approximate evaluation of contaminant transport through vertical barriers

Ž .Journal of Contaminant Hydrology 40 2000 311–333www.elsevier.comrlocaterjconhyd

Approximate evaluation of contaminant transportthrough vertical barriers

Hillel Rubin 1, Alan J. Rabideau )

Department of CiÕil, Structural and EnÕironmental Engineering, 207 JarÕis Hall, State UniÕersity of NewYork at Buffalo, Buffalo, NY 14260, USA

Received 2 September 1998; received in revised form 28 May 1999; accepted 9 June 1999

Abstract

Containment of groundwater contamination using physical barriers can be an importantelement of a subsurface remediation program. This work presents simple analytical tools forpredicting the performance of barriers in terms of the steady-state contaminant flux across thebarrier, the duration of the transient period following barrier installation, and the time-dependentcontaminant concentration distribution within the barrier. The analytical expressions are developed

Ž .from approximate boundary layer BL solutions to the advective–dispersive equation subject toconservative fixed concentration boundary conditions. Critical ranges of important dimensionlessquantities are identified for use in barrier performance assessment, for both steady-state andtransient conditions. Comparative calculations made with the BL equations and more exactsemi-analytical solutions are used to characterize the accuracy and applicability of the BLapproach. q 2000 Elsevier Science B.V. All rights reserved.

Keywords: Contaminant transport; Vertical barriers; Boundary layer

1. Introduction

A common strategy for the containment of groundwater contaminant plumes is theŽ .use of physical barriers e.g., Rowe et al., 1995; Rumer and Mitchell, 1996 , particularly

vertical barriers such as soil–bentonite cutoff walls. The design of such barriersnormally emphasizes the achievement of low hydraulic conductivity to reduce advective

) Corresponding author. Tel.: q1-716-645-2114; fax: q1-716-645-3667; E-mail: [email protected] On leave from Department of Civil Engineering, Technion - Israel Institute of Technology, Haifa 32000,

Israel.

0169-7722r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved.Ž .PII: S0169-7722 99 00060-1

( )H. Rubin, A.J. RabideaurJournal of Contaminant Hydrology 40 2000 311–333312

contaminant transport. However, an effective barrier must be designed to restrict allrelevant contaminant transport processes, including molecular diffusion. In evaluatingthe potential for contaminant transport associated with a proposed barrier design,

Ž .Rabideau and Khandelwal 1998a recommended the use of the one-dimensional advec-tive–dispersive–reactive equation with conservative fixed-concentration boundary con-ditions. Relatively simple solutions are available for the steady-state contaminant fluxunder these conditions. However, in many cases, the transient behavior of the barrier is

Ž .of interest e.g., Shackelford, 1990 , either due to the temporary use of the barrier or theneed for an appraisal of the time period in which the site remediation should becompleted prior to the release of significant quantities of the contained materials. Theprediction of transient behavior for fixed concentration boundaries requires the applica-

Žtion of straightforward but tedious semi-analytical solutions e.g., the series expansion of.Owen, 1925 . The work explores the application of an alternative approach for generat-

ing approximate solutions.In analyzing contaminant transport in porous vertical barriers, it is common to

assume steady flow conditions and apply the one-dimensional advective–dispersiveŽ .equation. The study of Rubin and Buddemeier 1996 indicated that the boundary layer

Ž .BL approach can be useful for cases where one-dimensional flow is associated withone-dimensional or two-dimensional contaminant transport, provided that adequaterepresentation of the system boundary conditions is feasible. For many cases, theapplicability of the BL approach can be extended by distinguishing between regions of

Ž . Ž .uniform or negligible contaminant concentration and the region of interest ROI thatcontains the migrating contaminant front. In general, the BL approach is based on thedirect integration of the contaminant transport equation over these distinct regions.

2. Conceptual approach and formulation

ŽWe consider the common one-dimensional conceptual model of a barrier e.g., Rowe.and Booker, 1985; Rabideau, 1996 as shown in Fig. 1. The barrier separates two

Fig. 1. Schematic description of the barrier.

( )H. Rubin, A.J. RabideaurJournal of Contaminant Hydrology 40 2000 311–333 313

Ž .completely mixed regions CMRs that are subject to significant advection and disper-sion. Conversely, only minor quantities of the contaminant are subject to advection anddiffusion within the barrier. The effectiveness of the barrier is evaluated by its ability tominimize the migration of contaminant between the CMRs. Although the primary

Ž .applications considered in this work are vertical barriers e.g., cutoff walls , theŽ .conceptual model may also be applicable to horizontal barriers e.g., landfill liners .

In performing design calculations, it is normally assumed that the piezometric headon each side of the barrier is either controlled or fixed at a known value, thus enablingthe assumption of steady flow across the barrier. Contaminant transport by advectionacross the barrier therefore depends on the hydraulic conductivity of the barrier material,the thickness of the barrier, andror the change in hydraulic head across the barrier. Theamount of dispersive transport depends on the effective contaminant dispersion coeffi-cient, which will be dominated by molecular diffusion for low permeability barriers, andthe barrier thickness. Both advective and diffusive transport can be mitigated byengineered decay reactions andror sinks within the barrier. If significant quantities offluid are removed from within the barrier, the contaminant velocity will vary spatially.However, with little loss of generality, we consider only cases involving uniform flow.

Contaminant transport under the above assumptions can be represented by theone-dimensional advective–dispersive–reactive equation:

EC EC E2 CR qÕ sD ylCyq 1Ž .2Et Ex Ex

where C is the volume-averaged contaminant concentration, R is the retardation factor,which implies the assumption of linear equilibrium sorption, t is time, x is the distancefrom CMR 1, Õ is the fluid velocity, D is the dispersion coefficient, which incorporatesboth hydrodynamic mixing and molecular diffusion, l is the contaminant decayconstant, and q represents a zero-order contaminant sink.

Ž .Eq. 1 can be subject to a variety of boundary conditions. Following Rabideau andŽ .Khandelwal 1998a , we assume an initially uncontaminated barrier and adopt the

conservative approach of prescribed contaminant concentrations at the edges of thebarrier:

C x ,ts0 s0 2aŽ . Ž .

C xs0,t sC 2bŽ . Ž .0

C xsL,t s0 2cŽ . Ž .

where C is the concentration within the contained area and L is the barrier thickness.0

Although the conditions within the contained area may change with time, a reason-able and conservative first approximation is to assume a constant source concentrationŽ Ž ..Eq. 2b . For low permeability barriers, the assumption of constant zero concentration

Ž Ž ..at the barrier exit Eq. 2c is conservative because it maximizes the driving force fordiffusion across the barrier. The physical interpretation of this boundary condition is thatthe contaminant exiting the barrier is removed by advectionrdispersion at a much higherrate than diffusion-dominated transport occurs within the barrier. However, as noted by


Ž . Ž .Rabideau and Khandelwal 1998a , differences between the solutions of Eq. 1 subjectto different types of exit boundary conditions become less significant as the amount ofadvection across the barrier increases, as discussed below.

If the contaminant source remains constant over time, a conservative design may bedeveloped by considering the steady-state behavior of the barrier. However, predictionof the time needed to arrive at steady-state may also be desired. In particular, acommonly discussed remediation strategy is the amendment of the barrier materials witha sorbing additive, which delays the time to reach steady-state but does not reduce thesteady-state flux in the absence of decay. With the assumption of a zero concentration atthe barrier exit, the primary variables of interest are the contaminant concentration

Ž .within the barrier and the contaminant flux per area CF , which is given by:

ECCFsf ÕCyD 3Ž .

Ex

where f is the porosity of the barrier material.

3. Dimensionless equations

We introduce the following dimensionless variables for concentration, distance, andtime, respectively:

CUC s 4aŽ .

C0

xUx s 4bŽ .

L

tDUt s . 4cŽ .2RL

Ž . Ž . Ž . Ž . Ž .Substituting the dimensionless grouping into Eqs. 1 , 2a , 2b , 2c and 3 , weobtain:

ECU ECU E2 CU

U U UqPe s yl C yq 5Ž .U U U 2Et Ex Ex

CU xU ,tU s0 s0 6aŽ . Ž .

CU xU s0,tU s1 6bŽ . Ž .

CU xU s1,tU s0 6cŽ . Ž .L ECU

UFsCF sPeC y 7Ž .UfDC Ex0


with:

ÕLPes 8aŽ .

D

lL2U

l s 8bŽ .D

qL2Uq s 8cŽ .

DC0

where F is the dimensionless contaminant flux, Pe is the barrier Peclet number, and lU

and qU refer to the dimensionless decay rate and sink term, respectively.

4. Steady-state conditions: the exact solution

ŽIn the following developments, we consider only the case of a conservative non-de-.caying contaminant, with no extraction of the contaminant along the barrier thickness.

However, the methods presented in this paper are readily extended to other scenariosinvolving more complex interactions.

If the domain is at steady-state, the quantity F is preserved at each cross-section ofŽ . Ž . Ž . Ž .the barrier. According to Eqs. 6a , 6b , 6c and 7 , under steady-state conditions:

ECU ECU

FsPey sy . 9Ž .U Už / ž /U UEx Exx s0 x s1

Ž . Ž . Ž . Ž .By direct integration of Eq. 7 subject to Eqs. 6a , 6b and 6c , we obtain:U1yexp yPe 1yxŽ .

UC s 10Ž .1yexp yPeŽ .Pe

Fs . 11Ž .1yexp yPeŽ .

Ž . Ž .Eqs. 10 and 11 indicate that, under steady-state conditions, the contaminant fluxand concentration distribution within the barrier depend on a single parameter, the

Ž .barrier Peclet number Pe . Fig. 2 shows the calculated steady-state contaminantdistribution along the barrier thickness for various values of Pe.

As determined by the assumed boundary conditions, the contaminant may be subjectto both advection and dispersion at the entrance of the barrier, but only dispersion at the

Ž .exit. The commonly applied semi-infinite condition Cs0 at infinity leads to asteady-state solution in which only advection contributes to the contaminant flux.

Ž .Conversely, the finite exit boundary condition Cs0 at xsL has no effect on thedegree of advection within the barrier, but may lead to greater overall contaminant fluxacross the barrier. This impact can be quantified by defining a diffusion enhancement

Ž .function G that represents the amount of calculated contaminant flux through thebarrier attributable to the imposed exit boundary condition:

GsFyPe. 12Ž .


Fig. 2. Exact solutions for the steady-state contaminant concentration distributions.

Thus, the zero concentration exit boundary condition influences contaminant trans-port through the barrier when G is significantly larger than zero. The portion of thecontaminant flux across the barrier due to the exit condition is given by:

Gfs . 13Ž .

FFig. 3 shows the dependence of G and f on the barrier Peclet number. As expected,

the assumed zero concentration exit condition has a significant effect on the predictedŽ .steady-state contaminant flux across the barrier for low values of Pe less than 1 . For

Pe)4, the impact of the exit boundary condition on the calculated steady-state flux issmall.

5. Steady-state conditions: approximate solutions

Although the exact solution for the steady-state contaminant distribution representedŽ .by Eq. 10 can be calculated with no difficulty, it is useful to consider even simpler


Ž . Ž .Fig. 3. Diffusion enhancement function G and fraction of flux attributable to exit boundary condition f .

approximations, as these expressions are used in the unsteady solutions developedŽ .below. These approximations are obtained for different ranges of Pe by: 1 series

Ž .expansions of the various terms, or 2 considering a control volume of unit cross-sec-tion and length identical to the barrier thickness.

( )5.1. Very small barrier Peclet number Pe<1

Ž . Ž .As Pe™0, application of L’Hospital’s rule to Eqs. 10 and 11 leads to:

CU s1yxU 14Ž .Fs1. 15Ž .

( )5.2. Small barrier Peclet number Pef1

Ž .As shown by the control volume of Fig. 4 and expressed by Eq. 9 , the flux ofcontaminant at the entrance of the barrier incorporates advection and diffusion, whereas


Fig. 4. Schematic of barrier control volume with boundary fluxes.

at the barrier exit, the contaminant is subject only to diffusion. If the barrier Pecletnumber is on the order of one, we assume that the contaminant concentration deviates

Ž .from the linear relationship of Eq. 14 and can be represented by a second orderpolynomial expression:

CU sa qa xU qa xU 2 16Ž .0 1 2

Ž .where a is0, 1, 2 are adjustable coefficients.iŽ Ž . Ž . Ž ..By applying the auxiliary conditions Eqs. 6a , 6b and 6c and the definition of

Ž Ž ..flux Eq. 7 , we obtain the following expressions:

Pe PeU U U 2C s1y 1y x y x 17Ž .ž /2 2

PeFs1q . 18Ž .

2

Ž .Fig. 5 shows that the contaminant profiles represented by Eq. 17 are nearly identicalŽ .to those of Eq. 10 in the range of barrier Peclet numbers close to one.

( )5.3. Large barrier Peclet number Pe41

Ž . Ž .If Pe™`, Eqs. 10 and 11 can be approximated, respectively by:

U UC s1yexp yPe 1yx 19Ž . Ž .

F™Pe. 20Ž .

These expressions are relevant when the steady-state diffusive flux at the barrierentrance is much less than the advective flux. Furthermore, if the barrier Peclet number


Ž .Fig. 5. Comparison between exact solution and approximate solution for steady-state conditions Pef1 .

is large, the steady-state concentration distribution can be approximated by the followingexpression:

CU s1yxU P e . 21Ž .Fig. 6 shows that, for barrier Peclet numbers larger than approximately 2, the

description of contaminant concentration distribution given by the power law model ofŽ . Ž .Eq. 21 is almost identical to that given by the exact solution of Eq. 10 .

6. Unsteady-state conditions

Consideration of transient contaminant transport is useful in evaluating the effectivelife of the barrier andror the need for additional mitigation measures. In this section, weconsider the time-dependent spatial contaminant distribution, again for the case of nocontaminant sink and negligible decay. Following the general outline of Section 5, we


Ž .Fig. 6. Comparison between exact solution and approximate solution for steady-state conditions Pe41 .

develop BL expressions applicable to three ranges of barrier Peclet number, with eachexpression convergent to the relevant steady-state approximation.

( )6.1. Very small barrier Peclet number Pe<1

If Pe<1, contaminant transport through the barrier is dominated by molecularŽ .diffusion. Under such conditions, the advection term of Eq. 5 can be neglected, to

obtain:

ECU E2 CU

s . 22Ž .U U 2Et Ex

We assume that the development of the contaminant distribution in the barrierŽ .proceeds in two stages: 1 development of a BL primarily affected by the upstreamŽ .boundary condition, and 2 adaptation of the BL to the downstream boundary condition.


During the first stage, the contaminant distribution is affected only by the upgradientconstant concentration boundary condition. The affected region is smaller than thethickness of the barrier and is considered a BL of thickness d . The contaminantconcentration distribution in the BL region is given by:

nU 0C s 1yh 23Ž . Ž .where n is a constant power coefficient and hsxUrd .0

Ž . UEq. 23 is a typical BL equation that enforces the desired limiting behavior of C atŽ .the boundaries hs0,1 . Readers interested in details of the BL approach are referred to

Ž . Ž .Ozisik 1993 and Rubin and Buddemeier 1996 .Ž . Ž .We introduce Eq. 23 into Eq. 22 , apply Leibnitz’s theorem, and integrate over the

BL to obtain:

E n1 0n0d 1yh dh s . 24Ž . Ž .HUEt d0

Ž .After performing the integration of Eq. 24 and integrating again with respect totime, we obtain:

d 2 s2n n q1 tU . 25Ž . Ž .0 0

The first stage is completed at tU s tU , when the BL thickness is identical to the0

barrier thickness, which implies:

1Ut s . 26Ž .0 2n n q1Ž .0 0

During the second stage, the BL thickness remains constant at ds1, due to the exitŽ .boundary condition. We again apply Eq. 23 , but the value of the power coefficient,

represented by n , decreases from the value of n sn until it obtains the steady-state1 1 0

value of n s1, leading to:1

1 dn1y s1. 27Ž .U2 d tn n q1Ž .1 1

Ž .By separation of variables and direct integration of Eq. 27 , and considering that attU s tU the value of n is equal to n , we obtain:0 1 0

n n q1 1 1Ž .0 1U Ut y t s ln y q . 28Ž .0 n n q1 n q1 n q1Ž .1 0 1 0

Steady-state conditions are established at tU s tU , at which time n s1. Therefore,1 1Ž .according to Eq. 28 , we obtain:

2n 1 10U Ut y t s ln y q . 29Ž .1 0 ž /n q1 2 n q10 0

Ž .The solution of Eq. 22 by the BL approach is well known for a constantconcentration entrance and semi-infinite exit condition. Under this scenario, it is usually


Žassumed that the value of the power coefficient is approximately 2 e.g., see discussion.by Rubin and Buddemeier, 1996 . Adopting this value, we obtain:

1Ut s 30aŽ .0 12

tU f0.20. 30bŽ .1

The BL method developed above is applicable for very small values of the barrierPeclet number. However, comparisons with the semi-analytical solution indicate that theabove approximation is reasonable for barrier Peclet numbers approaching one.

( )6.2. Intermediate barrier Peclet number Pef1

In cases of barrier Peclet numbers on the order of 1, we identify three stages in thedevelopment of the contaminant concentration profile. In the first stage, a BL develops

Ž .at the upstream edge of the barrier. By applying Leibnitz’s rule and integrating Eq. 5over the contaminated region, we obtain:

ECU d d U UPey s C d x . 31Ž .HU Už / UEx d t 0x s0

Ž .We again assume that the contaminant concentration is described by Eq. 23 .However, we apply a constant power coefficient n that is not necessarily identical to n3 0

Ž . Ž .of Eq. 23 . By introducing this assumption into Eq. 31 , we obtain:

n 1 dd3Peq s . 32Ž .U

d n q1 d t3

After separation of variables and time integration, and considering an initial BLthickness of ds0, we obtain:

d n Pedqn3 3Un q1 t s y ln . 33Ž . Ž .3 2 ž /Pe nPe 3

When tU s tU , the BL occupies the entire thickness of the barrier. Introducing this0Ž .condition into Eq. 33 , we obtain:

1 n Peqn3 3Un q1 t s y ln . 34Ž . Ž .3 0 2 ž /Pe nPe 3

The second stage of the contaminant profile development follows the arrival of theBL at the downstream edge of the barrier. Here the BL thickness is assumed constant,but the power coefficient decreases until it reaches 1. We consider a variable power


Ž .coefficient n , whose initial value is equal to n , and again apply Eq. 31 , with ds1.4 3

After separation of variables and integration, we obtain:

dn4Ud t sy . 35Ž .2Peqn n q1Ž . Ž .4 4

Ž .We integrate Eq. 35 for barrier Peclet numbers different from 1, while consideringthat at tU s tU , the power coefficient n is equal to n , to obtain:0 4 3

1 Peqn n q1Ž . Ž .3 4U Ut y t s ln0 2 Peqn n q11y2 PeqPe Ž . Ž .4 3

1 1 1q y . 36Ž .ž / ž /Pey1 n q1 n q14 3

Ž .If the barrier Peclet number is equal to 1, the integration of Eq. 35 yields:

1 1 1U Ut y t s y . 37Ž .0 2 22 n q1 n q1Ž . Ž .4 3

The second stage of the contaminant profile development is completed at tU s tU. At1Ž . Ž .that time, n s1. Introducing this condition into Eqs. 36 and 37 , we obtain:4

1 2 Peqn 1 1 1Ž .3U Ut y t s ln q y1 0 2 ž / ž /Peq1 n q1 Pey1 2 n q11y2 PeqPe Ž . Ž .3 3

38Ž .1 1 1

U Ut y t s y . 39Ž .1 0 22 4 n q1Ž .3

Ž . Ž . U UEqs. 38 and 39 describe the dependence of t y t on the barrier Peclet number1 0Ž .as a continuous decreasing function. Eq. 38 is applicable if the barrier Peclet number is

Ž .different from 1, while Eq. 39 is applied if the barrier Peclet number is equal to 1.

Table 1Summary of important features of the approximate models

Pe Stage Equation for Time Equation for the Equation for thecontaminant dependent calculation of the calculation ofprofile parameter parameter stage completion

Ž . Ž . Ž .<1 1st Eq. 23 d Eq. 25 Eq. 26Ž . Ž . Ž .<1 2nd Eq. 23 n Eq. 28 Eqs. 30a1

Ž .and 30bŽ . Ž . Ž .f1 1st Eq. 23 d Eq. 33 Eq. 34Ž . Ž . Ž .f1 2nd Eq. 23 n Eqs. 36 Eqs. 384

Ž . Ž .and 37 and 39Ž . Ž .f1 3rd Eq. 40 g Eq. 44 –Ž . Ž . Ž .41 1st Eqs. 45a b Eq. 49 Eq. 50Ž .and 45b


Ž .Fig. 7. Comparison between exact and BL transient solutions Pe<1 .

In the third and last stage, the contaminant concentration converges towards theŽ .steady-state profile of Eq. 17 . Under this constraint, we assume that the contaminant

concentration and flux across the barrier exit can be approximated, respectively, by:

g gU U U 2C s1y 1y x y x 40Ž .ž /2 2

ECUg

y s1q . 41Ž .Už / UEx 2x s1

where g is a time-dependent coefficient.When the third stage of the contaminant profile development starts, gs0. At the end

of the last stage, when steady-state conditions are established, gsPe. We apply theŽ .conservation of mass principle by direct integration of Eq. 5 over the entire barrier:

ECU ECU d 1 U UPey q s C d x . 42Ž .HU U Už / ž /U UEx Ex d t 0x s0 x s1


Ž .Fig. 8. Comparison between exact and BL transient solutions Pef1 .

Ž . Ž .We introduce Eq. 40 into Eq. 42 , while considering the time-dependence of thecoefficient g , to obtain:

1 dgsPeyg . 43Ž .U12 d t

After separating variables and integrating, and considering that at tU s tU the value1

of g is zero, we obtain:U U

gsPe 1yexp y12 t y t . 44� 4Ž . Ž .1

( )6.3. Large barrier Peclet number Pe41

If the barrier Peclet number is large, we assume that the contaminant concentrationprofile can be approximated by the following expression:

U b P exU UC s1y at 0Fx Fb 45aŽ .ž /b

CU s0 at xU )b 45bŽ .


where b is a time dependent coefficient that varies between the initial value of zero andŽ Ž ..final value of one as required by the steady-state solution of Eq. 21 .

Ž . Ž .The approximation of Eqs. 45a and 45b ignores the effect of contaminantdiffusion at the barrier entrance and assumes that steady-state conditions are established

Ž .in the domain when bs1. By direct integration of Eq. 5 , or reference to a finitecontrol volume of the barrier, we obtain:

d d U UPes C d x . 46Ž .HUd t 0

Ž . Ž . Ž .By introducing Eq. 46 into Eqs. 45a and 45b and considering the time-depen-dence of the coefficient b , we obtain:

bPe bPeq2 dbŽ .Pes . 47Ž .U2 d tbPeq1Ž .

Fig. 9. Correspondence between critical times predicted by BL method with effluent flux predictions generatedŽ . Ž .by exact solution for a Pe<1 and b Pes1.


Ž .Fig. 9 continued .

Ž .By separation of variables and integration of Eq. 47 , and considering bs0 attU s0, we obtain the following second order expression:

b 2 yb tUPey tU s0. 48Ž .

The contaminant profile development is completed at tU s tU. At that time, the0Ž . Ucoefficient b obtains the final value of one. The solutions of Eq. 49 for b and t ,0

respectively, yield:

1 2U U U(bs t Peq t Pe q4 t 49Ž . Ž .2

1Ut s . 50Ž .0 Peq1


Ž .Eq. 50 indicates that at high barrier Peclet numbers, an approximation of steadycontaminant transport through the barrier is established after a dimensionless time periodinversely proportional to the barrier Peclet number. This time period is approximatelyequivalent to the time needed to advect the contaminant front through the barrier

Ž .thickness LrÕ .

7. Discussion

The models developed in this work are motivated by the growing possibility that, forsome sites, vertical walls might be proposed as the primary engineered measure tomitigate subsurface contamination. We are not aware of any reported field measure-ments of contaminant concentrations within installed walls, and the design of suchsystems must therefore be strongly influenced by theoretical considerations. While thewisdom of such a remediation strategy may be subject to debate, a rational andconsistent approach to design can be developed by first considering the limiting

Žbehavior of idealized systems. It has been argued elsewhere Rabideau and Khandelwal,.1998a that the fixed-concentration boundary conditions can lead to conservative

predictions of barrier performance using analytical and numerical solutions to theone-dimensional advective–dispersive–reactive equation. The current work extends thisconcept by developing even simpler equations from BL theory.

We evaluate the applicability of the proposed approximations by comparing resultsŽ .obtained using the BL models summarized in Table 1 with the ‘‘exact’’ solutions to

Ž . ŽEq. 5 obtained by numerical and semi-analytical procedures Rabideau and Khandel-.wal, 1998a,b . The intended application is a typical engineered barrier such as a

soilrbentonite slurry wall, for which the barrier Peclet number may be interpreted as:

KiLPe; 51Ž .

fD

where K is the barrier hydraulic conductivity, i is the hydraulic gradient across thebarrier, f is the barrier porosity, L is the barrier thickness, and D is approximated bythe effective porous media diffusion coefficient. For typical slurry wall parametersŽ y6 2 y7 .L;1 m, f;0.4, D;5=10 cm rs, K;10 cmrs; e.g., Rabideau, 1996 , Eq.Ž . Ž U50 reduces to Pe;5i. The corresponding unit of dimensionless time t s1, Eq.Ž .. Ž .4c is approximately 63 years for a nonsorbing Rs1 barrier.

Ž .Two features of the model are considered: 1 ability of the model to predictŽ .contaminant spatial distributions at different stages of the barrier life, and 2 correspon-

dence between the duration of the various BL stages and the temporal evolution ofbarrier flux. Knowledge of the spatial distribution could be used to determine thelocations of sampling points for field verification of barrier performance, while keypoints in the temporal distribution of barrier flux may be used to delineate the effectivelife of the barrier. Because the results are considered in dimensionless terms, they can beextended to evaluate the impact of amending the barrier materials with sorbing additivesŽ .i.e., by increasing R .


In the examples that follow, the different ranges for the barrier Peclet number reflectpossible differences in the placement of the barrier relative to the direction of regionalgroundwater flow. If the placement of a vertical barrier disrupts the regional flow onlyminimally, the governing hydraulic gradient across the barrier may be similar to the

Ž .natural gradient, which suggests a very small Pe <1 due to the low hydraulicconductivity of the wall. Also, the placement of extraction wells within the contained

Ž .area entrance side may lead to a very small net hydraulic gradient. Conversely, theoccurrence of hydraulic mounding on the upgradient side of the barrier could lead to

Žlarger gradients and Pe on the order of 1 personal communication, confidential.industrial source . Higher Pe scenarios are less likely for field installations of vertical

walls.Fig. 7 provides comparisons between the semi-analytical and the BL results for low

ŽPe. Calculations of the BL solutions were made by assuming n s2 e.g., Rubin and0.Buddemeier, 1996 . Although the BL solutions assume negligible flux exiting the

Ž U UFig. 10. Comparison between the exact and BL solutions for Pes5 for t s0.10, b s0.654; for t s0.20,.b s1 .


barrier, the semi-analytical and BL spatial contaminant distributions are very similar forboth stages. Similar results are shown in Fig. 8 for the case of Pes1, with n s2.3

Again, the predicted contaminant spatial distributions are in good agreement.In Fig. 9, the flux calculated by the semi-analytical model is shown, along with the

times corresponding to the duration of the BL stages. Depending upon the definition ofŽ . Ž . Ž . Ž .‘‘acceptable’’ barrier flux, Eqs. 30a and 30b or Eqs. 39 and 40 may be used to

provide quick estimates of the transient period duration. A drawback to the BL approachis that, except for high Pe, an expression for the flux at the barrier exit is not provided.However, under the assumed fixed-concentration boundary conditions, the transientperiod is characterized by a relatively sharp increase in flux, and it may be sufficient for

Ž Uengineering purposes to delineate the critical times in the approach to steady-state t0U .and t . For a nonsorbing barrier described by the parameters given above, the1

calculated dimensionless times indicate that the sharp increase in flux occurs during theU Ž . Ž . Uperiod t s0.08–0.20 years 5–12 for the low Pe scenario Fig. 9a and t s0.06–0.12

Ž U UFig. 11. Comparison between the exact and BL solutions for Pes10 for t s0.06, b s0.688; for t s0.10,.b s1 .


Ž . Ž .years 4–8 for Pe;1 Fig. 9b . These estimated times would be multiplied by theretardation factor for the case of a sorbing barrier. For example, the experimental results

Ž .of Mott and Weber 1992 suggest that R values as high as several hundred would applyto soilrbentonite barrier materials amended with fly ash, and the theoretical barrier lifewould therefore increase from years to centuries.

As shown in Figs. 10 and 11, the predictions generated using the two approachesdeviate at high Pe. For the BL approach, neglecting diffusion during the initial stages ofthe barrier life has a significant effect on the predictions for those stages. However, thedifferences between the two methods decrease over time.

8. Summary and conclusions

This paper summarizes the development of quick methods for evaluating the perfor-mance of vertical barriers, based on the conservative assumption of prescribed contami-nant concentrations at both boundaries. For the case of a non-decaying contaminant,knowledge of the dimensionless barrier Peclet number is sufficient for quick calculationof the steady-state flux across the barrier, as well as the spatial distribution ofcontaminant within the barrier during the transient period. Three solutions were devel-

Ž .oped, corresponding to: 1 low Peclet numbers, for which diffusion is expected toŽ .dominate transport across the barrier, 2 Peclet numbers on the order of one, where both

Ž .diffusion and advection are significant transport mechanisms, and 3 high Pecletnumbers, for which advection is the dominant transport mechanism.

The most promising immediate application of the tools developed through this workis for the case of low permeability barriers where diffusion is a significant transportmechanism. For these scenarios, the BL methods can provide a simple and accuratecharacterization of the transient period of the barrier, enabling the barrier designer toquickly assess the effects of changes to the barrier properties, including the thickness,hydraulic conductivity, hydraulic gradient or sorption capacity.

9. Notation

a , a , a Coefficients of polynomial expression1 2 3

BL Boundary layerw y3 xC Contaminant concentration ML

w y3 xC Contaminant concentration at xs0 ML0

CU Dimensionless contaminant concentrationw y2 y1 xCF Contaminant flux per unit cross-section ML T

CMR Completely mixed regionw 2 y1 xD Coefficient of dispersion L T

f Portion of diffusive flux at the barrier upstream edgeF Dimensionless contaminant fluxG Diffusion enhancement function


i Hydraulic gradientw y1 xK Barrier hydraulic conductivity LT

w xL Thickness of the barrier Ln , n , n , n Power coefficients of the BL0 1 3 4

Pe Barrier Peclet numberw y3 y1 xq Local contaminant sink ML T

qU Dimensionless contaminant sinkR Retardation factorROI Region of interestSO Second order

w xt Time Tw xt Characteristic time scale T0

tU Dimensionless timetU , tU , tU Dimensionless times characterizing consecutive stages of con -0 1 2

taminant profile build-upw y1 xÕ Interstitial fluid velocity LTw xx Longitudinal coordinate L

xU Dimensionless longitudinal coordinatexU Moving dimensionless longitudinal coordinate1

xU Dimensionless longitudinal coordinate in the upstream direction2Ž . Ž .b Time dependent coefficient defined in Eqs. 45a and 45bŽ .g Time dependent coefficient defined in Eq. 40

d Dimensionless BL thicknessD tU Dimensionless time intervalD xU Dimensionless longitudinal intervalh Coordinate of the BL

w y1 xl Decay coefficient Tl

U Dimensionless decay coefficientf Porosity

References

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approximate evaluation of contaminant transport through vertical barriers

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