An empirical model for the evaluation of the degree of saturation of shallow...

An empirical model for the evaluation of thedegree of saturation of shallow soils in relation torainfalls

Roberto Valentino, Lorella Montrasio, Gian Luca Losi, and Marco Bittelli

Abstract: The evaluation of shallow soils water content is very important in many fields, and different hydrological modelsare widely applied to simulate field-scale water flow and soil water content. The degree of saturation of a shallow soil is atime-varying variable, depending on different weather conditions. In many applications, it is useful to directly correlate thesoil degree of saturation to time series of rainfall amount. This paper presents a simplified empirical model, which allowsfor computation of the soil degree of saturation using readily available climate data on air temperature and rainfall depths.The model is tested with in situ measurements of soil water content collected at three sites in the Emilia Romagna region innorthern Italy. The experimental data are compared with the results obtained from the simplified model over an observationperiod of almost 5 years. The model has been used to generate soil water content time series at different depths and requiresa separate calibration for each depth where the degree of saturation is computed. A discussion on the model’s calibration isalso carried out to clarify the model sensitivity to the different parameters and the choice of input data.

Key words: degree of saturation, water content, rainfall, shallow soils, empirical model, time domain reflectometry (TDR)measurements.

Résumé : L’évaluation de la teneur en eau dans les sols peu profonds est très importante dans plusieurs domaines, et diffé-rents modèles hydrologiques sont couramment utilisés pour simuler l’écoulement de l’eau et la teneur en eau du sol à l’é-chelle du terrain. Le degré de saturation d’un sol peu profond est un paramètre qui varie selon le temps et qui dépend desconditions météorologiques. Pour plusieurs applications, il est souhaitable de corréler directement le degré de saturation dusol avec les quantités de pluie tombées. Cet article présente un modèle empirique simplifié qui permet de calculer le degréde saturation du sol à partir des données climatiques facilement disponibles de température de l’air et de hauteur des précipi-tations. Le modèle est testé avec des mesures in situ de la teneur en eau du sol pour trois sites dans la région Emilia Roma-gna, dans le nord de l’Italie. Les données expérimentales sont comparées aux résultats obtenus à l’aide du modèle simplifiépour une période d’observation de près de 5 ans. Le modèle a été utilisé pour générer les teneurs en eau du sol à travers letemps pour des profondeurs différentes, et nécessite un calibrage séparé pour chaque profondeur où le degré de saturationest calculé. Une discussion sur le calibrage du modèle est aussi présentée, afin de démontrer la sensibilité du modèle auxdifférents paramètres et au choix des données de base.

Mots‐clés : degré de saturation, teneur en eau, pluie, sols peu profonds, modèle empirique, mesures de réflectométrie dansle domaine temps (RDT).

[Traduit par la Rédaction]

IntroductionThe spatial and temporal measurement of the volumetric

water content or, alternatively, the degree of saturation (Sr)of shallow soils (up to nearly 0.8 m below the ground level)assumes a basic importance in many fields: in agricultural

sciences, for water resources management and control (Mar-shall et al. 1996; Dingman 2002); in environmental sciencesand soil chemistry, for the evaluation of soil pollution (Mar-aqa et al. 1999; Alimi-Ichola and Gaidi 2005; Mantovi et al.2006); in soil sciences, for the study of rainfall-induced shal-low landslides (Fredlund and Rahardjo 1993; Tsai et al.2008); and in hydrological science and modeling, where thesame hydrograph can be obtained by multiple combinationsof state parameters, often making the models ill-defined, andrequiring distributed soil water content measurements formodel testing (Wooldridge et al. 2003).In the case of shallow landslides, in particular, for an ap-

propriate modeling of the triggering mechanism of instabilityphenomena both at the local (a single slope) and regionalscale (order of hundreds of square kilometres), it is useful todirectly obtain the soil degree of saturation from rainfallamounts (Montrasio and Valentino 2008a, 2008b). To de-velop an alert system against shallow rainfall-induced land-

Received 28 January 2010. Accepted 2 November 2010.Published at www.nrcresearchpress.com/cgj on 6 May 2011.

R. Valentino, L. Montrasio, and G.L. Losi. Department ofCivil, Environmental, Territory Engineering and Architecture,University of Parma, Viale G.P. Usberti 181/A, 43100 Parma,Italy.M. Bittelli. Department of AgroEnvironmental Science andTechnology, University of Bologna, Via Fanin 44, 40127Bologna, Italy.

Corresponding author: R. Valentino (e-mail:[email protected]).

795

Can. Geotech. J. 48: 795–809 (2011) doi:10.1139/T10-098 Published by NRC Research Press

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slides, there has recently been an increasing demand for sim-plified models to be implemented in geographical informa-tion system (GIS) platforms for real-time territorymonitoring (Montrasio et al. 2009; Liao et al. 2010).To reach this goal, some models allow obtaining the de-

gree of saturation on the basis of atmospheric variables, suchas rainfall and temperature, disregarding the soil typology(Granberg et al. 1999). Other models, which are based onwater-balance equations, take into account the soil hydraulicproperties. Some of them evaluate the soil moisture, simulat-ing topsoil water balance and taking into account the precip-itation, runoff, gravity-driven infiltration, and actualevapotranspiration with a daily step (Pistocchi et al. 2008).Further models often take into account different solutions

of Richards’ equation and are widely applied to simulatefield-scale water flow, both on a local and regional scale, tosolve problems regarding solute transport (Jacques et al.2002) or shallow landslides triggering (Iverson 2000; Salciar-ini et al. 2006). Some authors developed effective numericalcodes to solve water flow equations (Simunek et al. 1998;Baum et al. 2002). Fully integrated numerical models havebeen developed, including all the processes necessary for acomplete quantification of the water balance and providingthree-dimensional spatial patterns of soil moisture (Van derKwaak and Loague 2001; Bittelli et al. 2010).Recently, such models are also used at large scales up to

hundreds of metres. It is worth noting that, as was pointedout by Vogel and Ippisch (2008), an increase of the scale istypically accompanied with an increase of the spatial discreti-zation scale for the numerical solution of the problem. Due tothe underlying assumption of local equilibrium betweenwater content and water potential, Vogel and Ippisch (2008)demonstrated that there is an upper limit of spatial discretiza-tion above which the solution is expected to be biased. Thisissue is common to all engineering analyses where discretiza-tion plays a key role, due to the numerical interpolation ofsoil variables by means of the assumed shape function. Fur-thermore, as stated by the same authors, in cases where theheterogeneities of soil properties are in the size range of thediscretization scale or larger, they may dictate a finer resolu-tion.Moreover, such models usually require a high number of

input parameters (soil physical properties, soil hydraulicproperties, crop parameters) that are often not available, espe-cially over large areas. The main limit of the available modelsis in their inadequacy to extend the soil moisture evaluationat a regional scale, on the basis of few soil parameters, whichare easily obtained.In this paper, we present a simplified empirical model that

can reconstruct the degree of saturation of the soil at differentdepths, taking into account only four parameters that arestrictly linked to soil physical properties and are easily evalu-able (Valentino et al. 2009). The model simulates complete,multiple annual cycles using the readily available climatedata on air temperature and precipitation as driving variables.A validation procedure is carried out by comparing simulatedfield site conditions with long-term time series of field meas-urements of water content in the unsaturated zone in typicalsoils of the Po River Plain and of the Emilian Apennines(northern Italy).

Experimental data

Field measurements considered in this study have beencollected in three different sites in the Emilia Romagna re-gion (northern Italy). The three sample sites were selected torepresent different soil properties, in particular with regard totextural aspects. The measurements consisted of air tempera-ture and soil volumetric water content.At each sample site, air temperature was measured by us-

ing a Vaisala HMP155A device, which was located at 2 mwith respect to the ground surface. Volumetric water contentwas acquired at different depths by using time domain reflec-tometry (TDR) devices. A TDR system was installed at eachsite (TDR100, Campbell Scientific Inc., Logan, Utah),equipped with a datalogger for automatic data collection(CR10X, Campbell Scientific Inc., Logan, Utah) and 30 cmlong probes. The system was multiplexed through a singleSDMX50 multiplexer (Campbell Scientific Inc., Logan,Utah). Soil temperature measurements were also performedat each depth by using thermistors to measure soil tempera-ture close to the TDR probes. The TDR waveform analysis,necessary to obtain the relative dielectric permittivity, andsubsequently the soil water content, was carried out with thedual-tangent approach (Heimovaara 1994; Wraith and Or1999). It is well known that the success of TDR consists inits ability to accurately determine the permittivity of a mate-rial from wave propagation, based on the strong dependenceof the permittivity of a material on its water content, as dem-onstrated in the works of Hoekstra and Delaney (1974), Toppet al. (1980), and Whalley (1993). With the awareness of thelimits of this measurement technique (Schneider and Fratta2009), we considered the field acquired data to be reliable.More details about the correction of TDR-based soil watercontent measurements for conductive soils are reported byBittelli et al. (2008).

Sample site of San Pietro CapofiumeIn August 2004, the Hydro–Meteo–Climate service of the

Regional Agency for Environmental Protection (ARPA) ofthe Emilia Romagna region installed a TDR system for soilwater content measurements at the experimental station ofSan Pietro Capofiume (Bologna, Italy) (Fig. 1). A TDR sys-tem with seven probes at 0.10, 0.25, 0.45, 0.70, 1.0, 1.35,and 1.8 m was installed. The site is typical of the agriculturalarea of Po River valley, and the soil surface is covered bynatural grass.The soil characteristics of the sample site, which were de-

rived from the pedological, chemical, and physical analysesof the soil profile, are reported by Tomei et al. (2007).Briefly, the stratigraphy can be considered as composed offour layers: in the first layer, from the ground level up to adepth of 0.80 m, sandy soil is prevalent; from 0.80 m up to1.20 m, the soil can be classified as clayey–silt; from 1.20 mup to 1.65 m, the soil is silty with a relatively small percent-age of clay; finally, from 1.65 m up to 2.0 m, the soil is prev-alently composed of sand. In this study, only four pointsbelonging to the first layer, respectively, at 0.10, 0.25, 0.45,and 0.70 m from ground level, have been considered. Table 1summarizes the soil characteristics at these four points, whichare representative of the first soil layer.

796 Can. Geotech. J. Vol. 48, 2011

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Sample site of OzzanoThe other two soil water content time series came from the

site of Ozzano, in the Emilian Apennine. The site is locatedin the Centonara watershed (44°24′N, 11°28′E), southeast ofBologna, Italy (Fig. 1). The elevation at the study site is200 m above sea level (a.s.l.). The geology is a fresh alluvialdeposition of the upper Pleistocene, with undifferentiatedclay moraine and recent, yellow sand (Farabegoli et al.1994). The geological settings of the area create stratigraphicdiscontinuities (Farabegoli et al. 1994), potentially facilitatinglocal perched water tables. In the same area, two soil mois-ture experimental stations were installed in two different sam-ple sites, having very distinct soil properties. The first site,which will be identified as Ozzano-C (Ozzano clay), is char-acterized by a shallow soil, from ground level down to 0.8 m,with a high percentage of clay, as reported in Table 1. At thesecond site, which will be identified as Ozzano-S (Ozzanosand), the shallow soil layer, from ground level down to0.6 m, is characterized by a prevalent fraction of sand, butalso a relatively high fraction of silt is present (Table 1). Thetwo sites are also characterized by different vegetation cover-age. The area at the Ozzano-C site is covered by a dense nat-ural vegetation (various species of shrubs and herbaceousplants), typical of the natural settings of the Apennine Moun-tains, resulting in high plant water uptake, with roots that canreach down to 1.5 m depth. The area at the Ozzano-S site isa maize-cultivated area. Detailed information about the exper-imental sites can be found in Pieri et al. (2007), Bittelli et al.(2008), and Pistocchi et al. (2008).

Field measurementsThe experimental measurements consisted of daily volu-

metric water content (q) at different depths and for a periodof almost 5 years: August 2004 – February 2009, with alack of measurements between August 2006 and July 2007,for the site at San Pietro Capofiume; January 2005 – Decem-ber 2009, with a lack of measurements between December2007 and August 2008, for Ozzano-C; February 2005 – De-cember 2009, with a lack of measurements between Septem-ber 2007 and December 2007, for Ozzano-S. The lack ofexperimental measurements were due to a bad functioning ofthe field devices.For the aim of our analysis, the daily effective saturation of

the soil (Se) was calculated on the basis of the measuredvolumetric water content by using the well-known equation:

½1� Se ¼ q � qr

qs � qr

where q is the volumetric water content, qr is the residualwater content, which is an empirical parameter, while qs isthe saturation water content, which represents the maximumvolumetric water content. In the present work, for simplicity,qr is assumed equal to zero. Considering the negligible effectof the approximation qr = 0, the degree of saturation (Sr) isconsidered equal to the effective saturation (Se).Figure 2 shows an example of the trend of the measured

degree of saturation at a depth of 0.10 m from the groundlevel at San Pietro Capofiume site for the period betweenAugust 2004 and July 2006.

Fig. 1. Localization of the sample sites where soil moisture experimental stations are installed: (a) map of Italy; (b) San Pietro Capofiume andOzzano dell’Emilia (Emilia Romagna region, northern Italy).

Table 1. Soil characteristics in the sample sites.

Sample siteDepth(m) ID

Sand(%)

Silt(%)

Clay(%) n

San Pietro Capofiume 0.10 SPC_10 62 24 14 0.4100.25 SPC_25 61 23 16 0.3800.45 SPC_45 67 18 15 0.3700.70 SPC_70 74 18 8 0.420

Ozzano-C 0.20 OZZ_C_20 16 24 60 0.6000.80 OZZ_C_80 15 27 58 0.610

Ozzano-S 0.10 OZZ_S_10 42 33 25 0.4100.30 OZZ_S_30 49 35 16 0.4250.60 OZZ_S_60 57 27 16 0.415

Note: ID, identification; SPC, San Pietro Capofiume; OZZ_C, Ozzano clay; OZZ_S, Ozzanosand; n, soil porosity.

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Figure 3 shows the trend of the air temperature measuredat the sample site of San Pietro Capofiume, as a function oftime, for the same period. The temperatures ranged betweena maximum of 35.7 °C in summer and a minimum of –3.4 °Cin winter, and their trend is clearly seasonal. For the SanPietro Capofiume sample site, daily groundwater tabledepths from the ground level were also available for the pe-riod considered, and their trend is reported in Fig. 4. It isclear that the four points analyzed in this study were abovethe groundwater table all the time. It is worth noting thatthe water table in this area is partially controlled by an irri-gation channel, whose level is determined by irrigationmanagement, although the experimental area is not irrigated.

Simplified modelIt is well known that a certain value of soil moisture is de-

termined by many complex physical mechanisms, such as in-filtration, redistribution, evapotranspiration, runoff, and deeppercolation. All these mechanisms are deeply influenced bymeteorological forcing, such as air temperature, wind speed,air relative humidity, and incident solar radiation, and can bereasonably described in terms of energy balance, taking intoaccount heat fluxes between soil and atmosphere. By affect-ing water and energy flows between the land surface and theatmosphere, the thermal and hydrological status of the soil isstrongly connected with atmospheric variations in severalways. A large number of land surface schemes (LSSs) havebeen designed to simulate land surface processes in a varietyof numerical ways. Such models include all the importantprocesses but might emphasize different ones, depending ontheir specific goals (Milly 1996; Luo et al. 2003).The model, which will be described in the following, may

be considered a simplified method aimed at obtaining the de-gree of saturation on the basis of the soil features and thewidely available micrometeorological observations, such asrainfall amount and air temperature. It has been planned toallow a simple and computationally fast evaluation of the de-gree of saturation, without introducing a large number of pa-rameters.We suppose that Sr could be expressed through the sum of

two main contributions, which for simplicity could bethought independent of each other: the first one is a functionof air temperature (T) (in °C) as a change in time; the secondone, which takes into account infiltration, is function of rain-fall depth (h), changing in time as well.

½2� Sr ¼ SrðTÞ þ SrðhÞGiven that T and h depend on time, both terms of eq. [2]

are also indirectly dependent on time (t). It is well knownthat a shallow soil undergoes drying and wetting processesthat are determined by changes in complex climate processes.Moreover, rainfalls, and consequently water infiltration, andair temperature are not really independent from each other,but in this simplified model the two contributions are bothaccounted explicitly. We propose to express the first part ofeq. [2] through an exponential law in which the exponentcan be represented by the air temperature, which in turn is afunction of time. Such a law allows reproducing the seasonaltrend of degree of saturation following the mean temperaturefluctuations. It must be considered, indeed, that the effect of

air temperature on soil moisture is an indirect effect becauseair temperature is a key variable in the evapotranspirationprocess. We can say that, globally, air temperature could beconsidered a good proxy for seasonal climate processes thataffect soil moisture, as suggested by the comparison betweenfield measurements shown in Figs. 2 and 3. Moreover, the airtemperature considered in the model is not at a level close tothe ground surface, where air temperature can be influencedin turn by soil moisture.Starting from an initial value of the degree of saturation Sr(in),

the exponential law that represents the first term of eq. [2]can be written as follows:

½3� SrðTÞ ¼ SrðinÞ expð�jTÞwhere the coefficient j (whose dimension is the inverse of atemperature, i.e., in this case °C–1) assumes the meaning of anumerical damping, and Sr(in) is a calibration parameterlinked to the initial state of the soil. It is expected that para-meter j should be correlated to the type of soil: in particular,j should assume higher values for sand than for clay or silt,since each soil type is characterized by a specific volumetricheat capacity and thermal conductivity (Farouki 1986; Wuand Nofziger 1999). Then, it can be supposed that the re-sponse, in terms of moisture, to temperature fluctuations ismore rapid for sands than for clays. Moreover, it is expectedthat j should be a constant, characterizing the decrease inthe effect given by air temperature with an increase in dis-tance from the soil surface.The value of Sr expressed by eq. [3] represents the daily

value of Sr without rainfall. Once the value of Sr(in) is as-sumed, Sr is updated daily according to the value of the dailytemperature T. It has been considered that soil response to airtemperature fluctuations is very loose, and the sign of corre-lation between the two variables can change, depending onclimatic conditions. The term given by eq. [3] predicts a cy-clic seasonal increase or decrease of degree of saturation.This cyclic fluctuation is due partly to evaporation, which isresponsible for the decrease of Sr, and partly to fluctuation ofthe water table. This seems consistent with experimental evi-dence: as an example, for one of the investigated samplesites, the fluctuation of the groundwater table depicted inFig. 4 is in phase with the field measurements of Sr inFig. 2, and both of them are in phase with the climatic cyclesummarized by the variation of temperature (Fig. 3). Indeed,since the water table reaches the maximum level of about1.0 m below the ground level, Sr of the soil layer down to0.8 m can be affected only by capillary effects.For computation, T is calculated as a mean of average

daily air temperatures on a previous time period, whoselength has been set at 30 days after a procedure of calibra-tion, as will be explained in the following. This choice ispartly because, as drying speeds are strongly related at sea-sonal scale, averaging allows the elimination of fluctuationsat shorter time scales, which can introduce unnecessarynoise.Moreover, evaporation from frozen soils is negligible, due

to the formation of ice both in the pores and in a thin layer ofthe top soil, as confirmed by experimental measurements inlaboratory soil columns (Caruso and Jommi 2010) and bythe analysis of heat fluxes in shallow soils (Kane et al.2001). Therefore, at T < 0 °C, when reasonably we can have



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frozen soils close to the soil surface, the rate of drying of thesoil can be assumed negligible. For this reason, when the cal-culated mean air temperature is negative, the model assumesj = 0.With regard to the second term of eq. [2], it is well known

that SrnH is the water contained in the depth, H, of a soil perunit area of ground (Rodriguez-Iturbe and Porporato 2004),where n is the soil porosity. Our interest is to evaluate thedegree of saturation after a rainfall event that can be de-scribed in terms of rainfall depth (h). Then, the average in-crease of Sr over the entire soil column of height, H, can beevaluated as follows:

½4� Sr ¼ S0 þ b�h�

nH

where S0 is the initial degree of saturation, before the rainfallevent, while b*h* is the contribution to the degree of satura-tion given by rainfall. It could be considered that the totalrainfall amount does not completely infiltrate into the soil,prevalently due to runoff. For this reason, in eq. [4], the por-tion of infiltrated rainfall, h*, has been considered and is de-fined as follows:

½5� h� ¼ h if h < ð1� SrÞnHh� ¼ ð1� SrÞnH otherwise

It is well known that runoff depends strongly on the exist-ing soil moisture at the time of the rainfall event because indryer conditions a higher amount of soil storage volume isavailable to be filled before runoff starts (Pistocchi et al.2008). But it is also true that for the kind of soil consideredin this study, a constant value of runoff coefficient can be de-termined (Pistocchi et al. 2008). On the basis of these consid-erations, it has been assumed that only a portion of theinfiltrated rainfall works in raising the degree of saturationof the soil. This portion of rainfall is expressed by the coeffi-cient b*, which can be reasonably considered as a constantcalibration coefficient that includes, for a certain soil and inthe same conditions, both runoff and leakage. The evapora-tion that affects rainfall immediately before infiltration, whichcan be large when air temperature is relatively high, is disre-garded in this term, but it is taken into account indirectly ineq. [3].If Sr is calculated by considering a time interval equal to

1 day, then h is the daily cumulated rainfall depth. But themost interesting aspect for a direct correlation between thedegree of saturation and rainfalls consists of obtaining a dy-namic evaluation of Sr as a function of time, over a periodmuch longer than 1 day, including at least one season.Moreover, it could be considered that the quota of the de-

gree of saturation given by a rainfall depth (h), which can becomputed through the second term of eq. [4], decreases dur-ing time as a consequence of a natural drying process orevapotranspiration. As stated by Saxton and Lenz (1967),the greater the time lapse between a rainfall event and agiven day, the less influence the rain has on the soil moisturecontent of that day. This reduction of influence can be ob-tained by a decreasing exponential relation in which the ex-ponent is represented by a numerical parameter. Therefore,by considering eq. [4], Sr(h) can be expressed as a modifica-tion of the antecedent precipitation index (API) model (Saxton

and Lenz 1967); then, by expressing explicitly also the de-pendence on time (t), Sr(h) becomes as follows:

½6� SrðhÞ ¼ b�h�

nHexp½�xðt � t0Þ�

In eq. [6], the numerical coefficient x assumes the meaningof a damping coefficient (whose dimension is the inverse oftime, i.e., in this case day–1), and t0 is the starting time inter-val, i.e., the first day of computation. It is expected that pa-rameter x should be different for each type of soil in relationto soil permeability, since a higher hydraulic conductivity al-lows a more rapid decrease of accumulated water. On theother hand, it must be considered that the rainfall infiltrationmechanism is deeply affected by macroporosity, which mayhave an opposite effect with respect to the saturated soil per-meability. Moreover, it is expected that parameter x shouldcharacterize the decrease in water discharge with an increasein distance from the soil surface. In fact, the considered shal-low soils are characterized by a relatively high permeability,due to fissures, gaps, and channels, which tend to close withdepth and obstruct the rapid transport of surface water togroundwater.In this way, one obtains the definition of Sr as a function

variable over time correlated to the amount of rainfall (h)within the time interval (t – t0) (in days).If instead of representing an instantaneous (or daily) varia-

tion of Sr we want to obtain a dynamic trend, consideringthat the variation of Sr at each time interval depends on pre-vious rainfalls, the function Sr(h) could be rewritten as thesum of a succession of terms corresponding to different timeintervals (t – ti). In other words, the expression of Sr(h) canbe discretized and finally expressed as a sum of terms includ-ing the rainfall depth, hi, over the previous time intervals, u,which correspond to the prior days. In short, under the as-sumption that porosity and fabric of the soil remain constantwith time, eq. [6] results as follows:

½7� SrðhÞ ¼ b�

nH

Xu

i¼1

h�i exp½�xðt � tiÞ�

By substituting eqs. [3] and [7] into eq. [2], one can derivethe following expression for Sr:

½8� Sr ¼ SrðinÞ expð�jTÞ þ b�

nH

Xu

i¼1

h�i exp½�xðt � tiÞ�

As said before, Sr represents the average degree of satura-tion over a soil column of height H, but after a separate cali-bration for each depth where the degree of saturation iscomputed, it can be considered the local value at depth H.Although it has been obtained empirically, eq. [8] allows arapid and simple evaluation of the degree of saturation of ashallow soil as a function of the rainfall depth and of the airtemperature during time.

Results and discussionThe model described in the section “Simplified model” has

been applied to the sample sites of San Pietro Capofiume,Ozzano-C, and Ozzano-S to compare the computed resultswith those obtained from the field measurements.



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The soil porosity (n), which has been considered constantin time, assumes the values reported in Table 1, on the basisof both laboratory analyses on soil samples, collected at dif-ferent depths in the three sites, and experimental measure-ment of the maximum volumetric water content qs (Tomei etal. 2007; Polenghi 2009). The porosity values are character-istic of the type of soils that are present at each point(Table 1) and are on average characteristic of those reportedby the scientific literature for the same kind of soils (Fred-lund and Rahardjo 1993; Lancellotta 2004).

Parameters j, x, b*, and Sr(in) have been determinedthrough a procedure of adjustment so as to get the best fitbetween the measured values of Sr and those calculated fromrainfall by using eq. [8]. Moreover, the parameters have beenchosen to avoid the prediction of the degree of saturationabove 1, which is incorrect from a physical point of view,but possible given the mathematical form of the model.In particular, parameter b* has been assumed for each

depth according to the values reported in Table 2 over theentire period considered, disregarding the intensity of rainand the current degree of saturation of the soil. It must beemphasized that parameter b* is not partitioned betweenlayers because a separate analysis has been carried out foreach investigated soil depth. It has been assumed that theportion of infiltrated rainfall that works in raising the degreeof saturation for a thick layer is higher than for a thin one,depending on a greater volume of voids that can be filled bywater. That is why the deeper the point, the higher the valueof b* for the corresponding layer (Table 2), except for thesite Ozzano-S, where the values of b* do not show signifi-cant dependence on depth.

For all simulations, to consider the delay of the soil re-sponse to the temperature fluctuations, the input daily valueof the air temperature has been calculated as a mean of dailytemperatures over a period of 30 days. The averaging periodof 30 days, which depends on both the latitude and the cli-mate of each site, has been chosen after a procedure of cali-bration of the model. The effect of a different averaging timeperiod on the model response will be shown in the followingsection “Sensitivity analysis”. The best fitting for field meas-urements can be obtained for an averaging period of 30 days.When the calculated mean value for 30 days is negative, themodel assumes j = 0, as previously said.Figures 5–8 show the results of calculated values of the

degree of saturation, compared with experimental measure-ments at San Pietro Capofiume site. Modeled and measuredvalues of the degree of saturation are compared in terms oftrend in time, in correlation with the amount of rainfall, andin terms of coefficient of determination, for four differentpoints: at a depth of 0.10 m (Fig. 5), 0.25 m (Fig. 6),0.45 m (Fig. 7), and 0.70 m (Fig. 8) from the ground level.The parameters of the model are determined through a

curve fitting by using the first measurements’ cycle, corre-sponding to one full year, from August 2004 to July 2005(Figs. 5a, 6a, 7a, and 8a); the same parameters, except for Sr(in),are used to predict the degree of saturation in the next twomeasurements’ cycles: the following year, i.e., from August2005 to July 2006 (Figs. 5b, 6b, 7b, and 8b) and from Au-gust 2007 to February 2009 (Figs. 5c, 6c, 7c, and 8c).From August 2006 to July 2007, there were no field meas-urements.Table 2 summarizes the values of the parameters that have

been assumed for the model’s application to the sample site.Only the parameter Sr(in) needed a recalibration before thenext cycles.Figures 5–7 demonstrate the agreement between the values

of the degree of saturation obtained through the model andfield measurements, with relatively high coefficients of deter-mination for the first three layers analyzed, down to 0.45 mbelow the ground level, for both calibration and predictionanalyses. It is worth noting that, notwithstanding a fair num-

Fig. 2. Measured degree of saturation for San Pietro Capofiume siteduring the monitoring period at a depth of 0.10 m from groundlevel. Dates are shown as day/month/year.

Fig. 3. Mean daily air temperature for San Pietro Capofiume siteduring the monitoring period. Dates are shown as day/month/year.

Fig. 4. Daily depth of groundwater table from ground level at SanPietro Capofiume site. Dates are shown as day/month/year.



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ber of assumptions involved in the evaluation of Sr, there is aclose agreement between the model and experimental meas-urements. On the contrary, for the point at 0.70 m fromground level, the model seems to be less efficient, achievinga value of the coefficient of determination R2 = 0.741 onlyfor the calibration analysis (Fig. 8). This is probably due tothe effect of the groundwater fluctuations on the soil mois-ture dynamics. As said previously, the water table in thisarea is partially controlled by an irrigation channel; therefore,its depth may change independently from rainfall recharges,affecting the soil moisture at the lower depths.Table 2 also summarizes the results obtained for the four

layers after the calibration analysis (set) and the predictionanalyses (pre) in terms of the main statistical parameters.Notwithstanding a lack of experimental measurements in thefall of 2005, due to a bad functioning of the field devices, theresults referred to the shallow layers up to 0.45 m can beconsidered satisfactory, with a standard error of the estimatelower than 11.4%.With regard to the Ozzano-C site, the main results are re-

ported in Figs. 9 and 10. Also in this case, modeled andmeasured values of the degree of saturation are compared interms of trend in time, in correlation with the amount of rain-fall, for two different points: at a depth of 0.20 and 0.80 mfrom ground level. Unfortunately, a TDR probe installed at0.40 m from ground level did not provide reliable measure-ments, due to a bad functioning. The model results of thecalibration analysis, considering the period between January2005 and December 2005, for the two depths, are reported

in Figs. 9a and 10a, respectively. The results obtainedthrough the prediction analysis, considering the period be-tween January 2006 and December 2007, for the two depths,are reported in Figs. 9b and 10b, respectively. The furtherprediction analysis, considering the measurements’ cycle be-tween August 2008 and December 2009, for the two depths,are reported in Figs. 9c and 10c, respectively. Field measure-ments were missing for the period between January and July2008. The values of the input parameters, which have beenassumed for the model’s application to this sample site, andthe statistical parameters obtained are summarized in Table 2.It can be noticed how the best performance of the model cor-responds only to the shallow layer at 0.20 m from groundlevel, whilst for the deeper layer at 0.80 m the model ismuch less efficient, achieving very low values of the coeffi-cient of determination (Fig. 10).The model has been also applied to the same kind of field

data coming from the Ozzano-S sample site. Table 2 summa-rizes the values of the input parameters that have been as-sumed for the model’s application to this sample site.Figures 11–13 show the results of calculated values of thedegree of saturation, compared with experimental measure-ments, as their trend in time, in correlation with the amountof rainfall, for three different points: at a depth of 0.10 m(Fig. 11), 0.30 m (Fig. 12), and 0.60 m (Fig. 13) from theground level. In particular, the model results of the calibra-tion analysis, considering the period between the 18 February2005 and the 17 February 2006, for the three depths, are re-ported in Figs. 11a, 12a, and 13a, respectively. The results

Table 2. Values of parameters assumed for the model.

Sample site Period (day/month/year) ID j (°C–1) x (day–1) b* Sr(in) R2 s sest

San PietroCapofiume

01/08/04–31/07/05 (set) SPC_10 0.048 0.079 0.35 0.76 0.904 0.108 0.057SPC_25 0.036 0.051 0.43 0.93 0.918 0.081 0.054SPC_45 0.035 0.025 0.48 0.91 0.806 0.142 0.090SPC_70 0.024 0.022 0.63 0.92 0.741 0.114 0.084

01/08/05–31/07/06 (pre) SPC_10 0.048 0.079 0.35 0.79 0.842 0.123 0.066SPC_25 0.036 0.051 0.43 0.94 0.831 0.087 0.063SPC_45 0.035 0.025 0.48 0.98 0.817 0.082 0.062SPC_70 0.024 0.022 0.63 0.92 0.663 0.104 0.075

01/08/07–25/02/09 (pre) SPC_10 0.048 0.079 0.35 0.65 0.740 0.164 0.081SPC_25 0.036 0.051 0.43 0.79 0.739 0.150 0.090SPC_45 0.035 0.025 0.48 0.81 0.557 0.182 0.114SPC_70 0.024 0.022 0.63 0.71 0.255 0.176 0.116

Ozzano-C 01/01/05–30/12/05 (set) OZZ_C_20 0.023 0.077 0.39 0.86 0.696 0.117 0.081OZZ_C_80 0.022 0.054 0.48 0.81 0.400 0.267 0.134

01/01/06–04/12/07 (pre) OZZ_C_20 0.023 0.077 0.39 0.86 0.639 0.100 0.064OZZ_C_80 0.022 0.054 0.48 0.75 0.259 0.215 0.108

20/08/08–31/12/09 (pre) OZZ_C_20 0.023 0.077 0.39 0.88 0.748 0.093 0.065OZZ_C_80 0.022 0.054 0.48 0.78 –0.198 0.219 0.125

Ozzano-S 18/02/05–17/02/06 (set) OZZ_S_10 0.025 0.050 0.31 0.78 0.522 0.136 0.097OZZ_S_30 0.005 0.056 0.25 0.75 0.154 0.129 0.095OZZ_S_60 0.006 0.049 0.26 0.82 0.235 0.134 0.098

18/01/06–08/09/07 (pre) OZZ_S_10 0.025 0.050 0.31 0.60 0.517 0.164 0.085OZZ_S_30 0.005 0.056 0.25 0.75 0.320 0.175 0.097OZZ_S_60 0.006 0.049 0.26 0.87 0.204 0.118 0.084

15/12/07–31/12/09 (pre) OZZ_S_10 0.025 0.050 0.31 0.46 0.518 0.262 0.125OZZ_S_30 0.005 0.056 0.25 0.64 0.413 0.214 0.108OZZ_S_60 0.006 0.049 0.26 0.73 0.346 0.228 0.128

Note: set, calibration analysis; pre, prediction analyses; R2, coefficient of determination; s, standard deviation; sest, standard error of estimate.



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obtained through the first prediction analysis, considering theperiod between January 2006 and September 2007, for thethree depths, are reported in Figs. 11b, 12b, and 13b, respec-tively. To allow the model to reach a certain reliability, thefirst month of this time series overlaps the last month of theperiod previously considered. The further prediction analysis,considering the 2 year time interval between December 2007and December 2009, for the three depths, are reported inFigs. 11c, 12c, and 13c, respectively. Also, in this case, therewas a lack of field measurements corresponding to the periodbetween October and November 2007. It is worth noting that,in this case, the agreement between the model results and ex-perimental measurements is reasonable only for the analysisat 0.10 m below ground level (Fig. 11). For the analyses car-ried out at 0.30 and 0.60 m, the model appears inefficient,

with very low values of the coefficient of determination(Figs. 12 and 13).The very different performance of the model in the three

sample sites and at different depths is prevalently due to thepresence of a different kind of vegetation, which also changesduring different seasons. In fact, the effect given by the vege-tation on the degree of saturation is not taken into account bythe proposed model. At the San Pietro Capofiume site, thesoil is covered by natural grass, which is kept cut at 10 cmin height; at the Ozzano-C site, the soil is covered by densenatural vegetation (various species of shrubs and herbaceousplants); at the Ozzano-S site, the soil is covered by cultivatedmaize. It is worth noting that, in this latter case, the dryingprocess, which is evident from the field measurements duringsummer, is prevalently due to plant water uptake from the

Fig. 5. San Pietro Capofiume site. Degree of saturation (Sr) versustime at 0.10 m below ground level (SPC_10): (a) calibration analy-sis; (b), (c) prediction analyses for two different periods. Dates areshown as day/month/year. R2, coefficient of determination.

Fig. 6. San Pietro Capofiume site. Degree of saturation (Sr) versustime at 0.25 m below ground level (SPC_25): (a) calibration analy-sis; (b), (c) prediction analyses for two different periods. Dates areshown as day/month/year.



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maize plants. Plant water uptake depends on many variablessuch as air temperature, solar radiation, wind velocity, plantspecies, and roots depth. Such a complex phenomenon can-not be adequately described only by the trend of air temper-ature and then by a unique set of the model parameters. Forthis reason, especially during summer, the model is less effi-cient, and probably a different set of parameters should bechosen for different seasons. The effect of vegetation is alsoevident by comparing results obtained for a soil covered bygrass, such as at San Pietro Capofiume, and those coveredby a vegetation characterized by deep roots, such as atOzzano. On the other hand, even at Ozzano_C and atOzzano_S, the model seems to be still efficient if applied tothe shallow soil layer, up to nearly 0.20 m. Overall, the anal-yses carried out for all the sample sites reveal that the model

seems to be more suitable to reproduce the real trend of thedegree of saturation of bare soils not deeper than 0.45 mfrom ground level and, if any deep rooted vegetation ispresent, for soils not deeper than 0.20 m.

Sensitivity analysisTo clarify how the numerical parameters of the model have

been chosen, it is worthy to discuss the calibration procedure.The model requires a separate calibration for each point

where the degree of saturation is to be computed becausesoil layers at growing depths are affected in different waysby the weather conditions. Nevertheless, the main parametersseem to have a certain trend with soil depth. A sensitivityanalysis on the effect of different parameters on the computedresults has been carried out only for the sample site of San





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Pietro Capofiume, for the first two points at 0.10 and 0.25 mfrom the ground level. For each analysis, only one parameterhas been changed with respect to the data set reported in Ta-ble 2. For each parameter, two further values, one higher andone lower than the corresponding value reported in Table 2,have been considered (bold values in Table 3). For each anal-ysis, the value of the coefficient of determination R2 was alsocalculated (Table 3).Figures 14–18 show the results of the model application,

changing the parameters one at a time, both to the layers0.10 and 0.25 m deep. In particular, Fig. 14 shows the effectof different values of parameter j, and Fig. 15 shows the ef-fect of two values of parameter x. The effects of two valuesof parameters b* and Sr(in) are shown in Figs. 16 and 17, re-

spectively. In all cases, the chosen values for the input pa-rameters summarized in Table 2 result in higher values ofthe coefficient of determination R2.A further sensitivity analysis regards the time interval con-

sidered for the calculation of the average daily air tempera-tures T(t). Figures 18a and 18b show the results obtained byconsidering a time interval equal to 1 and 60 days for thepoints at 0.10 and 0.25 m, respectively. It appears that atime scale equal to 1 day is too short and shows an unrealis-tic fluctuation of the degree of saturation; on the other hand,a wider time scale of 60 days implies a trend of temperaturesthat is too smooth. Moreover, as the R2 value obtained for atime scale equal to 30 days was the highest, this can be con-sidered the optimum period recommended at the latitude of

Fig. 9. Ozzano-C site. Degree of saturation (Sr) versus time at0.20 m below ground level (OZZ_C_20): (a) calibration analysis;(b), (c) prediction analyses for two different periods. Dates areshown as day/month/year.

Fig. 10. Ozzano-C site. Degree of saturation (Sr) versus time at0.80 m below ground level (OZZ_C_80): (a) calibration analysis;(b), (c) prediction analyses for two different periods. Dates areshown as day/month/year.



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the investigated sample sites. It can be noticed that the fur-ther each parameter is from the mean chosen value (Table 2),the smaller the value of R2 (Table 3).To give some indication about the appropriate values of

different parameters of the model, it seems useful to com-ment about the values reported in Table 2. By comparing theresults obtained, it can be noticed, for example, how the pa-rameter j decreases with depth, as expected, and belongs toa rather narrow range of values, approximately between0.005 and 0.05, which can be considered to be bounded by alower and an upper limit. The trend of j values for San Pie-tro Capofiume site, where the soil was prevalently sandy,seems to form an upper limit, while the trend of j valuesfor Ozzano-S seems to form the lower limit. It is useful toremember that at Ozzano-S, both silt and sand were preva-

lent. Indeed, the values assumed for parameter j at the siteOzzano-C, where clay was undoubtedly prevalent, are not sofar from those assumed for the other two sites. The slight de-crease of j with depth allows the model to take into account,in a certain way, the decreasing influence, with depth, of airtemperature on the degree of saturation.Table 2 also summarizes the values of parameter x that

have been assumed at different depths for the three samplesites. Even in this case, parameter x can be consideredslightly decreasing with depth and belonging to a rather nar-row range of values, included between two trends, represent-ing a lower and an upper limit. In particular, the lower limitis given by the trend of the values corresponding to a sandysoil (San Pietro Capofiume site), while the upper limit isgiven by the trend of the values corresponding to a clayey

Fig. 11. Ozzano-S site. Degree of saturation (Sr) versus time at0.10 m below ground level (OZZ_S_10): (a) calibration analysis;(b), (c) prediction analyses for two different periods. Dates areshown as day/month/year.




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soil (Ozzano-C). The values of parameter x for a mixed soil(Ozzano-S) seem to be between the two limits, even thoughin this case different evaporation phenomena affected bymacroporosity must be considered, as previously said.With regard to the values of parameter b*, the reason for

their increase with depth has been previously explained. Acertain correspondence with a particular kind of soil can bealso drawn by the trend of the values reported in Table 2.

Concluding remarks

This paper deals with the in situ measurements of soilwater content and the description of a simplified empiricalmodel, which allows one to obtain the degree of saturationof soil as a function of available climate data such as air tem-perature and rainfall depths.

The field measurements of the soil water content comefrom three sample sites in Emilia Romagna region, northernItaly. They were compared with the results of the simplifiedmodel over an observation period of almost 1 year to obtain


Table 3. Set of parameters used for the sensitivity analysis for SanPietro Capofiume site.

ID j (°C–1) x (day–1) b* Sr(in) t (days) R2

SPC_10_set 0.028 0.079 0.35 0.76 30 0.5920.068 0.079 0.35 0.76 30 0.7450.048 0.029 0.35 0.76 30 0.2310.048 0.129 0.35 0.76 30 0.8160.048 0.079 0.15 0.76 30 0.6880.048 0.079 0.55 0.76 30 0.7630.048 0.079 0.35 0.61 30 0.6630.048 0.079 0.35 0.91 30 0.6960.048 0.079 0.35 0.76 1 0.7410.048 0.079 0.35 0.76 60 0.855

SPC_25_set 0.016 0.051 0.43 0.93 30 0.2500.056 0.051 0.43 0.93 30 0.5730.036 0.021 0.43 0.93 30 0.6820.036 0.081 0.43 0.93 30 0.8690.036 0.051 0.23 0.93 30 0.8310.036 0.051 0.63 0.93 30 0.8620.036 0.051 0.43 0.87 30 0.8830.036 0.051 0.43 0.99 30 0.8880.036 0.051 0.43 0.93 1 0.7100.036 0.051 0.43 0.93 60 0.893

Fig. 14. Effect of different values of parameter j on the trend of thedegree of saturation (Sr) during the period used for the calibrationanalysis: (a) at 0.10 m below ground level (SPC_10); (b) at 0.25 mbelow ground level (SPC_25). Dates are shown as day/month/year.



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Fig. 15. Effect of different values of parameter x on the trend of thedegree of saturation (Sr) during the period used for the calibrationanalysis: (a) at 0.10 m below ground level (SPC_10); (b) at 0.25 mbelow ground level (SPC_25). Dates are shown as day/month/year.

Fig. 16. Effect of different values of parameter b* on the trend ofthe degree of saturation (Sr) during the period used for the calibra-tion analysis: (a) at 0.10 m below ground level (SPC_10); (b) at0.25 m below ground level (SPC_25). Dates are shown as day/month/year.

Fig. 17. Effect of different values of parameter Sr(in) on the trend ofthe degree of saturation (Sr) during the period used for the calibra-tion analysis: (a) at 0.10 m below ground level (SPC_10); (b) at0.25 m below ground level (SPC_25). Dates are shown as day/month/year.

Fig. 18. Effect of different values of time interval (t), considered foraveraging air temperature, on the trend of the degree of saturation(Sr) during the period used for the calibration analysis: (a) at 0.10 mbelow ground level (SPC_10); (b) at 0.25 m below ground level(SPC_25). Dates are shown as day/month/year.



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a reasonable calibration of the model parameters. Then, thesame parameters were used to carry on a prediction analysisfor two further time intervals for all sample sites.The model seems to be suitable to simulate complete, mul-

tiple annual cycles of field measurements of water content inthe unsaturated zone, at different depths with respect to theground level. In particular, it seems to be more suitable forshallow layers, up to 0.45 m below ground level, than fordeeper ones. At two of the three sample sites, the phenom-enon of plant water uptake, given by the presence of deeplyrooted vegetation such as shrubs or maize, led to the obser-vation that the complex mechanism of evapotranspirationcannot be adequately described through a unique set of themodel parameters. For this reason, the model performance,in the proposed formulation, is not suitable for deep soillayers (more than 0.20 m) of vegetated soils.A discussion on the calibration of the model parameters

has been carried out to clarify how the input data have beenchosen. In particular, it has been pointed out how the modelrequires a separate calibration for each point where the de-gree of saturation is to be computed. An attempt to givesome indication about the appropriate values attached to dif-ferent parameters of the model, according to different kindsof soil and depths, has been also made. The authors areaware that, notwithstanding the promising results, the modelproposed requires more tests with experimental data fromseveral sites for further validation and for an effective appli-cation on different scales.

AcknowledgementsExperimental data of San Pietro Capofiume site were pro-

vided by the Hydro–Meteo–Climate service of the RegionalAgency for Environmental Protection (ARPA) of the EmiliaRomagna region. The authors would like to express theirgratitude to Dr. Fausto Tomei and Dr. Fiorenzo Salvatorellifor their cooperation.

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An empirical model for the evaluation of the degree of saturation of shallow...

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