs

� �þ @2c2

@u2rTuDru

þ 2@2c2

@u@KrTuDrK þ @2c2

@K2rTKDrK: ðC7Þ

Since K is a function of s, the last two terms in (C7) can bewritten as

@2c2

@u@KrTuDrK ¼ @2c2

@u@srTuDrs ðC8Þ

@2c2

@K2rTKDrK ¼ @2c2

@K2

@K

@s

� �2

rT sDrs: ðC9Þ

Substituting (C8) and (C9) into (C7) and noticing that

@2c2

@s2¼ @c2

@K

@2K

@s2þ @2c2

@K2

@K

@s

� �2,

we finally obtain

r

n¼ @2c2

@s2rT sDrsþ @2c2

@u2rTuDruþ 2

@2c2

@u@srTuDrs: ðC10Þ

Appendix D: Analytical Solution for theOverall Reaction Rate in a Binary System

[72] Here we provide the analytical expression of theoverall rate, rW (equation (30b)), in the case of largedispersion effects (large dimensionless volumes, eVd).

[73] The derivation relies on the fact that when eVd is largewe can write an approximate expression for the local rateupon disregarding terms of order larger than 2 (in 1/eVd) as

r

n¼ 2K

u20 þ 4K� �3=2 1

2t

18

2pð Þ1=2ueeVd

" #2r2 exp �r2

; ðD1Þ

where r is defined by (27).[74] Recalling that

Z10

r4 exp �r2 dr ¼ 3

8

ffiffiffipp

; ðD2Þ

integration of (D1) over the domain renders rW (equation(32)) as

rW ¼ VeK

t

u2eeVd

27

4p1=2 c10 þ c20ð Þ3 : ðD3Þ

Appendix E: General Expression for theReaction Rate

[75] Here we detail the steps needed to obtain (38),expressing the reaction rates for the general case of multi-component transport. We start from the transport equationsfor the secondary species (equation (37)) and recall that c00a issolely a function of u and K (equation (36b)). Let us firstconsider a single secondary species’ concentration, c00am.Using the chain rule, we express the time derivative of c00amas

@ n c00am u;Kð Þ @t

¼ nXNu

i¼1

@c00am@ui

@ui@tþ n

XNr

p¼1

@c00am@Kp

@Kp

@t; ðE1Þ

where Nu = Ns � Nc � Nr is the number of primary species.[76] Application of the advective operator, Lt_adv

(equation (B3a)), yields

Lt adv c00am� � ¼XNu

i¼1

@c00am@ui

�V � ruið Þ þXNr

p¼1

@c00am@Kp

�V � rKp

� �:

ðE2ÞThe diffusive operator, Lt_d (equation (B3b)), acting on c00amis expressed as

Lt d c00am� � ¼XNu

i¼1

@c00am@uir � Druið Þ þ

XNu

i¼1r @c00am

@ui

� �� Druið Þ

þXNr

p¼1

@c00am@Kp

r � DrKp

� �þXNr

p¼1r @c00am

@Kp

� �� DrKp

� �:

ðE3Þ

Upon noting that

r @c00am@ui

� �¼XNu

j¼1

@2c00am@ui@uj

ruj þXNr

q¼1

@2c00am@ui@Kq

rKq; ðE4Þ


274

equation (E3) can be rewritten as

Lt d c00am� � ¼XNu

i¼1

@c00am@uir � Druið Þ þ

XNr

p¼1

@c00am@Kp

r � DrKp

� �þXNu

i¼1

XNu

j¼1

@2c00am@ui@uj

rTuiDruj

þ 2XNu

i¼1

XNr

q¼1

@2c00am@ui@Kq

rTuiDrKq

þXNr

p¼1

XNr

q¼1

@2c00am@Kp@Kq

rTKpDrKq:

ðE5ÞUsing (E2) and (E5), the linear operator (7) acting on c00amcan be written as

Lt c00am� � ¼ n

XNu

i¼1

@c00am@ui

�V � rui þr � Druið Þ½ �(

þXNr

p¼1

@c00am@Kp

�V � rKp þr � DrKp

� � þXNu

i¼1

XNu

j¼1

@2c00am@ui@uj

rTuiDruj

þ 2XNu

i¼1

XNr

q¼1

@2c00am@ui@Kq

rTuiDrKq

þXNr

p¼1

XNr

q¼1

@2c00am@Kp@Kq

rTKpDrKq

): ðE6Þ

Substituting (E1) and (E6) into (37), when written for asingle species, c00am, and recalling (11), leads to

rm

n¼XNr

p¼1

@c00am@Kp

� @Kp


� ��

þXNu

i¼1

XNu

j¼1

@2c00am@ui@uj

rTuiDruj þ 2XNu

i¼1

XNr

q¼1

@2c00am@ui@Kq

rTuiDrKq

þXNr

p¼1

XNr

q¼1

@2c00am@Kp@Kq

rTKpDrKq: ðE7Þ

Finally, the rate of reaction of the entire system is

1

nr ¼

XNr

p¼1

@c00am@Kp

� @Kp


� ��

þXNu

i¼1

XNu

j¼1

@2c00am@ui@uj

rTuiDruj þ 2XNu

i¼1

XNr

q¼1

@2c00am@ui@Kq

rTuiDrKq

þXNr

p¼1

XNr

q¼1

@2c00am@Kp@Kq

rTKpDrKq: ðE8Þ

[77] Acknowledgments. The second and third authors acknowledgefinancial support by the European Union and Enresa through projectFUNMIG. The authors are thankful to three anonymous reviewers for theircomments on a preliminary version of this paper.

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��J. Carrera and X. Sanchez-Vila, Department of Geotechnical Engineering

and Geosciences, Technical University of Catalonia, Gran Capita S/N, E-08034 Barcelona, Spain.

M. De Simoni and A. Guadagnini, Dipartimento di Ingegneria Idraulica,Ambientale, Infrastrutture Viarie, Rilevamento, Politecnico di Milano,Piazza L. Da Vinci 32, I-20133 Milano, Italy. ([email protected])


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Stream-Stage Response Tests and Their JointInterpretation with Pumping Testsby Tobias S. Rotting1,2, Jesus Carrera2, Jose Bolzicco2, and Josep Maria Salvany2

AbstractHydraulic head response to stream-stage variations can be used to explore the hydraulic properties of stream-

aquifer systems at a relatively large scale. These stream-stage response tests, also called flooding tests, are typi-cally interpreted using one- or two-dimensional models that assume flow perpendicular to the river. Therefore,they cannot be applied to systems that are both horizontally and vertically heterogeneous. In this work, we use thegeostatistical inverse problem to jointly interpret data from stream-stage response and pumping tests. The lattertests provide flow data (which are needed to resolve aquifer diffusivity into transmissivity and storage coefficient)and may supply supplementary small-scale information. Here, we summarize the methodology for the design,execution, and joint numerical interpretation of these tests. Application to the Aznalcollar case study allows us todemonstrate that the proposed methodology may help in responding to questions such as the continuity of aqui-tards, the role and continuity of highly permeable paleochannels, or the time evolution of stream-aquifer inter-action. These results expand the applicability and scope of stream-stage response tests.

IntroductionNaturally occurring or controlled stream-stage varia-

tions have been used successfully to study large-scalehydraulic properties of fluvial aquifers (e.g., Pinder et al.1969; Grubb and Zehner 1973; Loeltz and Leake 1983;Carrera and Neuman 1986c; Reynolds 1987; Sophocleous1991; Tabidian et al. 1992; Genereux and Guardiario1998; Bolster et al. 2001). Aquifer reaction to tidal fluctu-ations has also been employed to calculate aquifer param-eters (e.g., Erskine 1991; Jha et al. 2003). A large numberof one-dimensional (1D) and mostly analytical solutionsare available for interpreting the tests (e.g., Ferris 1951;Rorabaugh 1960; Rowe 1960; Hantush 1961; Cooper andRorabaugh 1963; Pinder et al. 1969; Hornberger et al.1970; Singh and Sagar 1977; Onder 1997; Mishra andJain 1999; Zlotnik and Huang 1999; Singh 2003; Swamee

and Singh 2003). Some authors have also interpreted thetests in a plane perpendicular to the stream (Loeltz andLeake 1983; Carrera and Neuman 1986c; Moench andBarlow 2000) to account for two-dimensional (2D) het-erogeneity.

However, these interpretation methods suffer fromthree limitations. (1) They assume that flow is perpendic-ular to the river, thus ignoring heterogeneity (irregularaquifer geometry, paleochannels, silt layers, etc.).(2) They use only head data, which only reveal informa-tion about aquifer diffusivity (Carrera and Neuman1986b). Accurate flow rate data, which are difficult toobtain during high-flow events, are needed to resolveaquifer diffusivity into transmissivity and storage coeffi-cient. (3) They do not allow direct incorporation of othertypes of hydraulic data (e.g., cross-hole pumping tests).These additional tests can yield the aforementioned flowrate data and supplementary small-scale information.They also create additional flow conditions to better iden-tify the transmissivity distribution (‘‘multi-directionalaquifer stimulation’’: Snodgrass and Kitanidis 1998;‘‘hydraulic tomography’’: Yeh and Liu 2000, Liu et al.2002). Joint interpretation of different tests or of differentdata types provides an integrated picture of the systemand can narrow the range of likely parameter sets (e.g.,

1Corresponding author: Department of Geotechnical Engi-neering and Geosciences, Technical University of Catalonia, 08028Barcelona, Spain; [email protected]

2Department of Geotechnical Engineering and Geosciences,Technical University of Catalonia, 08028 Barcelona, Spain

Received October 2004, accepted July 2005.Copyright ª 2005 National Ground Water Association.doi: 10.1111/j.1745-6584.2005.00138.x

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Carrera and Neuman 1986c; Anderman et al. 1996; Weissand Smith 1998; Saiers et al. 2004). This can be best ach-ieved by automatic calibration of a numerical aquifermodel (Poeter and Hill 1997; Hill et al. 1998; Carrera etal. 2005).

When data from several tests are interpreted jointly,some parameters may change between tests (e.g., severefloods may erode streambed sediments, thereby alteringstream-aquifer interaction). The flexibility of a numericalmodel should allow accounting for these changes.

The aim of this article is to summarize a methodologyfor the design and execution of stream-stage responsetests and their joint interpretation with pumping tests. Aquasi–three dimensional (3D) geostatistical inversioncode is used for parameter estimation. The methodologyis illustrated through application to the Aznalcollar casestudy. Here, an alluvial aquifer became contaminated byheavy metal–laden acidic water from a mine tailingsspill. Pumping tests did not suffice to characterize theaquifer on a sufficiently large scale, so that a number ofquestions were left unanswered regarding (1) the con-tinuity of a silt layer that was found while building a per-meable reactive barrier (PRB) across the floodplain; (2)the role and extent of deep gravel channels; (3) the timeevolution of the stream-aquifer connection; and (4) theimpacts of the PRB construction. A stream-stage re-sponse test was performed to answer these questions.

Methodology

Stream-Stage Response Test Design and ExecutionA stream-stage response test is a hydraulic test that

consists of observing the head response of an alluvialaquifer to a controlled change of the water level ina stream or channel. This response is measured in piez-ometers and can be used to determine aquifer characteris-tics and parameters. As the source of stress is a line andnot just a point, the area of influence is much greater thanthat of pumping tests. The aquifer can be stressed eitherby raising or by lowering the water level in the stream.The methods presented here can be applied to both kindsof situations.

The three key parameters in the design of a stream-stage response test are (1) the magnitude of water levelchange in the stream; (2) the test duration; and (3) themeasurement intervals. They are discussed subsequently.

Stream stage should change as much as possible tomaximize head response, but the water level shouldremain within the channel (i.e., should not inundate thefloodplain) to simplify interpretation. The water levelchange depends on river channel characteristics and flowrate, which can be specified if the stream-stage variationis generated by opening the gates of a dam.

Appropriate test duration and measurement intervalsdepend on aquifer parameters. Test duration should belong enough for most wells to react significantly to thestage change (i.e., longer than equilibration time, teq).Measurement intervals should be short enough to definethe transient part of the test (i.e., shorter than the wellreaction time, tr). For planning purposes, teq and tr can be

approximated for each observation well as (e.g., Crank1975)

teq’SL2

Tð1Þ

tr’ 0:1SL2

Tð2Þ

where L is the distance between well and stream and Tand S are estimates of aquifer transmissivity and storagecoefficient, respectively. Ideally, test duration shouldbe longer than the teq of all the wells involved in the test.Estimates of T and S can be obtained from pumping testsor from textural data and geology. However, room shouldbe left for surprises as deep observation wells mayrespond faster than derived from Equations 1 and 2(Carrera and Neuman 1986c) and T may grow with scale(e.g., Sanchez-Vila et al. 1996).

The network of piezometers should cover the wholearea of interest. Well screens should be short whereververtical variations in aquifer parameters or responsesare expected. All relevant features (e.g., abrupt lateralchanges in aquifer characteristics, formation boundaries,layers) should be monitored. Preliminary information onthe approximate position of such boundaries may be ob-tained from standard hydrogeological exploration meth-ods (boreholes, trenches, geophysical methods, hydraulictests, etc.) and from the observation and qualitative inter-pretation of piezometric surfaces under different flowconditions (especially aquifer response to previousstream-stage variations, i.e., flood events, if such data areavailable).

The river stage should be measured at several points,especially if stream width or section changes along thetest reach. In general, the propagation of the hydraulicpulse is much faster in the channel than in the aquifer.However, if the test reach is long, upstream wells close tothe river will have equilibrated by the time the pulse rea-ches the downstream end of the reach. Therefore, it isnecessary to measure stream stage along the river. More-over, actual stage variations are sensitive to river width.Therefore, stage measurement points should be locatedso as to make sure that river width and section variabilityare properly sampled.

The test may be performed by recording the responseto a naturally occurring flood or to an artificial variationin stream stage (e.g., by opening or closing the gates ofa dam). However, Equation 2 implies that measurementsmay have to be made at short intervals. Continuousrecording of heads at such intervals may saturate therecording devices. Therefore, if possible, an artificialstream-stage variation should be preferred. In this way,not only the heads but also other parameters (e.g., con-centrations or temperature) may be monitored. Here, wewill deal only with head measurements.

Test data are best interpreted in terms of head varia-tions (rather than actual values). Head variations aredefined as changes in head caused by the stream-stageresponse test compared to the natural evolution of headsin the aquifer system, similar to drawdowns in pumpingtests. River-stage variations are defined analogously. The

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advantages of this procedure are explained subsequently.Measurements must start prior to the test in order to beable to filter out natural trends of head variability.

It may be advisable to use automatic recording devi-ces in piezometers and river-stage measuring points, par-ticularly if many points have to be monitored at shortintervals.

Numerical InterpretationEmphasis is placed on estimation of spatially varying

parameters. Otherwise (i.e., homogeneous medium),a model perpendicular to the river should suffice. More-over, many conceptual models need to be tested. There-fore, the use of geostatistical inversion methods isrecommended (Meier et al. 2001). This reduces the effortdevoted to calibration and allows the modeler to concen-trate on conceptual issues. Here, we used the quasi-3Dfinite-element code TRANSIN II (Medina et al. 1995).This code calculates both direct and inverse flow andtransport problems in a stationary or transient regime.The inverse problem is solved by minimizing a maximumlikelihood criterion (Carrera and Neumann 1986a), whichincludes prior information on aquifer parameters. Fora geostatistical interpretation, it can include the spatialcorrelation structure (covariance matrix) of the hydraulicconductivity of the aquifer system. The code also com-putes several model identification criteria (Akaike, modi-fied by Akaike, Hannan, and Kashyap; details are givenby Carrera and Neuman 1986a), which allow one to eval-uate the quality of different model structures. Accordingto Carrera and Neumann (1986c), Kashyap’s (1982) is thebest criterion because it takes into account the goodnessof fit while penalizing overparameterization.

The modeling procedure consists of five steps: (1)definition of the conceptual model; (2) definition of a geo-statistical model for transmissivity; (3) discretization; (4)nonlinear maximum likelihood estimation of the modelparameters; and (5) revision of the conceptual model andrepeated calibration until obtaining a good fit betweenobserved and modeled data. The general principles of thismethodology have been described by Meier et al. (2001).The details specific to the joint interpretation of stream-stage response and pumping tests with parameters varyingin time are described subsequently.

Conceptual Model

To simulate stream-aquifer interaction, a ‘‘mixed-type’’ boundary is used, relating flow across the riverbedto the difference between external head He (river stage)and aquifer head h according to:

q ¼ aðHe2hÞ ð3Þwhere q is flux (m/d), i.e., flow rate per unit area, and a isthe leakance (d21), also called conductance, which repre-sents the hydraulic conductivity of the riverbed dividedby its thickness. More complex approaches to describestream-aquifer interaction have been developed (fora review, see Sophocleous 2002). Nevertheless, here weapply this simple approach in order to keep the number ofparameters low.

Aquifer heads and stream stage are expressed interms of variation with respect to the natural evolution ofthe system. A hydraulic test can be considered as a signalthat is added on the natural evolution of the studied sys-tem. Therefore, when the test data are expressed in termsof variations, the model only needs to simulate thechanges induced by the test (in our case, pumping ratesor variations in river stage and their effect on aquiferheads) but not, for example, the natural base flow in theaquifer. This simplifies boundary and initial conditions.Only the boundary conditions that are driving variablesof the test (pumping rates, river-stage variations) have tobe expressed as time functions. All other boundary condi-tions of parameters that are not changed by the test arehomogeneous (zero values). Initial aquifer head varia-tions and stream-stage variations are also zero because bydefinition, before the start of the test, aquifer heads andsteam stages are equal to the ‘‘natural’’ values of the sys-tem. Additionally, systematic measurement bias (e.g., dueto erroneous reference levels) is eliminated by workingwith head variations.

Geostatistical Model for Transmissivity

Transmissivity can be treated deterministically orstochastically. In the latter case, one must specify thecovariance structure of transmissivity. In layered aquifers,the different alluvial layers that were deposited at differ-ent times can often be assumed as statistically indepen-dent of each other. The same applies to any othercomponent of the aquifer (i.e., anthropogenic structuressuch as PRB). This must be reflected in the covariancematrix, where each formation is represented by a blockthat describes intercorrelation within the formation orstructural unit, while the different blocks are not corre-lated with each other.

Discretization

In the quasi-3D approach implemented in TRANSINII, aquifer layers are represented by 2D triangles and rec-tangles, while aquitards are represented by 1D linear ele-ments. If equilibration time in the aquitard with respectto head changes in the aquifers is quick compared to theduration of the hydraulic tests and time steps of themodel, then transient head variations within the aquitardcan be neglected and the aquitard can be represented bylinear elements. Otherwise, the linear elements have to berefined by intermediate nodes (Chorley and Frind 1978).In order to include both data of small-scale pumping testsand those of larger-scale stream-stage response tests,numerical accuracy demands that the mesh be highlyrefined between sources of stress (rivers and pumpingwells) and observation points. However, the mesh can becoarser far away from these zones.

Different events that occurred during a period whereaquifer parameters did not change can be interpretedwithin the same mesh. To do so, the data sets and timefunctions of each event have to be separated by‘‘dummy’’ periods with zero stress that allow aquiferhead and stream-stage variations to recover to initiallevels (Figure 1).

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Calibration

Calibration is best performed automatically to avoidthe need for performing a large number of model runs toattain a manual fit by trial and error. In essence, one hasto specify the covariance matrix of all the data. Here, weassume head errors to be independent, so that one onlyneeds to define their standard deviation. As for the modelparameters, the covariance matrices of transmissivity arederived from the geostatistical model. All other parame-ters are assumed independent, so that one only needs todefine their standard deviation.

All model parameters (aquifer transmissivities, aqui-tard hydraulic conductivities, storage coefficients, andleakance) are calibrated simultaneously using the data ofall the tests. As a result, the problem may be ill posed,resulting in instability, high uncertainty, and poor conver-gence. To deal with these problems, one may choose tofix some parameters or to artificially increase the weightassigned to prior estimates. Here, we propose theapproach of Carrera and Neuman (1986c), which consistsof initially assigning a high weight to prior informationin order to facilitate convergence. In successive runs,prior information weight is lowered stepwise in orderto give the model more freedom in adjusting parametervalues.

When different model configurations are tested inorder to explore the structure of the stream-aquifer sys-tem, they can be compared using a number of criteria(Carrera et al. 1993). We use residual analysis (examiningthe fit between measured and computed heads at everyobservation point) and parameter assessment (evaluating

whether computed parameters are reasonable) for pre-liminary screening of conceptual models. For quantitativecomparison, we use overall model fit, as measured by theheads objective function (sum of squared weighed re-siduals; for details, see Carrera and Neuman 1986a andMeier et al. 2001) and, better, Kashyap’s model structureidentification criterion (equation 30 in Carrera andNeuman 1986a).

The Aznalcollar Case Study

Site and GeologyThe aim of the Aznalcollar case study was to charac-

terize the River Agrio alluvial aquifer situated in theAznalcollar mining district in the Seville province ofsouthwest Spain (Figure 2a). This aquifer had becomecontaminated by heavy metal–laden acidic water from themine tailings spill that occurred on April 25, 1998 (e.g.,Grimalt et al. 1999). As one of the remediation measures,a PRB was built across the floodplain (Carrera et al.2001). As the pit for barrier emplacement was excavated,the aquifer turned out to be more complex than expected(Salvany et al. 2004) in several aspects, which will beexplained subsequently.

The River Agrio is a tributary of the River Guadia-mar (Figure 2a) and forms an alluvial valley made up offour Quaternary terraces (Salvany et al. 2004):

d The upper terrace (T3) is hydraulically separated from the

other terraces by the bedrock and will therefore not be

considered in this study.d The intermediate terrace (T2) forms a wide flat area below

the hills. It is composed of three layers (Figure 3): a lower

layer of coarse gravels and sands, an intermediate layer

of silts containing subordinate levels of sands and small

gravels, and an upper layer of sandy gravels, which always

lies above the water table and will not be considered here-

inafter. The deepest gravels form a lower paleochannel

that flows obliquely to the surface terrace and river trends.d The lower terrace (T1) is a single deposit of sandy gravels

a few meters below the T2 terrace. It forms an upper pale-

ochannel that follows the river trend and occasionally cuts

the T2 deposits and even the bedrock.d The current floodplain (T0), 1 to 1.5 m below the T1

terrace, is an erosive terrace that cuts through T1.

The Miocene Blue Marls Formation forms the imper-vious base of the aquifer system.

To sum up, in the vicinity of the PRB (Figure 3,cross section B-B9), the hydrogeologic system consistsof the rivers Agrio and Guadiamar and of three layers: aconfined aquifer formed by the lower T2 deposits, anaquitard consisting of the T2 intermediate silts, anda phreatic aquifer formed by T1 deposits. In this study,we suppose that the phreatic aquifer is cut off ~400 m up-and downstream of the PRB because the silt layer out-crops right on top of the Blue Marls in the channel of theRiver Agrio (crosshatched areas in Figure 2b). In thefollowing, the T1 deposits will be referred to as ‘‘uppergravels’’ and the T2 lower gravels as ‘‘lower-layergravels’’ or ‘‘paleochannel.’’

a)

b)

c)

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0.4

0.8H

ead

varia

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(m)

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(m)

0

1000

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0 10 1000 1005Pum

ping

rat

e (m

3 /da

y)

Time (days) Time (days) + 1000

5

Figure 1. Example of input processing in order to interpretdata from one pumping test and one stream-stage responsetest in one mesh: (a) observed head variations in an observa-tion well (‘‘S-5’’ of the case study), (b) stream-stage varia-tion, (c) pumping rate of a pumping well (‘‘S-3’’ of the casestudy).

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The Aznalcollar PRB spans the floodplain 2 kmdownstream of the tailings pond. It consists of three reac-tive modules (Carrera et al. 2001). Each module is 30 m

wide, 1.4 m thick, and penetrates into the Blue Marls(on average 6 m deep). The three modules are separated bytwo 10-m wide nonreactive sections of low permeability.

Uncertainties remain about the degree of hydraulicconnection between the two aquifers and about the lateralextension of the aquifer layers. It is not clear if the siltlayer is a continuous feature in the area where the uppergravels overlie the lower gravels. Likewise, the course ofthe T2 paleochannel is only known in the area wherewells have been drilled.

The hydraulic connection of the two aquifer layersto the River Agrio is also unclear: remediation of thetoxic spill led to an artificial river channel, which wasmerely a few meters wide and very shallow. Therefore,it is believed that it was only connected to the upperwater table aquifer. However, between December 2000and January 2001, severe floods reshaped the floodplainmorphology, increased the width of the channel to>10 m, and deeply eroded the riverbed. This may haveenhanced stream-aquifer interaction between River Agrioand the upper aquifer by changing the hydraulic con-ductivity of the streambed sediments. The lower aquifermay have also become connected to the upper aquiferand to the river by partial erosion of the silt layer.

As all these features might severely affect PRB effi-ciency, a number of questions about this hydrogeologicsystem were raised, regarding (1) the continuity of thesilt layer; (2) the geometry of the lower gravel layer andits hydraulic connection to the river; (3) the evolution ofstream-aquifer interaction; and (4) the impact of PRB

Figure 2. (a) Geological map of the studied site showing the River Agrio terraces and their bedrock. The dotted-line rectangleshows the extent of the numerical model. (b) Location of observation wells used during the flooding test (filled circles) andof other wells and trenches (open circles). The crosshatched areas show where the Blue Marls appear in the bed of the RiverAgrio (a and b adapted from Salvany et al. 2004). (c) Observation wells around the permeable reactive barrier (‘‘PRB’’). Suf-fixes ‘‘-u’’ and ‘‘-l’’ stand for wells screened only in the upper or lower gravel layer, respectively. The stars represent piezome-ter nests inside the three reactive PRB modules (‘‘RMB, CB’’, and ‘‘LMB’’ stand for right margin, central, and left marginbarrier module).

Figure 3. Cross sections of River Agrio deposits (adaptedfrom Salvany et al. 2004). See Figure 2b for location.

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construction on the flow system. A stream-stage responsetest promised to be an adequate instrument to study theseissues.

Stream-Stage Response and Pumping TestsPrior to stream-stage response test execution, reaction

and equilibration time were calculated for several wells inthe upper and lower gravel layers using Equations 1 and 2and prior estimates of T and S as obtained from pumpingtests (see four examples in Table 1). Reaction and equili-bration times are quite short, especially for wells in theconfined layer. Therefore, automatic sensors wereinstalled in 14 wells and at one point in the river. Due totechnical limitations of the data-logging system, the mea-suring intervals used during the test were somewhat largerthan the lowest calculated reaction times (5 min in fivewells connected to the water table aquifer, 1 min in allother wells). All other points were measured manually.

The Aznalcollar drinking water dam was opened at12:00 h on June 5, 2002 and closed at 12:00 h on June 6,2002. The flow rate was 11 m3/s, a value that had produceda large rise in river stage without causing excessive flood-plain inundation in previous high-flow events. Water levelrose by up to 64 cm near the PRB (Figure 4) and up to 97cm in a narrow part of the riverbed near well A-1bis (notshown). The evolution of water levels was monitored in 54wells (Figures 2b and 2c) and at 11 river-stage measure-ment points. Manual measurements were continued untilthe evening of June 7; the automatic sensors kept on mea-suring in 15-min intervals until July 22. Almost all the wellsin the floodplain and beneath the T2 terrace responded veryquickly (within <1 h) to changes in the river stage.

Prior to the stream-stage response test, three series ofcross-hole pumping tests had been performed at the PRBsite. Wells S-1, S-3, and S-6 (cf. Figure 2c) were pumpedsequentially in each series, and drawdowns were mea-sured in the surrounding wells. Two such series were per-formed prior to PRB construction (in January and March2000). They differ somewhat in pumping rates and dura-tion. One test series (in March 2001) was performedafter PRB construction and after the riverbed had beenaffected by the winter 2000/2001 floods. During this lasttest series, the stream stage of the River Agrio variedtwice due to changes in the water release rate of the Az-nalcollar drinking water dam, so this undesired variationhad to be taken into account for the interpretation of thetest (cf. Figure 1a).

Numerical InterpretationThe numerical model simultaneously interprets the

stream-stage response test and the three series of cross-hole pumping tests.

Conceptual Model

The aquifer system consists of two aquifer layers(Figures 3 and 5a) separated by a silt aquitard asdescribed in Site and Geology. The lower confined aqui-fer consists of the lower gravels in the area of the T2 pale-ochannel and of T2 intermediate silts in the remainingarea between the margins of the alluvial valley. Thecourse of the paleochannel beyond the area where wellshad been drilled was extrapolated parallel to the river(Figure 5a). The model area extends up to the miningcompound upstream of the PRB (dotted-line rectangle inFigure 1a). Downstream of the PRB, the model boundaryis formed by the River Guadiamar. All other boundariesare assumed to be of prescribed flow type and will betreated as no-flow because we are working with head var-iations (see Methodology).

In order to explore the structure of the aquifer systemand to test the benefits of the geostatistical approach, dif-ferent model configurations were tested. After prelimi-nary screening model runs, three conceptual models werebelieved reasonable. They differ in leakance zone struc-ture and treatment of transmissivity (geostatistical ordeterministic), as summarized in Table 2.

Model 1 contains four leakance zones (‘‘Upstream a,’’‘‘Upper two-layer a,’’ ‘‘Downstream a,’’ ‘‘Guadiamar a’’;cf. Figure 5b). It assumes that river-aquifer interactiondoes not change in time. The rivers are hydraulically

Table 1Estimated Reaction and Equilibration Time for Two Wells in the Upper Gravel Layer and for

Two Wells in the Lower Gravel Layer Using Prior Estimates of T and S

Upper Gravel Layer (water table aquifer) Lower Gravel Layer (confined aquifer)

T (estimated) 300 m2/d 3000 m2/dS (estimated) 0.2 0.0001Well S-3-u S-6-u S-23 S-25

Distance to river (L) 7 m 50 m 150 m 400 mReaction time (tr) 5 min 4 h 0.1 min 0.8 minEquilibration time (teq) 50 min 40 h 1 min 8 min

34.4

34.6

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5 jun 020:00

6 jun 020:00

7 jun 020:00

8 jun 020:00

9 jun 020:00

10 jun 020:00

11 jun 020:00

Date + time

Riv

er s

tage

(m

a.s

.l.)

Figure 4. River Agrio stage close to the PRB during thestream-stage response test.

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connected only to the uppermost layer in each segment.Therefore, in the two-layer zone, the River Agrio has nohydraulic connection to the lower-layer gravels (‘‘Lowertwo-layer a’’ is not assigned).

Model 2 acknowledges that the 2000/2001 floodsmay have changed the aquifer connection with the RiverAgrio. This is represented by assigning different leakan-ces to each segment of the River Agrio in the 2000 and2001/2002 tests. Additionally, the River Agrio may alsobe connected to the lower layer in the two-layer zone(Lower two-layer a is also assigned in both meshes).River Guadiamar is assumed to remain unchanged by thefloods. This totals nine leakances (four leakances forRiver Agrio in 2000, four leakances for River Agrio in2001/2002, and one leakance for River Guadiamar, whichis identical in both periods).

Model 3 has the same structure of leakance zones asmodel 2, but in order to examine what is gained by

geostatistical inversion, transmissivity is treated deter-ministically: the transmissivity fields of the upper aquiferlayer, the lower aquifer paleochannel, and the three bar-rier modules are replaced by single zones, resulting ina model with only 11 independent transmissivity zonesthat are all statistically independent from each other.

As no quantitative data on leakances were available,prior estimates and initial values of all leakances of thethree model configurations were set to an intermediatevalue of 1 d21. The variances were set to 5 orders of mag-nitude (r2 log a ¼ 5) in order to allow the models toassign both very good and very poor hydraulic con-nections to the different leakance zones.

Discretization

The two aquifer layers are represented by 2Dtriangles and squares (Figures 6a through 6c). The aqui-tard is represented by 1D linear elements without inter-mediate nodes because preliminary evaluation of theequilibration time in the silt aquitard indicated thatequilibration is fast. Intermediate nodes are used only atwells with continuous screens that connect both aquifers.These nodes distribute the total pumping rate betweenthe aquifers.

The total model domain is 5120 by 1536 m. Thelargest 2D elements have a side length of 256 m, thesmallest elements, 0.7 m. Close to the PRB, the RiverAgrio is represented as an area having the real width ofthe river. Further away it is represented as a line. Stream-aquifer interaction is represented using Equation 3, sothat total inflow at each node is equal to the flux, given byEquation 3, times the nodal area. The latter is taken as thelength of river represented by the node times a fixedwidth of 12 m in the portions where the river is repre-sented as a line.

Two unconnected meshes are used for simulating thefour tests: one (‘‘2000 mesh’’) representing conditionsprior to the PRB construction and the winter 2000/2001floods, and one (‘‘2001/2002 mesh’’) representing condi-tions after both events. The latter is a bit more refined inorder to adequately represent the barrier and the addi-tional wells. Two tests series are interpreted in eachmesh. The first test series is separated from the secondseries by a 1000-d dummy period to allow heads torecover to zero (cf. Figure 1).

Time discretization is variable and was adjusted untilfurther refinement in time did not change the solution.

Table 2Overview of Tested Model Configurations: Treatment of Transmissivity and Number of Estimated

Parameters in the Three Model Configurations

Model 1 Model 2 Model 3

Treatment of transmissivity Geostatistical Geostatistical DeterministicTransmissivity zones 102 102 11Storage zones 4 4 4Leakance zones 4 9 9Sum of estimated parameters 110 115 24

Figure 5. (a) Transmissivity zones (the subdivisions of thegeostatistical transmissivity fields are not shown) and (b) lea-kance zones of the conceptual and numerical model. Onlythe downstream half of the model area is shown; zones con-tinue homogeneously upstream.

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Geostatistical Model for Transmissivity

As stated in the Methodology, the different alluviallayers and each module of the PRB are statistically inde-pendent. Prior information about aquifer transmissivitieswas calculated as saturated layer thickness times an ini-tial estimate of hydraulic conductivity (Table 3) that wasobtained from grain size analyses. Saturated thickness

was obtained from boreholes and trenches. The silt layerwas assumed to be continuous, with a constant thicknessof 1 m. From these point values, the transmissivity valueswere interpolated by block kriging using exponentialvariograms. The kriging parameters (Table 4) were esti-mated assuming longer ranges parallel to paleochannelsthan laterally and choosing relatively short ranges toavoid prior information to force too much continuity.

The transmissivity field of the upper layer has 35zones, that of the lower aquifer paleochannel has 49zones, and the lower layer silts are divided into threeindependent zones. The silt layer is represented by a sin-gle zone. For the PRB, each reactive module is dividedinto four correlated zones, plus two zones for the separat-ing low-permeability modules. This totals 102 transmis-sivity zones. Each aquifer (cf. Figure 5a) is divided intotransmissivity zones that are correlated according to thegeostatistical model (cf. Table 4). The size of the zones isadapted according to distance to test zones and density ofwells, with zones being smaller where the density ofobservation points is higher.

Calibration

Standard deviations of head data of the differentwells (required for defining the relative weights of headdata) were 5 cm for the six fully penetrating wells closeto the barrier where the mesh and T zones were refinedand 15 cm elsewhere because of increased discretizationerrors. Pumping well drawdowns were excluded from cal-ibration to prevent discretization and in-well effects suchas skin from affecting the aquifer calibration.

ResultsEstimated parameters of the three model configura-

tions are represented in Tables 5 and 6. Calibrated heads(Figures 7 and 8) of most wells are shown for model 2,which achieves the smallest residuals (Table 5). Fits aregood for most wells during the stream-stage response test(Figures 7 and 8d), except at the end of the recoveryperiod. Fits for the pumping test data are reasonable(Figures 8a through 8c). Since fits are quite similar in the

Figure 6. Finite-element mesh in plane view, showing onlythe 2001/2002 mesh. (a) Total modeled area, (b) zoom ofthe zone with two layers. In a and b, the two-layer zone withthe smaller upper layer is shaded in gray. (c) Zoom of thesurroundings of the PRB.

Table 3Prior Estimates and Variances of Hydraulic Conductivity (m/d) and Storage Coefficient (2) or

Storativity (m21, for the silt layer) of the Different Model Zones

Prior Estimateof Hydraulic

Conductivity, K (m/d)

Variance orSill r2 of Log T or

Log K

Prior Estimate ofStorage Coefficient, S (2), or

Storativity, Ss (m21)

Variance r2

of Log Sor Log Ss

Upper gravel layer 200 11 0.2 1Silt layer 0.05 2 5 3 1023 1Lower-layer gravels 1000 11 1023 1Lower-layer silts 0.1 1 1023 1Right margin barrier module 0.5 41 0.2 1Left margin barrier module 1 41 0.2 1Central barrier module 1 41 0.2 1Nonreactive sections 0.2 1 0.2 1

1Variance for model 3; sill for models 1 and 2, where covariances are determined by block kriging (see text and Table 4).

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three PRB modules, only results from the central barriermodule are shown. The residuals of model 1 are similar tothose of model 2 in most wells as expressed by the headsobjective function (Table 5). The major difference can befound in well S-2 during the second pumping test beforePRB construction (Figure 9), where model 1 performsmuch worse than model 2, and model 3 actually achievesthe best fit of all models. However, residuals of model 3are much larger in wells S-3, S-5, S-17 (not shown), andA-1bis (Figure 10). This causes the heads objective func-tion to be more than twice as high as those of the othermodels. Kashyap’s model identification criterion is best(most negative) for model 2 (Table 5). Despite the smallernumber of calibrated parameters (cf. Table 2), models 1and 3 achieve poorer scores.

Calibrated upper-layer transmissivities are similar formodels 1 and 2 (Figures 11b and 11c, right-hand side)and generally higher than prior estimates (Figure 11a),except at the upstream end. In the lower-layer paleo-channel, the calibrated transmissivities of models 1 and 2(Figures 11b and 11c, left-hand side) tend to be lowerthan the prior values, especially close to the southern lim-its of the displayed area. Toward the northern limits,model 1 assigns transmissivities that are higher than priorestimates, while model 2 assigns lower values. Model 3calibrates lower transmissivities than the geometricmeans of models 1 and 2 (Table 5) for both the uppergravel layer and the lower-layer paleochannel. Thehydraulic conductivities of the PRB modules in the threemodels remain similar to their prior estimates (Table 5),

Table 4Geostatistical Parameters Used for Block Kriging of the Different Transmissivity Regions of the Model

Upper LayerLower-Layer

Gravels (Paleochannel)Reactive Modulesof the PRB

Number of zones 25 49 4 zones/moduleDirection of anisotropy (clockwise relativeto the river axis near the PRB)

0� 45� Isotropic

Range in principal direction (m) 100 300 10Range in secondary direction (m) 30 100 10Sill (r2log T) 1 1 4

Figure 7. Measured (dots) and calculated (continuous line) head variations of model 2 for the stream-stage response test. Forclarity, only selected wells and part of the measured data points are shown.

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Table 5Results of the Model Evaluation Criteria, Prior Estimates and Calibrated Values of Transmissivities (T) of

Upper Gravel Layer and Lower-Layer Gravels, Hydraulic Conductivities (K) of PRB Modules andSilt Layer, and Storage Coefficients for the Different Model Configurations (for models 1 and 2

geometric mean of each transmissivity field or PRB module)

Prior Estimate

Calibrated Values

Model 1 Model 2 Model 3

Heads objective function — 224.0 209.5 535.0Kashyap (1982) — 25093 25209 24643Transmissivities and hydraulicconductivitiesUpper gravel layer T (m2/d) 166.3 ¼ 102.22 492 ¼ 102.69 416 ¼ 102.62 266 ¼ 102.42

Lower-layer gravels T (m2/d) 1718 ¼ 103.24 1634 ¼ 103.21 1451 ¼ 103.16 1399 ¼ 103.15

Right margin barrier K (m/d) 0.50 0.47 0.46 0.0066Central barrier K (m/d) 1.0 11 1.1 0.84Left margin barrier K (m/d) 1.0 1.5 2.2 1.2Silt layer K (m/d) 0.20 0.066 0.11 0.19

Storage coefficients (2)Upper gravel layer 0.20 0.23 0.26 0.20Silt layer 0.005 0.034 0.049 0.084Lower-layer gravels 0.001 0.0004 0.0059 0.0011Lower-layer silts 0.001 0.0007 0.0008 0.0009

1For the silt layer: storativity (m21)

Figure 8. Measured (dots) and calculated (continuous line) head variations of model 2 in the wells of the central PRB modulefor (a) the first and (b) the second pumping test before PRB construction, (c) the pumping test after PRB construction, and(d) the stream-stage response test (only wells not shown in Figure 7). For clarity, not all measured data points are shown.

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with two exceptions: the hydraulic conductivity of theright margin barrier module of model 3 is 2 orders ofmagnitude lower than the prior estimate, and in the cen-tral barrier module of model 1, the hydraulic conductivi-ties of the two K zones next to the river increase to 214and 40 m/d, raising the geometric mean of this module to11 m/d. In all the three models, the silt layer hydraulicconductivity is slightly lower than the prior estimate andthe silt layer storage coefficient is 1 order of magnitudehigher, while the remaining storage coefficients are simi-lar to the prior values (Table 5).

Many calibrated leakances (Table 6, cf. Figure 5b)change considerably with respect to their prior estimates.In model 1, the calibrated leakance for the connection ofRiver Agrio with the upper layer (Upper two-layer a)rises 6 orders of magnitude, while Guadiamar a drops by1 order of magnitude. In models 2 and 3, the general pat-terns in the leakance zones are similar, even though actualvalues differ. In the two-layer zone, the highest leakanceis assigned to the upper layer in 2001/2002 and a moder-ate value is assigned to the lower layer during the sameperiod, while both values are lower in 2000. Upstream ain the 2001/2002 mesh is very low, while it remainsalmost unchanged in the 2000 mesh, as does Downstream

a in both meshes. The biggest difference between the twomodels is found in Guadiamar a, which is lower than theprior estimate in model 2 (similar to model 1) but higherin model 3.

Discussion

Silt Layer ContinuityThe hydraulic conductivity of the silt layer is much

lower than that of the upper and lower gravel layers inthe three models. This shows that the models need to sep-arate these layers hydraulically in order to reproduce theobserved data. In the pumping tests (especially before thefloods), the presence of the silt layer enables hydraulicpulses to pass underneath the river through the lowergravel layer, while the very good connection between theriver and the upper gravel layer leads to a fast recoveryafter the pumping has stopped or the stream-stage peakhas passed. Therefore, the silt layer seems to be a continu-ous feature.

Stream-Aquifer InteractionModels 2 and 3 consistently display a marked differ-

ence between the leakances of the 2000 and the 2001/2002 meshes. In 2001/2002, they assign higher leakancesthan in 2000 to both layers of the two-layer zone. At leastin the 2001/2002 mesh of both models, the River Agrioalso has a moderate connection to the lower layer in thisarea. Even model 1, which was not allowed to assigna leakance to the lower-layer gravels within the two-layerzone, yields high hydraulic conductivities in two sectionsof the central PRB module, probably to hydraulicallyconnect the river to the lower layer.

This shows that the winter 2000/2001 floods changedstream-aquifer interaction in this aquifer and that at leastsince then, the River Agrio is hydraulically connected tothe lower aquifer close to the PRB. This is consistentwith the observation that the River Agrio deepened andbroadened its channel during the winter 2000/2001floods. The river increased its hydraulic connection to theupper layer and probably excavated the silt layer in someparts, creating or enhancing the connection to the lowerlayer. As a result, pumping during the test performedafter the winter 2000/2001 floods is barely noticed acrossthe river.

Role and Geometry of the Lower Gravel LayerMany wells at a considerable distance from the river

react quickly to the stream-stage variation. This impliesthat (1) the gravels found beneath the western T2 terraceare well connected to the River Agrio and that (2) a con-fining layer exists in this part of the aquifer (low storagecoefficients assigned by all models, cf. Table 5). Thisexplanation is consistent with the silt layer continuity andriver-aquifer interaction discussed previously.

As to paleochannel geometry, the proposed easternand western boundaries can satisfactorily explain theobserved data, especially the high contrast in reaction tothe stream-stage response test between wells that are only

-0.3

-0.2

-0.1

0.0

0 2

Time (days)

Hea

d va

riatio

n (m

)

Measured data

Model 1

Model 2

Model 3

1 3 4

Figure 9. Measured and calculated head variations of thedifferent model configurations in well ‘‘S-2’’ for the secondpumping test before PRB construction.

0.0

0.2

0.4

0.6

0.8

0 2 4

Time (days)

Hea

d va

riatio

n (m

)

Measured data

Model 1

Model 2

Model 3

1 3

Figure 10. Measured and calculated head variations of thedifferent model configurations in well ‘‘A-1bis’’ for thestream-stage response test.

287

short distances apart (Figure 7, i.e., S-26, S-24, S-28 vs.S-25, S-26 at the western border and S-5, S-6 vs. S-7 atthe eastern border). At the southern end of the paleo-channel, both models 1 and 2 assign a low leakance toRiver Guadiamar and lower paleochannel transmissiv-ity. Model 3, which has to assign a homogeneous

transmissivity to the whole paleochannel, cannot selec-tively lower transmissivity close to Guadiamar River.Instead, it even raises Guadiamar a, achieving fits in thewells of the southern paleochannel (S-23, S-24, S-28; notshown), which are similar to those of models 1 and 2.This seems to contradict the results of the other models,

Figure 11. Transmissivity fields (left-hand side: lower layer; right-hand side: upper layer) of the geostatistical models. Forthe lower layer, only the central part is shown; transmissivity zones continue homogeneously up- and downstream. (a) Priorestimate of models 1 and 2. (b) Calibrated values of model 1. (c) Calibrated values of model 2.

Table 6Prior Estimates and Calibrated Leakances (d21) for the Different Model Configurations

Leakance ZonePrior Estimates Model 1

Model 2 Model 3

All Models and Meshes Both Meshes 2000 Mesh 2001/2002 Mesh 2000 Mesh 2001/2002 Mesh

Upstream a 1.0 0.87 1.2 0.0033 0.97 0.00027Upper two-layer a 1.0 121,200 7.6 129 0.39 24Lower two-layer a 1.01 na 0.014 7.8 0.67 5.8Downstream a 1.0 1.0 1.0 0.99 1.0 1.0Guadiamar a 1.0 0.098 0.088 8.1

na ¼ not assigned.1Except model 1 (leakance zone not assigned).

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but presumably, model 3 uses River Guadiamar as aninjection zone (analogously to an image well) in order toimitate the effect of the low-permeability zone assignedby models 1 and 2. Therefore, we conclude that the paleo-channel is only poorly connected to River Guadiamar.This conclusion is consistent with independent ob-servations not used during modeling and not shown here.First, wellS-28 was not contaminated, as it should if the paleo-channel had been connected to River Guadiamar. Second,natural head gradients in the paleochannel point to theRiver Agrio, supporting that this area is not well con-nected to River Guadiamar.

Nevertheless, it is less clear how well and where thelower paleochannel is connected to River Agrio upstreamof the studied area. The situation is ambiguous in two as-pects. First, it appears contrary to the aforementioned the-ory on stream-aquifer interaction that Upstream a ishigher before the winter 2000/2001 floods in models 2and 3. This is probably only an artifact of the lack of datain this zone during the pumping tests (no measurementswere taken in well A-1bis). Therefore, the models are notvery sensitive to this leakance in the 2000 mesh andmaintain its value close to the prior estimate. Second,none of the three model configurations achieves a satis-factory fit for well A-1bis (Figure 11). All modelsoverestimate both peak height and peak arrival time.Additional wells and river-stage measurement points dur-ing the stream-stage response test would have helped toincrease model sensitivity in this area.

Impact of PRB ConstructionThe PRB construction has created a low-permeability

feature in the aquifer. Observation wells now react muchless to pumping in wells on the opposite side of the PRBthan before PRB construction. Head variations in thesewells during the last pumping test series are due tochanges in river stage rather than to pumping (cf. Figures6a and 6b and 8a through 8c).

Modeling ApproachThe present numerical model allows us to simulta-

neously interpret pumping and stream-stage response testdata and to resolve aquifer diffusivities into transmissiv-ities and storage coefficients. In our case study, thestream-stage response test alone would not reveal infor-mation about PRB characteristics because ground waterflow during the stream-stage response test is parallel tothe barrier.

A large number of observation points were used inthe calibration. Therefore, large errors in single pointslead to relatively small changes of the heads objectivefunction. As a result, the fits have to be verified individu-ally in all observation points in order to evaluate the con-ceptual model (analysis of residuals). This means thatautomatic calibration helps to test alternative conceptualmodels rapidly, but it does not eliminate the need forthorough, and often tedious, qualitative analysis of re-sults. This applies to any model with many observationpoints, not only to the interpretation of stream-stageresponse tests.

Models 1 and 2 obtain almost equal fits for mostobservation points. Model 2 uses somewhat more plausi-ble parameter values that are consistent with independentobservations (changes in stream channel geometry due tothe winter 2000/2001 floods) so it appears more realistic,but other models could be developed that can explain theobserved data equally well.

ConclusionsA methodology for the design, execution, and joint

numerical interpretation of stream-stage response testswas summarized and successfully applied to a test per-formed in the River Agrio (southwest Spain). The objec-tive of the test was to complement three sets of cross-holepumping tests in the characterization of the layeredstream-aquifer system. Two geostatistical models and one‘‘conventional’’ model with homogeneous transmissivitieswere tested in order to investigate the trade-off betweenmodel complexity and achieved fit.

The results show that

d The combination of pumping and stream-stage response

test data allows calibrating both transmissivities and stor-

age coefficients. The pumping tests yield small-scale

information about the barrier and its surroundings and pro-

vide the flow data necessary to resolve diffusivity into T

and S, while the stream-stage response test gives informa-

tion on a larger scale.d River-aquifer interaction can be simulated using lea-

kances.d The geostatistical approach obtains better overall results

than the more conventional deterministic approach.d The numerical model used in the case study allowed us to

explore (1) aquitard continuity; (2) degrees and temporal

changes of stream-aquifer interaction; (3) the impact of the

construction of a PRB on the hydraulic system; and—to

some degree—(4) aquifer geometry.

In summary, the stream-stage response test provideshydraulic information at a relatively large scale and isrelatively easy to perform. Therefore, its use is recom-mended. However, a thorough interpretation requires cou-pling to pumping tests or other types of flow data, whichcan be achieved satisfactorily using geostatistical inver-sion codes.

AcknowledgmentsThis study was funded by the EU PIRAMID project

(EVK1-1999-00061P) and by the Spanish CYCIT pro-gram (HID99-1147-C02). The authors wish to thank theHydrographical Confederation of the Guadalquivir andC. Mediavilla from the Geological and Mining Instituteof Spain in Seville for making possible the execution ofthe stream-stage response test, and J. Jodar and C. Knud-by for their help in the initial design of the numericalmodel. T. Rotting’s work was supported by the GermanAcademic Exchange Service DAAD (grant number D/01/29764) and the Gottlieb-Daimler and Karl Benz Founda-tion (grant number 02-18/01). The authors also thank

289

Steffen Mehl, Marios Sophocleous, and one anonymousreviewer for their constructive comments that helped toimprove the present paper.

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Yeh, T.-C.J., and S.Y. Liu. 2000. Hydraulic tomography: Devel-opment of a new aquifer test method. Water ResourcesResearch 36, no. 8: 2095–2105.

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Dealing with spatial heterogeneity

Gh. de Marsily · F. Delay · J. Gon�alv�s · Ph. Renard ·V. Teles · S. Violette

Abstract Heterogeneity can be dealt with by defininghomogeneous equivalent properties, known as averaging,or by trying to describe the spatial variability of the rockproperties from geologic observations and local mea-surements. The techniques available for these descriptionsare mostly continuous Geostatistical models, or discon-tinuous facies models such as the Boolean, Indicator orGaussian-Threshold models and the Markov chain model.These facies models are better suited to treating issues ofrock strata connectivity, e.g. buried high permeabilitychannels or low permeability barriers, which greatly af-fect flow and, above all, transport in aquifers. Geneticmodels provide new ways to incorporate more geologyinto the facies description, an approach that has been welldeveloped in the oil industry, but not enough in hydro-geology. The conclusion is that future work should befocused on improving the facies models, comparing them,and designing new in situ testing procedures (includinggeophysics) that would help identify the facies geometryand properties. A world-wide catalog of aquifer faciesgeometry and properties, which could combine site gen-esis and description with methods used to assess thesystem, would be of great value for practical applications.

R�sum� On peut aborder le probl�me de l’h�t�rog�n�it�en s’effor�ant de d�finir une perm�abilit� �quivalentehomog�ne, par prise de moyenne, ou au contraire en d�-crivant la variation dans l’espace des propri�t�s des ro-ches � partir des observations g�ologiques et des mesureslocales. Les techniques disponibles pour une telle des-cription sont soit continues, comme l’approche G�osta-tistique, soit discontinues, comme les mod�les de faci�s,Bool�ens, ou bien par Indicatrices ou GaussiennesSeuill�es, ou enfin Markoviens. Ces mod�les de faci�ssont mieux capables de prendre en compte la connectivit�des strates g�ologiques, telles que les chenaux enfouis �forte perm�abilit�, ou au contraire les faci�s fins de bar-ri�res de perm�abilit�, qui ont une influence importantesur les �coulement, et, plus encore, sur le transport. Lesmod�les g�n�tiques r�cemment apparus ont la capacit� demieux incorporer dans les mod�les de faci�s les obser-vations g�ologiques, chose courante dans l’industrie p�-troli�re, mais insuffisamment d�velopp�e en hydrog�o-logie. On conclut que les travaux de recherche ult�rieursdevraient s’attacher � d�velopper les mod�les de faci�s, �les comparer entre eux, et � mettre au point de nouvellesm�thodes d’essais in situ, comprenant les m�thodes g�o-physiques, capables de reconna�tre la g�om�trie et lespropri�t�s des faci�s. La constitution d’un cataloguemondial de la g�om�trie et des propri�t�s des faci�saquif�res, ainsi que des m�thodes de reconnaissance uti-lis�es pour arriver � la d�termination de ces syst�mes,serait d’une grande importance pratique pour les appli-cations.

Resumen La heterogeneidad se puede manejar por me-dio de la definici�n de caracter�sticas homog�neas equi-valentes, conocidas como promediar o tratando de des-cribir la variabilidad espacial de las caracter�sticas de lasrocas a partir de observaciones geol�gicas y medidas lo-cales. Las t�cnicas disponibles para estas descripcionesson generalmente modelos geoestad�sticos continuos omodelos de facies discontinuos como los modelos Boo-lean, de Indicador o de umbral de Gaussian y el modelode cadena de Markow. Estos modelos de facies son masadecuados para tratar la conectvidad de estratos geol�gi-cos (por ejemplo canales de alta permeabilidad enterradoso barreras de baja permeabilidad que tienen efectos im-portantes sobre el flujo y especialmente sobre el trans-porte en los acu�feros. Los modelos gen�ticos ofrecen

Received: 17 April 2004 / Accepted: 15 December 2004Published online: 25 February 2005

Springer-Verlag 2005

G. de Marsily ()) · J. Gon�alv�s · S. VioletteUniversit� Pierre et Marie Curie, Laboratoire de G�ologieAppliqu�e,Paris VI, Case 105, 4 place Jussieu, 75252 Paris Cedex 05, Francee-mail: [email protected].: +33-144-275126Fax: +33-144-275125

F. DelayUniversit� de Poitiers, Earth Sciences Building,40 Avenue du Recteur Pineau, 86022 Poitiers cedex, France

P. RenardUniversity of Neuchatel, Centre of Hydrogeology,Rue Emile Argand 11, 2007 Neuchatel, Switzerland

V. TelesLaboratoire des Sciences du Climat et de l’Environnement,UMR CEA-CNRS,Orme des Merisiers, bt. 709, 91191 Gif sur Yvette Cedex, France

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nuevas formas de incorporar m�s geolog�a en las des-cripciones de facies, un enfoque que est� bien desarolladoen la industria petrolera, pero insuficientemente en lahidrogeolog�a. Se concluye que los trabajos futuros de-ber�an estar m�s enfocados en mejorar los modelos defacies, en establecer comparaciones y en dise�ar nuevosprocedimientos para pruebas in-situ (incuyendo la geof�-sica) que pueden ayudar a identificar la geometr�a de lasfacies y sus propiedades. Un cat�logo global de la geo-metr�a de las facies de los acu�feros y sus caracter�sticas,que podr�a combinar la g�nesis de los sitios y descrip-ciones de los m�todos utilizados para evaluar el sistema,ser�a de gran valor para las aplicaciones pr�cticas.

Introduction

The word “Hydrogeology” can be understood as a com-bination of “hydraulics” and “geology.” Hydraulics is arelatively simple science; we know, at least in principle,the governing hydraulic equations and can solve them,analytically or numerically, given the geometry of thesystem, boundary conditions, etc. Geology is more com-plex: it refers not only to the description of what thesystem looks like today, its properties in space, etc., butalso to the history of its formation, because geologistshave been trained to accept that one needs to understandthe succession of complex processes involved in thecreation and modification through time of the naturalobjects that one is trying to describe. To understand thiscomplexity, geologists have a limited number of clues ordata, whose interpretation requires several assumptionsand may lead to alternative solutions. Geology also in-cludes a set of disciplines whose contribution is needed tostudy or describe the system: sedimentology, tectonics,geophysics, geochemistry, age dating, etc. “Hydrogeolo-gy” is thus the science where the two are combined:finding the solution of the flow (and transport) equationsin a complex, only partly identified, geologic system.

If the world were homogeneous, i.e. if the rock prop-erties were constant in space, and/or easy to determine,hydrogeology would be a rather boring job: solving well-known equations in a perfectly identified medium. For-tunately, the world is heterogeneous, with highly non-constant properties in space, and “dealing with hetero-geneity” is what makes the work fascinating. Everybodyhas heard of in situ experiments conducted in the field byexperimentalists trained to work in the lab.: the first thingthey do is to physically “homogenize” the site, by me-chanically mixing the superficial horizons. “Otherwise itis too complex and one cannot understand what is goingon,” they say, referring to Occams’razor. The first thoughtthat comes to mind is that they miss part of the fun byignoring spatial heterogeneity, the second one is that theirresults are mostly useless, because the world is hetero-geneous, and to understand “hydrogeology,” one has toacknowledge this and deal with it.

Hydrogeology has been too much inclined towards“hydraulics” and the solving of the flow equations, and

not enough towards “geology” and understanding/de-scribing the rock structure, facies and properties in a ge-ologically realistic manner, thus proposing “exact” solu-tions, but to poorly posed problems.

This article first provides a brief history of how hy-drogeologists have dealt with heterogeneity so far, andthen an attempt is made to give a personal view of howhydrogeologists may be dealing with it in the future.Making predictions is quite difficult, and a French sayingadds “especially for the future”! These predictions arevery likely to be wrong, but it is hoped that these sug-gestions may trigger additional work, foster discussions,generate controversy, and that, in the long term, bettermethods will be developed to deal with heterogeneity.

Four important issues are not addressed here: (i) thetransition from Navier-Stokes’ equations to Darcy’s law,which, at the pore scale, is the first scale of heterogeneityof the velocity vector in natural media; (ii) the multi-plicity of scales of heterogeneity, discussed in Carrera(this issue), and also Noetinger (this issue); and (iii) themultiple processes involved in flow and transport innatural media (flow, transport, diffusion, biogeochemicalreactions, etc.). The focus will rather be on the methodsused to account for heterogeneity for any of these pro-cesses, to represent it and to model it. Finally, (iv) thereasons why the Earth is heterogeneous will not be ex-amined either, i.e.: sedimentation processes, the formationof crystalline rock, tectonics, diagenesis, etc, because it isassumed that all geologists are familiar with this.

Brief history of the methods used to dealwith heterogeneity

The early approachThe first practical field hydrogeologists dealt with het-erogeneity by trying to locate “anomalies” and make useof them. Water is not present everywhere in the ground;only those who can locate the highly porous and perme-able strata in the ground are able to decide on well lo-cations or discover springs. For instance, Brunetto Latini(1220–1295) wrote: “The earth is hollow inside, and fullof veins and caverns through which water, escaping fromthe sea, comes and goes through the ground, seeping in-side and outside, depending on where the veins lead it,like the blood in man which spreads through his veins, inorder to irrigate the whole body upstream and down-stream” (in Poir�e 1979). Famous hydrogeologists, likeParamelle (1856), used their geologic knowledge to detectheterogeneities and discover springs. Others use diviningrods... Darcy (1856) however, who was probably the firstto attempt to quantify flow, developed his theory for ar-tificial homogeneous sand filters and did not have to dealwith heterogeneity. He produced however cross sectionsof an “aquifer,” tapped by an artesian well, thereforeimplicitly differentiating between an aquifer and an“impervious layer.” It was Dupuit (1857, 1863) who firsttried to apply Darcy’s law to natural media, which were ofcourse heterogeneous. It is not clear if Dupuit was con-

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scious of the magnitude of heterogeneity. To extendDarcy’s law to natural media, his approach was intuitive:assume the existence of a “homogeneous” layer and useDarcy’s law. He did not discuss in detail how he woulddefine the permeability of this homogeneous layer, butsince Darcy had developed an apparatus to measure thepermeability, he would most likely have taken samples onwhich to measure it. Had he taken several samples, andmeasured different values, nothing is there to give a clueon how he would have averaged them, in order to definean “equivalent” homogeneous layer. In fact, Dupuit was amathematician, he developed the steady-state flow equa-tion, and proceeded to find analytical solutions in varioussettings (like the classical Dupuit–Forscheimer solutionaround a well, Dupuit 1863). Had he been aware of het-erogeneities, his analytical tools would not have allowedhim to take them into account, and he would have ignoredthem altogether. Theis (1935) was the first to produce asimple tool to directly “estimate” the equivalent homo-geneous property of an aquifer; this was the new well testthat automatically “averages” the local (variable) prop-erties, such as permeability and storativity. Before Theis,it was already possible to derive in situ values for per-meability, by means of the steady-state solutions around awell (Thiem 1906), or local injection tests (Lefranc, Lu-geon, etc) and the local values were in general averagedwithout any consideration for the type of average to use.This quotation is by Morris Muskat (1949), speaking ofoil reservoirs: “it appears extremely unlikely that actualunderground strata will be of strictly uniform perme-ability over distances or areas associated with oil-pro-ducing reservoirs. However, as such random lateral vari-ations as undoubtedly occur are literally impossible ofexact determination, they must be considered as averagedto give an equivalent uniform-permeability stratum.Moreover, even if the nature of these variations wereknown, the difficulties of exact analytical treatmentwould still force the use of averaging procedure and re-duction to an equivalent permeability system.” He thengoes on to mention the Cardwell and Parsons (1945)bounds for averaging, and treats some special cases wheresimple geometric trends in permeability are known.

The simple distinction between an aquifer and a sub-stratum or confining layers seems to have emerged rela-tively early. It is a first recognition of heterogeneity, butwith a very simple “layered cake” conceptual model,which can be traced back to many early earth scientists,like the British geologist Whitehurst in 1778, or theFrench chemist Lavoisier in 1789 (who was interested ingeology). This conceptual model is deeply inscribed inour discipline, and hard to abandon. It has been the causeof the use of transmissivity rather than permeability. Theclassification of the confining layers into aquitards andaquicludes came much later, when leakage was studied,and is probably due to Paul Witherspoon in the 1960s, seee.g. Neuman and Witherspoon (1968).

The rules for averaging permeability in hydrogeologywere discussed in detail by Matheron (1967), although theissue of averaging had already been addressed earlier in

other fields of physics, e.g. by Landau and Lifschitz(1960), or by Cardwell and Parsons (1945). Matherondealt with the issue by considering permeability as amagnitude varying randomly in space according to aspecified distribution. He showed that for two-dimen-sional parallel flow, in steady state, for a spatial distri-bution that is statistically invariant by a rotation of 90 and identical for the permeability k/E(k) and the resis-tivity k�1/E(k�1), the correct average is the geometricmean. An invariance of the spatial distribution impliesthat not only the univariate (pdf) and bivariate (covari-ance) statistics are identical, but the complete spatialdistribution function must remain the same for k and k�1.In particular, this applies to the multi-Gaussian log-nor-mal distribution, which is often mentioned as correctlyrepresenting the distribution of permeabilities in naturalmedia. Note that the physical reason why permeabilitiesare distributed log-normally has never been explicitlystated, and, on the contrary, examples are found wherethis assumption is not correct. One has to remember,however, that the flow equation is a diffusion equationwhich can be written, for heterogeneous media:

@ ln Kð Þ½ �=@x @h=@xþ @2h=@x2 ¼ . . .

Assuming that the parameter ln(K) has a Gaussian dis-tribution is quite frequent for the same equation in otherfields of physics. Matheron also recalls the arithmetic andharmonic bounds for averaging, which apply to all media,already published by Cardwell and Parsons in 1945.Surprisingly, he also states (without complete demon-stration) that there is no averaging in radial flow insteady-state, which is against the standard practice ofsteady-state well testing; he argues that the average per-meability can vary anywhere between the arithmetic andharmonic means, depending on the value at the well, andon the boundary conditions. It has however been shownempirically that, in transient state at least, the geometricmean is the long-term average that a well test produces(see e.g. Meier et al. 1999). This issue of averaging willbe discussed again below, and its role and some of itsproperties will be specified. It must be emphasized thatpumping tests are still a key research topic, since theyremain the most useful way to investigate the propertiesof the underground. In other words, hydraulic tests inboreholes still have a long life ahead; for instance, newanalytical solutions such as those by Barker (1988) orChang and Yortsos (1990), Acuna and Yortsos (1995), fornon-integer or fractal spatial dimensions, which do notapply only to fractured rocks, show that new pumping testanalyses can help to better characterize heterogeneity,even if the fractal dimension itself is not a predictiveparameter that can yet be linked with the geometry,spatial permeability distribution or observable connec-tivity of the medium. But other techniques, as suggestedbelow, may be developed.

Did this simple equivalent homogeneous medium ap-proach work, with the tools available at the time (ana-lytical solutions, the image theory, conformal mapping,etc...)? Yes, somehow it did, since early hydrogeologists

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were able to develop well fields, to adapt the exploitationto the resources and to solve local engineering problemswithout major catastrophes.

Heterogeneity and numerical modelingThe breakthrough in quantitative hydrogeology came inthe early sixties with numerical modeling. It is hard topinpoint who built the first digital model of an aquifer, itcame after several attempts (started in the oil industry) touse electric analogs, the digital models being initiallynumerical versions of the analog, see e.g. Walton andPrickett (1963), Prickett (1968). The names that come tomind are Tyson and Weber (1964) in Los Angeles, Zaoui(1961) at Sogr�ah and the Chott-el-Chergui in Algeria,Prickett (1975) for a review, and others.

The analogs and, better still, the digital models, offeredthe possibility of making the aquifer properties variable inspace while adapting the model to the geometry of thedomain and changing the boundary conditions, the re-charge, etc. Heterogeneity and the way in which it wasaccounted for in models will be the focus here. Assumingthat there are a few pumping test values for the aquiferpermeability, they are nevertheless much fewer than thenumber of cells in the model: the model offers a possi-bility of “dealing” with heterogeneity in a much moreprecise way than the well test data alone would permit.The initial approach was to assume that a permeability (ortransmissivity) given by a pumping test was a “local av-erage” of the true local value, over an area equivalent tothat of the mesh of the model. In other words, the issue of“support” in Geostatistics, which concerns the size of thedomain on which a measurement is made, was not yetclearly understood, but the idea was firmly establishedthat the permeability measured by a well test was to beassigned to the mesh where that well was located. Be-tween wells, the value assigned to each mesh was to beinterpolated from the adjacent wells. This could be doneby zoning (e.g. Vorono�, or Thiessen, polygons sur-rounding each well), or by polynomial trend fitting, or byhand contouring, etc. Heterogeneity was thus defined by aset of “measured” averaged values at local wells, inter-polated in space. This did not work very well: since themodel could calculate heads and flow rates, these valuescould be compared with observations, and the modelcould be “verified.” Model calibration was the answer,initially through trial-and-error and soon to be followedby automatic inverse procedures (see Carrera, this issue).The important point here is that heterogeneity is repre-sented (at a certain scale) in the models, but heterogeneityis neither defined nor prescribed: the heterogeneity de-scription is based on the local (averaged) measurements,and its distribution in space is inferred by the inverseprocedure. As the modelers soon realized that the inverseshave an infinite number of solutions, they tried to restrictthe range of “inferred heterogeneities” by assigningconstraints to the inverse: regularity, zoning, algorithmicunicity, etc, see Carrera (this issue).

Did that work? Yes, it did, and it is still often usedtoday, a calibrated model is a reasonable tool for aquifer

management and decision making, see e.g. a discussionwith Konikow and Bredehoeft (1992a, 1992b) in Marsilyet al. (1992). Where the approach started to fail or at leastto show its limitations was first in the oil industry, wherethe predictions of oil recovery and water-cut (i.e. water-to-oil ratio in a producing well) made by calibratedmodels proved very unreliable. Had such models beenused to predict groundwater contamination problems,their limitations would also have become apparent, butsuch problems were still new in hydrogeology at that time(seventies–eighties). The reason is that oil production andwater-cut, as well as contaminant transport, are verysensitive to conductive features such as high-permeabilitychannels, faults or, on the contrary, low permeabilitybarriers that are by definition intrinsically “averaged” inthe well-testing method and in the inverse inference ap-proach. In a comparison of seven inverse methods, Zim-merman et al. (1998) tried to “embed” in artificiallygenerated media such highly conductive channels, andasked the seven inverses to identify them, based on cali-bration on head measurements. The results showed thatmost missed such heterogeneous features.

In fact, the head variations due to heterogeneity aresmall, whereas those of velocities and travel time arelarge, as can be seen intuitively when considering a lay-ered aquifer with flow parallel to the bedding: the verticalheterogeneity does not produce any variation in the headdistribution over the vertical, while velocities can varytremendously from layer to layer. Trying to infer theheterogeneity from the head data alone was a quasi-im-possible task. At present, head and concentration data(e.g. environmental tracers or contaminants) are moreoften available and used jointly in the inversion.

In the oil industry, to overcome these limitations, im-portant efforts were devoted to “reservoir characteriza-tion” in the 1980s, following the introduction of “se-quential stratigraphy,” see e.g. Vail et al. (1991). Thisinvolved advanced geologic analysis of depositional en-vironments, combined geophysics and well logging, de-tailed characterization of modern outcropping deposi-tional analogs of deep buried reservoirs, in order to learnhow to describe and represent them. This advanced geo-logical analysis of reservoirs and of organics-rich source-rocks, to characterize their heterogeneity, was at the baseof the development of new tools, such as Geostatistics andBoolean methods, which will now be described. Unfor-tunately, these “reservoir characterization” efforts did notreally catch on in hydrogeology, which was (and still is)lagging behind reservoir engineering in this matter.

Stochastic methods and geostatisticsConsidering an aquifer property as a random variable wasdiscussed, e.g. by Matheron already in 1967, but for thepurpose of averaging, not to describe heterogeneity. In theoil industry, Warren and Price used a stochastic approachas early as 1961 to assign permeability values to a 3-Dreservoir model, also with the problem of averaging inmind, but this did not evolve into significant changes inthe treatment of heterogeneity. The origin of stochastic

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hydrogeology can be traced to two schools: Geostatistics,developed by Matheron (1963, 1965), for mining esti-mates, and first applied to hydrogeology by Delhomme(1976, 1978, 1979), and to Freeze (1975), Gelhar (1976)and Smith and Freeze (1979). In 1975, Freeze assignedrandom values of permeability to cells of a 1-D model,drawn from a log-normal distribution, without consider-ing any spatial covariance between the random values, asWarren and Price (1961) had done. Delhomme (1976,1978, 1979) and Smith and Freeze (1979) included such acovariance. At the same time, Gelhar (1976, 1993) waspioneering stochastic hydrogeology, taking the covarianceinto account; his work was more theoretical and usedanalytical methods, as did the subsequent works by Dagan(1985, 1989).

Geostatistics, and in particular the definition of thecovariance (or the variogram) of the permeability distri-bution, is the first appearance of the new concept thatheterogeneity can be described by a “structure,” i.e., thatthe geological processes that created the medium haveimposed a pattern on the spatial distribution of the inho-mogeneous values. It is no longer possible to produce a“plausible” map of the heterogeneity of a medium withouthaving some underlying “structure” in mind. This struc-ture is, in Geostatistics, defined by a new function, thespatial covariance (or variogram) which is a tool tocharacterize the heterogeneity. The inference of thisstructure from the data is the compulsory first step of aGeostatistical analysis, with the tacit assumption that allrelevant heterogeneities and “structures” can be capturedor represented by the covariance, which will be chal-lenged later. See e.g. Chiles and Delfiner (1999), orMarsily (1986).

The Geostatistical approaches to dealing with hetero-geneity were initially considered as continuous processes.These can produce three results:

1. The first is a better estimate of the spatial distributionof the parameter values in the cells of a model, basedon a set of “measured” values. Kriging provides an“optimal” (in the sense of minimum variance of theestimation error, not necessarily “best approach”)method for assigning permeability (or transmissivity)values to the meshes of a model, significantly superiorto zoning, or to arbitrary interpolation (see e.g.,Marsily 1986; Chiles and Delfiner 1999). However,when highly correlated structures such as buriedchannels or fractures are present, it is not possible todirectly represent them with a single variogram, and,if they have been identified, geologically informedzoning is preferable. Furthermore, an extension ofKriging, co-Kriging, can be used to estimate perme-ability based on both well-test results and additionalmeasurements (e.g. specific capacity, electrical resis-tivity, etc.). See e.g. Aboufirassi and Marino (1984),Ahmed and Marsily (1987, 1988, 1993). Lastly,Kriging provides a rigorous framework to address the“support” issue, i.e. to consider data that are repre-sentative of different averages over space, e.g., slug-

test data that are only local values and well test data,which are averages over areas depending on the du-ration of the test. Kriging can incorporate these dif-ferent data and produce estimates that are represen-tative of averages over the precise area of a mesh, thatcan vary in size inside the domain, see e.g., Chiles andDelfiner (1999), Roth et al. (1998), Rivoirard (2000).In several instances, it has been shown that a Krigeddistribution of the permeability or transmissivity willproduce a model that is almost calibrated and need notbe adjusted, see e.g. Raoult (1999), or to some extent,in basin modeling, Gon�alv�s et al. (2004b). It is clearhowever that Kriging does not always perform well,as will be seen later. One also needs to apply Krigingwith some rigor on the log-transformed transmissivityvalues, in order to estimate geometric mean valuesand not arithmetic means. Back-transforming theKriged lnT into T values must also be done correctly,i.e. simply as T= exp[lnT] without any additional termusing the variance of the estimation error to suppos-edly correct an assumed estimation of the medianrather than of the mean, as is sometimes erroneouslydone: see e.g. Marsily (1986).

2. The second is that Geostatistics provides a clearconcept to constrain an inverse calibration procedure,to infer the spatial distribution of the parameter value.This is discussed at length in Carrera (this issue) andwas the basis for the “Pilot Point” inverse approach(see e.g. Marsily 1978; Certes and Marsily 1991;Ramarao et al. 1995; Lavenue et al. 1995; Marsily etal. 2000; Lavenue and Marsily 2001).

3. Monte-Carlo simulations of aquifer models where theparameters are conditional realizations of the (un-known) spatial distribution of the parameters is onemethod to assess the consequences on flow andtransport of the uncertainty on the heterogeneity of thesystem, as represented by the Geostatistical approach.Assuming that the conditional realizations display thefull range of uncertainty on the heterogeneity of theaquifer, and assuming that the covariance is actuallycapable of capturing all of the relevant heterogeneity,the Monte-Carlo flow simulations provide an estimateof the resulting uncertainty on the flow and transport(head, flow rate, concentration, etc); see e.g., Del-homme (1979), Ramarao et al. (1995), Lavenue et al.(1995), Zimmerman et al. (1998).

Geostatistics has helped greatly, and the method is widelyused to deal with heterogeneity. It is conceptually simple,the additional degree of freedom is very small (a variance,a range and a type of covariance model), the tools areavailable and the data requirements are standard ones: tobuild a groundwater model, well tests are necessary, andGeostatistics make better use of the data without askingfor more. Additional data such as specific capacity,electric resistivity, etc., can also be used jointly. To thequestion: “how do we know, in a very practical sense, thatthe generated statistics are correct?,” the answer is thatthis description of reality is a model, not an exact de-

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scription; natural heterogeneity is much more complexthan any model can account for. The only value of usingsuch a model is if it provides a better description (closerto the unattainable reality) than previous models, and thusgives better predictions than its competitors. Through theexamples outlined above, simple geostastistics has shownthat it provides better descriptions of continuous param-eter fields than any alternative method. This is not to saythat it is correct, only that it is useful.

Stochastic shalesGeostatistics represented a first attempt at describing thestructure (in a statistical sense) of heterogeneities. Thetool used for that, the variogram (or the covariance), hastwo major components: the range, which is proportionalto the average size of heterogeneous bodies, and the sill,which is a measure of the magnitude of the changes (e.g.,in permeability) from one heterogeneous body to another.These “bodies” were thought of as “hills” and “holes,”circular or ellipsoidal (in the case of an anisotropic vari-ogram), distributed randomly in space. On a Kriged map,or better on a conditionally simulated one, it is easy to seethe range of the structure, which is close to half the av-erage distance between two “hills” or two “holes” in themap. The sill is close to half the average squared differ-ence between the values measured on the top of the hillsand those measured at the bottom of the holes. The mapwas representing a continuous function. It is clear how-ever that this description of reality is only a model.

In 1986, Haldorsen and Chang initiated a breakthroughin the oil industry with their “stochastic shales” approach,where the “objects” are discontinuous. Their concept ofheterogeneous bodies is also statistical, but founded ongeometrical concepts, not covariances. They considerheterogeneous sand bodies as “objects,” whose shape isperfectly known (e.g., shoe boxes, see Fig. 1) but the sizeof the box and the position of its center are drawn ran-domly from a prescribed distribution. These objects areembedded in a continuous matrix, which for Haldorsenand Chang, was a shale, the “objects” being sand lenses.This type of model is called Boolean. The shape of theobject is extremely flexible and left to the decision of themodeler, see many examples in e.g. Haldorsen and

Damsleth (1990). The required statistics concerning theshapes and positions of the objects are estimated fromavailable data, such as borehole logs, outcrop mapping,seismic surveys, etc. Several sets of objects with differentshapes and properties can be used together (e.g. sandlenses and limestone bodies inside a shale matrix, ordifferent types of sand structures). Once the distributionin space of the objects is drawn, they are sometimes calledfacies, or “flow units”; it is still necessary to discretize thedomain into cells and to assign properties to each facies:permeability, porosity, etc. This can be done by a directrelation (each facies is described by one set of parame-ters), or by a random sampling of parameter values, de-fined by a distribution function and sometimes by a co-variance, for each facies. The model is by definitionstochastic, and many realizations can be produced.

This approach allowed abrupt changes in facies andpermeability when Geostatistics was generating continu-ous fields. This is a fundamental step toward modelingactual geological media, where discontinuities and abruptchanges are widely present (e.g. sand bars vs. matrix,channels vs. floodplain, turbidites vs. continental margindeposits, etc). Note that the concept of roughness definedhere for a surface could be defined for a 3-D medium as a“texture” parameter. This parameter has not been con-sidered so far, since it is not relevant for the flow prob-lem, but it might be interesting to investigate it whendealing with dispersion.

Did this approach work? It was indeed a breakthrough,essentially in the oil industry, as it empirically popular-ized the underlying concept of connectivity. If two sandlenses are not hydraulically connected, fluids will notreadily flow from one to the other. This lack of connec-tivity happens in nature, and so far the models had beeninept at dealing with this concept, Geostatistics intro-duced quite smooth transitions in space, not abruptchanges, whereas in reality nature does not varysmoothly. So the stochastic shales concept was veryuseful, although Boolean models addressing the connec-tivity issue had been developed much earlier, e.g. byMatheron (1967); Marsily (1985) and Fogg (1986) alsoemphasized the connectivity issue. In hydrogeology, the“stochastic shale” concept has not been widely used (seehowever e.g. Desbarats 1987; Pozdniakov and Tsang2004) except that the same type of Boolean model wascaptured by the fractured rock community, where the“objects” were discrete fractures (Fig. 1). The concept offracture connectivity (sometimes named percolationthreshold, in the framework of the percolation theory)makes a lot of sense. Many versions of the discretefracture model are available and have been tested, vali-dated and compared, see e.g., Marsily (1985), Long andBillaux (1987), Cacas et al. (1990a, b), Long et al. (1991).They also completely renovated the vision of how frac-tured systems behave by introducing this concept ofconnectivity, which will be discussed later together withthe upscaling issue. These Boolean tools are available andused, but require a new type of data on the geometry ofthe “objects” which they represent. Some of these data are

Fig. 1 Examples of Boolean models: (left) sand lenses, from Hal-dorsen and Damsleth (1990), and (right) circular discrete fractures,from Billaux (1990)

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available (well logs) and were in fact ignored in earlierapproaches. But some are not easily accessible, e.g., thefracture shapes and sizes, and have to be “guessed” orassumed constant in space, so that what is seen on anoutcrop can be taken as representative of what lies kilo-meters below the surface, in the real aquifer. One addi-tional drawback of the Boolean models is that they areextremely difficult to condition to observed data. In otherwords, it is very hard to make the stochastically generated“objects” occupy the exact positions of the observed ones,e.g., along a borehole (sand lenses, fractures, etc.). Someattempts at conditioning have been made (see e.g. An-dersson 1984; Chiles and Marsily 1993; Lantu�joul1997a, b, 2002, for fractured rocks and facies). Condi-tioning to objects without a volume (i.e. a fracture) iseasier than to an object with a volume (i.e. a facies of agiven thickness). Conditioning may be iterative, and is apriori only possible for a finite number of conditioningpoints. For facies, it is therefore necessary to discretizethe facies, e.g. along a borehole. More fundamentally,some scientists question the utility of these models andclaim that continuous Geostatistical models can managejust as well as discontinuous Boolean ones (e.g. Anda etal. 2003), but the authors of the present paper stronglydisagree. Boolean models have made an extremely valu-able contribution in dealing with heterogeneity by intro-ducing the facies concept and addressing the connectivityissue.

Geostatistics fights back:discontinuous facies modelsWith their stochastic shales concept, Haldorsen andChang (1986) in fact started a controversy with the con-tinuous Geostatistical approach. The tenants of Geo-statistics immediately fought back, and introduced dis-continuous Geostatistical models. The concept of dis-junctive Kriging was introduced by Matheron as early as1973 (Matheron 1973, 1976), but not commonly used;Journel (1983), Journel and Isaaks (1984), Journel andAlabert (1990) and Journel and Gomez-Hernandez (1993)developed the Indicator Kriging approach, where an in-dicator can take (at any given point in space) the valuezero or one, depending on whether the point is inside oroutside a given facies. Matheron and the Heresim Groupat the French Institute of Petroleum developed theGaussian Threshold model, where a continuous Gaussianrandom function in space generates a given facies if thefunction value at a point in space falls between twosuccessive prescribed thresholds (Matheron et al. 1987,1988; see also Rivoirard 1994; Chiles and Delfiner 1999;Armstrong et al. 2003, or Marsily et al. 1998, for a reviewof such stochastic models). With the new method, “ob-jects” are also produced as in the Boolean one, but theseobjects are defined by applying a threshold value to theresult of a continuous Geostatistical simulation. This de-scription of the various methods is a drastic simplificationof the complexity of these models and ignores the sig-nificant differences between them. But the final outcomeof the simulations is a series of “facies” in 3-D space, the

shapes of which are indirectly defined by the covariancefunction of the underlying continuous Geostatisticalfunction (indicator or Gaussian).

An alternative is the multiple-point Geostatistical ap-proach developed by Journel, see Strebelle (2002). Theapproach allows simulations of facies maps using condi-tional probabilities that describe the exact geometry of thesurrounding data. In practice, the conditional probabilitiesare calculated from a training image that can be derivedeither from outcrop observation, expert knowledge orgeophysics (see an example in Caers et al. 2003). Themethod makes it possible to simulate complex geometriessuch as channels, meanders or lenses and can preserve therelations between the facies.

An infinite number of simulations of a given aquifercan be generated with these methods. Both the Gaussianthreshold and the multiple-point methods are very pow-erful, and the 3-D facies structures that they generate areconsistent with what geologists have observed on out-crops or inferred from imaging the subsurface, as illus-trated by Fig. 2 (Beucher personal communication 2004).They look realistic, much more so than the Boolean onesand have many features of real geologic structures.Contrary to the Boolean ones, these models can easily beconditioned to observations and thus provide the correctgenerated facies at the location where the true facies hasbeen observed, e.g., along a borehole. When a series offacies distributions in 3-D space has been defined, it isstill necessary to assign properties to each facies (per-meability, porosity, etc). As in the Boolean models, thiscan be done deterministically or by sampling from aprescribed distribution. The fitting of such models isdiscussed below.

Datawise, this approach offers the hydrogeologist whocarries out the study the opportunity to use the geologicinformation that is, in general, available, or that could becollected at little additional cost by examining the geol-ogy (adequate description of the boreholes, better inter-pretation of the geophysics, comparison of the site withexisting well-known sites, etc...). By contrast, these dataare more appreciated and used in the oil industry, where agood geological description is made of each borehole,sometimes with cores taken from the formations, and al-ways including well logging. In that respect, hydrogeol-ogists are clearly lagging way behind the reservoir engi-neers in terms of use of geologic data. The fitting of thefacies covariance function for the truncated Gaussianmodel is thus possible, and this approach is increasinglybeing used. Seismic data can also be used, either to defineseismic facies that are correlated with “real” facies, or asindicators of the proportion of each facies in a verticalprofile (Beucher et al. 1999; Fournier et al. 2002), or as atraining image for the multiple-point approach. Makinguse of additional data is certainly a better way to deal withheterogeneity than trying to extract new results from thesame old data.

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Markov chain modelsIn the same vein as the Indicator facies models, a thirdapproach was developed in the 1990s to address the sameproblem of building facies distributions from the geologicinformation, to describe the heterogeneity. It is based onthe use of transition probability and Markov chains, todescribe facies discontinuous in space. Following Carle etal. (1998), the transition probability from a facies k to afacies j is defined as:

tjk hð Þ¼ Probability that k occurs at x

þ h given that j occurs at x

where x is a point in space and h a lag vector. It can beshown that the transition probability can be related to theindicator cross-variograms gjk(h) of facies j and k throughthe relation:

2gjk hð Þ ¼ pj 2tjk 0ð Þ � tjk hð Þ � tjk �hð Þ where pj = E[Ij(x)] is the volumetric proportion of facies j,assuming stationarity in space, and Ij(x) is the indicatorvariable of facies j, i.e. Ij=1 if point x is inside facies j,and zero otherwise. Note that the transition probability isnot symmetric in h and –h, whereas the cross-variogramis. Defining such a transition probability is, by definition,a Markovian approach: the probability of k occurring atlocation x+h is only dependent of what happens at loca-tion x, and nothing else; by contrast, a non-Markovianapproach would say that the transition probability mayalso depend on what happens at other points in thevicinity of x. For those more familiar with Markovianprocesses in time, the assumption is that the probability ofwhat happens at time t+Dt is entirely defined by the initialcondition, i.e. what happens at t.

Given this definition, the continuous-lag Markov chainmodel simply defines the function which relates thetransition probability tjk(h) to the lag h. This function isassumed exponential with distance in the example givenby Carle et al. (1998), which can be seen as choosing agiven functional form for a set of variograms and cross-variograms in the Indicator approach: tjk hð Þ¼ exp rjk h

.

The coefficients of the exponential, rjk, are the unknownsto be calibrated, called the conditional rates of changefrom category j to category k per unit length of the lagdistance h. Furthermore, this rate of change can be made afunction of the direction of the lag vector h. The exactMarkov chain is in fact written in matrix form, since thereare n2 transition probabilities if there are n facies; these n2

rates are however not independent, and only (n�1)2 co-efficients have to be calibrated on the available data. Inpractice, these coefficients can be directly related tofundamental interpretable properties of a geologic medi-um, such as proportions of each facies, mean length,asymmetry and juxtapositional tendencies. The first stepof the Markov model is thus a calibration of the param-eters on the data, just as the first step in Geostatistics is tocalibrate the variogram models on the data. Then, con-ditional simulations can be generated, as in the indicatoror Gaussian threshold approach.

Despite its simplicity, the Markov model seems to bequite flexible and to better account for spatial cross-cor-relation, such as juxtapositional relationships, e.g. fining-upward tendencies of different facies, than the indicatorapproach. The approach seems also internally consistent,as is the Gaussian threshold method, since the full matrixof the transition probabilities between all facies is con-sistently calibrated simultaneously from the data, whereaswith the Indicator method, the direct and cross-vari-

Fig. 2 Example of a four-faciesGaussian Threshold model us-ing the HERESIM code(Matheron et al. 1987, 1988).a Cross-section; b proportionof each facies over the vertical;c 3-D representation. From(Beucher, personal communi-cation, 2004)

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ograms are generally calibrated one by one and may notalways be consistent. Their authors claim that it is easierto fit to the data than the Indicator variograms, and that itcan produce simulations by the sequential simulationmethod and simulated annealing. It seems that the majoradvantage of the Markov transition probability approachmay be to better insure that a given facies is found closeto another one, as occurs in nature, due to sedimento-logical principles.

Additional presentations of the Markov chain model,and its comparison with the Indicator approach, can befound in Chiles and Delfiner (1999). Additional devel-opment and applications to real facies systems in hydro-geology can be found in Carle and Fogg (1996, 1997),Carle et al. (1998), Fogg et al. (1998, 2000), Weissmannand Fogg (1999), Weissmann et al. (1999, 2002), LaBolleand Fogg (2001).

UpscalingBoth the Boolean models and the facies models requireupscaling. Upscaling is in fact the new terminology foraveraging, a concept already discussed and initially usedto find an equivalent property for an entire aquifer, whenanalytical methods made it necessary to have one singleparameter for an entire system. But here the issue is dif-ferent. The Boolean and facies models are pixel or vauxelmodels, i.e. they represent reality as a series of small cellsor volumes. Typically, several tens or hundreds of mil-lions of cells are generated by the models, still far toomany to use them directly as calculation cells for the flowmodels, even if parallel computing and the continuousincrease in computing power have today raised thenumber of cells acceptable in a model to millions. Fur-thermore, uncertainty evaluation requires that many sto-chastic realizations of the same problem are run, thusprohibiting the treatment of very large problems with theflow models.

Upscaling is the grouping of these elementary cells, towhich a permeability has been assigned, as discussedabove, into larger blocks that define the upscaled grid of aflow model. There are numerous methods of upscaling,generally with the objective that the flow and transportcalculations made at the upscaled level, with the upscaledparameters, provide a calculated solution as close aspossible to the one which could have been calculated, ifthe small scale meshes had been kept, with their originalparameter values. Upscaling is not a problem for porosity(except if the upscaled volume includes aquitards inwhich solute can diffuse), as porosities add up arith-metically, but the real issue is the permeability, see e.g. areview in Renard and Marsily (1997). For infinite media,averaging has been resolved at least theoretically, asshown above (e.g. the geometric mean in 2-D, Matheron1967, or a power mean in 3-D, e.g. Noetinger 1994, 2000;Abramovich and Indelman 1995; Indelman and Rubin1996). But these theoretical results are only valid if rig-orous assumptions are satisfied, on the distribution func-tion of the permeability (in general, log-normal), on thetype of flow (in general, parallel), etc. One particularly

striking example of how wrong such generally unques-tioned assumptions can be is given by Zinn and Harvey(2003), who compare three random permeability fields in2-D, all with nearly identical log-normal distributions andisotropic covariance functions. The fields differ in thepattern by which the high- or low-conductivity regions areconnected: the first one has connected high-conductivitystructures; the second is multivariate log-Gaussian and,hence, has connected structures of intermediate value; andthe third has connected regions of low conductivity. Theauthors find substantially different flow and transportbehaviors in the three different fields, where only the log-Gaussian case behaves as predicted by the stochastictheory. This example stresses again the importance ofconnectivity in dealing with heterogeneity, as pointed outby e.g. Fogg (1986), or Western et al. (2001). Given theseresults, one must consider that upscaling is still a veryimportant research subject, where the detailed propertiesof the elementary cells must be taken into account in amuch more precise way than just by their average values.However, the need to upscale may diminish with theconstant increase in computing power of modern equip-ment. On the other hand, an important issue is the in-creasing use of unstructured grids within the numericalmodels while Geostatistics requires a regular grid (sup-port effect). To preserve the coherence of the Geostatis-tical model, it will still be necessary to transfer the het-erogeneity model onto the unstructured grid using bothupscaling techniques, when the mesh of the numericalgrid is larger than the Geostatistical one (He et al. 2002);downscaling techniques will also be needed when theopposite situation occurs e.g. around flow singularities.Upscaling of geochemical parameters, such as distributioncoefficients, kinetic constants, etc, is an almost virginissue, see e.g. some preliminary work by Pelletier (1997),Glassley et al. (2002) or discussion in Delay and Porel(2003), Carrayrou et al. (2004).

Fitting a facies modelWhen a facies model has been built, either with a Booleanor a Geostatistical approach, and when permeability val-ues have been assigned to each small-scale cell, and up-scaled to the flow model cells, the ideal situation wouldbe that the model is (by chance) perfectly calibrated.Unfortunately, this almost never occurs, and the modelhas to be calibrated. There are two possible ways of doingthis: keep the geometry of the facies, and change thevalues of the permeabilities within the facies. Or alter-natively, keep the values of the permeabilities, andchange the geometry of the facies. The first option isstandard: the facies geometry can be seen as zoning in aclassical inverse (see Carrera, this issue) and the perme-abilities can be adjusted within each facies. However,some recent work on the adjustment of the geometry hasbeen published, called the “gradual deformation” (Hu2000, 2002; Hu et al. 2001a, b). This offers the possibilityof gradually and continuously changing the shape of bothBoolean objects (e.g. the position of a discrete fracture, itssize, etc, or the shape of a sand lens) and Geostatistical

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facies. This is a breakthrough in the calibration of faciesmodels that could be combined with a more classicalinverse on the parameter values.

Genetic/genesis modelsThe next breakthrough in dealing with heterogeneitycame in 1992 with Kolterman and Gorelick. Despite someearly attempts at recreating the spatial distribution of rockproperties by modeling the rock formation processes (e.g.by basin modeling, which will be discussed later), thispaper was the first real attempt at generating sedimentfacies not by Boolean or Geostatistical methods, but byusing a sedimentation model, that of Tetzlaff and Har-baugh (1989), and simulating river flow and sedimenttransport, deposition and erosion over a period of 600,000years, for the San Francisco Bay. This required consid-erable computing power and CPU time, as well as thereconstruction of the climate history of the region over thesame period of time, day by day, and of the evolution ofthe streams, the bay and the movement of the HaywardFault, which crossed the area. Fig. 3 is a cross-section ofthe outcome of this effort, taken from the authors, andcompared with observations.

The authors of this paper believe this is a breakthroughbecause, for the first time, the major effort is put intobuilding a model whose sole purpose is to represent theactual geologic processes that created the sediments; fromthe outcome of this model, the properties of these sedi-ments are derived, in particular their heterogeneity. Ofcourse this is a very challenging task, with immensedifficulties: which process is represented, how to recon-struct the necessary data (e.g. climate records), how canthe embedded uncertainty be quantified, are there randomcomponents in the modeling (there are some randomdecisions made by the model through time in Kolterman’sand Gorelick’s 1992 approach). But the power of thegenetic approach is that the modeling of processes can inprinciple represent features that statistical methods wouldnever have captured. For instance, a meandering deposi-tional environment would never be amenable to a simplebivariate Geostatistical approach, only multiple-pointGeostatistics (still under development, see e.g. Strebelle2002; Krishan and Journel 2003) could approach that, orbivariate Geostatistics where the correlation structure it-self is a correlated random field function of the meanderproperties, see e.g. Carle et al. (1998). Although a firstattempt at comparing a genetic and a Geostatistical ap-

Fig. 3 Observed and simulatedcross-section of the San Fran-cisco Bay, using the geneticapproach. From Kolterman andGorelick (1992)

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proach on the same site has been published recently(Teles et al. 2004, see below), there is still much work tobe done to compare the various genetic and Geostatisticalmodels; this is a very important area of new research.Another striking example can be seen in Fig. 3: Kolter-man and Gorelick introduced the movement of the Hay-ward fault in their model; the result is that the mouth ofthe sediment outlet in the bay (where the coarse sedimentsaccumulate) moves from North to South during the sim-ulation, as can be seen on the North–South cross-section.This would never have been found in a statistical ap-proach.

It is worth noting that very early attempts at devel-oping genetic models had been launched by Matheron inthe late 1960s, he called them at the time “the Ambar-zoumian processes,” see e.g. Matheron (1969) or Jacodand Joathon (1971, 1972). But they had not been used inhydrogeology. A number of genetic models have beendeveloped since, mostly for the oil industry, where sedi-ment transport and deposition are approximated by adiffusion equation, which can be shown to approximatelyrepresent sediment transport, in a marine or deltaic en-vironment; see e.g. Paola et al. (1992), Heller and Paola(1992), or Grandjeon (1996), Grandjeon et al. (1998),Grandjeon and Joseph (1999), Doliguez et al. (1999),Euzen et al. (2004), or Quiquerez et al. (2000). Webb(1995), Webb and Anderson (1996), have developed asediment model, partly empirical, partly genetic, to re-present fluvial deposits. Kolterman and Gorelick (1996)present a review of such models. Teles et al. (2001) havebuilt a genetic/genesis model of fluvial sedimentation,representing both meandering and braiding patterns,which uses empirical rules to represent sediment transportand erosion. The model does not represent water flow andsediment transport per se, but rather the result of the al-luvial processes by using geometric rules. It is consideredthat these geometric empirical rules embed the fluvialprocesses that create the structures. Some of these geo-metric rules can be seen as in part Boolean. A series ofdifferent facies are thus assembled, see e.g. Fig. 4.

As for Boolean or Geostatistical facies models, eachfacies is then assigned a property value, e.g. a perme-ability, to be introduced, after upscaling, into a flowmodel. Teles et al. (2004) showed that the genetic modelis able to represent the connectivity of buried channels inthe alluvium (or the role of barriers of such channels,when they are filled with clay), whereas the bivariateGeostatistical approach ignores such features, with sig-nificant influence on the prediction of solute transport. Sofar, these genetic models have not been conditioned tohard data. They cannot generate a facies at the locationwhere it has been observed or, at best, with great diffi-culty. It is conceivable to combine multiple-point Geo-statistics and genetic models where the outcome of thelatter would be used as training images for the former;other ideas have also been suggested, e.g. to prevent anon-observed facies from being deposited at a given lo-cation; this approach has been used to generate condi-tional Boolean fracture sets by not keeping random

fractures which do not appear at the conditioning points,see e.g. Andersson (1984), Chiles and Marsily (1993).

At a different scale, basin models can also be consid-ered as genetic models: Betheke (1985), Ge and Garven(1992), Garven (1995), Burrus (1997), Person et al. (1996,2000). Indeed, they represent the succession of sedimentdeposition, they can include high resolution sedimenttypes for a given facies and they explicitly model com-paction and porosity–permeability reduction as a functionof the effective stress. They also represent thermal evo-lution, organic matter maturation and sometimes, sometemperature-dependent diagenetic processes (porosityclogging). The 3-D Paris basin model is perhaps one ofthe more detailed examples of this type. It is based onmore than 1,100 litho-stratigraphic well data (Gon�alv�set al. 2004a), see Fig. 5.

Some genetic models try to produce faults as a result oftectonic stresses, see e.g. Quiblier et al. (1980), Renshawand Pollard (1994, 1995), Taylor et al. (1999), Bai andPollard (2000), Wu and Pollard (2002), Person et al.(2000), Revil and Cathles (2002). Karstic systems can alsobe studied with a genetic approach, by trying to model thecarbonate dissolution mechanism; a number of attemptshave been made in this area, see Bakalowicz (this issue).So far, the genetic approach has been focused on detritalor alluvial sediments rather than on calcareous deposits,although some attempts have been made to represent theevolution of limestone (Grandjeon and Joseph 1999). Butmuch work remains to be done on various types of rocks.

Does this approach work? The answer is yes, and it isdeveloping rapidly. In terms of data, it requires a solidscrutiny and interpretation of the litho-stratigraphic re-cord. Sedimentologists have to study, in detail, theavailable sediment samples, sometimes to date them(with14C on organic debris for recent alluvial sediments,or with micropaleontology, isotopic geochronology...), toinvestigate the depositional environment, and interpolatethe information in space keeping a genetic concept inmind. For the Aube valley, as described in Teles et al.(2004), for instance, a series of 44 auger holes was suf-ficient to build a reasonable genetic model of the plain,bearing in mind the existence of studies of the successionof climate periods during the Holocene, where each pe-riod is characterized by a type of sediment, a depositionpattern (braided or meandering), which is valid for a largearea, not only for a given alluvial plain. Generic or re-gional geologic knowledge is thus available, and vastlyincreases the value of the collected data. An example ofthe value of such knowledge can be found in Herweijer(1997, 2004). He studied the famous MADE experimentalsite in the Mississippi valley, where very well character-ized tracer tests had been made (see e.g. Harvey andGorelick 2000). His approach was to construct a con-ceptual model of the stratigraphy of the site, based on theavailable geologic observations and the results of thepumping tests. Previous authors had tried to use theGeostatistical approach at MADE, and to infer the spatialcovariance structure of the alluvial sediments in order todescribe its heterogeneity, without paying much attention

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to the geology (“all this is alluvial material”). WhatHerweijer (1977, 2004) observed is that within the allu-vium, two successive sedimentation structures existed;one from a braided stream, one from a meanderingstream. Geostatistics cannot easily incorporate such in-formation. Calibrating a unique covariance structure forthe entire thickness of the alluvial aquifer was just liketrying to find a compromise between apples and oranges,whereas incorporating some genetic knowledge woulddefinitely have improved the “dealing with heterogene-ity,” even with the Geostatistical approach and without agenetic model. An alternative approach could have beento use “variogram maps” (or covariances) which can in-corporate anisotropy and statistical non homogeneity.Much more complex structures than those achievable witha simple variogram can be obtained in this way, see e.g.Anguy et al. (2001, 2003).

Very recently, ANDRA, the French Nuclear WasteDisposal Agency, presented some on-going work on theconstruction of a flow and transport model for two car-bonate aquifers, in the Dogger and Oxfordian of the Parisbasin, where these two aquifers surround a clay layerwhose feasibility as a repository is being studied. Themodel is to be used to represent the transport to the outletsof radionuclides that would eventually leak out of the clayformation. To calibrate the flow model, ANDRA and itscontractor, the French Institute of Petroleum, used thegenetic model DIONISOS (Grandjeon 1996; Euzen et al.2004), which solves the diffusion equation to representtransport and deposition of sediments, to define theproperties of the two aquifers. This work will be pub-lished in 2005 (Houel et al. 2005). Genetic models are nolonger only theoretical research tools for academics!

Fig. 4 An example of sedimentpattern in the Rh�ne Valley inFrance, as created by a geneticmodel (Teles et al. 2001).a Plan view of the sediments inthe plain. Meshes are200�200 m, the alluvial domainis 20 km long. b Cross section(North-West–South-East atVilleurbanne, as marked on a.The colors show the sedimentunits ages from 15,000 years topresent, as given in b. Thelithology of each sediment isassociated with each episode.From Teles et al. (2001)

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Different genetic models may be more appropriate fordifferent scales. Those representing small-scale processes,such as Webb (1995) or Teles et al. (2001, 2004) arebetter suited to account for stratigraphic details that arenot generally recorded in classical hydrogeologic studies,e.g. an alluvial meandering plain over a small distance.On the contrary, the genetic models based on the diffusionequation to represent transport and deposition are bettersuited for large scale problems, like marine sedimenta-tion, alluvial fans, even marine carbonates. The Kolter-man and Gorelick (1992) approach may lie in between.

The future

When does heterogeneity matter?The first question is: “when does heterogeneity matter?”As has been shown earlier, methods to deal with hetero-geneity exist and will certainly be improved in the future,but they generally need additional data, on top of theusual ones, or at the very least, increased scrutiny of thegeologic data; they also need a good deal of thinking...!An attempt to list the areas where “dealing with hetero-geneity” is obligatory, and those where ignoring it will dojust as well is given below.

Ignoring heterogeneityIt is impossible to completely ignore heterogeneity, butone can use methods that “deal” with heterogeneity au-tomatically and allow the user to ignore it. A simple ex-ample is permeability measurements. Any hydrogeologistknows that if he/she needs to know the permeability of anaquifer, making measurements on core samples will notbe a good solution. One should rather perform a pumpingtest, of a reasonable duration, and extract from it theaverage permeability of the aquifer, which can then beused to make predictions, or build a model. It is the “tool”(the pumping test) that “deals” with the heterogeneity,and automatically produces a reasonable and useful valuefor practical purposes.

Again and again, it has been found that if a watershedmodel is built, to represent surface and subsurface flow,making local measurements of the soil permeability, oncores or even with double-ring infiltrometers, produces apermeability value much too low to represent the ob-served behavior of the watershed (flow, water levels,...see e.g. Carluer and Marsily 2004). The same observationhas been made e.g. on a deep sandstone aquifer, the PierreShale, see Neuzil (1994). Is this due to an insufficientnumber of measurements? To an inappropriate measure-ment system? To unidentified processes acting at a dif-ferent scale than that of the measurements? If the flow in

Fig. 5 Basin modeling of the Paris basin: calculated horizontalpermeability and storativity distributions of three aquifer layers atpresent. The heterogeneity is the result of the sediment facies de-scription, kriged from the borehole lithostratigraphic data, definedby the percentage of three poles: 1. clay; 2. limestone; 3. sand. Themaps include the effect of the compaction history with different

compaction rules for each pole, and different porosity–permeabilityrelations. The calculated permeability in each cell is the geometricweighted average of the permeability of each pole. The depth of thecenter of each layer is given on the right. From Gon�alv�s et al.(2004b)

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the aquifer is to be modeled accurately, and the onlyavailable information is a set of local values measured oncores, then it is indeed necessary to “deal with hetero-geneity” and derive a method that can construct the spa-tial pattern of the local permeabilities, and identify theconnectivity of special features and channels that governthe flow, which are, it is believed, not accessible at thelocal measurements scale.

But, in general, when the appropriate measurementsare available at the appropriate scale, then it is indeedpossible to ignore, to some extent, heterogeneity and usethe global measurements directly to make predictions.This would apply first to flow, not to transport. Theconclusion is thus that where regional aquifer manage-ment is concerned, for optimization of well discharge,estimates of available resources, then a large-scale viewof the heterogeneity is sufficient, and aquifer propertyestimates can be derived by means of the usual tools:pumping tests and aquifer model calibration by trial-and-error or inverse procedures. Of course, as emphasizedearlier, selecting an inverse which adopts a given pa-rameterization of the permeability field already amountsto making some assumption on the nature of the hetero-geneity (e.g. zoning, Pilot Points based on Kriging, etc),but again, the tool (inverse) will “deal” with heteroge-neity, and most likely two different inverses will, in theend, provide predictions that are not vastly different, evenif the permeability fields are far from identical. Just togive an example, Besbes et al. (2003), see also OSS(2003), recently built a flow model of the multilayeraquifer system of northern Sahara, extending over threecountries (Algeria, Libya, Tunisia) and an area of morethan 1 million km2, probably the largest flow model everbuilt. The objective was to agree on a multinational policyof exploiting this huge aquifer, which receives little re-charge and is currently being mined by increasing with-drawal. How long can the situation last, are there riskslinked with over-exploitation, is pumping in one countryaffecting its neighbors? The distance between “adjacent”wells is sometimes several hundred kilometers; even if thegeology is well-known in terms of horizons, local het-erogeneity is obviously not a concern. Average trans-missivity, vertical permeability of aquitards, local dis-charge into the sea and into playas, present level ofwithdrawal, present recharge, and finally values of thespecific yield in the unconfined sections of the aquifersare the most important parameters that will govern thelong-term response. The precise nature of the boundaryconditions, present and future recharge (with climatechanges) are also very important issues to be resolved insuch large aquifers (see e.g. Jost et al. 2004). Whereshould the money be spent? Most likely on large scalereconnaissance: geology, pumping tests when available orfeasible, collection of seismic surveys (made for oil ex-ploration) that give the thickness of each layer, use ofGeostatistics (Kriging) to interpolate the scarce data inspace, and then model calibration using the 50 years ofexisting records. Depending on which lithologies arepresent, large-scale sedimentation models or basin models

could be used to determine the general trend in theproperties of the system, after a state-of-the-art hydros-tratigraphic conceptual model has been built. This wouldbe the recommendation. There is a small incentive tobracket the uncertainty in this example. If a withdrawallimit is given to a particular country, not to jeopardizeanother country’s resources, is this known with a 5%, or50% bracket? To address that issue, it is necessary torecognize that the basic parameters of the model are un-certain, and to do one of the following:

i Simple sensitivity study. Arbitrarily assume that thecalibrated parameters have a given estimated range ofuncertainty (e.g. 20%, 100%...) everywhere and runthe model with different sets of parameters, and see ifthe consequences are different.

ii Post-optimal residual sensitivity study. This can bedone if a stochastic inverse has been run, which gives,after calibration and linearization, the residual un-certainty on the parameters, which can be different ineach area of the aquifer. This defines the range ofparameter uncertainty, which was “guessed” and as-sumed uniform in approach (i). See e.g. Cooley(1997).

iii Full-scale post-optimal uncertainty analysis using astochastic inverse and Monte-Carlo simulations, seee.g. Ramarao et al. (1995), Lavenue et al. (1995).

In this huge aquifer, there may be some local problems,however, where local heterogeneity can be important.One is the exact situation around playas. At present, theplayas are outlets for the aquifers. In the future, it hasbeen shown by the huge model that the lowering of thepiezometric surface will dry up this flux: the playas,containing salt brines, may start salt water infiltration intothe aquifers, when they receive storm flow, or return flowfrom irrigation. Predicting the rate of salinity increase inthe aquifers, the time needed for near-by wells to becomebrackish, or possibly engineering measures to prevent thisfrom occurring, will require a better understanding of theflow and transport at a small scale around the playas, thustaking local heterogeneity into account. Another instanceis the vertical upward leakage from deep brackish aquifersinto superficial freshwater aquifers. Assigning a singlepermeability to one aquitard is likely to be much toosimplistic, and a detailed study of the aquitard heteroge-neity would be necessary. But this is unlikely to happen inpractice today, as (i) it would be quite expensive; (ii)there is no well established method so far to deal withaquitard heterogeneity, and (iii) the bad consequenceswould be felt in tens of years, thus reducing the incentiveto assemble funds. However, this problem of aquitardheterogeneity should be considered as an area for futurework, see below.

Taking heterogeneity explicitly into accountOne opposite example when “dealing with heterogeneity”is obligatory is in transport. Consider a remediation case,where surfactants have to be injected to dissolve and re-

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cover DNAPL’s in a superficial aquifer. A large amountof money will be spent on injecting, recovering andtreating the water. Having an optimally functioning sys-tem is economical, and spending money to decipher theheterogeneity will pay off, as this will make it possible tooptimally locate the wells, adjust the pumping rates, andinsure that the system is hydraulically closed and that nosurfactant coupled with DNAPL’s will leak out of it. Thescale of the problem is meters, at most hundreds of me-ters. What can be done? It would be useless to apply the“ensemble stochastic approach,” and try to characterizethe heterogeneity by a covariance function: in order toapply the stochastic theories, some averaging must bedone. The pollution plume becomes “ergodic” when thetravel distance is several times (on the order of 10) thecorrelation length. One is not interested in an average orensemble behavior, but in a detailed local description ofthe real system. One needs imaging of the underground.Any geophysical imaging technique (radar, electric pan-els, electromagnetic surveys, NMR, etc, see Gu�rin, thisissue) will be the starting point. An initial borehole surveywill then describe the lithology, and provide clues tounderstand the structure of the aquifer, together with thegeophysics. The technique needed here to describe theheterogeneity is definitely a facies model, one needs alithological description (nature and shapes of facies), asprecise as possible, on the basis of which properties willbe assigned. This is not just permeability, but porosity,clay content, sorption parameters, etc, all properties thatare estimated facies by facies, and extrapolated in spaceto represent the site. Modeling must be very detailed, andupscaling may not be needed, what is still a challenge isto determine the number of samples that are needed perfacies, their size, measurement method (on samples or insitu) and the grid size that can represent them. A first cutof the facies model would be obtained from a geneticmodel. A state-of-the-art geologic characterization of thesite must first be completed, as commonly made in pe-troleum work but less often in hydrogeology. Then, agenetic model (such as Kolterman and Gorelick 1992;Webb 1995; or Teles 2001, 2004) should be built. It willdescribe which different stratigraphic units are present atthe site, the likelihood of erosion processes, the type ofsedimentation pattern (meandering or braided stream, if inan alluvial deposit, etc). Since these models are presentlynot conditioned to the data, it will be necessary to go to aGeostatistical facies model, not a Boolean one which isdifficult to condition. How to “fit” the Geostatistical fa-cies model to the genetic one is still unclear (this dis-cussion is about the future!), but the basic idea would beto select the covariances (in case of bivariate Geostatis-tics) or the multiple-point pattern by fitting them on theimages produced by the genetic model. Another approachto fitting a facies model on the data is the transitionprobability and Markov chain method, as briefly de-scribed above, see e.g. Carle and Fogg (1997), Carle et al.(1998), Weissmann and Fogg (1999), Weissmann et al.(1999). The final facies model would then be conditionedon all the observation points, and the geophysics. The

fine-tuning of the model would be made by calibration onexperiments made on the site (pumping tests, see e.g. anexample of 3-D interpretation of a pumping test with aninverse in Lavenue and Marsily 2001). Preliminary tracertests would also be used to calibrate the model. Multiplerealization models may be necessary to provide an esti-mate of the uncertainty of the answers, e.g. the duration ofthe remediation period.

The grey areas when dealing with heterogeneityIn the two above examples, one real (Sahara) and onesemi-real (DNAPL), the answer to the question “does oneneed to address the heterogeneity issue?” is a clear-cut“no” and a “yes,” the difference is, in part, related to thedifference in scale of the two problems. But there are greyareas, where the answer is not that simple. There is indeedno universal answer to the method of treatment of het-erogeneity. The transport problems will in general requirethat more attention be paid to heterogeneities than theflow problems since for transport, connectivity is mostimportant. The scale of the problem must also be con-sidered: a small-scale problem requires detailed under-standing of the heterogeneous pathways, whereas for alarge-scale problem, flow and transport will be averagedby the crossing of several heterogeneous structures, andan “ensemble” average will emerge (the behavior be-comes ergodic, in the stochastic language), averages canbe used, or averaging tools, such as the covariancefunction.

Finally, the situation is also influenced by the mediumin which the study is performed. There is a plethoricamount of work in the recent literature that concludes onthe non-ergodic behavior of fractured rocks regardingboth flow and transport. This would mean that heteroge-neity (in this case mostly its influence on connectivity)should be carefully accounted for. On the other hand,densely fractured aquifers have been managed success-fully for a long time, assuming the continuity of flow and,to some extent, a pseudo-homogenization scale. For suchmedia, there is no clear answer except for theoreticalcases or synthetic cases. Further work is needed to clas-sify the behavior of fractured rocks with respect totopologic parameters such as connectivity, fracture lengthdistribution, mass density of the network, etc. See initialattempts in e.g. Aupepin et al. (2001), Rivard and Delay(2004).

The type of answer to be provided is also a criterion.Suppose that a leachate plume from a landfill is to bestudied. If the objective is to determine the flux to a river,the detailed pathway that the plume follows to reach thatriver is not important, it is the rate of mass transfer thatmatters. Average properties can be used. If, on the con-trary, the question is the maximum concentration that canbe attained in a downstream well, the pathways, the di-lution and dispersion along that pathway, etc, require amore detailed analysis of the heterogeneity at the site.Finally, the cost of data collection must also be consid-ered: deciding to launch a more sophisticated study and

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collect the necessary data requires that a realistic budgetis available.

This question pertains to an on-going controversy inhydrogeology, the parsimony issue, which is related tothis article through the selection of the method to dealwith heterogeneity: should simple and robust models (e.g.equivalent homogeneous media, continuous Geostatistics,zoning...) be preferred, as much as possible, or should onerather try to represent heterogeneity with the more com-plex methods? The authors of this paper think that usingsimple models has been taken too far in hydrogeology andis no longer justified, powerful models exist and computertime is no longer an issue. Making simplifying assump-tions such as averaging ignores the connectivity issue;treating a problem in 2-D may not be justified if the flowand transport processes include a 3-D component (e.g.vertical heterogeneity of the aquifer, density effects, etc).Even if the 3-D structure of the system is poorly known,making “educated guesses” on this 3-D structure may beless erroneous than ignoring the 3-D structure, and usingaverages. Problems like scale-dependent dispersion, ex-treme tailings, etc, are in fact the results of homogeni-zation rules that do not apply at the scale of interest, andthus do not allow the use of the classical governingequations, which assume ergodic behavior. Many inef-fective “pump and treat” operations are due to this mis-guided approach. This is however not to say that simplemodels do not have a role to play, in many instances: seee.g. a discussion on this issue in Voss (1998). But todetermine if a simple model is a satisfactory answer to agiven problem, there is no alternative than to compare itsoutput with that of a more complex one!

Areas for future workIn the following, attempts are made to indicate wherefuture research efforts should be concentrated in orderbetter to handle heterogeneity.

1. Transport problems generally require a better under-standing of heterogeneity than flow problems. Aver-age properties as given by a well test include theheterogeneities, for flow calculations, but do not de-scribe adequately the transport. And, in most cases,the time required for a tracer test to provide repre-sentative answers is prohibitive. The type of tracerexperiments carried out at Borden in Canada, CapeCod in Massachusetts or MADE in the Mississippivalley is scientifically of great interest for testing as-sumptions, models and methods, but too costly to beused regularly for transport problems. Something elseis needed.

2. Even for transport problems, the approaches maydiffer if the source term is diffused or at one point. Fordiffuse sources, e.g. nitrate or pesticide contamina-tion, an average description of the medium may beenough; what is of interest here is the mass conser-vation, possible decay by biological processes (linkedto the average velocity and transfer time), and generalflow direction. The major problem is rather the source

term, the amount used by the farmers, and the mod-eling of the consumption/release/decay of the prod-ucts in the plant root zone. The exact concentration ina given well downstream need not be precise, it is theorder of magnitude with respect to the norms thatmatters. This would apply to relatively well-definedaquifers with simple structures, like the Chalk aquiferin northern France and Great Britain. Aquifers withcomplex multilayered structures or with interbeddedaquitard lenses may require more refined treatment, ifbreakthrough concentrations at different horizonsneed to be predicted, see e.g. Weissmann et al. (2002).

3. Two-phase flow is probably more demanding thansolute transport, in terms of describing heterogeneity(see Noetinger, this issue). This is why many methodssummarized above were developed by the oil industry,for predicting oil recovery and water-cut. This wouldalso apply, in hydrogeology, for NAPL and DNAPLstudies, and for the unsaturated zone. Many moreparameters than just permeability and porosity arerequired (relative permeabilities, wetability, capillarypressure, sorption, etc). All these must be treated si-multaneously, since they are correlated. One of thefacies approaches seems the best option. In soil sci-ence, genetic models should be developed to betterdescribe the structure, in terms of facies or horizons,of the unsaturated zone. For sea water intrusion, or forsoil salinization by capillary rise, although transport isnot multiphase, the problem is similar with strongdependence on heterogeneity. Heterogeneity is also afundamental issue for superficial hydrological pro-cesses occurring in the vadose zone (infiltration, sat-uration, overland run-off) and their description. It haslong been recognized that the simple Horton modeldoes not fit all observations, but it is still widely usedfor lack of other simple models. Heterogeneity is animportant aspect to consider since it can explain someof the observations such as “saturation from below.”This phenomenon might be due to local hetero-geneities that lead to local perched saturated zonesand, in the end, to overland flow.

4. Dealing with heterogeneity is a three-part issue: theproblem to solve/the methods to use and models tobuild/the data to collect, in that order, and the relevantscales of each of these. But very often the problem isill-posed: the difficulty is there, the data are there, andthe hydrogeologist is asked to select and use the bestmethod to treat the data and solve the problem! Ingeneral, however, there is some flexibility in design-ing an additional data collection phase, which pro-vides a measure of freedom.

5. Many research programs have, at times, collectedlarge amounts of data without any model in mind, orelse an inappropriate one. In general, these data areuseless, even leading to the wrong type of model. Away to remedy that is to develop experimental siteswhere measurements and experiments are set up andinterpreted by joint research groups involved withboth experimental and modeling work. This was the

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case for the famous experiments in e.g. Borden, CapeCod, MADE, Mirror Lake in North America, theNeckar alluvial site in Germany, or the fractured rocksite of Fanay-Aug�res in France. One may expect abetter bridging because experiments are built to feedmodels and models evolve to account for new mea-surements.

6. Subsurface imaging is obviously an asset, as it canhelp describe the geometry of the heterogeneoussystem. Geophysics could be used more systemati-cally; so far, good examples of a successful use ofgeophysical images in the treatment of practicalproblems where the heterogeneity plays a role arelacking. Additional geological reasoning, as requiredby the genetic approach, has the merit of putting theproblem into a broader context, and to bring in sour-ces of information relevant to a site, but generic innature: e.g. succession of climate states, from whichthe morphology and nature of the sediments can bederived. On the North American continent, fluvio-glacial sediments are probably the first target wheregenetic studies would be of interest, see e.g. Anderson(1989).

7. Except for geophysical imaging, the range of explo-ration methods to display heterogeneity is poor. Mostof the standard tests are integrating tests, not explor-atory tests (pumping tests, tracer tests, etc). Newmethods must be developed. In the detailed fieldstudies for transport (e.g. Borden, Cape Cod, MADE,etc), one interesting tool was the borehole flow meterused, to assess the local permeability variations of theformation. But this tool was applied in a Geostatisticalframework, to assess the spatial variability (covari-ance) over the vertical and the horizontal of theaquifer, treated as a continuous stochastic field. It maybe possible to use the same tool in a facies-type ofapproach, to identify the local properties of each fa-cies. Tsang and Doughty (2003) used temperature orsalinity changes by logging boreholes which they hadflushed and filled with freshwater to detect flowingfractures. A related example of assigning facies per-meability from the results of in situ tests is given inReis et al. (2000). They built a Geostatistical faciesmodel of a petroleum reservoir. They used this modelat the finest scale grid (without upscaling) to representradial flow around a well and thus interpret well testsin four different wells. Rather than interpreting eachwell test independently with a standard Theis ap-proach, which would provide four different integratedtransmissivities, they decided to identify the individ-ual permeability of each facies, assumed to haveidentical properties at each well location, so that thefour tests could be simulated with the same perme-ability values assigned to each facies. Of course, ateach well, the facies distribution was different, butone nice feature of the Geostatistical facies model isthat it can easily be conditioned at the wells, thereforeat each tested well, the exact vertical distribution ofthe observed facies was represented in the radial flow

model. One needs to rethink the interpretation of the“classical” data in terms of describing the heteroge-neity. To quote C.V. Theis (1967): “I consider itcertain that we need a new conceptual model, con-taining the known heterogeneities of natural aquifers,to explain the phenomenon of transport in groundwater.” New tests may have to be invented, that em-phasize the role of heterogeneity rather than averagingit (e.g. injection tests in packed sections of a wellrather than a full well pumping test, over the wholethickness of the aquifer). Today, such tests are notthought to be very useful because there is no way toexploit their results. But if one thinks in terms of afacies model, which has been built for the site, andneeds parameters for each facies, then there are tens oftests that could be designed and interpreted with thefacies model, just as Reis et al. (2000) did. The ex-perimental sites whose interest was emphasized inSection 4 above should be used for that kind of de-velopment. Major breakthrough in experimental workhas already been achieved in many areas in nuclearwaste Underground Research Laboratories, such asLac du Bonnet, Canada, Yucca Mountain, US,Grimsel and Mont Terri, Switzerland, Stripa and�sp�, Sweden, Mol, Belgium, Asse and Gorleben inGermany, Tournemire in France, etc, thanks toavailable funding, creative ideas, and the need tocharacterize in detail the formations, even if the de-sign of new measurement methods was not the pri-mary objective.

8. There is a need to create a catalog of aquifer andaquitard properties and descriptions, with a summaryof the methods used in the reconnaissance, and themodeling of the site (as well as the purpose of thestudy). For one thing, there is a great similarity (evenif there are differences) in geologic objects. In tec-tonics, the study of the Himalayas has a lot to learnfrom the results of the study of the Alps or RockyMountains, and vice versa, as the processes at workare the same. It should be the same for hydrogeology.The structure of the sediments in a river along thecoast of Peru has probably a lot of features in commonwith a similar river in the Pyrenees. This becomesparticularly true when considered in genetic terms, theprocesses are the same everywhere, it is only the localconditions (climate, geometry, geology...) that makethe two aquifers different. But that difference is sec-ondary to describing the shapes, the texture, andmostly the methods used for studying the “objects.”Furthermore, one study in Texas may provide “de-fault” values for a parameter in France that was notmeasured. Take the case for instance of a local dis-persivity value in a given facies. If a good descriptionof that facies is available (grain size analysis, type ofsediment, etc), and if there is a similar facies at adifferent site, one may start by assuming that theTexan value is a first guess for the new case. This“default” value would be all the more credible if, byassembling the catalog, similar facies are found to

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have similar properties at different locations. The linkbetween grain size and hydraulic properties needs tobe re-evaluated and used (note that German hydro-geologists make much more use of grain size data thanany others). The creation of a catalog has already beenadvocated (e.g. Voss, personal communication 1988;Dagan 2002), but never put into practice. One reasonmay be the cost, and the exact focus of the catalog,which should be clear: if integrated values are to betransposed from site to site, this will never work (e.g.a transmissivity measured in an aquifer in Texas willbe of little use for a case study in France). But if faciesare considered, sedimentary structures, rock history,methods used to characterize sites, then the need forthe catalog may be more evident. One would need anInternational Organization like UNESCO to developsuch a project on a world-wide basis.

9. The upscaling problem may need additional work, seeNoetinger (this issue). The problem that Zinn andHarvey (2003) emphasized is a major one. When onedoes upscaling, the small-scale boxes that need to be“averaged” into a big box may have internal structuresthat do not permit them to be treated as a “homoge-neous” small-scale medium defined by one singlevalue (of permeability, even with anisotropy), asshown by these authors. The small-scale boxes havethemselves complex internal structures, with differentconnectivity properties. In other words, there is anested scale problem to consider. The upscaling in thevertical direction also needs more attention, the scaleof variability is finer and the measurements may notalways provide an adequate description of the truestructure in the vertical direction (there is no easymethod to measure vertical permeability in bore-holes). There is a need to look again at upscaling.Until the large-scale permeability of an aquifer can bereconstructed from small-scale measurements, therewill be a credibility problem for hydrogeology.

10. The connectivity issue is of utmost importance, notjust for upscaling, but for correct flow and transportpredictions. Just to give an example, the Rhine Valleyaquifer in France has been heavily polluted by saltbrines in the area upstream from Mulhouse byleaching from potash mine tailings. This started in theearly 1900s, and in the 1980s, the salt plume hadmoved several kilometers downstream, and was get-ting closer to the water supply wells of the city ofMulhouse. Detailed surveys had been made of theposition of the plume, its spreading and its progres-sion, until it was decided to move the wells laterally toa non-polluted area. Alas! When the new wells weredrilled, it was discovered that a second plume, totallyundetected, and connected to the first one only in thesource area, was also present, and that the selected“non-polluted” location was no better than the initialone! In this case, the distribution and connectivity ofhigh-permeability facies should have been studied; inother cases, it is the low-permeability barriers con-nectivity that would matter. Is the facies approach the

best one to tackle this issue? Or is percolation theorythe right tool? Or the fractal approach? Again, there isa scale problem; percolation thresholds are valid forensemble averages, i.e. large-scale problems, anddetailed facies description is perhaps better suited forthe small scale. This is a very important researchtopic. The multiple-point geostatistical approach maybe able to respect the connectivity pattern of a trainingimage; the definition of a connectivity function whichcan be imposed on a simulation by simulated an-nealing may be another, see e.g. Allard (1994) orWestern et al. (2001).

11. Heterogeneity of low-permeability layers and aqui-tards is almost an untouched subject, although veryrelevant for waste storage or brine or pollutant leakagethrough aquitards to aquifers. Heterogeneity can belinked to sedimentological processes, such as thosestudied e.g. by Ritzi et al. (1995) on till and lacustrineclays of glacio-fluvial aquifers; it can also be linkedwith tectonic processes in the underlying or overlyingaquifers. It has indeed frequently been observed, whena multilayered aquifer model is calibrated, that theleakage factor needs to be increased in the vicinity offaults, see e.g. Castro et al. (1998). Other factorsshould also be considered.

12. In a genetic approach, rock properties evolutionthrough time due to geomorphological and mechani-cal processes, to weathering at the surface, or to dis-solutions/precipitation by geochemical processes atdepth are still almost virgin topics, where predictivemodeling is concerned.

13. It has been emphasized here that hydrogeology has sofar failed to adequately apply fundamental geologicmethods and principles to aquifer characterization. Inthe future, this will be an important field of activity.Rather than bringing in merely “static” geologicalcharacterization, it seems appealing to considermodeling approaches used in other Earth sciencesfields such as geomorphology, marine or continentalsedimentology to generate particular landforms andfeatures. One interesting approach was developed byNiemann et al. (2003) to simulate topographies fromsparse elevation data. Their model embeds the effectsof evolutionary processes (tectonic uplift, fluvial in-cision and hillslope erosion). The simulated surface isiteratively adapted to observation points by spatiallychanging the erodability field. This leads to surfacesthat are consistent with the heterogeneity, roughnessand drainage properties of fluvially-eroded landscapesas well as being in exact agreement with observationpoints. Following this approach, one could considerembedding empirical laws describing the 3-D struc-ture of sediments or features in Geostatistical methodsor interpolators.

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Conclusion

In conclusion, the authors of this paper believe that thefuture of “dealing with heterogeneity” in hydrogeologydepends largely on a conscious decision to better char-acterize, describe and model the geology of the sites ofinterest, similar to what has been done for the last 25years in the oil industry. In many instances, the homog-enization or averaging approach that has been commonlyused so far in hydrogeology has shown its limits. Detailedstudies are needed of outcropping analogs, catalogs ofwell-characterized study sites, improved geophysical im-aging at the depth of interest with dedicated methods, asthe seismic methods used in the petroleum industry arepoorly suited to superficial aquifers. There are already aseries of facies models that seem adequate to model theobserved structures, they should combine genetic ap-proaches and Geostatistical, Boolean or Markovianmethods. There is certainly room for improvement inthese methods, but what is most needed at this time is athorough inter-comparison of these facies modeling ap-proaches (and of simpler models too) on a set of well-characterized study sites, where the geologic descriptionis properly done, and where transport experiments are run,like in the old Borden, Cape Cod, MADE sites. When asystematic comparison of methods is done, as for theinverse in the 1990s (Zimmerman et al. 1998), the out-come can really show the possibilities and limits of eachapproach, whereas, at present, most authors have shownhow good their own method is, on theoretical or realexamples, but have not convincingly shown how itcompares with others.

A second area of development is to derive methods toidentify the material properties at the scale of the faciesstructures. All the facies need to be identified, from thehighly permeable to the low, including the intermediatemixed texture facies such as sandy mud, silty clay, etc,that are presently often ignored. The methods to identifythe facies properties can go from newly designed in-situtests at the proper scale, to correlating these propertieswith indirect data like grain size, geophysical properties(electric, mechanic, magnetic, etc) or a combination of allthese (joint geophysical inversion). On that aspect, thegeophysicists need to start thinking about defining theunderground by complex facies models, with which theywould try to identify by inversion the physical properties,and stop offering “layered cake” structures where theyhave identified equivalent properties that are no longerwanted.

A third area of development is to derive techniques toassess the connectivity of facies. The geometric propertiesof the facies are the key ingredients, which can resultfrom geometric assumptions (directional transition prob-abilities, length distribution, etc). But they can also resultfrom field testing, by interference tests, environmentaltracer studies, temperature measurements, etc. As wasshown here, the upscaling issue, which often remains anecessity, is very much dependent on the correct assess-ment of the connectivity of facies.

Acknowledgments The authors wish to express their gratitude toPr. Graham Fogg and Pr. Gedeon Dagan for their detailed reviewon an earlier draft of this paper, which helped them immensely toimprove the text. A third anonymous reviewer is also thanked forhis/her helpful and wise comments. The comments of two col-leagues, Jean-Paul Chiles (Paris School of Mines) and PierreDelfiner (Total), are also gratefully acknowledged. Finally, theauthors wish to thank the Editor in Chief, Dr. Clifford Voss, forinviting them to write this paper, and for numerous comments andsuggestions throughout the preparation of this paper

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Stochastic methods for groundwater resource protection zones1

Dr Adrian Butler

Department of Civil and Environmental Engineering, Imperial College London, Exhibition Road, London, SW7 2BU, United Kingdon.

E-mail [email protected]

1. Introduction

Groundwater resources in arid and semi-arid regions are extremely important for the viability of these areas. Therefore, ensuring their long-term protection, and in particular the protection of abstraction wells is essential. Furthermore, the technical, legal, social and financial difficulties that can arise from restoring contaminated groundwater mean that the protection of aquifers is preferable. Regulations for the protection of drinking water wells usually require the delineation of areas that define a prescribed minimum groundwater residence time and the entire catchment area of the well. On the one hand there are existing or planned activities, which represent hazards or risks for a particular groundwater resource, and on the other hand, the aquifers exhibit a certain degree of vulnerability to a wide range of chemical and biological pollutants.

The protection of a groundwater source (Figure 1) is approached by considering the risks to the quality of that source. Because a source is an active abstraction, it is presumed that the maximum level of protection is required. The initial approach is to identify all pathways to the source, irrespective of relative vulnerability or any potential hazard. The concept is one of reducing risks of contamination to the source to a minimum. Thus, this concept usually implies the determination of an area, known as the Zone of Contribution or ZOC (see Table 1 for a list of terms and definitions), about the source, from which most hazards are to be excluded, through land use planning. The ZOC represents an ‘all risks’ source protection zone (SPZ).

Figure 1 An illustration of the effect of a pumping well on groundwater flow.

1 Material derived from Stauffer, F., Guadagnini, A., Butler, A.P., Hedricks Franssen, H-J., van de Wiel, N., Bakr, M.I., Riva, M. and Guadagnini, L., Delineation of source protection zones using statistical methods, Water Resources Management, (2005), 19, 163–185.

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However, the SPZ may be modified (i.e. the area reduced) by accepting very low risks to the source arising from unspecified potential contamination originating from certain sub-zones of the ZOC (Figure 2). The sub zones are usually delineated by determination of those pathways with potential travel times to the source above a certain limit (e.g. 50 days). Thus certain activities, in terms of land use, may be permitted which represent hazards that do not significantly increase the risk to the source when combined with long pathways.

Table 1. Terms and definitions commonly used in well capture zones

TERM DEFINITION

Capture Zone

The area around the well from which water is captured within a certain time t.Sometimes the time dependent aspect is emphasised by using the word time-related capture zone, but in this report the word capture zone will automatically imply time dependence.

Well catchment The capture zone for infinite time. It encompasses the entire area over which the well draws the water pumped from it. Another word that can refer to both a capture zones and a well catchment is the zone of contribution (ZOC).

IsochroneThe boundary of a capture zone. It is a contour line of equal travel time to the well.

Well head protection area (WHPA)

The area around a pumping well, which is associated with a specified level of protection. Its delineation is a political consideration based on a risk analysis, which may or may not involve an actual capture zone delineation.

Source Protection zones around a borehole

I – Inner zone (time related) II – Outer zone (time related) III – Total source catchment

III II IIII II I

Figure 2 Illustration of groundwater source protection zones

Basic elements for the protection of groundwater from the point of view of water quality are (Figure 2):

Well/spring capture zone (time-related): Normally it is required that the residence time of any groundwater or pollutant should exceed a prescribed value (e. g., between and 10 and 50 days according to the regulations on water resources protection in many countries). The idea behind this regulation is that pathogenic microbes are generally eliminated within this time span, and that in the case of hazards there would be enough time for interventions (abstraction of polluted water, establishment of hydraulic barriers) or other remediation measures.

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Well/spring recharge area: Any groundwater within this zone, and therefore any pollutant, would eventually reach the well, provided the flow field within this zone is at steady state. Regulations for the protection of water resources require the designation of recharge areas of pumping wells, which are endangered by pollution. The recharge area is related to the well/spring catchment or wellhead protection zone. Springs are normally treated in a similar way as wells.

Necessary measures and restrictions with respect to land use and restrictions of human activities within established protection zones are defined by the regulations. The protection of aquifers should be accompanied by a monitoring of the piezometric head and of the groundwater quality by an appropriate network of observation wells and periodical sampling. To assess the yield and potential capacity of groundwater resources, and to calculate the extent of the capture zones and contaminant travel times, the flow and transport characteristics of the aquifer are needed. This includes also an assessment of the temporal development (seasonal development, long term development) of the groundwater flow.

2. General procedure for the delineation of protection zones

The following types of groundwater protection zones are considered here: the time-related capture zone of a well, and the recharge area of a well. Well capture zones for a prescribed groundwater residence time can be determined by evaluating the corresponding isochrones, which are the contour lines of equal groundwater residence time. These isochrones can be calculated analytically for a system of wells in a uniform base flow (e. g. Bear and Jacobs, 1965). They can also be computed numerically for arbitrarily shaped flow fields (Kinzelbach et al., 1992), for example by back-tracking a fluid particle starting at the well until the prescribed residence time is reached. Well catchments can be determined numerically by back tracking a fluid particle starting near the stagnation point of the well. In both cases the detailed velocity field is required, assuming that advection is the dominant process. For the evaluation of the velocity field the following parameters and conditions are generally required:

(1) The flow geometry: This information is obtained from hydrogeological investigations. The prevailing flow field can often be approximated by a horizontal two-dimensional (2D) flow and transport model. Moreover, compared to three-dimensional (3D) flow the formulation and numerical implementation of 2D models is usually much simpler than in the 3D case. Nevertheless, it should be kept in mind that 3D effects may be important in practice. For instance, the evaluation of a 3D capture zone or catchment, at least in the vicinity of the well, is (in principle) required when dealing with partially penetrating or partially screened pumping wells.

(2) The pumping rate of the well: the given or planned schedule of the pump should be taken into account.

(3) The groundwater recharge rate: the rate is estimated on the basis of hydrological considerations.

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(4) The infiltration rate from rivers and creeks: the rate can be estimated on the basis of hydrological considerations, or by calibration of a flow model using nearby head and/or concentration data.

(5) The levels of the bottom and of the top of the aquifer formation: this information is generally obtained from borehole and/or geophysical investigations.

(6) The piezometric head of the aquifer: this information is generally obtained from boreholes and/or geophysical investigations.

(7) The location of the boundary of the flow domain to be investigated: this information is obtained from a regional hydrogeological and hydrological investigation. The boundaries are often chosen in such a manner that a feasible formulation of the boundary conditions (fixed head or streamline) can be obtained.

(8) The boundary conditions: this information consists of the heads at the boundary (or portions of it) or of the water flux through the boundary (or portions of it). This information can be obtained from hydrological and hydrogeological investigations.

(9) The hydraulic conductivity (or transmissivity) of the aquifer: this information can be obtained from pumping test evaluation or other procedures.

(10) The porosity of the aquifer: this information is relevant for proper isochrones prediction and can be deduced, for instance, from tracer tests.

3. Impact of parameter uncertainty

Relatively small capture zones can usually be determined in a reasonable manner using the above stated principles. However, problems arise for larger capture zones or recharge areas due to the impact of parameter uncertainty. Evers and Lerner (1998) asked the question "How uncertain is our estimate of a wellhead protection zone?" Therefore, the above list of parameters and conditions should be discussed in a qualitative manner with respect to the associated parameter uncertainty:

(1) The extent of the flow domain is subject to uncertainty, mainly due to the extrapolation and interpolation of data.

(2) The pumping rate of the well is probably the less uncertain of all information. Often, long term averaged pumping rates can be used. However, the pumping schedule can affect the capturing mechanism by the well by the time-dependent velocity field.

(3) The groundwater recharge rate can, in general, only be indirectly determined. It depends on the rainfall rate, on the evaporation and transpiration rate, and on the subsequent flow processes in the unsaturated zone. Overall, the recharge rate is time dependent and more or less spatially variable. Often these effects can hardly be assessed precisely. Even the temporally and spatially averaged recharge rate may show considerable uncertainty.

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(4) The infiltration rate from rivers, creeks, and lakes is difficult to assess since it cannot be measured directly, in general. It depends on the local infiltration conditions, which can be affected by clogging. The rate is in general time dependent and spatially variable.

(5) The level of the bottom and the top of the aquifer are usually based on local borehole information and can be obtained by interpolation. Consequently, some uncertainty remains. The situation may be improved by a combination of geophysical techniques.

(6) The piezometric head of the aquifer is based on local borehole measurements and represents valuable data used for a calibration of the flow model. The piezometric head essentially dominates the flow directions. Consequently, transient effects in the head field can be of utmost importance. It is usually vertically averaged information while in some cases can be known at different intervals along a vertical.

(7) The location of the boundary of the aquifer is based on a regional hydrogeological and hydrological assessment and is always subject to uncertainty.

(8) Fixed head boundary conditions are subject to uncertainty caused by data interpretation and interpolation. The transient behaviour of these conditions can hardly be assessed in detail. Flux boundary conditions are also difficult to estimate. They can often be determined in a satisfactory manner with the help of flow models, provided that reliable data of hydraulic conductivity and piezometric head are available. Nevertheless, some uncertainty inevitably remains. The averaged value and the transient behaviour of both types of boundary conditions can be important for the flow field, and, therefore, for the location of the capture zone or catchment.

(9) Hydraulic conductivity always shows a more or less pronounced spatial variability due to the heterogeneous nature of aquifers. Therefore a thorough investigation of the field scale hydraulic conductivity is advisable. The local values can never be known in detail everywhere. Spatial variability can considerably affect the uncertainty of the location of the capture zone or catchment. In addition, the scale at which the measurements have been taken has to be carefully considered in the evaluation of the measurement.

(10) Aquifer kinematic porosity directly affects the flow velocity and therefore the residence times, which subsequently determines the location of the capture zone. Moreover, a spatial variability of field scale and local porosity may exist also in unconsolidated aquifers. However, the effect of a spatial variability can be smaller than that of the hydraulic conductivity.

Many of the above listed items concern local information, which is typically measured in boreholes. Therefore, the quality of the overall information in a particular flow domain very much depends on the spatial and/or temporal density of the available data. However, due to economic and logistic reasons, the information is often sparse. The location of data points is normally restricted to particular regions

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within the aquifer. Similarly, the temporal frequency of measurements is often limited. Moreover, experimental data are sometimes corrupted by measurement and interpretive errors. Overall, the combined effects of the uncertainty of all parameters and conditions can considerably affect the precision of the calculated capture zone or catchment. For small areas a more simplified and intuitive assessment of the uncertainty is often possible, which can be taken into account in the delineation of the protection zone. However, for larger areas the uncertainty can be quite large. Depending on the economical and ecological importance of the protection zone the implications of the degree of uncertainty associated with its predicted location can be prohibitive. Therefore, methods are needed to quantify the uncertainty and provide guidance in the acquisition of site data to reduce it. In general, uncertainty can be reduced by increasing investigations (borehole investigations, parameter estimation, etc.). However, since resources are limited the task should consist of a methodological approach, which is pragmatic in the sense that it optimises efficiency. Consequently, there is a need for knowledge and tools, which enable a conceptual and quantitative assessment of the impact of parameter uncertainty on the location of existing or planned protection zones.

The consequence of the uncertainty of the essential parameters, which determine capture zones or catchments of a pumping well, is that the location of these zones cannot be determined with certainty. Therefore the location of the protection zones can only be defined in a statistical manner. Consequently, the best we can do is to offer a probability map, rendering the probability with which a particular location belongs to the capture zone or catchment. Such concepts can be fed directly into a risk-assessment of a particular groundwater resource.

4. Importance of data in the analysis of well protection zones

The uncertainty about the location and the extent of protection zones can generally be reduced by an increase of direct measurements (for example by increasing the amount of hydrogeological data, such as hydraulic conductivity, or direct measurements of state variables, such as piezometric head). Furthermore, geological data from field investigations (such as borehole descriptions, cone penetration tests, etc.) can, in principle, also be used to constrain predictions, resulting in a global reduction of uncertainty in the delineation of protection zones. The most important hydrogeological data are listed in Section 2. These data may be used:

To establish deterministic flow and transport models of the aquifer. Here, we use the wording "deterministic" in order to identify a model, which does not provide, by its nature, a quantification of uncertainty associated to predictions other than by means of a traditional sensitivity analysis. Deterministic models, based on calibrated averaged parameters (using the concept of equivalent parameters for different aquifer zones), are often obtained using various manual or automated optimisation procedures.

To deterministically delineate protection zones of pumping wells for various conditions (boundary conditions, pumping rates, recharge conditions, etc.), using

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the calibrated flow and transport models. However, a considerable uncertainty may remain; the latter cannot be quantified, by definition, via the use of a deterministic model.

To evaluate the statistical and stochastic parameters characterizing spatial variability.

To establish a stochastic model (Section 5) based on the deterministic (i. e., certain, or known with rather reasonable certainty) and stochastic parameters, forcing terms and boundary conditions.

5. Modelling spatial variability and uncertainty of variables

The uncertainty of the parameters may be on the one hand due to measurement errors inherent in a specific evaluation method, and on the other hand due to the more or less strong spatial variability of many parameters, like hydraulic conductivity K(x), which can never be known in detail everywhere. A viable way out of the dilemma may be, for example, to cast the problem in a probabilistic framework and considering the aquifer as one of many stochastic realizations. Stochastic variables such as hydraulic conductivity do not behave like a white noise but show a distinct spatial correlation structure with the correlation between two values, depending on their distance. This correlation structure can be described by, e. g., a two-point (auto-) covariance function. A further important feature is the probability density function of the parameter under consideration.

A common approach in the practical application of prediction models is to formulate equivalent parameters, thus replacing the real heterogeneous system by a homogeneous equivalent model (e. g., Renard and de Marsily, 1997). Therefore one task consists of finding adequate equivalent parameters, such as equivalent hydraulic conductivity, as a function of quantities investigated.

The investigation of the impact of spatial variability of flow parameters concerns the evaluation of the expected (mean) location (first statistical moment) of the capture zone or catchment and its associated variance (second statistical moment). Such moments can be theoretically based on the ensemble of all equally possible heterogeneous realizations of the considered aquifer, honouring some sets of measured data. The formulation of second moments is referred below by the use of moment equations (Section 7). Widespread numerical procedures to the solution of this problem are, for example, Monte Carlo – based techniques (Section 6), in which space-dependent or also time-dependent parameter values of numerical models are generated in a statistical manner, followed by a subsequent (numerical) solution and analysis of each of the corresponding deterministic systems. Usually, the following statistical or geostatistical parameters are required for a stochastic method:

The stochastic variable(s), e. g., the hydraulic conductivity K(x), or its natural log-transformed counterpart, Y(x)=ln(K(x)).

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The probability density function, pdf, of the stochastic variable: This is often approximated by a normal or a log-normal probability distribution. For log-normally distributed variables a log-transform of the variable is applied.

The ensemble mean value of the stochastic variable, e. g., <Y(x)>, or the geometric mean value, Kg.

The variance of the stochastic variable, e. g., Y2; and the related standard

deviation.

The two-point covariance function CY(x1,x2). Often, a particular and convenient invariant covariance model is selected to express the spatial correlation, e. g. the exponential covariance model CY(x1,x2)= Y

2 exp(-|x1-x2|/IY), where the correlation length, IY, has to be evaluated on the basis of available data.

When considering randomness in more than one parameter, cross-correlations or cross-covariance amongst different parameters might be needed. The parameters have to be evaluated a priori, based on measurements or on experience. For spatially distributed variables, like hydraulic conductivity, a variogram analysis (de Marsily, 1986) may be used.

6. Use of Monte Carlo techniques

Monte Carlo (MC) techniques are general and versatile tools that allow one or several parameters of a model to be uncertain. The idea is to generate many realisations of synthetic aquifer flow (and eventually transport) models in such a manner that they reflect the observed (experimental) parameter uncertainty. The results are subsequently analysed in a statistical manner to quantify the uncertainty inherent in the expected result. However, MC techniques are often very time consuming and it not always clear what number of realizations is necessary for the convergence of the method (e. g., Ballio and Guadagnini, 2004). Nevertheless, they represent rather general and versatile tools for the investigation of ranges of spatially variable parameters in the context of both linear and non-linear problems.

Consider a randomly heterogeneous flow domain, where, for simplicity, we assume the hydraulic conductivity, K(x), to be the only source of uncertainty, and the objective is to condition prediction and associated uncertainty to both hydraulic conductivity and head measurements. For this domain a log-transformed hydraulic conductivity field Y(x)=ln(K(x)) is generated (realization i), conditional to hydraulic conductivity measurements in boreholes. Statistics of Y to be employed as input to the generation process are usually derived from the available data set and are a critical aspect of the entire procedure. The corresponding flow field is then calculated using a forward numerical model and the computed hydraulic heads are compared with available measured hydraulic head values. The misfit between the model predictions and the measurements can be used to obtain a better estimate of the hydraulic conductivity field using a numerical inverse modelling technique, thus effectively conditioning the i-the realization of the MC process on the available hydraulic head data. The ensemble of the results of all realizations (i =1, …, NMC) is then statistically

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analysed in order to obtain predictions and quantify the uncertainty of the expected results. Solving the inverse groundwater flow and/or transport model for each realization is many times more intensive in computing time than solving the forward flow and/or transport model and the computational time increases with the number of conditioning data. A possible procedure for applying conditional MC techniques to delineate well protection zones (or catchments) under steady state-flow conditions can be summarized as follows:

(1) Generation of NMC realisations (e. g., NMC = 500) of a random log-transformed hydraulic conductivity field Yi(x)=ln(Ki(x)) (i = 1, ..., NMC), conditional to available Kj(xj) data at locations xj , where j = 1,…, NK (see below), NK being the number of conditioning K measurements.

(2) Solution of the groundwater flow problem for each of the realizations Ki(x)using a numerical inverse modelling technique, conditional to head data hk(xk),with k=1,…,Nh or other type of data (see below). This results in updated realizations Ki(x). Alternatively, conditioning on hk(xk) data can also be performed simultaneously with Kj(xj) data.

(3) For each flow field hi(x) one or several idealized tracer particle are released at each grid cell of a regular grid. Numerical particle tracking is performed to calculate the well capture zone (for the given residence time) or the well catchment for each realization using an advective transport model. The end points of all particles are recorded.

(4) The statistical analysis of the ensemble of particle trajectories over all NMC

realizations provides the probability distribution P(x) of the capture zone or the catchment. In other words, one obtains a map showing the spatial distribution of the probability P that a fluid particle (or idealized tracer particle) released at a particular location x is captured by the well within the requested residence time, or the probability P that a particular location x belongs to the catchment.

Codes to generate random hydraulic conductivity fields Y(x)=ln(K(x))investigated by the authors during this work, were SGSIM (Deutsch and Journel, 1998), FGEN (Robin et al., 1993), GSTAT (Pebesma, 1999), or GCOSIM3D (Gómez-Hernández and Journel, 1993). GCOSIM3D is a follow-up version of SGSIM. It enables a better reproduction of the covariance function CY(x1,x2), especially in case of long correlation lengths IY.

The groundwater flow equation can be solved, for instance, with the finite difference code MODFLOW (McDonald and Harbaugh, 1988) and forward particle tracking can be performed using the computer code MODPATH (Pollock, 1994), whereby at least one particle is released at each grid cell.

One of the stochastic inverse modelling techniques investigated in this work is the Sequential Self-Calibrated method (Gómez-Hernández et al., 1997; Hendricks Franssen, 2001) for the inverse modelling of groundwater flow and mass transport, conditioned to hydraulic conductivity and head data. The method can also handle (in principle) transient head data with the joint estimation of spatially variable hydraulic conductivity and storativity fields and is formulated in three dimensions. The method

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has been extended to estimate jointly hydraulic conductivities and recharge (Hendricks Franssen et al., 2004a).

Another investigated inverse modelling technique is the Method of Representers (Valstar, 2001). This inverse algorithm considers spatially variable parameters explicitly. It can use both hydraulic head and concentration data to reduce the uncertainty of model parameters for three-dimensional, quasi-steady flow regimes. For conditioning on head measurements the Method of Representers has been implemented into MODFLOW (Van de Wiel et al., 2002) and in a modified version of the finite element code S-InvMan (Bakr, 2000). It has also been used to examine the worth of head and concentration data on groundwater remediation using the pump-and-treat method (Bakr et al., 2003).

The accuracy of the numerical estimate of the probability P(x) that a location xbelongs to a capture zone or catchment strongly depends on the number of Monte Carlo runs, NMC, and therefore on the convergence of the method. A minimum number NMC for which the estimated value of P(x) is practically independent of NMC

has to be identified. Van Leeuwen (2000) suggests that, at least in two dimensions, approximately NMC=500 realisations normally result in an acceptable convergence, and that the convergence after about 1000 realisations hardly improves.

For illustration, the results of an unconditional Monte Carlo analysis aimed at the evaluation of the catchment of a well is shown in Figure 3. At the western boundary a constant head boundary condition was applied. The remaining boundaries were chosen as impermeable. The geometric mean hydraulic conductivity Kg was 10 m/d, the variance Y

2 was 0.1, the correlation length IY=50m, adopting an exponential covariance function for Y. The pumping rate was 200 m3/d and the recharge rate 1 mm/d. The number of Monte Carlo runs was NMC=1000. The map shows the probability P(x), that a fluid particle at a given location x reaches the well.

Further numerical or analytical Monte Carlo techniques to deduce capture zones or catchments in a statistical manner were suggested, e. g., by Kunstmann and Kinzelbach (2000), Franzetti and Guadagnini (1996), Guadganini and Franzetti (1999), Van Leeuwen et al. (1999), Wheater et al. (2000), Hunt et al. (2001), Feyen (2001), Jacobson et al. (2002), or Feyen (2003). In addition, van Leeuwen et al. (1999) have used Monte Carlo analysis to evaluate the impact of uncertainty in the location of drift overlying a production aquifer on well capture zones at Wierden, Netherlands.

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-500 0 500 1000-500

0

500

Figure 3 Probability map of a well catchment; Monte Carlo results

(well located at x=0, y=0).

7. Use of moment equations

A major conceptual disadvantage of Monte Carlo approaches is that they do not provide a theoretical insight into the nature of the solution. Therefore there is a need for alternative approaches. One method is based on the use of moment equations.

A complete first-order (in the variance of Y) stochastic mathematical formalism that allows to obtain an estimate of the travel time and the trajectory (rendered by the first moments) together with the associated prediction errors (rendered by their second moments) for idealized tracer particles advected in a randomly heterogeneous aquifer has been derived (Guadagnini et al, 2003; Riva et al., 2004) and compared against numerical Monte Carlo simulations the formalism, solute particles are injected at a single point and travel along a (random) trajectory towards a given discharge point or line. Travel time mean and variance are functions of first and second moments and cross-moments of trajectory and velocity components. Trajectory mean and variance are functions of the statistical moments of the components of the velocity field. The equations were developed from a consistent first order expansion in Y. As such, they are nominally limited to mildly non-uniform fields, with Y

2 < 0.5, or more heterogeneous fields, in the presence of conditioning. The work has been developed in two dimensions and the extension to a more general three-dimensional scenario is straightforward. Furthermore, procedures were developed, which allow conditioning of the statistical moments of the flow field and, therefore, of travel time moments, on hydraulic heads measurements and / or aquifer architecture (e.g., Hernandez et al., 2003; Winter et al., 2003).

Stauffer et al. (2002) investigated the uncertainty in the location of two-dimensional, steady state catchments of pumping wells due to the uncertainty of the spatially variable hydraulic conductivity field by a semi-analytical Lagrangian technique. For the analysis it is assumed that the aquifer can be modelled as a steady-state horizontal, confined or unconfined system. The well discharge rate and the recharge rate are constant. The uncertainty bandwidth of the catchment boundary is

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approximated at first order (in the variance of Y) by formulating the transversal second moment of the tracer particle displacements along the expected mean boundary of the catchment, starting at the stagnation point in a reversed velocity field. A special approach is suggested for the estimation and conditioning of uncertainty in the location of the stagnation points. For illustration, the results for the estimated catchment boundary of a well together with its uncertainty bandwidth is shown in Figure 4. The conditions were the same as in Figure 3. A similar Langrangian approach for time-related capture zones in heterogeneous aquifers without recharge was presented by Lu and Zhang (2003).

0 500 1000 1500x [m]

300

800

1300

y [m

]

Figure 4 Uncertainty bandwidth (mean and 95% probability) of the location of a well catchment boundary estimated by the Lagrangian semi-analytical method

(Stauffer et al., 2002).

Bakr and Butler (2004) have used an alternative numerical approach, in which the original partial differential equation of flow and advective transport, is first discretized on a specified grid using finite elements, and then the resulting equation, the so-called space-time equation, is used to derive the statistical moments of the flow and mass transport quantities. The approach is based on a Taylor’s series approximation of the discrete system of equations and is often termed the vector space-state/adjoint state approach.

8. Impact of recharge uncertainty

The spatial variability of hydraulic conductivity is usually believed to be the main contributor to the uncertainty in the estimation of a well catchment and, therefore, its effects have been intensively studied (see, e. g., references cited above). The impact of the spatially variable recharge on the estimation of a well catchment is known to a lesser extent. In this case, the relevant question to be answered relates to whether recharge uncertainty does contribute significantly to the well catchment uncertainty in case there is at the same time a considerable spatial variability of hydraulic conductivity. In addition, another source of uncertainty may be the temporal variability of recharge.

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The recharge uncertainty has been subjected to a synthetic study (Hendricks Franssen et al., 2002). The study focused on a flow regime that is typical for a humid, temperate climate. The average yearly recharge is chosen as 360 mm and the seasonal variations in precipitation are not very strong. Furthermore, the aquifer materials have a relatively high permeability. The main practical conclusions from the Monte Carlo type study were:

The spatial variability of recharge has only a limited impact on the uncertainty of the well catchment. The impact is larger in case of a larger recharge correlation length. However, even in case of large recharge correlation length and unrealistically large spatial recharge variability, its influence on the well catchment is very limited for moderately heterogeneous hydraulic conductivity fields ( Y

2 1). It is expected that for a moderately or strongly heterogeneous hydraulic conductivity field the spatial variability of recharge is overruled by the spatial variability of hydraulic conductivity.

Nevertheless, the uncertainty in the mean recharge has an important impact on the well catchment uncertainty, also in case of a moderately or even strongly heterogeneous hydraulic conductivity field. For recharge it is therefore important to estimate the uncertain mean, while the detailed estimation of the spatially variable patterns of recharge is normally not needed. It may be necessary to estimate the mean areal recharge for multiple zones in case there is evidence of varying means between different areas (as is the case for differences in land use).

The temporal recharge variability can be important to consider in some specific situations. In case the groundwater residence time in the catchment is not clearly longer as compared to the time scale on which the recharge fluctuates, it may be necessary to investigate time series of recharge. In humid, temperate climate zones the variations between years are normally limited. However, the recharge varies significantly over a year. Normally a recharge minimum in summer and early autumn and a recharge maximum in winter and early spring is reached. In case the expected groundwater residence time in the catchment is not clearly larger than one year, effects of temporal recharge variability may be expected.

9. Impact of uncertainty in geostatistical parameters, average hydraulic conductivity, and boundary conditions

In most of the studies in stochastic subsurface hydrogeology the covariance function CY(x1,x2) and the geometric mean hydraulic conductivity Kg are assumed to be known exactly. In practice, they are estimated from a limited number of hydraulic conductivity data. As a consequence, these estimates are associated with a considerable uncertainty. A further important source of uncertainty, which normally is not addressed in hydrogeological studies, is related to the boundary conditions. The location of the boundaries and the prescribed flux or prescribed head values on the boundaries are normally assumed known.

The impact of the mentioned sources of uncertainty was tested in a synthetic study (Hendricks Franssen et al., 2004b). The studied two-dimensional domain had

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extensions of 5 x 5 km. The northern and southern boundaries were impervious and along the western and eastern boundaries fixed heads of respectively 0m and 5m were imposed. A pumping well was located 500m west of the domain centre. The area received a spatially uniform recharge of 363mm/year. Steady-state groundwater flow in an aquifer with constant thickness was simulated. A hydraulic conductivity field was generated with a mean value of 102 m/d and an exponential covariance function CY with a variance of Y

2=1, and a correlation length of IY=500m (1/10 of the domain). This reference field was considered as unknown reality of the study. As a consequence, a water divide along the eastern part of the area was present and the well pumps water from a considerable area located west of the water divide. Figure 5 shows the corresponding reference well catchment. The reference hydraulic conductivity field was sampled 100 times by selecting 10 measurements each. From the selected 100 random data sets 100 different mean hydraulic conductivity values and 100 different covariance functions of Y=ln(K) were estimated. The impact of these uncertainties on the estimation and estimated variance of the hydraulic conductivity field, the hydraulic head field and the catchment zone were quantified. The following practical conclusions could be drawn from this study:

(1) The uncertainty in the mean hydraulic conductivity, e. g., Kg, had a very limited influence on the characterisation of the hydraulic conductivity field, the hydraulic head field and the catchment zone.

(2) The uncertainty in the hydraulic conductivity covariance function CY(x1,x2) had a slightly larger impact on the characterisation of the hydraulic head field, the hydraulic conductivity field and the catchment zone than the uncertainty in the mean hydraulic conductivity, but was nevertheless quite small. However, this conclusion is valid only for the ensemble mean over all 100 samples of measurements. For the case that just one set of measurements is taken, as is of course the case in practical situations, the impact of the uncertainty of the covariance function can be much more pronounced.

(3) The uncertainty with respect to piezometric head values on prescribed head boundaries had a large effect on the characterisation of the hydraulic head field, the hydraulic conductivity field and the catchment zone.

(4) Inverse modelling (conditioning to error-free piezometric head data) was able to reduce the impact of the above mentioned three sources of uncertainty.

Note that the conclusions from the synthetic study cannot necessarily be generalized to any other case.

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0 1000 2000 3000 4000

x (m)

0

1000

2000

3000

4000

5000

y (m

)

Well

Figure 5 Reference well catchment of the synthetic study for the investigation of the impact of uncertainty in covariance function CY, the geometric mean hydraulic conductivity Kg, and the boundary conditions (Hendricks Franssen et al., 2004).

10. Sampling design and monitoring strategies

Incorporation of measurement data through conditioning is a requirement for reducing uncertainty in the location of the well capture zone or catchment. However, the success of this conditioning depends on the type of measurement, the amount and spatial pattern of the measurements and the measurement error (Bakr and Butler, 2003; Hendricks Franssen and Stauffer, 2004). Furthermore, the conditioning performance seems to be dependent on the type of aquifer (confined or unconfined), the correlation length IY and variance Y

2 and the well pumping rate (van Leeuwen et al., 2000; Bakr and Butler, 2003; Van de Wiel et al., 2002).

Extensive Monte Carlo – based analyses on the effect of hydraulic conductivity (or transmissivity) data and piezometric head observations (both separately and combined) in reducing capture zone uncertainty enabled to provide a set of basic rules about location and type of data and for the development of optimal measurement strategies for uncertainty reduction. The effect of conditioning is primarily measured in terms of the reduction in the width of the capture zone’s probability distribution.

In general, uncertainty related to well capture zones or catchments decreases with increasing incorporation of hydraulic conductivity and hydraulic head data (conditioning density). However, the conditioning effect for different sampling schemes with similar amounts of measurement data can be quite different. What is the relative worth of different types of measurements? Synthetic case studies indicate the greater importance of hydraulic conductivity measurements over head data in reproducing the reference well capture zone (Bakr and Butler, 2003; Feyen et al., 2003). It appears that it is not possible to get satisfactory results using head measurements alone, especially in highly heterogeneous aquifers (for Y

2>1).

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Although head measurements are capable of estimating hydraulic gradient quite accurately, they do not contain sufficient information on the variation of hydraulic conductivity which, in turn, leads to high variability in pore water velocity and hence the well capture zone location. Furthermore, results show that a combination of both head and hydraulic conductivity data can reduce the width of the capture zone distribution more significantly than either type of data alone (Bakr and Butler, 2003). Feyen et al. (2003) concluded that head observations are more effective in reducing the width of the capture zone distribution, whereas hydraulic conductivity measurements are more valuable in predicting the actual location of the unknown capture zone. Feyen et al. (2003) also incorporated tracer arrival times, hydraulic conductivity measurements and hydraulic head observations in the stochastic capture zone delineation. Their evaluation indicates that travel time data seem to be effective in reducing the overall uncertainty and to some extent in revealing large irregularities in the shape of the capture zone. In general, the incorporation of tracer and solute concentration enhances the aquifer characterization and the accuracy of flow and transport predictions, since such data are complementary to head and conductivity measurements (Medina and Carrera, 1996; Valstar, 2001; Hendricks Franssen et al., 2003).

Observation network of head and/or hydraulic conductivity measurements can be optimised in a systematic manner. Such methodologies can also provide a way to minimize the number of sampling locations, required to reduce capture zone uncertainty to an acceptable level. Van de Wiel et al. (2002) investigated several strategies for selecting the optimal additional location for a piezometric head observation in a synthetic confined aquifer with a single pumping well. The strategies are: a) Selecting the location, where the head variance is highest; or b) selecting the location where the sum of covariances between head at that location and at the other potential measurement locations is largest; or c) selecting the location that minimises the head variances summed over the model domain; or d) selecting the location where the reduction in capture zone uncertainty is highest. The last strategy clearly showed the best results in reducing the capture zone uncertainty. However, it is also the most time intensive, whereas the three other criteria do not require Monte Carlo simulations. Amongst these three design strategies, the strategy that minimised the summed head variance performed poorly in terms of minimisation of capture zone uncertainty in case of a relatively small number of selected head measurements.

Hendricks Franssen and Stauffer (2004) proposed optimisation procedures for selecting new locations for both head and hydraulic conductivity data in a synthetic confined aquifer with spatially uniform recharge and a single pumping well. The algorithm enables to place additional measurement locations nearly optimally. The true optimum can hardly be found since only a limited number of possible combinations of new measurement locations can be analysed realistically, since computation of all possible combinations, even for a few additional measurements, is extremely time consuming. Two selection criteria were implemented, a) the minimisation of the expected average log-transformed hydraulic conductivity variance, and b) the minimisation of the average hydraulic head variance. It was found that both strategies were successful. However, the differences between the optimal strategies

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and heuristic criteria, where the sampling points are distributed evenly over the whole aquifer, were small. This indicates that covering the domain of interest regularly with a measurement network is a close to optimal strategy in characterizing the general flow field. However, selecting measurement locations in zones with a capture probability of about 0.05<P(x)<0.95 seems to be a better option for characterizing a well catchment. Nevertheless, there is a trade-off between continuing adding measurement points in these zones and placing additional locations in the areas around the uncertain zones.

11. Application to a field case

Many of the methodologies developed for the stochastic characterization of well capture zones and catchments have been intensively tested in several synthetic numerical simulations. The next logical step was to apply these methodologies in a real world case study. For this purpose the Lauswiesen test site in the Neckar valley near the city of Tuebingen in southwest Germany was selected. This site has been intensively studied before and during the W- SAHaRA project (Martac et al., 2003a, b) including stochastic inverse modelling. For the sake of illustration, Figure 6 shows the estimated 50 days well capture zone for the Lauswiesen site during the tested pumping conditions, using a Monte Carlo approach based on the Representer Method. The main conclusion from this field case is that the stochastic methodologies are indeed capable of yielding reasonable stochastic estimates of the well capture zone or catchment. However, as compared with a synthetic case there exist several implications that point to the need for further research and development. In a synthetic test case the performance of an algorithm can be easily evaluated in a systematic manner. In practice one would like to use all available information, but then no data are available anymore to test how well the model predictions performed. Although detailed verification is in general not possible in practice, it is desirable to exclude some data and/or tests from the conditioning procedures and to use them for comparison with the model predictions. In the present test case we found that the model predictions do not deviate too much from the experimental results. The peak arrival time for three tracers was quite well predicted.

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Figure 6 Lauswiesen field case: Map of ensemble averaged hydraulic head (m) and the probabilistic 50 days well capture zone of abstraction well F0 calibrated for the

pumping stage. Locations of head measurements (H) and injection wells (F1-F6) are marked. Dark grey represents a high value for hydraulic head.

It can be stated that the results are generally compatible with the tracer tests. The most important conclusions were:

We found that the role of the river, and its interaction with the aquifer was crucial. Measured time series of groundwater and Neckar river levels indicate that the river level has a major impact on the hydraulic head distribution in the aquifer. Errors in modelling this boundary condition prohibit a more reliable characterisation of the aquifer hydraulic conductivity and of the capture zone. In order to directly handle the inverse stochastic modelling of river-aquifer interactions in an adequate and concise manner, further research is needed. In addition, in order to enable a successful inverse stochastic simulation of the river-aquifer flow conditions more measurement locations along the river are needed.

The estimation of the hydraulic conductivity covariance function had to be based on sparse data only. This was a further main source of uncertainty.

It was shown that inverse modelling is to a certain degree able to correct the estimation error.

Data from sieve analysis (soft information) allowed a more realistic reconstruction of the spatial distribution of aquifer hydraulic properties (Martac et al., 2003).

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The practical case of the Lauswiesen field site shows that there is a need for inverse modelling procedures that can handle three-dimensional aquifers with a large number of grid cells and transient flow conditions. In this sense, parallelization of Monte-Carlo type models can lead to an improvement in their performance. Ye et al. (2004) presented a comparative analysis, in terms of runtimes, of the computational efficiencies of parallel algorithms used in the context of recursive Moment Equation and Monte Carlo methods. In addition, the models should be able to condition to tracer test information on a routinely basis in order to further improve the predictions.

12. Conclusions

A result of incomplete knowledge of the essential parameters that determine well capture zones or catchment areas is that the location of these zones/areas cannot be determined with certainty, since the amount of available data is always limited. In particular, the location of data points is often restricted to specific regions within the aquifer (due to economic and logistic reasons). Furthermore, experimental data are always corrupted to some degree by measurement and interpretive errors. Consequently, the location of the protection zones can only be defined in statistical manner and should therefore, be represented using a probabilistic catchment/capture zone map (as illustrated in Figures 3, 4 or 5). This provides us the probability P(x), with which a particular location x belongs to the capture zone or well catchment (Section 3). This requires a stochastic analysis for well capture zone and catchment. The general procedure is depicted in Figure 7. The probabilistic capture zone or catchment map of a pumping well can then directly be used by the decision maker for the delineation of the protection zone, based on specific political, ecological, and/or economical reasons. Moreover, such a probabilistic representation naturally fits with the requirements for a risk-based assessment that are frequently required in groundwater management decisions.

The previous sections have described the development and evaluation of methods and tools to produce stochastic capture zone/catchment maps of pumping wells. However, it is important to emphasise that the selection of a specific method for a stochastic analysis of protection zones of pumping wells will depend on specific site conditions and data, i.e. the dimension of the flow problem, the flow geometry, the aquifer type (confined, unconfined, multi-aquifer system), the recharge conditions, the number of wells, the spatial variability of hydraulic conductivity, and the type, number and location of conditioning data.

The methodologies that characterize capture zones and catchments stochastically have reached the stage that they can be applied in practice, but further development is needed so that they can be applied more routinely. Semi-analytical Lagrangian methods without conditioning on measured data (Stauffer et al., 2002) are in many practical studies an option to get a quick "idea" of the uncertainty of a well catchment. They have the advantage that they need relatively little computing time. One of the few special requirements would be the estimation of a reliable hydraulic conductivity covariance function. Expert knowledge is needed to estimate such a covariance function. In case of only few measurements, multiple calculations with different

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covariance functions are possible, and not very time consuming. More general, methods based on moment equations, conditional to measurements, have shown their potential for interesting future applications.

Identify groundwater abstraction and potential sources of pollution

No

Yes

Probabilistic analysis: Identify sources of uncertainty (e.g. conceptual model,

hydrogeological parameters , boundary conditions)

Undertake preliminary unconditional simulation

Monte Carlo analysis

1

2

STOP

No

Select type and location of additional input measurements

Yes

Incorporate data for conditional simulation

Develop new/modified numerical model (2D/3D)

Yes

NoSTOP

Delineate capture zones

Is there a need to quantify model

uncertainty?

Have well capture/ catchment zones

been demarcated?

Existing numerical model (2/3D) e. g. MODFLOW, MODPATH

Method of derivation

Select method

Reduce uncertainty

Simple analytical solution (e.g. Bear and Jacobs)

Moment Equation analysis

Collect and/or collate data

Figure 7 Stochastic analysis of well capture zones or catchments.

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Numerical Monte Carlo type methods are already well developed for flow problems. They tend to be more flexible and can for example handle cases with a large log-transformed hydraulic conductivity variance and may also handle strongly non-linear systems. In addition, in case of inverse modelling they can also treat systems with various sources of uncertainty jointly, like hydraulic conductivities, spatial and temporal recharge and various boundary conditions.

The focus of this work has been primarily targeted at unconsolidated porous media. Although fractured rock systems have not been specifically considered, the techniques developed here may also be applied where such systems are treated as equivalent porous media.

Acknowledgements. The material presented in this study was undertaken within the European research project "Stochastic Analysis of Well Head Protection and Risk Assessment" (W-SAHaRA), Contract Nr. EVK1-CT-1999-00041.

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