Effects of Density and Viscosity in Modeling Heatas a Groundwater Tracerby Rui Ma1,2 and Chunmiao Zheng3

AbstractDecoupled simulation of groundwater flow and heat transport assuming constant fluid density and viscosity

is computationally efficient and simple. However, by neglecting the effects of variable density and viscosity,numerical solution of heat transport may be inaccurate. This study investigates the conditions under whichthe density and viscosity effects on heat transport modeling can be neglected without any significant loss ofcomputational accuracy. A cross-section model of aquifer-river interactions at the Hanford 300 Area in WashingtonState was employed as the reference frame to quantify the role of fluid density and viscosity in heat transportmodeling. This was achieved by comparing the differences in simulated temperature distributions with and withoutconsidering variable density and viscosity, respectively. The differences between the two sets of simulations werefound to be minor under the complex field conditions at the Hanford 300A site. Based on the same model setup butunder different prescribed temperature gradients across the simulation domain, a series of heat transport scenarioswere further examined. When the maximum temperature difference across the simulation domain is within 15◦C,the mean discrepancy between the simulated temperature distributions with and without considering the effects ofvariable density and viscosity is approximately 2.5% with a correlation coefficient of above 0.8. Meanwhile, thespeedup in runtime is roughly 225% when the maximum temperature difference is at 15◦C. This work providessome quantitative guidelines for when heat transport may be simulated by assuming constant density and viscosityas a reasonable compromise between accuracy and efficiency.

IntroductionHeat can be used as a groundwater tracer to solve the

inverse problem such as site characterization for hetero-geneous aquifers jointly with head and solute transportcalibration (Anderson 2005). For example, Bravo et al.(2002) used groundwater temperature data to supplementhead data to further constrain parameter estimation in

1Department of Geological Sciences, University of Alabama,Tuscaloosa, AL.

2MOE Laboratory of Biogeology and Environmental Geology,China University of Geosciences, Wuhan, China.

3Corresponding author: Department of Geological Sciences,University of Alabama, Tuscaloosa, AL; (205) 348-0579;fax: (205) 348-0818; czheng@ua.edu

Received June 2009, accepted November 2009.Copyright © 2009 The Author(s)Journal compilation ©2009NationalGroundWaterAssociation.doi: 10.1111/j.1745-6584.2009.00660.x

a groundwater flow model of a wetland system. Jiangand Woodbury (2006) combined groundwater flow andheat transport to improve the logarithm transmissivityestimates for a test case. Heat is also a good tracer todetect the interactions between groundwater and surfacewater, estimate groundwater recharge and discharge, anddelineate flow patterns and flow velocities in ground-water basins (Cartwright 1974; Constantz et al. 2003;Stonestrom and Constantz 2003; Conant 2004; Hatchet al. 2006; Lowry et al. 2007). For example, the workby Constantz et al. (2002) provided a detailed examina-tion of the basis for using heat as a tracer of stream andgroundwater exchanges and discussed several types oftemperature analysis techniques to determine streambedpercolation rates. Becker et al. (2004) quantified thegroundwater/surface water exchange along stream reachesusing heat transport modeling of measured temperaturegradients below the streambed in southwestern New YorkState. Due to the importance of heat as a groundwater

380 Vol. 48, No. 3–GROUND WATER–May-June 2010 (pages 380–389) NGWA.org

tracer along with the recent availability of improved tem-perature sensors and relatively inexpensive data loggers,as well as improved numerical codes for heat transportsimulation, using heat as a groundwater tracer has spurredrenewed and widespread interests (Anderson 2005).

In a groundwater flow system, fluid density andviscosity will vary spatially and temporally when thetemperature changes. Density differences cause buoyancyeffects, thus producing additional fluid motion. Densitygradients can introduce gravitational instabilities thatgive rise to recirculating groundwater systems, whilethe viscosity only alters the resistance to groundwaterflow (Phillips 1991). In a thermally driven flow field,the dimensionless Rayleigh number (Ra) is defined bythe ratio of buoyancy to viscous forces, and is used toquantify the potential for free convection, that is, heattransfer in response to flow driven by temperature-induceddensity differences (Anderson 2005). The free convectionappears in an infinitely extensive horizontal layer if acertain Rayleigh number of 4π2 is exceeded (Phillips1991; Kolditz et al. 1998; Simmons et al. 2001; Simmons2005). The dimensionless Peclet number is given by theratio of convection to conduction, analogous to that ofadvection to dispersion in solute transport studies, andis commonly used to assess the potential for forcedconvection as opposed to free convection (Phillips 1991;Anderson 2005).

A majority of hydrogeological studies assumed thatthe variation in density and viscosity across the flow fieldin a natural environment is small so that their effect ongroundwater flow may be considered negligible. Undersuch circumstances, the small thermal gradients in theaquifer can allow for the decoupling of the flow andheat transport equations. Moreover, the decoupling ofthe flow and heat transport will simplify the reactivetransport simulation affected by temperature changes(Prommer and Stuyfzand 2005). However, strong thermalcontrasts exist in some cases (e.g., deep well wasteinjection, geothermal systems), where temperature willplay a more important role in groundwater flow andsolute transport processes. In these cases, fluid densityand viscosity may be strongly affected by the temperaturedistribution, and heat transport needs to be consideredthrough coupled models that update groundwater flowvelocities to reflect different density and viscosity. Suchcoupled models usually require more computational timethan decoupled models where the groundwater flowvelocities are assumed to be independent of densityand viscosity. Thus, a question of significant interest isunder what conditions the effects of fluid density andviscosity on heat transport can be neglected withoutcausing excessive numerical errors.

This paper utilizes a cross-section model of aquifer-river interactions at the Hanford 300 Area in WashingtonState as the reference frame for comparison of heat trans-port simulation results that consider the effects of variabledensity and viscosity with those under the assumptionof constant density and viscosity. After it is shownthat the temperature distribution under the complex field

conditions can be approximated reasonably by numericalmodels, a series of heat transport scenarios with differ-ent prescribed temperature gradients across the flow fieldwere examined to quantify the discrepancy and correlationbetween the simulation results that consider and neglectthe density and viscosity effects, respectively.

Under the assumption of constant fluid density andviscosity, groundwater flow simulation is carried out usingthe MODFLOW code (Harbaugh et al. 2000), while heattransport simulation is conducted using the MT3DMScode (Zheng and Wang 1999; Zheng 2006). With thedecoupled approach, flow simulation is completed inde-pendent of heat simulation for greater computational effi-ciency. Heat transport under variable density and viscosityis simulated by SEAWAT Version 4 (Langevin et al.2007), which couples MODFLOW and MT3DMS forsimultaneous solutions of the flow and heat transportequations so that the effects of changing density and vis-cosity can be considered.

Using Solute Transport Simulator MT3DMSfor Heat Transport Modeling

Solute transport in aquifers is generally controlled bygroundwater advection, hydrodynamic dispersion, chemi-cal reactions, and sink/source mixing, while heat transportin aquifers is usually governed by heat convection alongwith the fluid phase, heat conduction in the aquifer sed-iments, and heat exchange between the aqueous phaseand the aquifer sediments. Due to the mathematical sim-ilarities between heat and solute transport, the multi-species transport model MT3DMS, originally developedfor solute transport, can be used in its present form tosimulate heat transport.

Several studies have already used the analogy betw-een solute and energy transport to apply MT3DMS torepresent heat migration in the subsurface. For example,Sethi and Molfetta (2007) used MT3DMS to simulateheat transport in an aquifer downgradient of a municipalsolid waste landfill in Italy to search for the best heatboundary condition to reproduce the thermal anomalyin the aquifer due to aerobic and anaerobic exothermicreactions occurring inside the waste. Bayer et al. (2008)and Hecht-Mendez et al. (in press) verified the MT3DMScode for heat transport simulation of closed shallowgeothermal systems by comparing the simulation resultsobtained by MT3DMS with those obtained by analyticalsolutions and the finite-element code FEFLOW.

Based on the analogy between solute transport andheat transport equations, variables for the MT3DMSsolute transport simulator can be reinterpreted for heattransport. The solute concentration becomes the tempera-ture. The distribution coefficient (Kd) for solute transportis replaced by the thermal distribution factor which canbe calculated for temperature using the following equation(Thorne et al. 2006; Langevin et al. 2007):

Kd = cs

cwρw(1)

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where ρw is the fluid density, cw is the specific heat capac-ity of the fluid, and cs is the specific heat capacity of thesolid.

The molecular diffusion in solute transport can berepresented by the thermal conduction in heat transport.Thus, the molecular diffusion coefficient (Dm) reinter-preted for heat transport is as follows (Thorne et al. 2006;Langevin et al. 2007):

Dm = κo

θρwcw(2)

where θ is porosity of the subsurface medium; and κo

is the bulk thermal conductivity of the rock matrix. Thethermal conduction term is also referred to as the bulkthermal diffusivity.

More detailed information on using solute transportsimulation for heat transport modeling can be found inKim et al. (2005), Langevin et al. (2009), and Hecht-Mendez et al. (in press). Additional information specificto the use of the MT3DMS code for heat transport isavailable in Zheng (2009).

Comparison of Field Applications withand without Density and Viscosity EffectsGroundwater Flow Model

A cross-section model of aquifer-river interactions isused in this work as the reference frame to evaluate theeffects of density and viscosity on heat transport model-ing under field conditions. The cross section is located inthe 300 Area (300A) of the U.S. Department of Energy(DOE) Hanford Site along the Columbia River in south-eastern Washington State. The hydrogeological settingof the cross section is shown in Figure 1 adapted fromMa et al. (2009). The unconfined aquifer is comprised ofhighly permeable Hanford Formation gravels and sands,underlain by upper Ringold Formation fine-grained sedi-ments. The unconfined aquifer is in hydrologic continuitywith the Columbia River which flows along the easternmargin of the 300A site. Generally, the river stage is highin the spring and early- to mid-summer, and low in the falland winter, superimposed by daily variations. The riverstage fluctuation causes seasonal and daily variations inwell water levels throughout the 300 Area (Zachara et al.2005; Williams et al. 2008), and is responsible for spa-tial and temporal differences in temperature and waterchemistry as a result of groundwater-river water mixing.

Three geologic units, namely Hanford Formation,Ringold Fine-grained Unit, and Ringold Unit 5, areincluded in the cross-section model (Figure 1). The Han-ford Formation consists of unconsolidated pebble-to-boulder gravels and fine-to-coarse sands, with occasionalfine-textured intercalations and has very high hydraulicconductivity, generally greater than 2000 m/d (Williamset al. 2007). Ringold Fine-grained Unit is a thick fine-grained silty sand interval and Ringold Formation Unit 5is composed predominantly of a fluvial fine-grained silt tosand interval, and a fluvial gravel to silty sandy gravel unit(Williams et al. 2007, 2008). Combined, these latter twounits comprise the lower and significantly less permeableportion of the unconfined aquifer beneath the 300A.

Groundwater flow along the cross section was simu-lated using the MODFLOW code (Harbaugh et al. 2000).The model grid consists of 272 columns and 29 layerswith irregular grid spacing ranging from 2 to 4 m inthe horizontal direction and 0.5 to 3 m in the verticaldirection.

The effect of recharge on the groundwater flowfield is negligible due to high permeability of HanfordFormation and a small recharge rate. Thus, the topboundary was treated as no-flow. The bottom boundarywas also treated as no-flow. The western boundary wassimulated as specified-head with measured hourly waterlevels in well 399-3-19 for a one-year period (October 27,2007, to October 27, 2008). The eastern boundary wassimulated using the MODFLOW River package withhourly river stages during the same period from a streamgauge (SWS-1) near the location where the cross sectionintercepts the Columbia River. According to the work ofZachara et al. (2005) and Yabusaki et al. (2008), a hourlytime interval is necessary to discretize the time-varyingboundary condition due to the highly frequent fluctuationof Columbia River. The flow model was spun-up toeliminate artificial transients resulting from an arbitraryset of initial head conditions as described by Ma et al.(2009). The flow model input parameters are listed inTable 1.

Heat Transport ModelFor the heat transport model, the western boundary

was prescribed by specified-temperature at an hourlyinterval in groundwater of well 399-3-19 and the easternboundary was also treated as specified-temperature atan hourly interval with river water temperature. Theinitial temperature in groundwater was interpolated by

Figure 1. Hydrostratigraphic units and the location of monitoring wells in the cross-section model at the Hanford 300A site.

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Table 1Input Parameters for the Groundwater Flow Model

Aquifer Unit

Hanford RingoldParameters Formation Fine-Grained Unit Ringold Unit 5

Kh (m/d) 7000 1 40Kz (m/d) 700 0.1 4Effective porosity 0.18 0.15 0.15Specific storage (1/m) 10−5 10−5 10−5

Specific yield 0.2 n/a n/a

Note: Kh, horizontal hydraulic conductivity; Kz, vertical hydraulic conductivity.

the measured temperature in wells. Figure 2 shows thetemperature changes in wells 399-3-19, 399-2-5, and 399-2-1 from October 27, 2007, to October 27, 2008, andin river water from October 27, 2005, to October 27,2006. Because of the lack of river water and groundwatertemperature data during an exactly same annual cycle,the river temperature data from October 27, 2005, toOctober 27, 2006, were used to approximate those forthe same dates two years later. An analysis of consecutivefour years’ temperature data records of Columbia River atthe gauge SWS-1 from June 2004 to March 2008 revealeda very similar annual pattern in river water temperaturevariations in terms of both timing and magnitude. Thus,the river temperature in an earlier year is considered areasonable proxy for that in a later year. It is noteworthythat the intent of this study was not detailed calibrationof a completely site-specific model. Rather, the primarygoal was to construct a realistic and reasonable heattransport model that can be used subsequently to evaluatehypothetical scenarios and establish general guidelineson the importance of density and viscosity effects ontemperature simulation.

The input parameters for the heat transport modelare listed in Table 2. According to Williams et al. (2008),the solid density for Hanford Formation and RingoldFormation is 2760 and 2650 kg/m3, respectively, and the

total porosity for Hanford, Ringold Fine-grained Unit,and Ringold Unit 5 is 0.2, 0.43, and 0.25, respectively.Based on the reference values of specific heat for Hanfordsediments as summarized by Ward (2007) for use at a sitein the Hanford 200 West Area and considering the massfraction of grain size distributions at the Hanford 300Asite, the specific heat capacity of the Hanford Formationused in this study is given in Table 2. The RingoldUnit specific heat capacity was estimated from referencedata on the basis of its lithology (Dim et al. 2002;Stonestrom and Constantz 2003; Langevin et al. 2007).No measured thermal conductivity values are availablefor the Hanford Site, and the thermal conductivity ofHanford and Ringold Formations is assigned to 2.2 and1.2 W/(m·K), respectively, based on the lithology of theseformations and literature reference values (Dim et al.2002; Anderson 2005; Langevin et al. 2007). The heatcapacity and thermal conductivity of water are taken fromthe SEAWAT Version 4 manual (Langevin et al. 2007).

The heat transport was simulated by MT3DMS andSEAWAT, respectively. As mentioned above, the use ofthe MT3DMS code (Zheng and Wang 1999; Zheng 2006)for heat transport modeling is based on the assumptionof constant fluid density and viscosity so that flowand transport simulations can be decoupled. On theother hand, SEAWAT (Langevin et al. 2007), which

Figure 2. Temperature changes at wells 399-2-1, 399-2-5, and 399-3-19 from October 2007 to October 2008, and in ColumbiaRiver from October 2005 to October 2006.

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Table 2Input Parameters for the Heat Transport Model

Aquifer Unit

Parameters Hanford Formation Ringold Fine-Grained Unit Ringold Unit 5

ρw (kg/m3) 1000 1000 1000ρs (kg/m3) 2760 2650 2650cw [J/(kg·K)] 4.186 × 103 4.186 × 103 4.186 × 103

cs [J/(kg·K)] 715 920 920Dispersivity α (m) αL = 1; αV = 0.01αL αL = 1; αV = 0.01αL αL = 0.5; αV = 0.01αL

Total porosity 0.2 0.43 0.25κs [W/(m·K)] 2.2 1.2 1.2κw [W/(m·K)] 0.58 0.58 0.58Dm (m2/s) 2.2408 × 10−6 5.1856 × 10−7 9.9857 × 10−7

Kd (L/mg) 1.7081 × 10−7 2.1978 × 10−7 2.1978 × 10−7

Note: ρs, density of solid; κs and κw, thermal conductivity of the solid and fluid phase.

couples MODFLOW and MT3DMSinto a single code,can account for the effects of variable fluid density andviscosity on heat transport through an iterative procedure.

Representation of Density and Viscosity EffectsThe fluid density required in the SEAWAT simulation

was calculated as the function of the changes in soluteconcentration, temperature, and water pressure (Langevinet al. 2007). Since no solute species was present in thismodel, the fluid density change was only the functionof temperature and water pressure changes, which aredetermined in SEAWAT by the following equation:

ρ = ρ0 + ∂ρ

∂T(T − T0) + ∂ρ

∂�(� − �0) (3)

where ρ0 is the fluid density at the reference temperature,and l is the height of water column of density ρ0.

The SEAWAT code in this study uses the followingequation to represent dynamic viscosity as a function oftemperature (Langevin et al. 2007):

μT (T ) = A1A

(A3

T +A4

)2 (4)

where μT is the viscosity at the temperate T ; A1, A2, A3,

A4 are the constants which are set equal to 2.394 × 10−5,10, 248.37, and 133.15, respectively (Langevin et al. 2007).

Model ComparisonFigure 3 shows the comparison of calculated temper-

ature variations at well 399-2-1 (top) and well 399-2-5(bottom) from October 2007 to October 2008. The solidline in blue indicates the simulation result by MT3DMSassuming constant fluid density and viscosity, while thedashed line in black indicates the simulation result bySEAWAT considering variable fluid density and viscos-ity. Also shown in Figure 3 as solid lines in red are thetemperature observations at the same wells.

It is noteworthy that the MT3DMS and SEAWATresults are nearly identical, indicating that the effects offluid density and viscosity on this field application are

negligible under the complex field conditions at the Han-ford 300A site. It is of interest to note that the simu-lated temperature variations capture the general trends ofthe observed variations. However, as pointed out in theproceeding section, the simulation model used the rivertemperature from October 2005 to October 2006 as anapproximation for the boundary condition from October2007 to October 2008. Thus, the comparison between sim-ulation results and field observations is presented hereonly as a qualitative indicator of general agreement. Theprimary purpose of this modeling application is to com-pare the results of MT3DMS and SEAWAT without andwith the effects of variable fluid density and viscosity.Detailed calibration of simulated temperature against fielddata is beyond the scope of this work.

Figure 4 shows the temperature contours along thecross section at simulation times of 150, 230, and360 days, as computed by MT3DMS and SEAWAT,respectively. Again, calculated temperature contours agreevery well between the two sets of simulations. Thus,the agreement of simulated temperatures by MT3DMSand SEAWAT, both temporally in the observation wells(Figure 3) and spatially along the cross section (Figure 4),indicates that the density and viscosity effects caused bytemperature changes can be neglected in this field appli-cation without any significant loss of accuracy. This isbecause the temperature variation across the flow domainis relatively small, rendering the density and viscosityeffects insignificant and negligible. In the next section,the conditions under which the effects of variable densityand viscosity are important will be quantified.

Quantifying the Effects of Densityand ViscosityMeasures of Model Discrepancy

To investigate the conditions under which heat trans-port can be simulated using a solute transport code,assuming constant density and viscosity, both MT3DMSand SEAWAT are used to simulate a series of scenarios

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Figure 3. Temperature changes simulated by MT3DMS and SEAWAT with composite boundary conditions (the specified-temperature at the eastern river boundary is based on data from October 2005 to October 2006 while the specified-temperatureat the western boundary from October 2007 to October 2008). The observed temperature data for well 399-2-1 (top) andwell 399-2-5 (bottom) are from October 2007 to October 2008.

Figure 4. Simulated temperature contours at 150, 230, and 360 days. The red solid lines represent the MT3DMS resultswhile the color zones represent the SEAWAT results.

for heat transport in which the flow model setting isidentical to that described in the section Comparison ofField Applications with and without Density and ViscosityEffects except that the length of stress periods is increasedfrom hourly to daily for the sake of simplicity. Accord-ing to the thermal Peclet number for this field setting,the heat transport in these scenarios is convection domi-nated. For the heat transport model in these scenarios, the

western boundary is specified-temperature with a fixedtemperature of 15◦C for the entire simulation. The east-ern boundary is also treated as specified-temperature witha fixed temperature for each scenario ranging from 10◦Cto 60◦C. The initial temperature is set to 15◦C everywherein the aquifer.

Two different statistical measures are used to quan-tify the difference between the MT3DMS and SEAWAT

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simulation results. These are the mean absolute temper-ature discrepancy and correlation coefficient. The meanabsolute temperature discrepancy is defined as follows:

Mean discrepancy(%) =1N

N∑k=1

|T1,k − T2,k|

T2× 100

k = 1, 2, 3 · · · N (5)

where N is the total number of active cells and T2 is givenby

T2 = 1

N

N∑k=1

|T2,k| (6)

where T1,k and T2,k is the temperature in cell k in theMT3DMS model and the corresponding SEAWAT model.The mean discrepancy of temperature results betweenMT3DMS and SEAWAT is normalized by the averagetemperature of all active cells in the SEAWAT model.

The correlation coefficient between all temperaturepairs calculated by MT3DMS and SEAWAT is defined asfollows:

r =

N∑k=1

(T1,k − T1)(T2,k − T2)√N∑

k=1(T1,k − T1)2

√N∑

k=1(T2,k − T2)2

(7)

where T1 and T2 are the mean temperatures in theMT3DMS and SEAWAT solutions, respectively. The r

value is equal to one if the two sets of results are perfectly,positively correlated, or zero if completely dissimilar.

Comparison of Model AccuracyBecause the groundwater flow field is highly transient

in the study site, the mean temperature discrepancy andcorrelation coefficient between the MT3DMS and SEA-WAT results are computed at six selected times of 60, 120,180, 230, 300, and 360 days, respectively. Figure 5 showsthe mean temperature discrepancy between the MT3DMSand SEAWAT simulation results at the six selected times.The normalized mean discrepancy increases with the max-imum temperature difference across the two boundaries.Furthermore, it varies with the simulation time. When theriver water intrudes inland further during the period ofhigh river stages (210 to 270 days), the river water tem-perature has affected a much larger domain of the aquifer,leading to a larger mean discrepancy. The area affected bythe river water decreases after river water recession dur-ing low river stages. Consequently, the mean discrepancybetween the two models at 230 days is much higher thanthose in other time periods. For the sake of discussion, ifit is assumed that a discrepancy error of 2.5% betweenthe two models is an acceptable accuracy requirement,then the MT3DMS code can be used for heat transportmodeling under the assumption of constant fluid density

Figure 5. Mean discrepancy of simulated temperaturesbetween MT3DMS and SEAWAT results at different sim-ulation times (the temperature difference between twoboundaries was calculated as the temperature at the riverboundary minus the temperature at the western boundary).

and viscosity when the maximum temperature differenceacross the flow domain is within 15◦C.

Figure 6 shows the correlation coefficient (r) betweenthe MT3DMS and SEAWAT simulation results. It canbe seen that the r value decreases when the maximumtemperature difference increases across the flow domainbetween the two specified-temperature boundaries. Whilea perfect match is indicated by r = 1.0, some levelof mismatch at a lower r value may be acceptablefor practical field applications. For example, if r = 0.8is considered an acceptable threshold, MT3DMS canprovide an adequate approximation of heat transportassuming constant density and viscosity, as long as themaximum temperature difference across the flow domainis within 15◦C.

From Figures 5 and 6, it is noteworthy that the largestdiscrepancy error and the smallest correlation coefficientdo not necessarily occur at the same time. The largestdiscrepancy occurs when the river water intrudes fur-ther into the aquifer (about 210 to 270 days) becausethe aquifer domain affected by the temperature changeis relatively large. However, during this period, the tem-perature plume pattern is most similar between MT3DMSand SEAWAT models. Based on a visual comparison ofthe simulation results, most of the temperature mismatch

Figure 6. Linear correlation coefficient r between the simu-lated temperature distributions by MT3DMS and SEAWATat different simulation times (the method for calculating thetemperature difference between two boundaries is the sameas that for Figure 5).

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Table 3Comparison of Computational Times for Different Scenarios

Maximum Temperature Difference Across the Model Domain 5◦C 10◦C 15◦C 20◦C

HCLOSE 10−5 10−5 10−5 10−5

RCLOSE (SEAWAT) 10−2 10−2 10−2 10−2

RCLOSE (MODFLOW) 10−5 10−5 10−5 10−5

Simulation time in MODFLOW + MT3DMS (min) 3.356 3.353 3.357 3.348Simulation time in SEAWAT (min) 4.493 4.580 10.912 21.716Relative efficiency [(SEAWAT − MT3DMS)/MT3DMS] (%) 33.9 36.6 225.1 548.6

between MT3DMS and SEAWAT occurs near the riverbank when the model domain affected by the tempera-ture change from river water intrusion is small. Whilethe river water recession or intrusion only affects a smallarea near the river bank, the mismatch is very noticeable,leading to a significantly lower correlation coefficient.Although the river stages generally stay higher during 210to 270 days, they fluctuate frequently during other peri-ods, causing repeated river water recession and intrusion.The river water recession/intrusion affects a small areanear the river bank zone at the times of 120, 180, 300,and 360 days, and the most mismatch occurs at 120 daysafter river water recedes temporarily. Thus, the discrep-ancy error between SEAWAT and MT3DMS is similarat 120, 180, 300, and 360 days, and the lowest correla-tion occurs at 120 days. From the above discussion, itis clear that both discrepancy error and correlation coef-ficient should be evaluated to assess the agreement (ormismatch) between two sets of simulation results.

To differentiate the relative importance of viscosityand density, the MT3DMS simulations assuming con-stant density and viscosity were compared with SEA-WAT simulations assuming variable density or variableviscosity, respectively. It was apparent that the densityeffect accounted for most of the discrepancy between theMT3DMS and SEAWAT simulations, especially whenthe temperature difference across the flow field was within30◦C. As the temperature difference increased, the relativeimportance of the viscosity effect increased, but remainedsignificantly lesser than that of the density effect.

The effect of groundwater flux on heat transport wasalso tested by comparing two simulations under exactlythe same conditions except for the groundwater flux acrossthe flow field. The results show that the discrepancy errorincreases and the correlation coefficient decreases as thegroundwater flux increases. Because the results describedin Figures 5 and 6 are based on a field model with ahydraulic conductivity of 7000 m/d, and Darcy flux of∼10 m/d, they represent the “worst-case” results validunder a wide range of field conditions.

Comparison of Computational EfficiencyUnder the same convergence criteria, the computa-

tional time for each MT3DMS run (along with MOD-FLOW) was compared with that for each corresponding

SEAWAT run to evaluate the relative computational effi-ciency of these two simulations. When running the SEA-WAT model with the VDF (variable-density flow) packagewhich accounts for the density effect, the RCLOSE resid-ual closure criterion for the PCG matrix solver is interms of the fluid mass balance rather than the fluid vol-ume balance (Langevin et al. 2003). This means for thecross-section model examined in this study, the RCLOSEvalue in SEAWAT can be specified three orders of mag-nitude larger than that for a standard MODFLOW sim-ulation (Table 3). The HCLOSE head closure criterionfor the two models is set exactly the same at 10−5.The computational time for MT3DMS (with MODFLOW)assuming constant density and viscosity is more than 30%faster than that for SEAWAT considering variable densityand viscosity, when the maximum temperature differenceacross the flow domain is within 10◦C (Table 3). Whenthe maximum temperature difference increases to 15◦C,the SEAWAT simulation time increases significantly to225.1% over that required by MT3DMS (along withMODFLOW) (Table 3). When the maximum temperaturedifference increases further to 20◦C, the difference in runtime between SEAWAT and MT3DMS rises to 548.6%.Thus, the use of MT3DMS is computationally efficientfor heat transport modeling if the effects of density andviscosity are negligible under sufficiently small tempera-ture variations across the simulation domain. It is worthyto note that SEAWAT has an option to update the flowfield only if the density difference at any cell changesby a user-specified density tolerance. Thus, the SEAWATefficiency could be substantially improved without a largeloss in accuracy by tuning the density tolerance parameter(Langevin et al. 2009).

Summary and ConclusionsBased on the analogy between solute transport and

heat transport equations, the solute transport simulatorMT3DMS can be used to simulate heat transport throughsimple variable conversion. While this approach is com-putationally efficient, it introduces numerical errors sinceit does not consider the effects of variable fluid den-sity and viscosity caused by temperature changes. Thiswork investigates the conditions under which the effectsof density and viscosity may be neglected for greatercomputational efficiency without any significant loss of

NGWA.org R. Ma, C. Zheng GROUND WATER 48, no. 3: 380–389 387

accuracy. This is accomplished by comparing the simula-tion results from MT3DMS with those from the SEAWATcode which explicitly accounts for variable fluid densityand viscosity.

The temperature variations induced by aquifer-riverinteractions along a cross section at the Hanford 300Asite were simulated to compare the difference betweensimulation results by MT3DMS and by SEAWAT. Thesimulated temperature distributions from MT3DMS andSEAWAT agreed closely, indicating that MT3DMS pro-vides an adequate approximation of heat transport as theeffects of variable density and viscosity appear to be neg-ligible in this field application.

Using the same model setup, a series of heat trans-port scenarios were evaluated to determine the thresholdvalue for the maximum temperature difference across themodel domain below which the density and viscosityeffects become negligible. For the model setup exam-ined in this study, the simulation results indicate that thethreshold value may set to around 15◦C if a 2.5% dis-crepancy between the MT3DMS and SEAWAT simulationresults with a correlation coefficient r = 0.8 is consid-ered adequate. Computationally, a MT3DMS simulationassuming constant fluid density and viscosity is signif-icantly more efficient than a variable-density SEAWATsimulation, by approximately 225% when the maximumtemperature difference across the model domain is as largeas 15◦C. The simulation results that considered the densityand viscosity effects separately show the discrepancy andcorrelation coefficient between MT3DMS and SEAWATsimulations mainly ascribed to the density effect whenthe temperature difference between the two boundariesis within 30◦C. However, the portion of the model dis-crepancy caused by viscosity greatly increased when thetemperature difference is greater than 30◦C. The discrep-ancy increases and correlation coefficient decreases whenthe groundwater flux is increased under a fixed tempera-ture difference across the flow field.

There are other reasons besides computational effi-ciency that we use a constant-density code such asMT3DMS, rather than a full-fledged variable-densitycode, for temperature simulation. For one thing, the datarequirement is greater for variable-density simulation. Foranother, a constant-density code may have many featuresthat a variable-density code lacks, such as fully coupledreactive transport. Thus, one would be able to incorporatethe temperature effect into reactive transport modelingreadily if heat transport can be simulated with adequateaccuracy while neglecting the density effect.

It is noteworthy that although this study is based on asite-specific model at the Hanford 300A site, the findingsand conclusions from this study have wider implications.First, it is confirmed that the effects of density andviscosity on heat transport modeling are insignificantunder a set of complex field conditions as constant-densityMT3DMS and variable-density SEAWAT simulations areshown to agree closely. Second, the Hanford 300A siteused as the reference frame in this study, with a fast flowvelocity and highly transient, hourly measured boundary

conditions, is more complex than a typical field problem.Thus, the guidelines derived from this study are notadversely affected by the simplicity of test models. Still,caution should be exercised when applying the guidelinesbeyond river-aquifer interaction problems as examined inthis study. Finally, the technical approach and quantitativecriteria presented here for evaluating the importance ofvariable density and viscosity in heat transport modeling(i.e., mean discrepancy and correlation coefficient) couldbe used to investigate other types of groundwater flowand heat transport problems.

AcknowledgmentsThis research was supported by the Integrated Field-

Scale Subsurface Research Challenge (IFRC) Project ofthe U.S. Department of Energy (DOE). We are gratefulto John Zachara, Mark Rockhold, and Chongxuan Liu ofPacific Northwest National Laboratory for their invalu-able assistance. We thank Christian Langevin, ChristopherLowry, and an anonymous reviewer for their constructivecomments which have significantly improved the presen-tation of this work.

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