Methods Note/

A Method of Accelerating Transport SimulationWhen Groundwater Pumping Is Simulatedby Jim Zhang1, Glenn Randall2, and Xinyu Wei3

AbstractIn solving groundwater transport problems with numerical models, the computation time (CPU processing

time) of transport simulation is approximately inversely proportional to the transport time-step size. Therefore,large time-step sizes are favorable for achieving short computation time. However, transport time-step size mustbe sufficiently small to avoid numerical instability if an explicit scheme is used (and to guarantee enough modelaccuracy if an implicit scheme is used). For a transport model involving groundwater pumping, a small transporttime-step size is often required due to the high groundwater velocities near the pumping well. Small grid spacingoften specified near the pumping well also limits the time-step size. This paper presents a method to increasetransport time-step size in a transport model when groundwater pumping is simulated. The key to this approachis to numerically decrease the groundwater seepage velocities in grid cells near the pumping well by increasingthe effective porosity so that the transport time-step size can be increased without violating stability constraints.Numerical tests reveal that by using the proposed method, the computation time of transport simulation can bereduced significantly, while the transport simulation results change very little.

IntroductionWith dramatically increased computer processor

speed and physical memory in the past decades, ground-water modeling has become much more sophisticated.Smaller grids have been used to achieve higher resolutionsin representing subsurface environments and to improvethe accuracy of numerical simulation. Thus, the size ofgroundwater models (i.e., the total number of modelgrids/elements) has increased significantly. As a result,the computation time of model simulation has remainedrelatively long for some large groundwater models. For

1Corresponding author: URS Corporation, 1333 Broadway,Suite 800, Oakland, CA 94612; (510) 874-3154; fax: (510) 874-3268; jim_zhang@urscorp.com

2URS Corporation, 335 Commerce Dr., Suite 300, FortWashington, PA 19034.

3HydroGeoLogic, Inc., 11107 Sunset Hills Road, #400,Reston, VA.

Received January 2011, accepted July 2011.© 2011, The Author(s)Ground Water © 2011, National Ground Water Association.doi: 10.1111/j.1745-6584.2011.00851.x

instance, computation of a large three-dimensional (3D)groundwater flow and transport model, with long trans-port simulation time and small transport time-step size,may require tens of hours or even days. Under this circum-stance, accelerating the transport simulation is favorable,especially if multiple transport simulations are required,such as in evaluating remedial alternatives in feasibilitystudies or performing multiple transport simulations tooptimize pump-and-treat systems.

When groundwater pumping is involved in a trans-port model, a much smaller transport time-step is oftenrequired to maintain the stability of the transport solution,leading to a much longer computation time. Two fac-tors often affect the transport time-step: (1) the increasedgroundwater velocities due to radial flow toward thepumping well, and (2) the smaller grid sizes often spec-ified near a pumping well for achieving the desiredaccuracy.

In this paper, we discuss a method which reducesthe simulation time by increasing transport time-stepsize when groundwater pumping is the key to limit thetime-step size. With this method, the groundwater seepage

NGWA.org GROUND WATER 1

velocities are numerically decreased by artificially increas-ing the effective porosity in grid cells near the pumpingwell, thus allowing an increase in time-step size with-out violating time-step constraints associated with solutionstability.

A practical solution as it is, this method will intro-duce arguments about its effects on model accuracy andreliability. In this paper, we address this method in theo-retical detail, and demonstrate the method in a numericaltest. Discussion about the model accuracy, model massbalance, and limitations will be presented in detail.

Solution Techniques in Transport SimulationThe general partial differential equation describing

the fate and transport of a chemical in a 3D, transientgroundwater flow system can be expressed as (Zheng andBennett 2002):

∂(θwRC)

∂t= ∂

∂xi

(θwDij

∂C

∂xj

)− ∂

∂xi

(qiC)

+ qsCs − λ1θwC − λ2ρbC (1)

where θw is the effective water content and θw = θeff × Sw

where θeff is the effective porosity of the subsurfacemedium [dimensionless] and Sw is the degree of watersaturation [dimensionless], R is the retardation coefficient[dimensionless], C is the solute concentration in the waterphase [M/L3], t is time [T], xi is distance along the respec-tive Cartesian coordinate axis [L], Dij is hydrodynamicdispersion coefficient tensor [L2/T], qi is the Darcy veloc-ity [L/T], qs is the volumetric flow rate via sources(positive) and/or sinks (negative) [1/T], Cs is the concen-tration of the source or sink [M/L3], λ1 is the reaction rateconstant for the dissolved phase [1/T], λ2 is the reactionrate constant for the sorbed phase [1/T], ρb is the solidbulk density of the porous media [M/L3], and C is thesorbed concentration [M/M].

Numerical methods for solving the above advection-dispersion-reaction equation can be classified as afinite-difference (FD) method and a finite-element (FE)method. For simplification, discussion about the proposedmethod is limited to FD method only; however, the prin-ciple of the proposed method should also work for anyFE method.

Three techniques are often used for the advectioncomponent in the FD method: the standard FD method,the particle-tracking-based methods (i.e., method of char-acteristics [MOC] [Konikow and Bredehoeft 1978], themodified method of characteristics [MMOC] [Cheng et al.1984], and the hybrid method of characteristics [HMOC][Neuman 1984]), and the high-order total-variation-diminishing (TVD) methods (Harten 1983, 1997). Thestandard FD method can be formed implicitly or explic-itly in time. The implicit FD schemes solve the transportequation iteratively and thus are not subject to any stabil-ity constraint, but they are not computationally efficient.The explicit FD method avoids iterative calculations and is

much more computationally efficient; however, it is sub-ject to the stability constraint. The particle-tracking-basedmethods and the high-order TVD methods are subject tothe same stability constraint. Zheng and Wang (1999) haveprovided a detailed explanation of each solution techniqueand their advantages and disadvantages.

For all nonadvection transport components (i.e.,dispersion, source/sink, and chemical reaction), eitherimplicit or explicit block-centered FD method can beused. However, as the explicit scheme is subject to vari-ous stability constraints on time-step size, the implicit FDscheme is much more commonly used in solving thosenonadvection components. For example, the widely-usedMT3DMS code (Zheng 1990; Zheng and Wang 1999) usesonly implicit FD scheme to solve the nonadvection termsin its version 5.0 and later versions. Since the implicitFD scheme is not subject to instability constraints, thefollowing discussion is limited to stability constraints ontransport time-step associated with the advection term.

Instability Constraints on Transport Time-Step

Courant Number ConstraintIn solving the advection term, the user can choose

any of the three solution techniques enumerated earlier.If the explicit FD method, any of the particle-tracking-based methods, or any of the high-order TVD methods isselected, the dimensionless Courant number (Cr ) definedbelow is often used to determine the allowable maximumsize of time-step (Zheng and Wang 1999):

Cri = |vi |�t

�xi

(2)

where i is the respective Cartesian coordinate axis (i.e., x,y, and z), vi is the groundwater seepage (linear) velocity[L/T], �t is the transport time-step size [T], and �xi isthe grid size [L] in respective Cartesian coordinate axis.Cr represents the number of cells (or a fraction of a cell)a groundwater particle is allowed to pass through in anydirection within one transport time-step.

To satisfy the stability constraint, Cr should notexceed unity in any direction over the entire transportmodel domain. If sorption is modeled, the maximumallowable size of the transport time-step is determinedas (Zheng and Wang 1999):

�tmax = Min

(�xi

|vi |)

× Rd (3)

where Rd is the retardation factor representing the ratio ofgroundwater velocity to the velocity of chemical migra-tion. Equation 3 indicates that in order to maintain thestability of the solutions, Cr cannot exceed the retarda-tion factor in any location over the entire model domain.That is, within one transport time-step, a contaminant par-ticle should not travel a distance greater than the lengthof any single grid cell in any direction.

2 J. Zhang et al. GROUND WATER NGWA.org

Groundwater Velocities and Cr near a Pumping WellTo show the effect of pumping on groundwater

velocity and Cr near a pumping well, consider asimple flow condition induced by a pumping well. Forsimplification, it is assumed that the aquifer is infinite,isotropic, homogeneous, uniform, and there is no initialhydraulic gradient across the aquifer. It is also assumedthat the aquifer has a transmissivity of 1,000 ft2/d(92.9 m2/d), and the well pumps at a constant rateof 100 gallon/min (gpm) (19,250 ft3/d or 545.10 m3/d).The transmissivity and pumping rate are selected suchthat the groundwater velocities are not too low andthe drawdowns are not too high near the well. Tomodel the groundwater conditions near the pumping well,a two-dimensional (2D) steady-state flow model was setup with MODFLOW (McDonald and Harbaugh 1988;Harbaugh et al. 2000; Hill et al. 2000). To minimize theeffects of model boundaries, the model domain was set tobe a 10 × 10 miles (16, 093 × 16, 093 m) square with thepumping well at the center. The well-cell (the cell wherethe pumping well is located) was set to be a 20 × 20 feet(6.1 × 6.1 m) square, and the grid size increases by afactor of 1.1 in both x and y directions toward themodel boundaries. A uniform hydraulic head of 100 feet(30.48 m) was specified over the entire model domain asthe initial condition, and assigned to the four model edgesas boundary conditions.

Assuming an effective porosity of 0.20 for the aquifer,the modeled groundwater seepage velocity is shown inFigure 1 (ft/d, blue arrows and values in blue). As bothgroundwater flow and grid cells are axisymmetric for thiscase, only the upper-left quarter of the grid cells areshown here. For discussion convenience, the grid cells

Figure 1. Grid cell index (black numbers in parentheses),simulated seepage velocity V (ft/d, blue arrows and valuesin blue) and calculated Cr values with time-step of 1.0 day(pink value in parentheses).

were indexed (black in parenthesis) as shown in Figure 1,based on their relative locations to the well-cell indexedas Cell (1, 1).

For discussion convenience, assuming a transporttime-step size of 1.0 day, the calculated Cr is also shownin Figure 1 (value in pink in parenthesis). Note that onlythe higher Cr value of x and y directions is shown in thefigure. Figure 1 indicates that both groundwater velocityand Cr increase quickly toward the well-cell, and thehighest groundwater velocity (120 ft/d) and Cr (6.0) occurat the well-cell.

Proposed MethodAs shown in the example presented in the previous

section, when a transport model involves groundwaterpumping, Cr values can increase by orders of mag-nitude in grid cells near the pumping well, especiallyif the pumping rate is high and grid sizes are small nearthe pumping well. Consequently, the maximum allowabletime-step �tmax is controlled by Cr value of the well-cell.The key to this approach is to increase θ eff in grid cellsnear the pumping well so that vi can be reduced in thosecells and subsequently �tmax can be increased withoutviolating stability constraints. For simplicity and conve-nience, the previously calculated vi and Cr distributionsaround the well-cell are used in the following analysisand discussion. Separate discussions of this method forcases of without and with modeling sorption are presentedbelow.

Without Modeling SorptionThe linear groundwater velocity vi is related to the

specific flux (Darcy flux) through the relationship of(Zheng and Bennett 2002):

vi = qi

θw

= −Ki

θw

∂h

∂xi

(4)

where qi is the Darcy flux [L/T], Ki is the hydraulicconductivity tensor [L/T], h is the hydraulic head [L] andθw and xi are defined previously. Equation 4 shows thatwithout changing the groundwater conditions vi can bedecreased by simply increasing θw (or θeff).

To numerically decrease vi near the pumping well,we can choose to increase θeff for different sets ofgrid cells. Take the previous artificial aquifer as anexample (Figure 1), we can choose the following gridcells centered at the pumping well: the well-cell only (Cell[1, 1]), the 2 × 2 grid cells (i.e., Cells [1, 1], [1, 2], [2,1], and [2, 2]), or more grid cells (i.e., 3 × 3 cells, 4 × 4cells, etc.).

To effectively increase �tmax, θeff should be increasedsuch that the Cr values of those grid cells are all decreasedto the highest Cr value of all grid cells for which θeff isnot changed. This will assure that θeff is not increasedexcessively. Specifically, if we choose to increase θeff forthe well-cell only, we should increase θeff such that theCr of the well-cell (Cr = 6.0) is dropped to 3.9 (i.e., Cr

NGWA.org J. Zhang et al. GROUND WATER 3

Figure 2. Increased θeff values (in pink) for 3 × 3 cells nearthe well-cell.

of Cells [1, 2] and [2, 1]), which is the highest Cr amonggrid cells for which θeff is not changed, (see Figure 1).Because θeff increased by a factor of 1.54 (=6.0/3.9) forthe well-cell, �tmax can be increased by a factor of 1.54.Similarly, if we choose to increase θeff for 2 × 2 grid cells,we should increase θeff for the well-cell and Cells (1, 2)and (2, 1) such that the Cr values of these three grid cellsare decreased to 1.65 (i.e., Cr of Cells [1, 3] and [3, 1],Figure 1). We may choose to increase θeff for 3 × 3 gridcells or more grid cells in a similar way. As an example,the increased θeff values for 3 × 3 grid cells are shown inFigure 2.

Note that increasing θeff in a grid cell will decreasevi and Cr in all directions evenly. Also note that althoughphysically θeff can never exceed unity, there is no limitfor increasing θeff in the numerical simulations.

With Modeling SorptionIn transport modeling, it is generally assumed that

equilibrium exists between the dissolved and the sorbedconcentrations, and that the sorption reaction is instan-taneous. The sorption isotherms are expressed using thedimensionless retardation factor defined as (Zheng andBennett 2002):

Rd = 1 + ρb

θw

∂S

∂C(5)

where ρb is the solid bulk density of the subsurfacemedium [M/L3], S is the chemical concentration sorbedon the solids [M/M], and the other parameters are definedpreviously. The most commonly used isotherm is thelinear isotherm in which the retardation factor is calculatedas (Zheng and Bennett 2002):

Rd = 1 + ρbKd

θw

(6)

where Kd is the distribution coefficient [L3/M], represent-ing the partition of a contaminant between a dissolvedphase and a sorbed phase.

Equations 5 and 6 show that when sorption is mod-eled, increasing θeff causes a decrease in retardation factorRd, which negates the increase of �tmax that resulted fromthe decreased linear velocity (see Equation 3).

This problem can be resolved by increasing both θeff

and ρb simultaneously by the same factor so that theRd value can be retained (Equations 5 and 6). Sincethe method of increasing both θeff and ρb for modelingsorption is exactly the same as increasing θeff in the caseof without sorption, no example of increasing both θeff

and ρb is provided here.Instead of increasing both θeff and ρb, we can

choose to increase θeff and other related parametersalternatively. For example, for a linear sorption isotherm,we can choose to increase the distribution coefficientKd (Equation 6). For Freundlich and Langmuir sorptionisotherms (Zheng and Wang 1999), we can increase theFreundlich constant and Langmuir constant, respectively.As increasing ρb is just as effective as increasing otherparameters and it works for all types of isotherms, nodiscussion is provided for other types of isotherms.

Effects of Proposed Method on Transport SimulationThe above analysis shows how the proposed method

can be used to increase �tmax. Note that the increased�tmax comes with a price: increasing θeff and ρb willchange the local properties of subsurface media andconsequently the transport simulation in grid cells nearthe pumping well. The potential effects on transportsimulation are discussed below.

When the pumping well is located inside a con-taminant plume, this method adds extra mass to theinitial mass. The amount of the artificially increased massdepends on the number of grid cells where θeff (and ρb)are increased, as well as the initial concentrations and thenet increase of θeff (and ρb) at those grid cells. The addedmass may have significant effects on transport simula-tion if the initial concentration is high in those grid cellsand/or if a great number of grid cells are increased withθeff (and ρb).

The increase in θeff will reduce the linear velocityvi while retaining the Darcy flux. Physically, increasingθeff near the pumping well is equivalent to replacingthe fast moving water/contaminant particles with moreparticles moving at a lower rate. As a result, the amountof contaminant mass transported by advection to thepumping well within a unit time is almost identical beforeand after the change in θeff. But the decreased vi inthose grid cells will cause less hydrodynamic dispersionin these grid cells (where mechanical dispersion dominatesover the molecular diffusion), which may affect theaccuracy of transport simulation.

However, those potential impacts are limited to gridcells where θeff (and ρb) are increased. The effects maynot be significant if only a small number of grid cells areincreased with θeff (and ρb) near the pumping wells and/or

4 J. Zhang et al. GROUND WATER NGWA.org

the initial concentrations are not high in those grid cells. Inaddition, the decreased hydrodynamic dispersion in thosegrids can be corrected by increasing the dispersivities bythe same factor of increasing θeff (and ρb). In the nextsection, a numerical test is presented to show the actualapplication of this method.

Application and Numerical Test

Application of the MethodFor practical flow and transport modeling, regional

groundwater flow often exists and the flow field isoften not uniform because of the heterogeneous aquiferproperties and other hydrologic features. When pumpingis conducted and if the pumping rate is relatively high, theradial groundwater flow toward the well often dominatesover the regional flow near the pumping well, thus thepreviously calculated vi and Cr distributions near the well-cell (Figure 1) are considered a good approximation inadjusting θeff (and ρb). If the radial flow is not dominantin grid cells near the well-cell (i.e., pumping rate is small),then the pumping will not cause a significant reduction in�tmax, and there is no need to use the proposed method.

It is noted that due to the assumed axisymmetric flowcondition, we only discussed the increase of θeff (and ρb)for gird cells in the upper-left quadrant. In practice, wehave to increase θeff (and ρb) for all grid cells around thewell-cell.

Numerical Test RunsTo demonstrate the effectiveness and efficiency of

the proposed method, the method was tested againsta simple flow and transport model developed for ahypothetical aquifer. The groundwater flow was modeledusing MODFLOW 2000 (Harbaugh et al. 2000; Hill et al.2000), and the chemical transport was modeled usingMT3DMS (version 5.3) (Zheng 2010).

Flow ModelFor simplification, the hypothetical aquifer is assumed

to be confined, isotropic, and heterogeneous, withspatially-variable hydraulic conductivities ranging from 2to 100 ft/d (0.61 to 30.5 m/d). In plan view, the aquiferis 6,000 feet (1,829 m) long (east-west direction) and4,000 feet (1,219 m) wide (north-south direction). Ver-tically, the aquifer has a uniform thickness of 30 feet(9.1 m). Regional groundwater flows from west to eastwith an average hydraulic gradient of 0.33%. The hypo-thetical contaminant plume is located at the center of theaquifer, and one well is pumping near the down-gradienttip of the plume.

The model domain covers the entire aquifer andwas discretized into 73 rows and 87 columns, with agrid-cell of 20 × 20 feet at the pumping well. The gridsize increases from the pumping well toward the modelboundaries by a factor of 1.1, and the maximum grid sizewas limited to 150 feet in length and 100 feet in width.The model domain and grid cells are shown in Figure 3.

Figure 3. Model domain, grid, boundary condition, welllocation, and simulated steady-state head distributions.

The well is assumed to pump at a constant rateof 10,000 ft3/d (283 m3/d) which causes dominant radialflow in grid cells near the well-cell. For simplification,no-flow boundary was specified to the north and southboundaries, and constant-head of 110 and 90 feet werespecified for the west and east boundaries, respectively.For simplification, it is also assumed that no other hydro-logic features (i.e., surface water recharge, evapotranspi-ration, etc.) exist inside the model domain. The pumpingwell location and the model boundary conditions are alsoshown in Figure 3.

Steady-state flow conditions associated with the spec-ified pumping and boundary conditions were simulated.The simulated steady-state hydraulic head distributions arealso shown in Figure 3.

Transport ModelThe flow model was augmented with a transport

model having the same model domain and grid cells. Ahypothetical contaminant plume with a maximum concen-tration of 500 μg/L was used as the initial concentrationconditions (Figure 4). It was also assumed that there wereno point or areal contaminant sources or sinks inside themodel domain. The contaminant transport was modeledunder the previously modeled steady state flow condition.

Since the method of increasing θeff for the no sorptioncase is the same as increasing both θeff and ρb for the case

Figure 4. Locations of pumping well and observation pointsand initial concentrations in transport model.

NGWA.org J. Zhang et al. GROUND WATER 5

of with sorption, only the transport model with sorption ispresented here. The following parameters were assumedfor the transport model: longitudinal and transversedispersivity of 50 and 5.0 feet, respectively, θeff of 0.25,ρb of 1.6 kg/L, and Kd of 0.22 L/kg. Consequently, theRd was calculated as 2.41. To make the transport problemmore general, contaminant degradation was also modeled.Specifically, a degradation rate of 0.00019/d (i.e., halflife of 10 years) was specified for a contaminant in bothdissolved and sorbed phases. In solving the advectionterm, the most commonly used third-order TVD schemewas chosen, which is subject to stability constraints withrespect to the transport time-step size.

The transport model was first run with the originalθeff of 0.25 and ρb of 1.6 over the entire model domain(baseline simulation), and then run by increasing θeff andρb for the following grid cells (test runs): (a) well-cellonly; (b) 3 × 3 cells centered on the well-cell; (c) 5 × 5cells centered on the well-cell; and (d) 7 × 7 cells centeredon the well-cell. Because of the dominant radial flow ingrid cells near the well-cell, increasing θeff and ρb for thetest runs was determined based on the Cr distributionsof the previous example (Figure 1). For baseline and testruns, transport simulation was run for a period of 20 years.

To show the effectiveness of the method, the �tmax

determined by MT3DMS and the CPU processing timeof transport simulation were recorded and compared forall model runs. In addition, the simulated concentrationsversus time were recorded at the pumping well and at twoselected observation points. To show the potential effectsof the proposed method on the simulation results, thoserecorded concentrations of the test runs were comparedwith those of the baseline simulation. The locations ofthe pumping well and the two selected observation pointsare also shown in Figure 4.

Results of Numerical TestFigure 5 shows the comparison of maximum time-

step sizes (�tmax) and the CPU processing times forthe baseline and the test runs. The comparison indicatesthat increasing the θeff and ρb for grid cells near the

Figure 5. Increased maximum time-step sizes and reducedCPU times from the proposed method.

well-cell can increase �tmax significantly, thus signifi-cantly decreasing the CPU processing time of transportsimulation. In this example, increasing θeff and ρb for7 × 7 grid cells leads to an increase of �tmax by a factorof approximately 10, and CPU processing time is reducedalso by a factor of approximately 10. Note that althoughCPU processing time is largely controlled by time-stepsize when other factors are unchanged, it is also affectedby the iteration numbers in solving the transport equation,and different time-step sizes often cause different iterationnumbers. Consequently, CPU processing time is approxi-mately inversely proportional to the time-step size.

Figure 6a shows the simulated concentration versustime at the pumping well for the baseline simulationand the four test runs. The comparison of the resultsshows that the modeled concentrations of test runs (a), (b),and (c) are practically identical to those of the baselinesimulation. The modeled concentrations of test run (d) areslightly higher than those of the baseline simulation forthe first five years and gradually become the same as thoseof baseline simulation after five years. This is because thepumping well is located inside the contaminant plume, andincreasing θeff and ρb for those grid cells also increasesthe initial contaminant mass slightly.

Model results also show that the four test runsmodeled the same temporal concentration distributions atthe two observation points. Figure 6b shows the simulatedtemporal concentration distributions at the observation

Figure 6. Comparison of modeled concentrations: (a) at thepumping well for baseline and the four test runs; (b) at thetwo observation points for baseline and test run (d).

6 J. Zhang et al. GROUND WATER NGWA.org

Figure 7. Modeled total mass error variation over time ofthe four test runs.

points for the baseline and the test run (d). The comparisonindicates that the temporal concentration distributions arepractically identical. We conclude that the effects ofincreasing θeff and ρb on transport simulation are mainlylimited to those grid cells where θeff and ρb are changed.

The difference between the total mass in baselinesimulation and a test run is calculated, as a percent error,using the following formula:

Error (%) = Total MassTest run − Total MassBaseline

Total MassBaseline×100

(7)

where “Total MassTestRun” and “Total MassBaseline” are themodeled total contaminant mass in the aquifer for thebaseline simulation and a test run, respectively.

Figure 7 shows the modeled mass error variation overtime for the four test runs. The plots show that the masserror increases as more grid cells are increased with θeff

and ρb. Specifically, the mass error of test run (a) ispractically zero, and mass error of test run (b) is also verysmall (initial mass error of 0.4% and mass error drops toless than 0.1% after three years). However, the initial masserror increases to 1.2 and 4.3% for test runs (c) and (d),respectively. Those mass errors drop to 0.2 and 0.8% afterapproximately 5 years.

The example test results show that in order tomaintain appropriate accuracy, increasing θeff and ρb for7 × 7 or more grid cells near the well is not suggested, butincreasing θeff and ρb for up to 5 × 5 grid cells is probablyacceptable. In modeling practice, for a more conservativeconsideration, increasing θeff and ρb for 3 × 3 grid cells(actually increasing θeff and ρb for the well-cell and thefour grid cells immediately next to the well-cell) maybe a good choice, which can still increase �tmax byapproximately four times.

Summary and ConclusionA method is proposed to accelerate transport simu-

lations involving groundwater pumping by increasing the�tmax. The key of this method is to numerically decrease

the groundwater velocity in grid cells near the pumpingwell, thus allowing larger �tmax without violating stabilityconstraints.

For a transport model without adsorption, decreasingthe groundwater velocity can be achieved by increasingθeff. However, for a transport model with adsorption,increase of θeff has to be accompanied by an increase in ρb

or Kd by the same multiplication factor because increaseof θeff alone results in a reduced retardation factor, andsubsequently negates the positive effect on �tmax.

The proposed method decreases computational timeeffectively, but a certain level of model accuracy issacrificed due to the change of transport properties at gridcells near the pumping well. Before applying this methodmodelers should consider the following:

1. When the pumping wells are located inside a contam-inant plume, the artificially added contaminant massmay affect the transport simulation significantly if theinitial concentration is high near the pumping welland/or a large number of grid cells (more than 5 × 5cells around one pumping well) are involved.

2. Hydrodynamic dispersion at the grid cells where θeff

is changed will be reduced due to the reduced linearvelocity.

A hypothetical numerical test demonstrated thatthe proposed method may increase �tmax significantly,while the transport simulation results remain literallyunchanged.

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