Comparison of Numerical Techniques Used for SimulatingVariable-Density Flow and Transport Experiments

Rohit R. Goswami1; T. Prabhakar Clement2; and Joel H. Hayworth3

Abstract: The writers investigated the relative performance of three different numerical techniques available in the SEAWAT/MT3DMScode by simulating two new variable-density-flow and transport experimental data sets. The experiments were designed to represent twodistinctly different variable-density configurations that involve transport of a sinking groundwater plume and a rising groundwater plume. Thenumerical techniques used for simulating these experiments included the method of characteristics approach (MOC), total-variation-diminishing scheme (TVD), and finite-difference scheme (FD). Both homogeneous and heterogeneous hydraulic conductivity fields wereemployed in the numerical simulations to explore the effects of heterogeneities. The results indicate that all three numerical methods havelimitations and were not able to satisfactorily reproduce the instabilities observed in the experimental data sets. The results show the needfor improving the accuracy of numerical techniques that are currently being used for solving variable-density groundwater flow problems.DOI: 10.1061/(ASCE)HE.1943-5584.0000428. © 2012 American Society of Civil Engineers.

CE Database subject headings: Groundwater flow; Numerical analysis; Experimentation; Plumes.

Author keywords: Density-dependent flow; Groundwater; Transport; Dense flows; Numerical methods; MODFLOW; SEAWAT.

Introduction

Variable-density-flow problems can be broadly classified intotwo categories: 1) problems involving stable-density configuration,and 2) problems involving unstable-density configuration. Stable-density configuration problems usually involve a less-dense layerof fluid overlying a more-dense fluid layer; examples include salt-wedge experiments (or analytical/numerical solutions) that con-sider lateral movement of salt water beneath a natural groundwaterflow system (Cheng et al. 1998; Simpson and Clement 2004;Goswami and Clement 2007; Abarca and Clement 2009), andsalt-pool experiments that consider the upward movement of salt-water under pumping conditions (Oswald and Kinzelbach 2004).Unstable-density configuration problems involve a more-densefluid overlying a less-dense fluid; examples include sinking denseleachate plumes in freshwater aquifers (Oostrom et al. 1992;Simmons et al. 1999; Simmons et al. 2002; Hogan 2006), risingfreshwater plumes in saline aquifers (Peterson et al. 1978;Brakefield 2008), and sinking saltwater plumes that evolve fromlarge-scale inundation events such as tsunamis and hurricanes(Ilangasekare et al. 2006; Carlson et al. 2008).

Numerical modeling of unstable-density configuration prob-lems is far more challenging than stable-density configurationproblems (Simpson and Clement 2003). This is because unstable-density configurations give rise to density-driven mixing where theonset of instabilities and their growth and decay are governed by

pore-scale variations in hydraulic conductivity and concentrationdistributions (List 1965; Wooding et al. 1997a, b; Oostrom et al.1992; Schincariol et al. 1994; Simmons et al. 1999). Researchershave attempted to employ dimensionless numbers to predictthe onset and magnitude of unstable plume behavior (List 1965;Oostrom et al. 1992; Wooding et al. 1997a). However, recent ex-perimental and theoretical studies indicate that dimensionless num-bers are limited in their ability to predict temporal and spatialvariations in unstable, variable-density plumes (Menand andWoods 2005; Simmons 2005; Simmons et al. 2001). Two types ofnumerical approaches are available for simulating the unstableplume patterns without introducing any artificial perturbations inthe flow-field or in the concentration field (Liu and Dane 1997).The first approach exploits the randomness introduced by particle-tracking algorithms, such as the method of characteristics (MOC),to generate instabilities. In this article the writers refer to this asthe “particle” approach. The second approach explicitly introducessmall-scale heterogeneities in the hydraulic conductivity field togenerate instabilities (Schincariol et al. 1997; Schincariol 1998;Simmons et al. 2001). The writers refer to this as the “hetero-geneity” approach. There are no guidelines available in the litera-ture to evaluate the applicability of these two approaches, althoughthe heterogeneity approach is intuitively more appropriate sinceit attempts to mimic the causal mechanism for unstable plumebehavior. Both approaches introduce some level of perturbationcontinuously, whereas other alternatives, such as the initial concen-tration perturbation or initial flow-field perturbation methods (Liuand Dane 1997), only introduce perturbations at the onset of asimulation.

The objective of this study is to illustrate the challenges involvedin modeling unstable, density-driven flow problems using numeri-cal models. The USGS code SEAWAT (Zheng and Wang 1999;Langevin and Guo 2006) was used to simulate two variable-density plume experiments (a sinking plume and a rising plume).These experiments were performed to provide two simple variable-density benchmark scenarios for comparison with modelingresults. A brief summary of the challenges encountered during the

1Engineer, Geosyntec Consultants, Boca Raton, FL.2Professor, Dept. of Civil Engineering, Auburn Univ., Auburn, AL

(corresponding author). E-mail: clement@auburn.edu3Associate Research Professor, Dept. of Civil Engineering, Auburn

Univ, Auburn, AL.Note. This manuscript was submitted on March 18, 2010; approved on

May 9, 2011; published online on May 18, 2011. Discussion period openuntil July 1, 2012; separate discussions must be submitted for individualpapers. This paper is part of the Journal of Hydrologic Engineering,Vol. 17, No. 2, February 1, 2012. ©ASCE, ISSN 1084-0699/2012/2-272–282/$25.00.

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modeling effort is provided. The modeling results are used todevelop a qualitative understanding of the benefits and drawbacksof various solvers available within the USGS code SEAWAT, whichis a commonly used numerical code for solving variable-density-flow and transport problems.

Details of Experimental Work

The laboratory setup and data analysis procedures used to performthe variable-density benchmark experiments were similar to thosedescribed by Goswami and Clement (2007) and Goswami et al.(2009). Dimensions of the flow tank used in this study were225 mm ðlengthÞ × 200 mm ðheightÞ × 12 mm ðwidthÞ. Constanthead chambers, installed on both ends of the tank, were used tocontrol the head gradient across the tank. Clear silica glass beads(mean diameter of 1.1 mm) were used as the porous medium. Theporosity of the medium was measured using both volumetric andgravimetric methods and the mean value was found to be 0.38. Theaverage in situ hydraulic conductivity of the porous media was cal-culated using Darcy flux experiments [see Goswami and Clement(2007) for details] and the mean value was found to be 11:57 mm=swith a variation of �5%. Salt water was prepared by dissolvingNaCl in water and the density was estimated to be 1:025 g=mL(salt concentration of 0:035 g=mL), as measured using AmericanSociety for Testing and Materials (ASTM) # 111H hydrometer andalso by using the gravimetric method. Red food dye [Federal Food,Drug and Cosmetic (FD&C) #4] was used to color the tracer sol-ution. Goswami and Clement (2007) performed screening studiesand observed no measurable sorption of this food dye onto the glassbeads at concentrations similar to those used in the experimentspresented here. Additionally, the differences between diffusion co-efficients for salt constituents and dye are negligible (Goswami andClement 2007), which allowed the dye levels to be used to indi-rectly assess salt concentrations. A Canon digital camera (6 MP)providing 3-channel, 8-bit data was used to record images of plumebehavior during each experiment. Processed images were croppedand presented at an appropriate scale to provide visualization. Errorstatistics, similar to those provided by Goswami et al. (2009), wereused to access the quality of the image analysis results. These re-sults indicate that mass balance was conserved within a 10% rangefor all the concentration data processed using the image analysisprocedure. Further details of the image analysis process are de-scribed by Goswami et al. (2009).

Although the flow configurations were new, the overallexperimental steps used for conducting the two variable-density ex-periments were not substantively different from numerous otherdensity-flow experiments previously completed in the writers’laboratory [e.g., Hogan (2006); Goswami and Clement (2007);Brakefield (2008); Goswami et al. (2009)]. The first experimentrepresents a rising freshwater plume in a saltwater aquifer. Thisscenario can be encountered when treated wastewater is disposedvia injection wells into deep saline aquifers (Merritt 1997). Tophysically model this rising plume scenario, the experimental flowtank was initially filled with salt water. A freshwater slug was in-jected into the lower part of the tank through a port. The experimentwas conducted in the absence of ambient flow to provide optimalconditions for generating instabilities. The second experiment wasdesigned to simulate the transport of a dense plume in a freshwateraquifer. Such transport scenarios can occur near a leaking landfilldischarging dense leachate into ambient groundwater (Oostromet al. 1992) or can occur when open dug wells (which are commonin developing countries) or poorly constructed deep boreholes areinundated by seawater during catastrophic events such as tsunamis

or hurricanes (Ilangasekare et al. 2006; Carlson et al. 2008). In thissinking plume experiment, a uniform freshwater flow field was firstestablished to represent the ambient groundwater flow condition. Asaltwater slug was then injected in the upper part of the experimen-tal flow tank. The ambient flow was adjusted (sufficiently in-creased) to inhibit instabilities.

Laboratory Observations from the Rising PlumeExperiment

The flow tank was flushed with approximately 10 pore volumes ofsalt water (solution density of 1:025 g=mL, salt concentration of0:035 g=mL) resulting in saline initial conditions. Ambient flowwas avoided by setting zero head gradient across the tank. A fixedvolume (5 mL) of dyed freshwater tracer (with dye concentration of4:0 mL=L and solution density of 1:000 g=mL) was injected intothe tank. The injection port was located approximately 114 mmfrom the left boundary and 17 mm above the bottom of the tank.The total time taken for the injection was approximately 10 s. Dig-ital images of the freshwater plume were acquired at four differenttimes after the injection. Both the preprocessed (raw) and processedimages of the freshwater plume captured at different times areshown in Fig. 1. The figure presents eight panels that representthe four different transport times and correspond to the prepro-cessed and processed images, respectively. For example, Fig. 1(a)is the preprocessed digital image recorded 10 s after injecting thefreshwater slug. The image-processed data (showing estimatedconcentration values) corresponding to this time (10 s after injec-tion) is shown in Fig. 1(e).

Fig. 1(a) indicates that the initial shape of the injected freshwaterplume is almost circular. It can be observed from the concentrationdata that this plume has a very narrow mixing zone [see Fig. 1(e)].As time progressed, the buoyancy forces induced by the densitydifference between the fresh water and salt water transported theplume upward. During this transport period, the presence of theunstable-density configuration (dense water overlying fresh water)generated instabilities at the boundary of the plume. Evolution ofthese instabilities can be observed in Figs. 1(b) and 1(f), whichwere recorded two minutes after injection. As the fresh water mi-grated upward, it mixed with the ambient saline water, causing thefingers to dissipate. Other researchers have observed such dissipa-tion processes, which can help coalesce fingers and smoothen theplume front (Schincariol et al. 1994).

Figs. 1(c) and 1(g) show the plume distribution four minutesafter injection; the fingers at this time are wider and are connectedto each other. The plume eventually attained a conical shape asit reached the top of the tank; this pattern can be observed inFigs. 1(d) and 1(h), recorded after six minutes of transport. Theimage-processed concentration data [Fig. 1(h)] shows that littlefresh water (colored blue) is left in the system after six minutes oftransport; the core of the plume is mixed and appears to containsaline water with concentrations ranging from 0.02 to 0:03 g=mL.

Laboratory Observations from the Sinking PlumeExperiment

The experimental tank was repacked with porous medium identicalto that used in the rising plume experiment, using similar packingprocedures. Colored fresh water with a density of 1:000 g=mL anda very low dye concentration of 0:5 mL=L was used to saturate theporous medium. Uniform flow conditions were established by forc-ing a head difference of 4 mm across the tank. Darcy flow experi-ments were conducted to verify the hydraulic conductivity of theporous medium; the average conductivity value was similar to thepacking used in the previous experiment with a variability of �5%.Approximately 5 mL of colored saltwater tracer (solution density of

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1:025 g=mL) was injected into the tank to initiate the experiment.The injection port was located approximately 54 mm from the leftboundary and 145 mm above the bottom boundary. The total timetaken for injection was approximately 10 s. Similar to the risingplume experiment, digital images were recorded to analyze thetransport patterns. Both preprocessed (raw) and processed imagesof the saltwater plume recorded at three different times are pre-sented in Fig. 2. The figure presents images representing differenttransport times that correspond to preprocessed (or original digital)and processed images, respectively.

Fig. 2(a) was recorded immediately after injection. The initialshape of the injected saltwater slug was circular. The image-processed concentration data show that initially the plume had asharp mixing zone [Fig. 2(d)], similar to the rising plume mixingzone just after injection. As time progressed, the plume migrateddownward and was also transported toward the right boundary bythe ambient groundwater flow. The shape of the transported plumewas elliptical, as shown in Figs. 2(b), 2(c), 2(e), and 2(f)). The onsetof unstable plume behavior in this configuration (dense saltwaterplume penetrating a less-dense saturated porous medium) woulddepend on the density contrast at the plume-groundwater interfaceand the flow rate of the ambient groundwater (Oostrom et al. 1992).For the experimental conditions used here, unstable plume behaviorwas suppressed by using relatively high ambient flow rate. Unlikethe rising plume experiment, where only a small portion of in-jected fresh water (blue-filled contour) reached the top boundary

[Fig. 1(h)], in the sinking plume experiment the core of theplume (red-filled contour), with a relatively high concentration(∼0:03 g=mL), remained intact as the plume approached the rightboundary [Fig. 2(f)].

Numerical Modeling Methods

Results of the rising and sinking plume experiments are novelvariable-density experimental data sets that can be used for com-paring the performance of numerical models. The goal of the mod-eling effort was to compare the relative performance of variousnumerical solvers available within SEAWAT/MT3DMS andprovide a qualitative assessment of their ability to simulate theseexperimental results. Two numerical approaches were employed,namely, the particle approach and heterogeneity approach, to sim-ulate the experimental observations. A uniform hydraulic conduc-tivity field (set at the average experimental value) and the MOCoption available in SEAWAT were used for simulating the particleapproach. The finite-difference (FD) scheme and the total-variation-diminishing (TVD) scheme along with a lognormally dis-tributed hydraulic conductivity field were used to generate simu-lation results for the heterogeneity approach. For comparisonpurposes, additional simulations were also completed using FDand TVD schemes using the homogeneous hydraulic conductivityfield and MOC simulations with a heterogeneous hydraulic con-ductivity field.

Fig. 1. (Color) Digital images of the rising plume experiment recorded at various times after injection: (a) preprocessed image at 0 min; (b) pre-processed image at 2 min; (c) preprocessed image at 4 min; (d) preprocessed image at 6 min; (e) processed image at 0 min; (f) processed image at2 min; (g) processed image at 4 min; (h) processed image at 6 min

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The numerical modeling setup used for simulating the physicalexperiments is summarized in Figs. 3(a) and 3(b) for the risingand sinking plumes, respectively. The two-dimensional modelingdomain was 225 mm long and 180 mm high. Two cells at theupper-left and right corners of the rising plume problem wereset to zero pressure [Fig. 3(a)]. These cells allowed fluid to freelyexit the system during and after injection; a similar type of zero-pressure boundary condition has been employed by Voss and Souza(1987) to simulate the Elder problem. The longitudinal dispersivityvalue was set to 0.1 mm and transverse dispersivity was assumed tobe 1=10th of the longitudinal dispersivity. Additional model sim-ulations were conducted by varying the dispersivity values within arange of 0.1 mm to 1 mm. Previous studies have shown that it isdifficult (and in most cases, impossible) to obtain grid-convergedsolutions for unstable variable-density-flow problems (Simmons2005). In the present study, a very fine grid of dimension 1 mm(in both directions) was used for all the simulations. A similar gridsize was used by Johannsen et al. (2006) while trying to match re-sults from a three-dimensional variable-density-flow experiment.

The final grid design used was an optimal compromise betweencomputational speed and grid-convergence level. Three stressperiods were employed to simulate the following three experimen-tal stages: (1) steady-state conditions prior to injection, (2) injectionof freshwater/saltwater slug, and (3) migration of slug through thetank. Time step size was set to an initial minimum value of 0.1 s anda maximum value of 1 s. The courant number constraint was set to0.5. The MT3DMS manual recommends the use of 16 particles percell for two-dimensional simulations; however, the results fromMOC simulations can be sensitive to the number of particles allo-cated per cell (Al-Maktoumi et al. 2007). Therefore, additional sim-ulations were also completed by varying the maximum number ofparticles (NPH) from 16 to 124. A base-case model was set up us-ing measured values of solution density, hydraulic conductivity,and porosity (Table 1). The sensitivity of numerical results to var-iations in a given parameter value was analyzed by maintainingother parameters at the base-case values given in Table 1.

Numerical Modeling Results

Results of MOC Simulations (Particle Approach) withHomogeneous Conductivity Field

MOC simulation results for the rising plume experiment are pre-sented in Figs. 4(a)–4(d). These results show that MOC was ableto generate complex fingering patterns similar to those observed inthe rising plume experiment. Fig. 4(a) shows that the mixing zonebetween freshwater and saltwater concentrations is fairly narrowand is similar to the level of mixing observed in the image-processed data [Fig. 1(e)]. Comparison of Fig. 4(d) (MOC simu-lation) with Fig. 1(h) (rising plume experiment) suggests that theMOC simulation did not capture the amount of mixing observed inthe benchmark experiment. The experimental data indicate that a

Fig. 2. (Color) Digital images of the sinking plume experiment recorded at various times after injection: (a) preprocessed image at 0 min; (b) pre-processed image at 2 min; (c) preprocessed image at 5 min; (d) processed image at 0 min; (e) processed image at 2 min; (f) processed image at 5 min

Fig. 3. Conceptual description of the numerical model used for simu-lating the (a) rising-plume and (b) sinking-plume experiments

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Table 1. Summary of Parameters Used in the Numerical Simulations

Parameter Value

Model domain Length 225 mm

Height 180 mm

Cell discretization 1 mm (all directions)

Layer saturation All confined

Flow Hydraulic conductivity 11:57 mm=s (mean) measured range: �5% (0:57 mm=s)

Porosity 0.38 (mean) measured range: �10% (0.03)

Head tolerance for solver 10�4

Stress periods 3

Transport Longitudinal dispersivity 0.1 mm in base case range: 0.1 mm to 1 mm

Ratio of longitudinal to transverse (horizontal and vertical) dispersivity 0.1

Solver option used for simulating dispersion and sink/source terms Implicit

Concentration tolerance for solver 10�6

Time stepping Initial time step 0.1 s

Maximum time step 1 s or Courant ¼ 0:5

Method of characteristics Minimum number of particles 8

Maximum number of particles 16 in the base case range: 16 to 124

Initial particle placement 0 (random)

Tracking algorithm Combination

Small relative concentration 10�6

Heterogeneity Distribution Lognormal

Variance 1% simulated range: 0.1% to 10%

Correlation length 10 mm simulated range: 2 mm to 20 mm

Fig. 4. (Color) Simulation results obtained with homogeneous packing and MOC for the rising plume experiment at various times: (a) 0 min;(b) 2 min; (c) 4 min; (d) 6 min; and the sinking plume experiment at various times: (e) 0 min; (f) 2 min; (g) 5 min; times were recorded fromthe end of the injection period

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portion of relatively fresh water (blue colored) is trapped in the coreof the rising plume, whereas the numerical results were not ableto reproduce this core; a small amount of fresh water is present onlyin the upper-outer edges of the plume.

MOC simulations for the sinking plume experiment are pre-sented in Figs. 4(e)–4(g). The simulation results show some levelof fingering; however, this fingering was not observed in theexperimental data. Also, the core of the simulated plume appearsto be more dispersed in the numerical simulation results [Fig. 4(g)]compared to the experimental data [Fig. 2(f)].

Dispersivity values were modified in the rising and sinkingplume simulations to determine whether unstable fingering couldbe generated in the rising plume simulations while avoiding them inthe sinking plume simulations. It was observed that the risingplume simulations ceased to produce instabilities for longitudinaldispersivity values beyond 0.3 mm, whereas the sinking plume sim-ulations produced weak instabilities at later time steps even for arelatively high dispersivity value of 0.3 mm. The effect of the num-ber of particles (NPH) was also assessed by varying NPH from 16to 124. It was observed that a larger number of particles yieldedfingering patterns for both scenarios; however, the degree of finger-ing was more pronounced in the rising plume simulations.

Results of Finite-Difference Simulations with aHomogeneous Conductivity Field

Both the upstream and central finite-difference weighting optionswere used to simulate the experiments using a homogeneous hy-draulic conductivity value. Simulation results from the centralfinite-difference scheme are presented in Fig. 5 and the results fromthe upstream scheme are provided in Fig. 6. The central finite-difference scheme predicted the presence of a narrow mixing zoneimmediately after injection in both rising and sinking plume experi-ments. The amount of salt water in the sinking plume (colored red)present at the end of the numerical simulation was approximatelythe same as that observed in the experimental data [compareFig. 5(g) and 2(f)]. However, the central finite-difference schemeover estimated the amount of fresh water (colored blue) presentwithin the core of the plume at the end of the rising plume experi-ment [compare Fig. 5(d) and 1(h)].

Fig. 6 presents simulation results using the upstream finite-difference scheme and a homogeneous field for both the risingand sinking plume experiments. The results show that the upstreamscheme predicted a large mixing zone at the beginning of bothexperiments [Figs. 6(a) and 6(e)]. The upstream scheme closely

Fig. 5. (Color) Simulation results obtained with homogeneous packing and central finite-difference scheme for the rising plume experiment at varioustimes: (a) 0 min; (b) 2 min; (c) 4 min; (d) 6 min; and the sinking plume experiment at various times: (e) 0 min; (f) 2 min; (g) 5 min; times were recordedfrom the end of the injection period

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estimated the low volume of fresh water (colored blue) that re-mained in the system at the end of the rising plume experiment[Fig. 1(h) and 6(d)]. On the other hand, the upstream method couldnot re-create the presence of a large amount of salt water (coloredred) that remained in the core of the plume at the end of thesinking plume experiment as shown in Fig. 2(f) and as observedin Fig. 6(g).

In general, predicted plume characteristics using either the up-stream or central finite-difference schemes were similar. These dif-ferences are interpreted to be primarily caused by numericaldispersion (Woods 2004). Unlike MOC simulations, both upstreamand central finite-difference schemes failed to simulate instabilitiesobserved in the rising plume data. The schemes also failed to re-produce the observed narrow mixing zones observed at the onsetof both benchmark experiments. Also, the finite-difference resultswere largely insensitive to changes in dispersivity values due to theinfluence of numerical dispersion.

Results of TVD Simulations Using a HomogeneousConductivity Field

TVD schemes use higher-order finite-difference approximationsdesigned to reduce numerical dispersion effects and to track sharpmixing fronts (Zheng and Bennett 2002). Simulation results ob-tained using the TVD scheme in SEAWAT are presented in Fig. 7.For the initial time period, the rising plume simulation results[Figs. 7(a)–7(d)] are similar to the central finite-difference scheme

simulation results [see Figs. 5(a) and 5(b)]; both indicate thepresence of a small mixing zone with a large finger. The TVDscheme was able to simulate unstable fingering at later time periods[Fig. 7(c) and 7(d)]; however, the fingering patterns differ fromthose observed in the experimental data (Fig. 1). The major dif-ference between TVD simulation and benchmark experimentalresults is the simulated pattern of the central finger, which waslarger and migrated faster compared to the rising plume benchmarkexperimental data.

The TVD scheme closely predicted the narrow mixing zone ob-served at initial times in the sinking plume benchmark experiment[Fig. 2(d) and 7(e)]. However, the TVD scheme predicted unreal-istic unstable plume behavior within the range of modeled param-eters compared to experimental results.

Sensitivity analysis conducted by varying longitudinal disper-sivity values yielded results similar to those of the base-casesimulations in both the sinking and rising plume simulations.Increasing the value of longitudinal dispersivity to 0.4 yielded astable sinking plume; however, a longitudinal dispersivity valueless than 0.3 was needed to simulate the instabilities observed inrising plume experimental data.

Finite-Difference Simulations with a HeterogeneousConductivity Field (Heterogeneity Approach)

The use of a heterogeneous hydraulic conductivity field is an alter-nate approach for simulating the fingering patterns observed in

Fig. 6. (Color) Simulation results obtained with homogeneous packing and upstream finite-difference scheme for the rising plume experiment atvarious times: (a) 0 min; (b) 2 min; (c) 4 min; (d) 6 min; and the sinking plume experiment at various times: (e) 0 min; (f) 2 min; (g) 5 min; times wererecorded from the end of the injection period

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variable-density experiments. Although the experimental tank usedin this study was packed with a nominally homogeneous porousmedium, minor small-scale variations in the hydraulic conductivityfield are difficult to eliminate. To examine the effect of these small-scale heterogeneities, it was assumed that the local variability inhydraulic conductivity can be distributed using a lognormal fieldhaving 1% variance about the average measured value. Othervariance values were also tested within the range of 0.1% to 10%.Preliminary test simulations indicated that no instabilities weregenerated with a variance of 0.1%, and a variance value of 10%generated instabilities with patterns similar to those generatedby a 1% variance. The turning bands code TUBA (Zimmerman andWilson 2005) was used to generate the heterogeneous conductivityfield. Hydraulic conductivity fields were generated using a corre-lation length of 10 mm (in horizontal and vertical directions). Othercorrelation lengths ranging from 2 mm to 20 mm were used to gen-erate conductivity fields; simulation results were not significantlydifferent from those obtained using the correlation length of10 mm. Other TUBA parameters were set at default values [thecode manual (Zimmerman and Wilson 2005) should be referredto for more details].

Numerical simulation results for the central finite-differencescheme employing the heterogeneous field are presented inFig. 8. Results for the upstream scheme are provided in Fig. 9.As shown in Figs. 8 and 9 (a–d), both simulation schemes gener-ated complex fingering patterns for the rising plume experiment.

The central-difference scheme generated more complex fingeringpatterns than the upstream finite-difference scheme. Sinking plumesimulations completed using both upstream and central-differenceschemes matched well with the experimental data. Unlike the MOCand TVD results for the homogeneous case, no spurious instabil-ities were generated in the numerical simulations.

For both the rising and sinking plume simulations, the upstreammethod introduced more numerical dispersion and simulated alarger mixing zone when compared to the central-differencescheme results (evidenced by the more diluted core). Althoughthe level of numerical dispersion is low for the central-differencescheme when compared to the upstream scheme, it cannot be com-pletely eliminated. The effects of numerical dispersion can be re-duced by using front-tracking schemes such as MOC or TVD. Tofurther examine the effects of numerical dispersion, both TVD andMOC schemes were applied with the heterogeneous hydraulic con-ductivity field. Results of these simulations are not presented herebecause the concentration profiles (obtained using the hetero-geneous field) were similar to those obtained using the homo-geneous conductivity field presented in Figs. 4 and 7 for MOCand TVD, respectively. Generally, a larger number of instabilities(fingers) were observed in the simulation results obtained using aheterogeneous conductivity field. This is an expected result sinceheterogeneities introduce additional flow-field perturbations andtherefore increase the chance for generating instabilities. Also, sim-ilar to the homogeneous field results, both MOC and TVD schemes

Fig. 7. (Color) Simulation results obtained with homogeneous packing and TVD method for the rising plume experiment at various times: (a) 0 min;(b) 2 min; (c) 4 min; (d) 6 min; and the sinking plume experiment at various times: (e) 0 min; (f) 2 min; (g) 5 min; times were recorded from the end ofthe injection period

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failed to simulate the sinking plume experiment without instabil-ities (Fig. 4 and 7).

Discussion of Numerical Simulation Results

A qualitative comparison of this study’s modeling results indicatethat none of the numerical schemes was able to simultaneously sim-ulate the plume patterns and the degree of mixing observed in bothexperiments. For the homogeneous conductivity field simulations,MOC was able to closely simulate the rising plume data. However,MOC produced spurious instabilities in the sinking plume experi-ment for the range of parameters considered in this study (Fig. 4).The finite-difference schemes were unsuccessful in simulating thedegree of mixing in the homogeneous conductivity field, primarilydue to numerical dispersion errors (Figs. 5 and 6). MOC andTVD (Fig. 7) were able to minimize numerical dispersion effectsand simulated the presence of the sharp mixing boundary betweenfresh water and salt water at earlier times in both rising and sinkingplumes. However, MOC and TVD results were unsatisfactorysince they contained anomalies in the locations of fresh waterand salt water at later time steps in both rising and sinking plumesimulations.

The heterogeneity approach enabled the FD solvers to predictthe generation (or absence) of instabilities in plume behavior forboth rising and sinking plumes (Figs. 8 and 9). However, since theFD solvers suffered from numerical dispersion effects, they could

not predict the observed mixing levels. Use of the TVD solver al-lowed the control of numerical dispersion errors, but it introducedspurious instabilities in the sinking plume predictions. Similar tothe homogenous simulations, both TVD and MOC results wereable to simulate the sharp mixing zones at early time steps in bothrising and sinking plume simulations, but were unable to reproducethe level of mixing observed at later times.

Conclusions, Limitations, and Future Work

Three numerical techniques available in the SEAWAT/MT3DMScode (MOC,TVD, and FD) were employed to simulate twovariable-density groundwater experiments involving rising andsinking transport conditions. While others have compared the per-formance of these transport solvers for analyzing stable, neutral-density systems (e.g., Mehl and Hill 2001; Zheng and Bennett2002), as per the writers’ knowledge this is the first time a rigorousanalysis has been completed for problems involving unstablevariable-density systems. The results show that the instabilitiesin the rising plume experiment can be simulated using either MOCor TVD using homogeneous hydraulic conductivity conditions, orby using the finite-difference solver with a heterogeneous hydraulicconductivity field. The sinking plume experiment can be simulatedusing the finite-difference method with a homogenous conductivityfield, or using MOC or TVDwith a homogeneous field and a highervalue of dispersivity (compared to the values used in the rising

Fig. 8. (Color) Simulation results obtained with heterogeneous packing and central finite-difference scheme for the rising plume experiment atvarious times: (a) 0 min; (b) 2 min; (c) 4 min; (d) 6 min; and the sinking plume experiment at various times: (e) 0 min; (f) 2 min; (g) 5 min; timeswere recorded from the end of the injection period

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plume simulation). However, none of the solvers was able tosatisfactorily simulate both experiments using similar modelparameters.

Repeated runs of the experiments would have helped to quantifythe variations in plume behavior with respect to the onset of insta-bilities and the degree of mixing. These expected variations are theresult of local (pore-scale) variabilities in the hydraulic conduc-tivity field and concentration distribution. Under unstable transportconditions, the pore-scale variations might trigger instabilities atdifferent locations and times, which are difficult to reproduce ex-actly. Thus, repeated experimental runs of both the rising and sink-ing plume experiments should allow establishment of an “average”plume behavior that is independent of pore-scale variations. There-fore, future experimental efforts should include multiple repetitionswith an aim to construct the average behavior of these two plumes.However, based on the writers’ past experience in conducting sev-eral preliminary/screening experiments, it is unlikely that the grosstransport behavior of these repetitive experiments would differgreatly from the results reported in this study (for example, the ab-sence of unstable plume behavior in the sinking plume experiment).

The present work was limited to a two-dimensional analysis andignores three-dimensional flow and mixing processes. Extendingboth laboratory and numerical modeling efforts to study variable-density systems involving three-dimensional systems would helpfurther understanding of the nature of variable-density flow occur-ring under field conditions. Future studies should also focus on us-ing other numerical schemes and rigorous calibration procedures

that might provide a better match between the numerical andexperimental results. Another possibility is to test the applicabilityof advanced numerical formulations that could include nonfickiantransport (e.g., Hassanizadeh and Leijnse 1995).

Acknowledgments

The laboratory facilities used for this work were developed aspart of a research effort funded by the Office of Science (BER),U.S. Department of Energy Grant No. DE-FGO2-06ER64213. Thiswork was partially funded by the Sustainable Water ResourcesResearch Center of Korea’s 21st Century Frontier ResearchProgram.

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