Analytical Solution of Two-dimensional Solute Transport in an Aquifer Aquitard

Journal of Contaminant Hydrology 107 (2009) 162–174

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Journal of Contaminant Hydrology

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Analytical solution of two-dimensional solute transport in anaquifer–aquitard system

Hongbin Zhan a,⁎, Zhang Wen b,c, Guanhua Huang b,c, Dongmin Sun d

a Department of Geology and Geophysics, Texas A&M University, College Station, TX 77843-3115, United Statesb Department of Irrigation and Drainage, College of Water Conservancy and Civil Engineering, China Agricultural University, Beijing, 100083, PR Chinac Chinese–Israeli International Center for Research in Agriculture, China Agricultural University, Beijing, 100083, PR Chinad Department of Environmental Sciences, University of Houston-Clear Lake, Houston, TX 77058-1098, United States

a r t i c l e i n f o

⁎ Corresponding author. Tel.: +1 979 862 7961; faxE-mail address: [email protected] (H. Zhan).

0169-7722/$ – see front matter © 2009 Elsevier B.V.doi:10.1016/j.jconhyd.2009.04.010

a b s t r a c t

Article history:Received 6 March 2008Received in revised form 21 April 2009Accepted 28 April 2009Available online 7 May 2009

This study deals with two-dimensional solute transport in an aquifer–aquitard system bymaintaining rigorous mass conservation at the aquifer–aquitard interface. Advection,longitudinal dispersion, and transverse vertical dispersion are considered in the aquifer.Vertical advection and diffusion are considered in the aquitards. The first-type and the third-type boundary conditions are considered in the aquifer. This study differs from the commonlyused averaged approximation (AA) method that treats the mass flux between the aquifer andaquitard as an averaged volumetric source/sink term in the governing equation of transport inthe aquifer. Analytical solutions of concentrations in the aquitards and aquifer and masstransported between the aquifer and upper or lower aquitard are obtained in the Laplacedomain, and are subsequently inverted numerically to yield results in the real time domain (theZhan method). The breakthrough curves (BTCs) and distribution profiles in the aquiferobtained in this study are drastically different from those obtained using the AA method.Comparison of the numerical simulation using the model MT3DMS and the Zhan methodindicates that the numerical result differs from that of the Zhan method for an asymmetric casewhen aquitard advections are at the same direction. The AA method overestimates the masstransported into the upper aquitard when an upward advection exists in the upper aquitard.The mass transported between the aquifer and the aquitard is sensitive to the aquitard Pecletnumber, but less sensitive to the aquitard diffusion coefficient.

© 2009 Elsevier B.V. All rights reserved.

Keywords:Aquitard diffusionAquitard advectionMass conservationSolute transportLaplace transform

1. Introduction

Interaction between aquifers and aquitards is an importantprocess affectingflowand transport in subsurfaceflowsystems.Most aquitards consist of silt and clay and are well capable ofstoring water and solute, due to their large values of porosity.Thus when a solute in an aquifer contacts a previously solute-free aquitard, a concentration gradient exists across the aquifer-aquitard interface; andmolecular diffusionwill drive the soluteinto the aquitard. Furthermore, leakage often exists across theaquitard, thus advection in the aquitard will be anotherimportant mechanism for solute transport there.

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Diffusion at the aquifer–aquitard interface is somewhat similarto diffusion at the matrix-fracture boundary (Tang et al., 1981;Sudicky and Frind, 1982, 1984; Fujikawa and Fukui, 1990; Liu etal., 2004). The difference is that the aperture of a fracture ismuchsmaller than theaquifer thickness and theflowvelocity inthe fracture is oftenmuch greater than that in the aquifer underthe same hydraulic gradient. Many studies on fractured mediahave shown that matrix diffusion is the primary factor forretarding contaminants in the fractures (e.g. Neretnieks, 1980;Rasmuson and Neretnieks, 1981; Neretnieks et al., 1982;Moreno et al., 1985; Liu et al., 2004). Aquitard diffusion wasshown to be a controlling factor affecting solute transport inlaboratory experiments by Sudicky et al. (1985), Starr et al.(1985), Young and Ball (1998), and in numerous field aquiferstudies such as Gillham et al. (1984), Johnson et al. (1989), Ball

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163H. Zhan et al. / Journal of Contaminant Hydrology 107 (2009) 162–174

et al. (1997a,b), Liu and Ball (1999), Hendry et al. (2003),Hunkeler et al. (2004), Parker et al. (2004) and others. Afterpassingof the solute front in the aquifer, back diffusion from theaquitard to the aquifer is the primary cause of the tailing effectobserved in the aquifer, which has caused great difficulty forcontaminant remediation (Liu and Ball, 2002). Another im-portant reason causing the tailingeffect in real aquifer–aquitardsystems is diffusion into and out of low-permeability lenseswithin the aquifer materials (Gillham et al., 1984).

Sudicky et al. (1985) have investigated the aquitard diffusioneffect in an artificial sandyaquiferwhose thickness is about 0.02to 0.03 m in the laboratory. Because the aquifer thickness is sothin in respect to the horizontal transport distance, they couldapproximate the diffusive mass flux at the aquifer–aquitardinterface as a volumetric source/sink term in the governingequation of solute transport in the aquifer. Chen (1985) andTang and Aral (1992a,b) have also adopted this approach tostudy dispersion–diffusion in an aquifer–aquitard system withradial and uniform flows, respectively. The implication of thisapproach is that the transverse mixing across the aquiferthickness is so rapid that a thickness-averaged concentrationcan be used. The same approximation has been broadly adoptedin studying matrix diffusion in fractured media (Neretnieks,1980; Rasmuson and Neretnieks, 1981; Tang et al., 1981;Neretnieks et al., 1982; Sudicky and Frind, 1982, 1984; Fujikawaand Fukui, 1990; Johns and Roberts, 1991; Liu et al., 2004).

Sudicky et al. (1985) have realized that in real aquifers, thetransverse mixing is probably not always rapid enough towarrant the usage of a thickness-averaged approach. There-fore, they proposed an alternative method of treating thediffusive flux at the aquifer–aquitard interface as a boundarycondition rather than a source/sink term in the governingequation of transport in the aquifer. But to make the workamenable to analytical solution, Sudicky et al. (1985) haveneglected the longitudinal dispersion and only considered thetransverse vertical dispersion in the aquifer. Such a treatmentmight provide a satisfactory interpretation of conservativesolute transport in a very thin artificial sandy aquifer used inthe experiment of Sudicky et al. (1985). However, for morerealistic thicker aquifers, Starr et al. (1985) showed thatlongitudinal dispersion is important and should be taken intoaccount. In a different paper, Johns and Roberts (1991) haveproposed a model for investigating solute transport in large-aperture fractures by considering lateral dispersion to thesmall aperture regions and diffusion to the rock matrix.Longitudinal dispersion is not considered in that study.

Advection in the aquitard is rarely considered in previousanalytical solutions although it can be dealt with in numericalsimulations (see for example Bester et al., 2005). Most con-fined aquifer systems are recharged by leakage through theoverlying aquitard, which means, in case of transport, thatthere will be advection across the interface. Advective massflux in the aquitard may be small, but may be at a similar orgreater scale when compared to diffusive flux across theinterface. Therefore, to make the analytical solution useful,both advective and diffusive fluxes in the aquitard are con-sidered in this study. It is also important to compare therelativemagnitudes of advective versus diffusive fluxes acrossthe interface under realistic conditions.

In this study, we will solve for two-dimensional transportin the aquifer and one-dimensional advection–diffusion in

the aquitard simultaneously for a fully penetrating, horizon-tally infinite source without using the averaged approxima-tion employed by Sudicky et al. (1985), Chen (1985), Tangand Aral (1992a,b), and others. Mass balance requirement isrigorously maintained. An example of such a two-dimen-sional transport scenario is the leaking of toxic materials thatare buried in a long trench or a large landfill site. There is nodoubt that three-dimensional numerical simulations of flowand transport in complex aquifer–aquitard systems can becarried out with the present-day's computational power. Forinstance, Martin and Frind (1998) have carried out a three-dimensional numerical simulation of groundwater flow andcapture zone description for a multiple layer aquifer–aquitardsystem in the Waterloo Moraine. Bester et al. (2005) havecarried out a numerical simulation of road salt impact on anurban wellfield located in an aquifer–aquitard system.

The purpose of this paper is to illustrate the essence of theaquitard effect on solute transport from an analytical per-spective. Such analytical solutions may serve the purpose ofvalidating the numerical simulations which may suffer fromvarious types of numerical errors. For instance, the numericalerrors in numerical models tend to be the largest at theaquifer–aquitard interfaces (Martin and Frind, 1998; Besteret al., 2005). For the sake of simplicity of presentation, weonly discuss a conservative solute.

2. Conceptual and mathematical models

2.1. Conceptual model

The system investigated is an aquifer bounded at the topand bottom by two aquitards, or an aquifer bounded at the topby an aquitard and at the bottom by impermeable bedrock.The aquifer–aquitard and the bedrock–aquifer boundaries areassumed to be horizontal. The aquifer is homogeneous andhorizontally isotropic with constant longitudinal and trans-verse vertical dispersivities. The aquitards are also homo-geneous and sufficiently thick so that solute diffusion is notaffected by their thicknesses. This assumption may be rea-sonable because the penetration depths of solute into theaquitards via diffusion are often limited; meaning that onlythose regions of the aquitards that are very close to theaquifer will be affected by solute transport. However, if one isdealing with a very thin aquitard, it is possible that solute canpenetrate through the entire thickness of the aquitard into theadjacent aquifer. If this is the case, one must consider theaffected adjacent aquifer as well. Such a circumstance is ofinterest but will not be discussed in this article. The aquifer,the aquitards, and the bedrock extend horizontally to infinity.

Fig. 1A–B show the schematic diagrams of two cases thatwill be investigated. They are an aquifer bounded by upperand lower aquitards, and an aquifer bounded by an upperaquitard and lower impermeable bedrock, respectively. Thebig arrows there show the groundwater flow directions. Thehydraulic conductivity of the aquitard is assumed to be a feworders of magnitude smaller than that of the aquifer, thusadvective flow in the aquitard is nearly perpendicular to theaquifer–aquitard interface. Flow in the aquifer is nearly hori-zontal. We set up the coordinate system as follows. The x- andz-axes are along the horizontal and vertical directions, res-pectively with the origin at the left boundary. We choose the

Fig. 1. Schematic diagram of: A) an aquifer bounded at top and bottom byaquitards; B) an aquifer bounded at top by an aquitard and at the bottom byimpermeable bedrock.

164 H. Zhan et al. / Journal of Contaminant Hydrology 107 (2009) 162–174

origin of the coordinate system at the center of the leftboundary of the aquifer in Fig. 1A, whereas it is at the bottomof the left boundary of the aquifer in Fig. 1B.

We will deal with two kinds of boundary conditions,namely the first-type and the third-type conditions for thesource. At first, a constant concentration (the first-type) ismaintained at the left boundary for the entire thickness of theaquifer. The length of the source in the direction transverse tothe x–z plane is assumed to be sufficiently large that dis-persion in that direction is not considered. The first-typeboundary condition has been commonly used in many conta-minant transport problems described by Huyakorn et al.(1987), Leij et al. (1991), Batu (1996), and others.

However, some investigators such as Kreft and Zuber(1978, 1979) and van Genuchten and Parker (1984) havepointed out that the first-type boundary condition might notcorrectly predict the resident concentration, although it cancorrectly predict the flux concentration. They have shownthat the third-type or the flux-type boundary condition isprobably more suitable for predicting the resident concentra-tion in a column test. The difference between the first-typeand the third-type boundary conditions is expected to besmall when the concentration gradient near the boundary issmall. In real aquifer contamination problems, both bound-aries seem possible, depending on the site-specific condi-tions. In this article, we will derive solutions for both types ofboundary conditions. Unless stated otherwise, the concentra-tion is the resident concentration. If needed, the flux concen-

tration can be easily calculated via the resident concentrationusing the method of van Genuchten and Parker (1984).

The solution is based on the assumption of a constanthorizontal velocity in the aquifer and constant vertical velo-cities in the aquitards. This will lead to constant dispersioncoefficients in the aquifer and the aquitards. In reality, velo-cities in the aquifer and the aquitards could be variable andso are the dispersion coefficients. A numerical simulation isprobably more useful when variable velocities are encoun-tered. It is assumed that the aquifer and the aquitard are freefrom solutes at the start of transport. Based on the conceptualmodel, the following mathematical model is established.

2.2. Solute transport in an aquifer bounded by upper and loweraquitards

The governing equations and the initial and boundaryconditions for the aquifer and aquitards of this case are asfollows. For the aquifer,

ACAt

= DxA2C

Ax2+ Dz

A2C

Az2− v

ACAx

; ð1Þ

C x = 0; z; tð Þ = C0; ð2Þ

C x = + ∞; z; tð Þ = 0; ð3Þ

C x; z; t = 0ð Þ = 0; for x N 0; ð4Þ

and for the upper aquitard,

AC1

At= D01

A2C1

Az2− v1z

AC1

Az; ð5Þ

C1 x; z = B; tð Þ = C x; z = B; tð Þ; ð6Þ

θ1v1zC1 x; z = B; tð Þ− θ1D01AC1 x; z = B; tð Þ

Az

= − θDzAC x; z = B; tð Þ

Az;

ð7Þ

C1 x; z = + ∞; tð Þ = 0; ð8Þ

C1 x; z; t = 0ð Þ = 0; ð9Þ

and for the lower aquitard,

AC2

At= D02

A2C2

Az2− v2z

AC2

Az; ð10Þ

C2 x; z = − B; tð Þ = C x; z = − B; tð Þ; ð11Þ

θ2v2zC2 x; z = − B; tð Þ− θ2D02AC2 x; z = − B; tð Þ

Az

= − θDzAC x; z = − B; tð Þ

Az;

ð12Þ

C2 x; z = − ∞; tð Þ = 0; ð13Þ

C2 x; z; t = 0ð Þ = 0; ð14Þ

where C, C1, C2 represent the resident concentrations in theaquifer, theupper aquitard, and the lower aquitard respectively;

Table 1Dimensionless variables used in this study.

xD = xB

ffiffiffiffiDzDx

q; zD = z

B ; tD = DzB2 t β1 = D01

Dz;β2 = D02

Dz;

CD = CC0;C1D = C1

C0; C2D = C2

C0; σ1 = θ1

θ ;σ2 = θ2θ ;

M1D = M1

C0B2ffiffiffiffiffiffiffiffiffiffiffiDx = Dz

p ; M2D = M2

C0B2ffiffiffiffiffiffiffiffiffiffiffiDx = Dz

p Pe = vBffiffiffiffiffiffiffiffiDxDz

p ; Pe1 = v1zBDz

; Pe2 = v2zBDz


C0 is the constant concentration at x=0; v is the averagepore velocity of groundwater flow in the aquifer, v1z and v2z arethe vertical pore velocities of groundwater flow in the upperand lower aquitards, respectively and are positive for upwardflow; Dx and Dz are the longitudinal and vertical hydrodynamicdispersion coefficients respectively for the aquifer, and Dx=αxv+D0⁎, Dz=αzv+D0⁎, where αx and αz are the longitudinaland transverse vertical dispersivities, respectively, D0⁎is theeffective molecular diffusion coefficient of the solute in theaquifer (Bear, 1972); D01 and D02 are the effective moleculardiffusion coefficients in the upper and lower aquitards,respectively; B is the half thickness of the aquifer; θ, θ1,and θ2 are the porosities of the aquifer, the upper aquitard, andthe lower aquitard, respectively; and t is time. The effectivemolecular diffusion coefficients in the upper and loweraquitards are D01=τ1D0, and D02=τ2D0, respectively, whereτ1 and τ2 are tortuosities of the upper and lower aquitards,respectively, andD0 is the freemolecular diffusion coefficient ofthe solute.

Eqs. (6) and (7) are the continuities of concentration andmass flux at the upper aquifer–aquitard interface at z=B,respectively, whereas Eqs. (11) and (12) are the continuitiesof concentration and mass flux at the lower aquifer–aquitardinterface at z=−B, respectively. Both aquitards are thoughtto be sufficiently thick for the transport problem discussedhere.

If using the third-type boundary condition, Eq. (2) will bereplaced by

vC−DxACAx

� �jx=0 + = vC0: ð15Þ

A point to note is that diffusion along the x-axis is notincluded in the governing equations of transport in theaquitards (see Eqs. (5) and (10)). This is based on twoconsiderations. First, one purpose of this study is to comparethe result with those obtained with the averaged approxima-tion which only consider diffusion along the z-axis in theaquitard as well (Sudicky et al., 1985; Chen, 1985; Tangand Aral, 1992a,b). Second, adding the diffusion term alongthe x-axis in Eqs. (5) and (10) will make the analyticalmodeling challenging and complex. From a rigorous physicalpoint of view, neglecting diffusion along the x-axis inthe aquitard will certainly introduce some errors to themodeling results. Now the question is: how large are sucherrors? At the moving front of a contaminant plume wherethe concentration gradient along the x-axis could be verylarge, the diffusion term along the x-axis could be comparablewith other terms in the equations, thus one is expected to seesome non-negligible errors. On the other hand, at regionswhere the concentration gradients along the x-axis are nearlyzeroes, such as at regions far behind the moving front of acontaminant front, the errors of neglecting the diffusionterm along the x-axis should be minimized and negligible.Fortunately, many problems associated with aquitard con-tamination often involve time scales of many decades andcontaminant moving fronts that have traveled long distancesfrom persistent sources of contaminant, thus the resultdeveloped in this study will be applicable for regions thatare close to those sources (or far behind the moving fronts).Nevertheless, one still hopes that an analytical or semi-

analytical solution can be developed to consider diffusions inboth the x- and z-axes in the future.

Defining the dimensionless terms in Table 1 where thesubscript “D” represents the dimensionless term, and Pe, Pe1,Pe2 are the Peclet numbers of the aquifer, the upper aquitard,and the lower aquitard, respectively (Bear, 1972). Pe1 or Pe2will be positive for upward advection and negative for down-ward advection. After transforming Eqs. (1)–(9) into di-mensionless form, applying the Laplace transform to thegoverning equations and boundary conditions will result inthe following equation groups.

pCD =A2CD

Ax2D+

A2CD

Az2D− Pe

ACD

AxD; ð16Þ

CD xD = 0; zD;pð Þ = 1 = p; ð17Þ

CD xD = + ∞; zD;pð Þ = 0; ð18Þ

and for the upper aquitard,

pC1D = β1A2C1D

Az2D− Pe1

AC1D

AzD; ð19Þ

C1D xD; zD = 1; pð Þ = CD xD; zD = 1;pð Þ; ð20Þ

σ1½β1AC1D xD; zD = 1; pð Þ

AzD− Pe1C1D xD; zD = 1; pð Þ�

=ACD xD; zD = 1; pð Þ

AzD;

ð21Þ

C1D xD; zD = + ∞;pð Þ = 0; ð22Þ

and for the lower aquitard,

pC2D = β2A2C2D

Az2D− Pe2

AC2D

AzD; ð23Þ

C2D xD; zD = − 1;pð Þ = CD xD; zD = − 1;pð Þ; ð24Þ

σ2 β2AC2D xD; zD = − 1; pð Þ

AzD− Pe2C2D xD; zD = − 1; pð Þ

" #

=ACD xD; zD = − 1; pð Þ

AzD;

ð25Þ

C2D xD; zD = − ∞; pð Þ = 0; ð26Þ

where CD̅, C1̅D, C2̅D are the Laplace transforms of CD, C1D, C2D,respectively, and p is the Laplace transform parameter inrespective to the dimensionless time. All the associated termsare explained in Table 1.


The procedure of solving the equation group of Eqs. (16)–(22) is given in Appendix A. The derived solutions for CD̅, C1̅D,and C2̅D are:

CD =X∞n=0

Anexp −

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPe2 + 4 p + ω2

n

� �q− Pe

2xD

24

35 cos ωnzD + μnð Þ;

− 1 V zD V 1; ð27Þ

C1D =X∞n=0

Anexp½− ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPe2 + 4 p + ω2

n

� �q− Pe

2xD

−

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPe1 =β1ð Þ2 + 4p = β1

q− Pe1 = β1

2zD − 1ð Þ� cos ωn + μnð Þ;

zD z 1:

ð28Þ

C2D =X∞n=0

Anexp½− ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPe2 + 4 p + ω2

n

� �q− Pe

2xD

+

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPe2 =β2ð Þ2 + 4p = β2

q+ Pe2 = β2

2zD + 1ð Þ� cos ωn − μnð Þ;

zD V − 1:

ð29Þ

where ωn and μn are the frequency and the phase termsthat are determined via Eqs. (A8) and (A9), respectively inAppendix A, and

An =4F1 ωn; μnð ÞpF2 ωn; μnð Þ ; ð30Þ

where F1(ωn, μn) and F2(ωn, μn) are given in Eq. (A11).If the third-type boundary condition is used at xD=0,

Eq. (17) will be replaced by

CD−1Pe

ACD

AxD

!jxD =0+

= 1= p: ð31Þ

The solutions for the third-type boundary condition areidentical to Eqs. (27)–(29) except that An is given by

An =4F1 ωn; μnð Þ

pF2 ωn; μnð ÞG ωnð Þ ; ð32Þ

where G(ωn) is:

G ωnð Þ =ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 + 4 p + ω2

n

� �= Pe2

q+ 1

� �= 2: ð33Þ

We call the results of Eqs. (27)–(29) the Zhan methodhereinafter.

For the special case in which the upper and loweraquitards have identical hydrologic parameters, μn=0.Furthermore, if the advective velocities in the upper andlower aquitards have the same magnitude but oppositedirection, then the system investigated becomes symmetricabout the line of z=0 which can be treated as a no-fluxboundary. This can also be seen easily fromEqs. (28) and (29).

2.3. Solute transport in an aquifer bounded by an upperaquitard and lower impermeable bedrock

If the aquifer is bounded by an upper aquitard and lowerimpermeable bedrock, which is treated as a no-flux boundary

with negligible diffusion into the bedrock (Fig. 1B), the con-centrations in the aquifer and aquitard can be easily obtainedby modifying the solutions derived in Section 2.2. This can bedone as follows. One can use the x axis or (z=0 line) as asymmetric line to make an image aquifer and an image loweraquitard from z=0 to −∞. The image aquifer and aquitardhave the same hydrological parameters as their counterparts,respectively. Furthermore, the advective velocities in theupper aquitard and the lower image aquitard have the samemagnitude but opposite direction. The image aquifer and theaquifer are now combined into an equivalent hypotheticalaquifer with thickness 2B. The system generated in such awayis a special case of Section 2.2 as mentioned after Eq. (33).Therefore, the problem shown in Fig. 1B can be regarded asthe upper half-plane of a hypothetical aquifer of thickness 2Bwith identical upper and lower aquitard parameters and thesame magnitude but opposite aquitard advective velocities.With such a modification, the solutions derived in Section 2.2can be directly applied for the case of Fig. 1B.

2.4. Numerical consideration

Now the solutions are derived in the Laplace domain.Analytical inverse Laplace transform is often difficult, if notimpossible for the problems discussed here. Previous studiesby Tang et al. (1981), Sudicky et al. (1985), and many othershave shown that even under simplified conditions, analyticalinverse Laplace transform of the transport problemmight endin multiple integrations that can only be calculated numeri-cally. Therefore, we will use numerical inverse Laplace trans-form to find the concentrations in the real time domain.

The numerical inverse Laplace transform can be carried outusing themethodsdevelopedeither by Stehfest (1970), or Talbot(1979), or de Hoog et al. (1982), among others. The Laplacetransform Galerkin technique (LGT) proposed by Sudicky(1989) is anotherhighlyefficient and accurate numerical inverseLaplace transform method for solving flow and transport.Algorithms by Talbot (1979), de Hoog et al. (1982), and Sudicky(1989) involve complex variables, which are difficult to usein our solutions involving a frequency termωn and a phase termμn.

The Stehfest (1970) algorithm, on the other hand, onlyrequires real values of p. It has been used successfully byseveral hydrologists in similar problems (Moench and Ogata,1981; Moench, 1991), but it sometimes suffers from oscilla-tion and convergence problems. Fortunately, this algorithmis found to work well for the problems investigated herewhen comparing the numerical solutions with analyticalsolutions under simplified situations. The number of termsused in the Stehfest method is N=12, which seems to yieldthe best result for the problems investigated here. Under thespecial condition that transport in the aquitard is negligible,closed-form analytical solution exists. The results obtainedfrom the numerical inverse Laplace transformmethod is thentested against the closed-form analytical solution and agree-ment has been reached. Nevertheless, such an agreementmaynot be enough to prove that the numerical inverse Laplacetransform method is always accurate for the general casethat the aquitard transport is non-negligible; it gives ussome confidence of using the numerical inverse Laplacetransform.

167inant Hydrology 107 (2009) 162–174
H. Zhan et al. / Journal of Contam
3. Comparison with previous studies

3.1. Comparison with analytical solutions

It can be shown that for no flow and nomass flux across the aquifer–aquitard interfaces, and constant source concentration, thesolution simplifies to the well-known one-dimensional transport solution by Ogata and Banks (1961). Without the interface flux,β1=β2=0 and ωn=nπ, where n=0, 1, 2, ....from Eq. (A6). The only non-zero An term from Eq. (A7) is A0=1/p. After this, it iseasy to verify that the inverse Laplace transform of (A4) will result in the following solution in real time domain:

CD =12

erfcxD − PetD

2ffiffiffiffiffitD

p� �

+ exp λxDð Þerfc xD + PetD2ffiffiffiffiffitD

p� �

; ð34Þ

where erfc ( ) is the complementary error function. Eq. (34) is the well-known one-dimensional solution derived by Ogata andBanks (1961).

For the third-type boundary condition at xD=0, A0=1/[pG(0)], and An=0 for n≠0, where G(0) is given by Eq. (33) by settingωn=0. It is not difficult to verify that the inverse Laplace transform of the general solution (A4) under this circumstance willbecome:

CD =12erfc

xD − PetD2ffiffiffiffiffitD

p� �

+12

Pe2tDπ

!12

exp − xD−PetDð Þ24tD

" #− 1

41 + PexD + Pe2tD� �

exp PexDð Þerfc xD + PetD2ffiffiffiffiffitD

p� �

ð35Þ

Eq. (35) is the well known one-dimensional solution for the third-type boundary derived by van Genuchten (1981) and severalothers. Detailed derivation of the inverse Laplace transform can be found from van Genuchten (1981) or from the author uponrequest.

3.2. Averaged approximation of aquitard advection and diffusion

In many previous studies, aquitard diffusion is approximated as a volumetric source/sink term in the governing equation oftransport in the aquifer, and aquitard advection is not taken into account (Neretnieks, 1980; Tang et al., 1981; Chen, 1985; Sudickyet al., 1985; Fujikawa and Fukui, 1990; Tang and Aral, 1992a,b). In the following, we are going to include advection as part of themass transport mechanism. The governing Eq. (1) under this approximation is modified as:

ACAt

= DxA2C

Ax2− v

ACAx

− C1

2θB+

C2

2θB; ð36Þ

where Γ1 and Γ2 are the mass fluxes across the upper and lower interfaces of the aquifer and they include both advective flux anddiffusive flux:

C1 = θ1v1zC1−θ1D01AC1

Az

� �jz=B

;C2 = θ2v2zC2−θ2D02AC2

Az

� �jz= −B

: ð37Þ

The governing equations of the aquitards are still the same as Eqs. (5) and (10). C and C1 are continuous at the upper aquifer–aquitard interface, whereas C and C2 are continuous at the lower aquifer–aquitard interface. Changing above equations into theirdimensionless forms and applying Laplace transforms, one can obtain the following solutions in the Laplace domain for the first-type boundary condition as:

CD =1pexp −

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPe2 + 4p + σ1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPe21 + 4pβ1

q+ Pe1

� �+ σ2


q− Pe2

� �s− Pe

2xD

266664

377775; ð38Þ

C1D = CD xD; zD = 1; pð Þ × exp −

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPe1 =β1ð Þ2 + 4p= β1

q− Pe1 = β1

2zD − 1ð Þ

24

35; ð39Þ

C2D = CD xD; zD = − 1;pð Þ × exp


q+ Pe2 = β2

2zD + 1ð Þ

24

35: ð40Þ

If ignoring the advective fluxes in the aquitards and assuming the upper and lower aquitards have identical parameters, thenthe solutions come back to those derived by Tang et al. (1981), Fujikawa and Fukui (1990), and others. The same procedures can be


applied for the third-type boundary condition and will not be repeated here. Inverse Laplace transform of Eqs. (38)–(40) will yieldthe concentrations in the real time domain. We call the results of Eqs. (38)–(40) the averaged approximation (AA) methodhereinafter. Now we compare the AA method with the Zhan method derived in Section 2.2.

4. Results analysis

The results of this study are formulated in dimensionlessforms which can be applied to any suitable real parameters. Forthe sake of helping readers to understand the range of thosedimensionless variables, a table of realistically possible valuesis provided in Table 2 and is briefly illustrated. The effectivemolecular diffusion coefficients chosen for the aquitard (D01 andD02) and the aquifer (D0⁎) are equivalent to a dilute NaCl solutionwith D01=D02=D0⁎=1.16×10−9 m2/s. The average porevelocity v is chosen to be 0.10 m/day or 1.16×10−6 m/s. Thesame values of effective molecular diffusion coefficient andaverageporevelocitywerealsoused inSudickyet al. (1985). Forafield-scale dispersion problem, we choose the longitudinaldispersivity αx=2 m. The transverse vertical dispersivity isexpected to be 0.0095 of the longitudinal one as αz=0.019 m.Such a choice is consistentwithfield-scale dispersion (Domenicoand Schwartz, 1998; Fetter, 1999) but is greater than the localdispersivity used in the laboratory experiment of Sudicky et al.(1985) and the field experiment of Sudicky (1986), which is0.001 m. Therefore, the corresponding dispersion coefficientsare Dx=αxv+D0⁎=2.32×10−6 m2/s, and Dz=αzv+D0⁎=2.32×10−8 m2/s. The aquifer thickness is 2B=4 m. Thecorresponding Peclet number Pe is 10. Two different leak-age velocities in the aquitard are v1z=1.16×10−10 m/s, and1.16×10−9 m/s, respectively, corresponding to Pe1 of 0.1, and 1.The positive sign of v1z (or v2z) indicates that water is movingupward. Theporosity difference among the aquitards and aquiferis expected to be a secondary effect, thus the identical value ofporosity 0.36 is assigned for all units. The same porosity is usedfor the aquitard in Sudicky et al. (1985). The first-type boundarycondition is used in the following discussion.

4.1. Transport in the aquifer–aquitard system

4.1.1. Concentration profiles in the aquifer–aquitard systemFig. 2A–C show the vertical concentration profiles in the

aquifer–aquitard system under conditions of σ1=σ2=1,Pe=10, xD=10, and tD=2. Fig. 2A is for a case with differenthydrologic parameters of the upper and lower aquitards withβ1=0.05, β2=0.1, Pe1=0.5, and Pe2=1. Fig. 2B is for a case

Table 2The default values used in this study.

Parameter name Symbol Default value

Half aquifer thickness in Fig. 1A or fullaquifer thickness in Fig. 1B

B 2 m

Effective molecular diffusion coefficientsof the aquitard

D01, D02 1.16×10−9 m2/s

Longitudinal dispersion coefficient ofthe aquifer

Dx 2.32×10−6 m2/s

Transverse dispersion coefficient ofthe aquifer

Dz 2.32×10−8 m2/s

Average pore velocity in the aquifer v 1.16×10−6 m/sAverage pore velocity in the aquitard v1z, v2z 1.16×10−10 m/s

1.16×10−9 m/sPorosity θ=θ1=θ2 0.36

with identical hydrologic parameters of the upper and loweraquitards with β1=β2=0.1, and Pe1=Pe2=1. A point to noteis that advective velocities in the upper and lower aquitardsare both upward in Fig. 2A and B. Fig. 2C is for a case that isidentical to that of Fig. 2B except that the advective velocitiesin the upper and lower aquitards have opposite direction, i.e.,Pe1=1 but Pe2=−1. Fig. 2C is the special case discussedbefore in Section 2.2 after Eq. (33) with a symmetric systemabout z=0. The case shown in Fig. 2C for zN0 would also bethe solution applying for the conceptual model of Fig. 1B.

Several interesting points can be made from these threefigures. Firstly, as can be seen from the figures, Fig. 2A and Bhave asymmetric shapes of concentration distributions alongthe z-axis since the advective velocities in the upper and loweraquitards are both along the upward direction. However,Fig. 2C has a symmetric shape of concentration distributionalong the z-axis as expected. For the asymmetric cases ofFig. 2A and B, greater discrepancy exists between the result ofthe AA method and the Zhan method in the upper aquitardthan that in the lower aquitard. Secondly, the AAmethod onlyyields a z-independent concentration within the aquifer anda declined concentration with z in the aquitard. Notice thatzD=1 is the interface between the aquifer and the upperaquitard. The Zhan method, however, provides a concentra-tion profile that continuously varies with z. Because of thedifference between the dispersion coefficient of the aquiferand the diffusion coefficient of the aquitard, concentrationgradients are not continuous at the aquifer–aquitard interface.Fig. 2A–B show that the AA method considerably over-estimates the concentration in the upper aquitard up to afairly large distances from the aquifer–aquitard interface.However, for the symmetric case of Fig. 2C, the degree of suchoverestimation is much less. Fig. 2C shows that for the sym-metric case and for the case with the impermeable bedrock,the difference between the AA solution and our solution isactually quite small. Thirdly, the AA method and the Zhanmethod seem to converge to negligible concentration atapproximately the same distances from the aquifer–aquitardinterface. For instance, at zD≈3 in Fig. 2A–C, the concentra-tions calculated from the Zhan method and the AA methodwill both become nearly negligible.

4.1.2. Breakthrough curves (BTCs) in the aquiferFig. 3 shows the comparison of BTCs in the aquifer calcu-

lated by the Zhan method at zD=0 and zD=0.5 under con-ditions of β1=β2=0.1, σ1=σ2=1, Pe=10, Pe1=Pe2=1,and xD=10. The result from the AA method, which is inde-pendent of the z-axis, is also shown in this Figure. Severalobservations can be made.

First, the concentration at zD=0 is always greater than thatat zD=0.5. This simply reflects the fact that a concentrationgradient exists to drive mass upward into the upper aquitard.Second, theAAmethodoverestimates theaquifer concentrationat early times but underestimates the aquifer concentration atlate times. Third, even under relatively long times, aquiferconcentrations cannot reach the source concentration C0 (or CD

Fig. 3. The breakthrough curves (BTC) at zD=0, and zD=0.5 underconditions of β1=β2=0.1, σ1=σ2=1, Pe=10, Pe1=Pe2=1, and xD=10from the Zhan method. The result of the AAmethod, which is independent ofthe z axis, is also shown in this figure.

Fig. 2. The vertical concentration profiles in the aquifer–aquitard systemunder conditions of σ1=σ2=1, Pe=10, xD=10, and tD=2. (A) β1=0.05,β2=0.1, Pe1=0.5, and Pe2=1; (B) β1=β2=0.1, and Pe1=Pe2=1; (C) β1=β2=0.1, Pe1=1, and Pe2=−1.


of 1). That is because in addition to the horizontal mass flux inthe aquifer, the vertical mass flux across the aquifer–aquitardboundary serves as a sink to keep aquifer concentration lower

than the source concentration. This phenomenon is somewhatsimilar to that observed for a radioactively decaying solute.

4.1.3. Comparison of the semi-analytical solutions and thenumerical simulations

To test the semi-analytical solutions obtained here, wehave conducted numerical simulations using the modelMT3DMS developed by Zheng and Bennett (2002). Thenumerical simulations are straightforward, thus are onlybriefly outlined here. A two-dimensional aquifer–aquitardsystem is generated and the parameters used are the same asthose outlined in Section 4. The idea is to reproduce thesituation of β1=β2=0.1, σ1=σ2=1, Pe=10, and Pe1=Pe2=1. The simulated domain includes an aquifer that is 4 mthick and 1000m long, and an upper and a lower aquitard. Thethickness of each aquitard is 50m,which is sufficiently large tosatisfy the boundary conditions of Eqs. (8) and (13). Thehydraulic conductivity of the aquitards is approximately threeorders of magnitude lower than that in the aquifer, so thatflow is essentially vertical in the aquitards. The vertical flowvelocity in the aquitards is 1/100 of the horizontal flowvelocity in the aquifer. The flow velocities in the aquifer andaquitards are obtained through Modflow after appropriatelyspecifying the boundary condition of flowand these velocitiesare used by MT3DMS to generate the concentration field(Zheng and Bennett, 2002). The grid spacing was madesufficiently fine to reduce the numerical errors. After tryingdifferent grid spacings,we decided to use a spacingof 0.2m forthe aquifer and the aquitards. Further refining the grid space to0.1 m did not produce noticeable changes to the output.Different solver options of MT3DMS had been tried except theexplicit finite difference method, and similar results wereobtained. The hybrid Method of Characteristic (HMOM) waseventually chosen because of its relative short computationaltime. A constant–concentration boundary condition wasused on the left-hand side of the aquifer in the numericalsimulation.

However, one must be careful with the nature of com-parison of the analytical and numerical modeling results.


Such a comparison is by nomeans to be exact in a quantitativesense because of a few reasons. First, diffusion along the x-axis is not considered in the governing equations of transportin the aquitard (see Eqs. (5) and (10)), but it is included inMT3DMS. In this sense, the numerical modeling seems tohave an advantage over the analytical modeling of this study.Unfortunately, our numerical exercises indicate that MT3DMSperforms poorly near the aquifer–aquitard interfaces. Suchpoor performance is caused by the abrupt change of aquiferparameters and concentration gradients near the aquifer–aquitard interfaces. This issue cannot be simply resolved byrefining the grid blocks near those interfaces. New numericalmodeling strategies should be used to accurately model thetransport behavior near the aquifer–aquitard interfaces. Thefinding of our numerical exercises seems to agree withprevious numerical simulation which indicated that thenumerical errors tended to be the largest at the aquifer–aquitard interfaces (Martin and Frind, 1998; Bester et al.,2005). In this regard, the analytical modeling seems to havesome advantages over the numerical modeling.

The idea of comparing the analytical and numericalmodeling results is not to show which one is better; rather, itis to show how different are those results in a qualitative sense.The default parameters used in the comparison come fromTable 2. As an example, Fig. 4 shows the concentration dis-tribution in the systemobtained from the numerical simulationagainst that obtained from the Zhan method at xD=2.5 (orx=50 m) at tD=2 or (t=3991 day). The advections in bothaquitards are upward thus Fig. 4 is for an asymmetric case. Thehorizontalflow velocity in the aquifer is 0.1m/day. This impliesthat the averaged travel distance of the solute is x=3991×0.1 m=399.1 m, which is considerably larger than the co-ordinate of the observation point in Fig. 4 (which is x=50 m).Thus the result in Fig. 4 can be regarded as the one far behindthe front of the solute. As can be seen from this figure, the resultof the numerical simulation is close to that of the Zhanmethod

Fig. 4. Comparison of the vertical concentration profiles in the aquifer–aquitard system obtained from the numerical simulation using MT3DMS andthe Zhan method. β1=β2=0.1, σ1=σ2=1, Pe=10, Pe1=Pe2=1, tD=2The profiles at xD=2.5 is shown here. zD=1 and −1 are the upper andlower aquifer–aquitard interfaces. The first-type solute transport boundarycondition is used on the left hand side of the aquifer.

.

in the lower aquitard but it differs considerably from that of theZhanmethod in the upper aquitard and in the upper portion ofthe aquifer. Large differences between the analytical andnumerical modeling results exist at both the upper and loweraquifer–aquitard interfaces. The asymmetric shapes of theconcentration distribution in Fig. 4 are apparently due to theasymmetric aquitard advections which are upward in bothaquitards here.

4.1.4. Mass transported into the aquitardIt is interesting to know howmuch mass was stored in the

aquitard due to advective and diffusive transport across theaquifer–aquitard boundary. The total masses per unit widthalong the direction perpendicular to the xz plane in the upperand lower aquitards are denoted as M1 and M2, respectively.Their dimensionless forms, denoted as M1D and M2D aredefined in Table 1. The Laplace transforms of M1D and M2D,denoted as M ̅

1D and M ̅2D respectively, can be obtained

through the following integration:

M1D =Z ∞

0dxD

Z ∞

1C1DdzD; M2D =

Z ∞

0dxD

Z −1

−∞C2DdzD

ð41Þ

This can be done straightforwardly. Using Eqs. (28) and(29), one has

M 1D =X∞n=0

4An cos ωn + μnð ÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPe2 + 4 p + ω2

n

� �q− Pe

� � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPe1 =β1ð Þ2 + 4p= β1

q− Pe1 = β1

� � ;

M2D =X∞n=0

4An cos ωn + μnð ÞffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPe2 + 4 p + ω2

n

� �q− Pe

� � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPe2 =β2ð Þ2 + 4p = β2

q+ Pe2 = β2

� � ;

ð42Þwhere An is given by Eq. (30) for the first-type boundarycondition at x=0 or Eq. (32) for the third-type boundarycondition at x=0. Inverse Laplace transforms of M 1̅D andM 2̅D in Eq. (42) will result in the solution in the real timedomain for the Zhan method.

If using Eqs. (39) and (40) for the AA method, one canobtain M 1̅D and M 2̅D as:

M 1D =4= pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Pe2 + 4p + σ1


q+ Pe1

� �+ σ2


q− Pe2

� �s− Pe

×1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Pe1 =β1ð Þ2 + 4p= β1

q− Pe1 = β1

;

M2D =4= pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Pe2 + 4p + σ1


q+ Pe1

� �+ σ2


q− Pe2

� �s− Pe

×1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Pe2 =β2ð Þ2 + 4p= β2

q+ Pe2 = β2

:

ð43Þ

Inverse Laplace transforms of M ̅1D and M 2̅D in Eq. (43)

will result in the solution in the real time domain for the AAmethod. As an example, a comparison of the M1D valuescalculated from Eq. (42) (the solid line) and Eq. (43) (thedashed line) is shown in Fig. 5 for β1=β2=0.05, σ1=σ2=1,Pe=10, Pe1=Pe2=0.1 and 1. Several observations can bemade from Fig. 5.

Fig. 6. The dimensionless mass transported to the aquitard per unit width(M1D) calculated from the Zhan method and that from the AA method versusthe dimensionless time (tD) under the conditions of β1=β2=0.1, σ1=σ2=1, Pe=10, Pe1=Pe2=0.1 and 1.

Fig. 7. The dimensionless mass transported to the aquitard per unit width(M1D) calculated from the Zhan method and that from the AA method versusthe dimensionless time (tD) under the conditions of β1=0.05, β2=0.1,σ1=σ2=1, Pe=10, Pe1=Pe2=0.1 and 1.

Fig. 5. The dimensionless mass transported to the aquitard per unit width(M1D) calculated from the Zhan method and that from the AA method versusthe dimensionless time (tD) under the conditions of β1=β2=0.05, σ1=σ2=1, Pe=10, Pe1=Pe2=0.1 and 1.


First, the AA method always overestimates the masstransported to the aquitard, and the error increases with thePeclet numbers. For example, at tD=2 and Pe1=Pe2=0.1,the error is about 18%, while at tD=2 and Pe1=Pe2=1, theerror is about 144%. This is understandable. In the AAmethod,the aquitard diffusion is regarded as a volumetric source/sinkterm in the governing equation of transport in the aquifer,thus every point from zD=0 to zD=1 within the aquifer iscapable of losing mass, resulting in a greater loss of mass tothe aquitard. This is contradictory to the realistic masstransfer that only occurs at the aquifer–aquitard interface.The Zhan method treats the mass flux to the aquitard as aninterface term rather than a volumetric term, thus it results inless total mass transported.

Second, the value of the aquitard Peclet number greatlyinfluences theM1D values and a greater Peclet number resultsin a greater M1D value when time is sufficiently large (tDN1).The reason is quite straightforward since a greater Pecletnumber indicates greater upward advective transport fromthe aquifer to the upper aquitard, thus more mass ends up inthe upper aquitard.

Third, the functionality of M1D versus tD is nonlinear. Infact, the slope of M1D versus tD increases with tD when tD isrelatively small and eventually approach a constant when tDis close to 2, indicating an advective transport mechanismat the late time. A tD of 2 is equivalent to a real time oft=4000 days or close to 11 years (see Tables 1 and 2).Therefore, both diffusion and advection will affect the early-time mass transport, but only advection will affect the latetime transport between the aquifer and the aquitard.

Fig. 6 is similar to Fig. 5 except that both the β1 and β2

values are increased to 0.1. It is interesting to see that whenthe β1 and β1 values are doubled, indicating that the aquitarddiffusion becomes stronger, the mass transported to theupper aquitard increases as well, as expected. However, thepercentage of increase is greater for a case with a smalleraquitard Peclet number. For instance, at tD of 2 and Pe1=Pe2=0.1, the M1D value increases 20% from Figs. 5 and 6,while at tD of 2 and Pe1=Pe2=1, the M1D value increases

only 0.4% from Figs. 5 and 6. This finding implies that themasstransported between the aquifer and the aquitard is relativelyinsensitive to the aquitard diffusion when the aquitard Pecletnumber is greater than unity.

Fig. 7 is similar to Figs. 5 and 6 except that the upper andlower aquitards have different diffusion coefficients. In thiscase, the upper aquitard diffusion coefficient is one half of thelower one. One can tell that by reducing the upper aquitarddiffusion, theM1D value is slightly reduced. But in general, theM1D value is less sensitive to the difference of the aquitarddiffusion coefficients. A great discrepancy has been observedbetween the Zhan method and the AA method whenPe1=Pe2=1.

Fig. 8 shows the results for different aquitard Pecletnumbers. In this example, the Peclet number of the upperaquitard is 1/10 of that of the lower aquitard. As can be seenobviously, the M1D value is sensitive to the difference of the

Fig. 8. The dimensionless mass transported to the aquitard per unit width(M1D) calculated from the Zhan method and that from the AA method versusthe dimensionless time (tD) under the conditions of β1=β2=0.05, andβ1=β2=0.1, σ1=σ2=1, Pe=10, Pe1=0.1 and Pe2=1.


Peclet numbers between the upper and lower aquitards,particularly at late times.

5. Summary and conclusions

This study deals with two-dimensional solute transport inan aquifer–aquitard system by maintaining rigorous massconservation at the aquifer–aquitard interface. Advection,longitudinal dispersion, and transverse vertical dispersion areconsidered in the aquifer. Vertical advection and diffusion areconsidered in the aquitards. The first-type and the third-typeboundary conditions are considered in the aquifer. This studydiffers from previous AA method that treats the mass fluxbetween the aquifer and aquitard as an averaged volumetricsource/sink term in the governing equation of transport inthe aquifer. Instead, it solves the transport governing equa-tions in the aquifer and the aquitards explicitly by maintain-ing continuity of concentration and vertical mass flux at theaquifer–aquitard interfaces. Analytical solutions of concen-trations in the aquitards and aquifers are obtained in theLaplace domain and are inverted numerically to yield concen-trations in the real time domain. The following conclusionscan be drawn from this study.

The BTCs and concentration profiles obtained in the Zhanmethod are drastically different from those obtained usingthe AA method. The AA method shows a sharp change ofconcentration profile near the aquifer–aquitard interfacewhereas the Zhan method shows a continuously decreasingconcentration near that interface.

Concentration distributions in the aquitards are closelyrelated to advection in the aquitards. When flow velocities inthe upper and lower aquitards have the same magnitude andboth are along the upward direction, an asymmetric con-centration distribution along the vertical direction will result.The AA method overestimates the aquifer concentration atearly times but underestimates the aquifer concentration atlate times. However, a greater discrepancy exists between theresult of the AA method and the Zhan method in the upperaquitard than that in the lower aquitard. The AA method

generally overestimates the concentration in the upperaquitard for this asymmetric case.

When flow velocities in the upper and lower aquitards havethe same magnitude but opposite directions, a symmetricconcentration distribution along the vertical direction will beobtained. For this case, the concentrations in the aquiferobtained by the two methods differ considerably. However,much less discrepancy has been observed in the aquitards.

Comparison of the numerical simulation using MT3DMSand the Zhanmethod indicates that the result of the numericalsimulation could differ considerably from that of the Zhanmethod for an asymmetric casewhere aquitard advection is inthe same direction. More specifically, in the aquitard wherethe advection is towards the aquifer–aquitard interface, thenumerical result agrees with the result of the Zhan methodreasonably well; in the aquitard where the advection isaway from the aquifer–aquitard interface, the numerical resultdiffers considerably from the result of the Zhan method.

Compared to the Zhan method, the AA method overesti-mates mass transported into the upper aquitard when anupward advection exists. The mass transported between theaquifer and the aquitard is sensitive to the aquitard Pecletnumber, but less sensitive to the aquitard diffusion coefficient,particularly at late times.

Acknowledgements

This study is partially supported by National Science Foun-dation of China (#50428907), the CONACYT-TAMU Collabora-tive Research Program, and the Advanced Research Program(ARP) from the Texas Higher Education Coordination Board,USA. We thank two anonymous reviewers for their con-structive comments. In particular, we thank the Editor-in-Chief E. O. Frind for his critical and detailed suggestions of thismanuscript. Dr. Frind's comments have helped us improve thismanuscript substantially.

Appendix A. Solute transport in an aquifer bounded bytwo aquitards

The following solution is proposed for CD̅:

CD =X∞n=0

AnCxD xD; p;nð Þ cos ωnzD + μnð Þ; ðA1Þ

where An is the coefficient that needs to be determined, Cx̅D

(xD, p, n) is the part related to the x coordinate, and ωn is thefrequency of the Fourier transform along the z-axis. Sub-stituting Eq. (A1) into Eq. (16) results in the following equa-tion for Cx̅D(xD, p, n):

d2CxD

dx2D− PedCxD

dxD− p + ω2

n

� �CxD = 0: ðA2Þ

Considering the boundary condition of Eq. (18) will yieldthe general solution of Eq. (A2):

CxD xD;p;nð Þ = exp −


n

� �q− Pe

2

0@

1AxD

24

35: ðA3Þ


Therefore, the proposed solution for the aquifer becomes:

CD =X∞n=0

Anexp −


n

� �q− Pe

2

0@

1AxD

24

35 cos ωnzD + μnð Þ;

− V zD V 1 ðA4Þ

The solution to Eq. (19) with the boundary conditions ofEqs. (20) and (22) is:

C 1D = CD xD; zD = 1; pð Þexp −


q− Pe1 = β1

2zD − 1ð Þ

24

35;

zD z 1:ðA5Þ

Similarly,

C2D = CD xD; zD = − 1; pð ÞexpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPe2 =β2ð Þ2 + 4p= β2

q+ Pe2 = β2

2zD + 1ð Þ

24

35;

zD V − 1: ðA6Þ

Substituting Eq. (A4) into Eqs. (21) and (25), andconsidering Eqs. (A5) and (A6) will lead to:

ωn tan ωn + μnð Þ = σ1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPe21 = 4 + pβ1

q+ σ1Pe1 = 2;

ωn tan ωn − μnð Þ = σ2


q− σ2Pe2 = 2:

ðA7Þ

From Eq. (A7), one has:

ðω2n − σ1σ2


q+ Pe1 = 2

� �

×ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPe22 = 4 + pβ2

q− Pe2 = 2

� �Þ tan 2ωnð Þ

= ωn½ σ1


q+ σ1Pe1 = 2

� �

+ σ2


q− σ2Pe2 = 2

� ��;

ðA8Þ

ðω2n + σ1σ2


q+ Pe1 = 2

� �

×ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPe22 = 4 + pβ2

q− Pe2 = 2

� �Þ tan 2μnð Þ

= ωn½ σ1


q+ σ1Pe1 = 2

� �

− σ2


q− σ2Pe2 = 2

� ��;

ðA9Þ

ωn is determined from Eq. (A8). After that, μn is calculatedfromEq. (A9). For the numerical calculation, accurate determina-tion ofωn and μn are important since these parameters have beenused in solutions (see Eqs. (27)–(33)). As can be seen from Eq.(A8),ωn is the rootof a function tan(2ωn)=αωn/(ωn

2−b),wherea and b are two dummy variables associated with all the otherparameters of Eq. (A8). Notice that function αωn/(ωn

2−b)has asingularity atωn =

ffiffiffib

p, and the function of tan(2ωn) has infinite

numbers of singularities at ωn=π/4+nπ/2, where n is aninteger varying from negative infinity to positive infinity. The

singularities ofωn=π/4+nπ/2 separate the entire domain intomany sections with an equal interval of π/2. It is worthwhile topoint out that near the singular point of ωn =

ffiffiffib

p, there are

always two roots of ωn within that π/2 section includingωn =

ffiffiffib

p. Only one root ofωn exists within each of the rest π/2

sections.After determining ωn and μn, one can determine An from

the boundary condition Eq. (17):

X∞n=0

An cos ωnzD + μnð Þ = 1= p: ðA10Þ

Conducting inverse Fourier transform of Eq. (A10) resultsin:

An =4p

×F1 ωn; μnð ÞF2 ωn; μnð Þ ; ðA11Þ

where

F1 ωn; μnð Þ = sin ωnð Þ cos μnð Þ;F2 ωn; μnð Þ = 2ωn + sin 2ωnð Þ cos 2μnð Þ:

ðA12Þ

Therefore, CD̅, C 1̅D, and C2̅D are derived after substitutingEq. (A11) into Eqs. (A4), (A5), and (A6), respectively. For thespecial case in which the upper and lower aquitards haveidentical parameters, μn=0.

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Documents

Analytical Solution of Two-dimensional Solute Transport in an Aquifer Aquitard