Groundwater pollution by organic compounds: a three-dimensional boundary element solution of contaminant transport equations in stratified porous media with multiple non-equilibrium

INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS

Int. J. Numer. Anal. Meth. Geomech., 23, 1733}1762 (1999)

GROUNDWATER POLLUTION BY ORGANICCOMPOUNDS: A THREE-DIMENSIONAL BOUNDARY

ELEMENT SOLUTION OF CONTAMINANT TRANSPORTEQUATIONS IN STRATIFIED POROUS MEDIA WITH

MULTIPLE NON-EQUILIBRIUM PARTITIONING

ABBAS H. ELZEIN* AND JOHN R. BOOKER

Centre for Geotechnical Research, Department of Civil Engineering, University of Sydney, NSW 2006, Australia

SUMMARY

Industrial contaminants and land"ll leachates, particularly those with high organic content, may migrateinto groundwater streams under conditions of non-equilibrium partitioning. These conditions may either beinduced by time-dependent sorption onto the soil skeleton and intra-sorbent di!usion in the soil matrix, orby heterogeneous advective "elds within the pore. These processes are known as chemical and physicalnon-equilibrium processes respectively, and may result in signi"cant deviations from the paths predicted bysteady-state partitioning assumptions. In addition, multi-directional soil properties, soil strati"cation andcomplex geometries of the pollution source may require a full three-dimensional analysis for accuratecontamination prediction.

A three-dimensional boundary element solution of the time-dependent di!usive/advective equation innon-homogeneous soils with both physical and chemical non-equilibrium processes is developed. Saturatedconditions and rate-limited mass transfer are assumed. The Laplace transform removes the need fortime-stepping and the associated numerical complexity, and the use of Green's functions yields accuratesolutions of in"nite and semi-in"nite domains such as soils as well as media with "nite dimensions. Thesolution requires boundary discretization only and can therefore be a valuable tool in bio-remediation andland"ll design where di!erent geometries, soil properties and pollutant loads may be analysed at low cost.The proposed technique is validated by comparing its predictions to analytical solutions obtained fordi!erent types of soil and contaminant sources. The scope of the method is illustrated by analysing thecontamination of multi-layered soils by a neighbouring river and a surface source. Copyright ( 1999 JohnWiley & Sons, Ltd.

KEY WORDS: contaminant migration; non-equilibrium sorption; di!usion advection; boundary element;Green's functions; porous media

INTRODUCTION

An increased interest in groundwater quality on the part of local and national authorities over thelast decade has led to increasing research on contaminant migration in soils.1~4. Environmentalregulators must identify critical pollution situations and decide on acceptable levels of contami-nation while industrial managers and environmental engineers must develop strategies for

*Correspondence to: A. H. Elzein, Centre for Geotechnical Research, Department of Civil Engineering, University ofSydney, NSW 2006, Australia

CCC 0363}9061/99/141733}30$17.50 Received 22 September 1997Copyright ( 1999 John Wiley & Sons, Ltd. Revised 8 June 1998

pollution prevention or remediation. On one hand, acceptable levels of contamination bydi!erent compounds are studied on the basis of their public health or ecological implications.On the other hand, mechanisms governing the transport and physico-chemical state of thecontaminants need to be quanti"ed. Contaminant migration models are developed and used astools for the prediction of pollution plumes, given a certain pollution source and a number ofparameters that can be measured on site. Processes of molecular di!usion, mechanical dispersionand advective transport as well as the partitioning of contaminants between the di!erent phasesand their radioactive or biological decay are incorporated in the models.

Signi"cant deviations from predicted breakthrough curves (i.e. curves of concentration versustime at selected points of the soil) may occur when the Local Equilibrium Assumption (LEA),resulting in instantaneous rather than time-dependent sorption, is used in the di!usion/advectionequation. Under conditions of high hydraulic conductivity, tailing of breakthrough curves havebeen observed and shown to be caused by physical or chemical non-equilibrium processes.5~9.Models containing a simple rate-limited non-equilibrium component of mass-transfer havesucceeded in accounting for such deviations.10~12 Organic compounds, particularly non-hydro-phobic ones, and inorganic solutes may be subject to sorption-related non-equilibrium processesas a result of rate-limited chemical sorption at some sites or slow di!usive mass transfer within thesoil matrix.13 Physical non-equilibrium processes may result from tortuous #ow paths and sharpdiscontinuities of the advective "eld in the individual pore. It may be simulated througha distinction between &mobile' and &immobile' regions of the soil, where advection transportoccurs in the "rst but not the second.14,15 The mobile/immobile bi-continuum appears to bea valid approximation of the non-uniformity of the advective "eld, particularly in aggregatedheterogeneous soils. The &immobile' fraction is meant to denote, in addition to immobile orquasi-stagnant pore water, residual non-aqueous phase liquids. The exchange between &mobile'contaminants and &immobile' regions is believed to be governed by di!usion laws. Thus, strongexperimental evidence suggests that a series of non-equilibrium processes, all amenable torate-limited "rst-order mass transfer, are an important factor in contaminant transport in soilsand must therefore be included in the modelling e!ort.

Analytical and numerical methods have been used to solve the di!usive/advective equationdescribing the transport of pollutants in porous media. Most computer models of soil contamina-tion are one-dimensional or two-dimensional solutions based on analytical, "nite-di!erence or"nite-element methods. Although one-dimensional and two-dimensional solutions may yieldaccurate or conservative results in many situations, three-dimensional solutions are often re-quired to account for soil anisotropy, multi-directional hydraulic #ow, complex geometries ofpollution source when the concentration "eld is required near the source, and complex soilgeometries and boundary conditions. Analysing the transport of contaminants from a land"llthrough an underlying clay liner, Rowe and Booker16 showed that one-dimensional solutionsmay yield non-conservative results compared to two-dimensional solutions, depending on theprevailing boundary conditions. This is likely to be true when two-dimensional predictions arevalidated against three-dimensional ones.

A number of three-dimensional solutions of the di!usion or di!usion/advection equation canbe found in the literature. A "nite-di!erence solution of three-dimensional dispersion problemswas formulated as early as 1967 by Shamir and Harleman.17 A number of "nite di!erence and"nite element solutions of the transport equation in multi-layered three-dimensional media havesince been developed by Gupta and Tanji,18 Frind and Verge19 and Huyakorn et al.,20 amongothers. Goltz and Roberts21 used a combination of Laplace and Fourier transforms to solve for

1734 A. H. ELZEIN AND J. R. BOOKER

Copyright ( 1999 John Wiley & Sons, Ltd. Int. J. Numer. Anal. Meth. Geomech., 23, 1733}1762 (1999)

semi-in"nite media with physical non-equilibrium processes. Semi-analytical solutions for simpletwo-dimensional and three-dimensional geometries of pollution source such as spheres and cylin-ders were derived by Rahman and Booker,22 while Leij et al.,23 proposed semi-analytical solutionsof transport equations with non-equilibrium mass transfer. More recently, Zou et al.24 developedan analytical technique for non-Fickian di!usion. However, to the best of the authors' knowledge,no general solutions of the non-equilibrium equations have been developed in two-dimensional orthree-dimensional space, that are applicable to arbitrary soil strati"cations and source shapes.

The complexity of "nite-element or "nite-di!erence three-dimensional solutions usually stemsfrom the requirement of three-dimensional mesh development which, given the soil's in"nite orsemi-in"nite dimensions, can be a highly time-consuming task. This is further complicated bynumerical requirements such as a gradual rather than an abrupt decrease of element size from onepart of the soil to the others. Corners and discontinuities in contaminant source are also di$cultto model as they must be represented with a large concentration of small elements. Simpler toolsof analysis which would, nevertheless, incorporate the accuracy of three-dimensional solutionsare therefore needed. The boundary element method BEM25~27 o!ers such a possibility becausea three-dimensional solution requires a boundary (2D) discretisation rather than a domain (3D)one. Using Green identities and known singularity solutions of the governing equations, a bound-ary integral statement of the governing di!erential equations can be developed and discretized.The BEM has been e!ective in solving linear, steady-state problems in elasticity and "eldproblems including #uid #ow and mass transfer.28 However, its application to non-linear andtime-dependent problems has been slower, primarily because of the di$culty of developingadequate singular solutions. The absence of such solutions has led to the introduction of domainintegrals into the formulation and thus greatly reduced the e$ciency of, and often the justi"cationfor, the BEM approach. Dual reciprocity methods and, more recently, a multiple-reciprocityapproach, have been explored as possible ways of avoiding these domain integrals in thesolution.29,30

E$cient boundary element solutions of time-dependent thermoelasticity have been developedby applying the Laplace transform to the governing di!erential equations in order to eliminatethe time variable.31~33 This approach has been used in combination with the Finite ElementMethod to model groundwater #ow and mass transport in fractured and non-fractured me-dia.34,35 Sudicky34 found concentration pro"les to be smooth, well-behaved functions, whichrequire a relatively small number of "nite elements. Leo and Booker36,37 combined a semi-analytical Laplace solution of the contaminant migration problem38 to a boundary elementsolution of the modi"ed Helmholtz equation and developed an e$cient technique for solving thetwo-dimensional contaminant transport problem with anisotropic dispersion properties in non-homogeneous media. Elzein and Booker39 extended the solution to include non-equilibriumpartitioning processes. The formulation applies the Laplace transform to the governing equationsthen eliminates the dispersion anisotropy and advection terms through a co-ordinate transforma-tion combined with an exponential representation of the solute concentration. A standardcomplex-valued modi"ed Helmholtz equation is thus obtained, which is solved by BEM andinverted back into time-domain numerically. The method only requires boundary discretizationand does not involve time stepping. Its e$ciency is further ampli"ed in multi-layered domains asthe computational cost of adding extra layers is relatively low.

The present paper extends this boundary element solution into three-dimensional space. Theproposed technique is validated by comparing its predictions to analytical, semi-analytical andnumerical solutions developed elsewhere. In addition, the scope of the method is illustrated

GROUNDWATER POLLUTION OF ORGANIC COMPOUNDS 1735


through the study of the contamination of a multi-layered soil by a neighbouring river anda surface pollution source; results from two-dimensional and three-dimensional simulations arecompared and their validity discussed.

BOUNDARY ELEMENT SOLUTION OF THREE-DIMENSIONAL TRANSPORTEQUATIONS IN SATURATED MEDIA WITH NON-EQUILIBRIUM PARTITIONING

The transport of pollutants in a saturated three-dimensional porous domain ) with boundary! is assumed to be governed by steady-state hydraulic #ow, Fickian di!usion, instantaneous andrate-limited sorption on soil skeleton and rate-limited partitioning between a mobile andimmobile fractions of the solid solution. The non-equilibrium model of the two-dimensionalsolution developed earlier39 by the authors is adopted here and is shown schematically inFigure 1. At every point of the soil, the mass of contaminant is divided among three phases:dissolved in mobile water (block A), dissolved in immobile water or non-aqueous liquid (block B)or sorbed onto the soil matrix (blocks C and D). A fraction F of sorption onto the soil mineral ororganic matrix is assumed to be instantaneous while the remaining fraction is governed byrate-limited mass transfer. The &sorbed' contaminant blocks C and D are clearly in parallel,indicating that &equilibrium' and &non-equilibrium' processes are simultaneous and independent.The model is simpli"ed by assuming that the &immobile solution' (block B) undergoes nopartitioning with the solid phase. Applying the mass conservation principle, Fickian di!usion lawand rate-limited non-equilibrium partitioning laws, assuming that the groundwater #ow isaligned with one of the major co-ordinate axes and neglecting biodegradation and radioactivedecay, the following coupled transport equations are obtained:

n.D

xx

L2c.

Lx2#n

.D

yy

L2c.

Ly2#n

.D

zz

L2c.

Lz2!n

.<!x

Lc.

Lx!n

.<ay

Lc.

Ly

!n.<!z

Lc.

Lz"(n

.#o

$FK

$)Lc

.Lt

#o$

Lq.

Lt#n

*.

Lc*.

Lt(1)

Figure 1. Model for non-equilibrium processes



Lq.

Lt"K

tMK

$(1!F)c

.!q

.N (2)

n*.

Lc*.

Lt"w (c

.!c

.) (3)

(at steady-state: q."K

$(1!F )c

., c

*."c

.)

where t(T) is time, x, y and z (L) denote a three-dimensional Cartesian co-ordinate system,c.(t, x, y, x) (M/L3) is the dissolved concentration of contaminant in mobile water, c

*.(t, x, y, z)

(M/L3) is the dissolved concentration of contaminant in immobile water, Dxx

, Dyy

and Dzz

(L2/T)are the coe$cients of hydrodynamic dispersion of contaminant in soil, in the x, y and z directions,respectively, <

ax, <

ayand <

az(L/T) are the values of groundwater velocity in the x, y and

z directions, respectively, n.

is the mobile-water-"lled porosity of the soil, o$(M/L3) is the dry

density of the soil, n*.

is the immobile-water-"lled porosity of the soil, q.

(M/M) is thenon-instantaneous fraction of sorbed material, K

$(L3/M) is the soil/contaminant partitioning

coe$cient, F is the fraction of steady-state sorption that is instantaneous, Kt(1/T) is the "rst-order

sorption rate constant, and w (1/T) is the "rst-order transfer coe$cient between mobile andimmobile water. Finally, the sorption retardation factor is de"ned as: R"1#o

$K

$/n

..

Two types of boundary conditions are used. The "rst type is a linear combination ofconcentration and #ux:

ac.#b f

./"u (4)

where c.

is the pollutant concentration in mobile water at the boundary, f./

is the #ux ofcontaminant dissolved in mobile water in the direction n normal to the boundary, and a, b andu are given coe$cients which may vary over the boundary.

The second boundary condition type is the waste-repository or ,nite-mass condition40 whichconserves a speci"ed mass of contaminant rather than maintaining a speci"ed #ux or concentra-tion. Assuming no retardation or decay occurs within the contaminant source and the concentra-tion in the repository is an unspeci,ed (i.e unknown) quantity that is uniform in space, the"nite-mass condition can be written as follows:

cr.!cr

.0#

1

Hf¸

r=

rP

t

0P!r

f r./

d!rdt"0 (5)

where !ris the boundary at which the condition applies, H

fis the equivalent-height of the source,

¸r

and =r

are the length and width of the repository's boundary, respectively, cr.(t) is the

concentration at !r

assumed uniform, cr.0

is the concentration at t"0 and f r.n

is the #ux ofcontaminant through !

r.

Introducing the following Laplace transforms:

(cN., cN

*., qN

., fM

.n)"P

=

0

e~st (c., c

*., q

., f

.n) dt (6)

Using Laplace transform properties and rearranging, equations (1)}(5) become

Dxx

L2cN.

Lx2#D

yy

L2cN.

Ly2#D

zz

L2cN.

Lz2!<

!x

LcN.

Lx!<

!y

LcN.

Ly!<

!z

LcN.

Lz"'cN

.!'

.c.0

!

n*.

n.

'*.

c*.0

(7)



where

'"A'.#

o$

n.

't#

n*.

n.

'*.Bs

'."1#

o$FK

$n.

'*."

w

n*.

s#w

't"

KtK

$(1!F)

s#Kt

qN."'

tcN., (8)

cN*."'

*.AcN .#n*.w

c*.0B (9)

acN.#b fM

.n"

u

s(10)

cN r."

cr.0s

!

1

Hf¸

r=

r

1

s P!r

fM r.n

d!, (11)

(c.0

and c*.0

are the initial concentrations in the mobile and immobile solutions, respectively, andare taken as equal to satisfy initial equilibrium).

Equation (7) can be written as a standard modi"ed Helmholtz equation by applying a dualtransformation.41 The "rst transformation is a change of co-ordinate system which removes theanisotropy of the di!usion properties. The second transformation introduces an exponentialrepresentation of cN

.which removes the advective terms from the di!erential equation. Assuming

di!usion and advection coe$cients are uniform within a given soil layer, the following equation isobtained:

D!+2cN

.!"(cN

.!

A+2"L2

LX2#

L2

L>2#

L2

LZ2B (12)

where for the "rst transform:

X"uxx, y"u

yy, Z"u

zz

ux"J(D

xx/D

!), u

y"J(D

yy/D

!)

uz"J(D

zz/D

!), D

!"(D

xxD

yyD

zz)1@3

<X"u

yuz<x, <

Y"u

xuz<y

<Z"u

xuy<z, <

N"<

X¸X#<

Y¸Y#<

Z¸

Z

fM.N

"<NcN.!D

!

LcN.

LN

fM.N

"(uyuzlx¸X#u

xuzly¸Y#u

xuylz¸Z) fM

.n(13)



for the second transform:

aX"

<X

2D!

, aY"

<Y

2D!

, aZ"

<Z

2D!

cN."cN

.!e (aXXàYYàZZ) , fM

.N"fM

.!Ne (aXXàYYàZZ)

("'#D!(a2

X#a2

Y#a2

Z) fM

.!N"

<N2

cN.!

!D!

LcN.!

LN(14)

lx, l

yand l

zare the direction cosines of the normal n to the boundary in the (x, y, z) co-ordinate

system, N is the normal direction to the boundary in the (X, >, Z) co-ordinate system, ¸X, ¸

Yand

¸Z

are the direction cosines of the normal N.Initial concentrations have been assumed to be identically zero in equation (12). The second

and third terms on the right-hand side of equation (7) would normally introduce domain integralsinto the integral equation. However, for simple initial distributions, particular solutions can befound and equation (12) can be written in terms of incremental concentration. For more elaborateor random distributions, di!erent techniques, such as the dual-reciprocity method,42,43 havebeen successfully used in both potential and elasticity problems to avoid domain integration. Inthe authors' opinion, the method can be easily extended to the present formulation. In this paper,as mentioned earlier, only problems with zero initial-concentrations will be considered.

Boundary element solutions of equation (12) are widely documented in the literature.25,26,28Assuming identically zero initial distribution, and using the Gauss divergence theorem andsingular solutions of equation (12), it is found that

e (r0)cN

.!0"P

!(cN

.!fM *.!N

!cN *.!

fM.!N

) d! (15)

where r0

is the position vector of the singularity source, cN.!0

is the value of cN.!

at the singularitysource, and e (r

0) is 1 when the source is inside the domain and 1/2 on a smooth boundary.

A fundamental solution McN *.!

(r) , fM *.!N

(r)N of the three-dimensional modi"ed Helmholtz equation inthe Laplace domain can be written as follows:25,26

cN *.!

(r)"1

4nD!

e~rJ((@D!)

r(16)

fM *.!N

(r)"<N2

cN *.!

!D!

LcN *.!

LN(17)

where r is the Euclidean distance between the source node and the integration point.The "elds of concentration and normal #uxes can be discretized by dividing the boundary of

the problem into n%triangular or quadrilateral elements j over which system variables cN

.!and

fM.!N

are represented as functions of their values cN i.!

and fM i.!N

at n1pre-selected nodes i, by means of

simple interpolation function uji, where for every element j

(cN.!

, fM.!N

)"n1+i/1

uji(cN i

.!, fM i

.!N) ( j"1, n

%) (18)

In the present implementation, a library of constant (n1"1), linear (n

1"3 for triangular

elements or n1"4 for quadrilateral elements) and quadratic (n

1"6 for triangular elements or



n1"9 for quadrilateral elements) elements is developed. The problem of multiply de"ned

normals to the boundary at corners is resolved by using partly discontinuous triangularor quadrilateral elements to eliminate multiple-de"nitions of fM

./. Typically, the nodes occurring

at the corner edge are located inside the element at some small, arbitrary distance from theedge. Interpolation functions for such elements are derived numerically rather than semi-analytically.

If N5total nodes are created over the entire boundary of the problem, then N

5equations can be

constructed by placing the singularity source point over each of these nodes and writing equation(15) accordingly. A self-adaptive Gaussian numerical integration technique is used to evaluate theintegrals. An element subdivision scheme automatically increases the number of integrationpoints when singularity source is close to integration element and integral near-singularity isstrong. Substituting variables cN

.!and fM

.!Nwith expressions in terms of cN

.and fM

.!, respectively

(equations (13) and (14)), and rearranging, the following set of equations is obtained:

[H]McN k.N"[G]M fM k

.nN (k"1, N

5) (19)

By applying boundary conditions of the form (10) or (11), system (19) can be reduced to a set of N5

algebraic equations with N5unknown that can be solved by a standard Gauss solution proced-

ure.44 Concentration at points inside the domain can be found by placing the source nodes atthese points, writing equation (15) and using the known boundary values of concentrationand #ux to evaluate the integrals numerically. Therefore, the number of internal points does nota!ect the number of degrees of freedom in the system and the order of the algebraic system ofequations (19).

The value of c.

at each node is obtained by inverting the solution into time-domain numer-ically using a complex-valued scheme proposed by Talbot.45 The relative merits of di!erentinversion schemes have been explored by di!erent authors.46,47 Notably, the Crump algorithm48

requires a single set of Laplace variables to compute the solution at a number of time stations.This would induce signi"cant savings in computational time when solution at more than one timestation is required.34 In the present paper, the Talbot approach, used because of the availability ofa previously developed computer program based on the algorithm, was found to yield very goodaccuracy.

Non-homogeneous soils made up of a number of homogeneous regions having di!erenthydro-geological and chemical properties are modelled as a collection of regions or zones.A separate set of equations and unknowns is developed for each zone. At the interface betweentwo zones, forcing the continuity of the unknowns cN

.and fM

.nprovides additional equations in lieu

of the boundary conditions applied at non-interface boundaries. Thus, sets of zone-matricescontribute to the construction of global matrices re#ecting the entire problem. Zoning is also ane$cient way of creating a computationally e$cient bandwidth in fully populated boundaryelement matrices. The extra cost of adding a zone to a problem, relatively low because it involvesonly additional boundary discretization, is further reduced by the implicit creation of a band-width. A medium made of in"nite or semi-in"nite layers of soil is therefore an especiallysuitable problem for simulation by this technique. While arbitrarily transient boundary condi-tions, non-linear sorption and randomly heterogeneous soils are not suitable for simulation bythis method, the proposed solution still covers a very wide range of contaminant transportproblems.

The algorithm has been implemented in FORTRAN 90 on UNIX and PC platforms (CON-TAN 3D), along with a text-based easy-to-use facility for data-input and a number of output "les



for viewing on a spreadsheet. Meshing is performed automatically subject to user speci"cations.Special algorithms for the automatic meshing of rectangular surfaces, spheres and land"lls madeof two zones, have been developed.

The algorithm is validated through the simulation of nine contaminant migration cases forwhich analytical, semi-analytical or numerical solutions can be found for comparison. The scopeof the method is illustrated by simulating the migration of contaminant in two layers of sandpolluted by a small surface source and an adjacent river.

RESULTS

Case 1 and 2 (Semi-in,nite media and multiple sources of non-equilibrium processes). The case ofa semi-in"nite homogeneous soil is "rst considered (Figure 2(a)). The three-dimensional meshshown in Figure 2(b), consists of four square elements representing a 20]20 m2 patch of thesurface of the soil. Data used in the simulation of the nine validation problems is shown inTable I. In this problem, two types of boundary conditions at the surface are used: a speci"edconcentration c

."10 mg/l for case 1, and a speci"ed #ux f

.n"10 mg/m2 y for case 2. This is

a one-dimensional problem for which analytical solutions have been derived by di!erentauthors.40,49 In this paper, the solution proposed by Rowe et al.40 has been extended toincorporate multiple sources of non-equilibrium and used as a reference for validating BEMpredictions. Concentration pro"les after 1 month are shown in Figures 2(c)}(f).

Figures 2(c) and 2(d) compare four di!erent assumptions of partitioning kinetics: LocalEquilibrium Assumption (LEA), immobile fraction (n

*."0)15, w"2./y), non-equilibrium sorp-

tion (F"0)5, K5"1./y) and, "nally, both an immobile fraction and non-equilibrium sorption. In

the LEA case, no immobile fraction exist in the soil and sorption is instantaneous. In all cases

Table I. Data for nine validation cases

Cases 1, 2 Cases 3, 4 Caes 5, 6 Case 7 Case 8 Case 9Semi- Buried Buried Two-layer clay Two-layer Three-layer

in"nite soil cylinder sphere sand/silt sand/silt

layer 1 layer 2 layer 1 layer 2 layer 1 layers 2, 3(clay) (clay) (sand) (silt) (sand) (silt)

Dx

(m2/year) 0 0)01 0)01 0 0 0 0 5 0)5D

y(m2/year) 0 0)01 0)01 0 0 0 0 0)5 0)05

Dz(m2/year) 5 0)01 0)01 0)01 0)02 5 0)5 0 0

vx

(m/year) 0 0 0 0 0 0 0 5 0)5vy(m/year) 0 0 0 0 0 0 0 0 0

vz(m/year) 5 0 0 0)01 0)01 5 0)5 0 0

n.

0)2 0)4 0)4 0)4 0)4 0)35 0)4 0)26 0)4n*.

0)15 0 0 0 0 0 0 0)1 0R 4)52 1 1 10 5 2 4 1 1)1F 0)5 na na 1 1 0)3 0)3 na 0)2K

$(m3/kg) 0)0004 0 0 0)00212 0)000941 0)000206 0)000709 0 2)33E-05

Kt(/year) 1 na na na na 1 1 na 1

w (/year) 2 na na na na na na 1 na

na: Not applicable



Figure 2(a). Semi-in"nite homogeneous soil: lateral view (cases 1 and 2)

Figure 2(b). Semi-in"nite homogeneous soil: 2]2 BEM mesh (cases 1 and 2)

Figure 2(c). Semi-in"nite homogeneous soil with speci"ed concentration at the surface: c.

along depth after 1 month, withand without non-equilibrium processes (case 1)



Figure 2(d). Semi-in"nite homogeneous soil with speci"ed #ux at the surface: c.

along depth after 1 month, with andwithout non-equilibrium processes (case 2)

Figure 2(e). Semi-in"nite homogeneous soil with speci"ed concentration at the surface with multiple sources of non-equilibrium processes: BEM convergence, c

.along the depth after 1 month (case 1)

Figure 2(f ). Semi-in"nite homogeneous soil with speci"ed #ux at the surface with multiple sources of non-equilibriumprocesses: BEM convergence, c

.along depth after 1 month (case 2)



where no immobile fraction is included in the simulation, n.

is taken to be the total porosityat 0)35.

Both analytical and BEM results with four constant elements are shown. Clearly, goodagreement between the two sets of results is demonstrated. While the existence of an immobilefraction reduces the concentration of contaminants in the mobile solution, non-equilibriumsorption increases that concentration as retardation is slowed and therefore reduced. This isespecially the case when concentration rather than #ux is speci"ed at the surface. Figures 5(c) and5(d) also show that surface boundary conditions have a signi"cant in#uence on the e!ect of non-equilibrium assumptions. Comparing the LEA curves to multiple-non-equilibrium curves in both"gures, it is clear that the di!erence is far more pronounced when #ux is speci"ed at the surfacerather than concentration. Further simulations are required to establish whether this pattern issustained under di!erent soil properties.

Figures 2(e) and 2(f ) assess the convergence of the BEM results in the case of multiple sourcesof non-equilibrium processes in both cases 1 and 2. While a 2]2 mesh yields high accuracy witha maximum error of 1)25 per cent, the predictions obtained from a 3]3 mesh lead to 0)28 per centmaximum error.

Cases 3 and 4 (Deeply buried cylindrical repository). Next, the case of a cylindrical repositorydeeply buried in a clayey soil is considered (Figure 3(a)). The cylinder has a radius of 1 m and isconsidered to be in"nite in length. Its centre is located at point (x

0, y

0, z

0). Two types of

boundary conditions are used: a speci"ed concentration of c."3000 mg/l for case 3 and a "nite

mass of contaminant with initial concentration cr.0

"3000 mg/l for case 4. Two mesh densitiesused in the simulations are shown in Figure 3(b) (16 rectangular elements) and Figure 3(c) (32rectangular elements). The respective aspect ratio of elements (ratio of width to depth), are around25 and 13.

Figures 3(d) and 3(e) show concentration pro"les along the positive y-axis half-way along thelength of the cylinder after 10 years. Results predicted by CONTAN 3D are compared toanalytical solutions proposed by Rahman and Booker.22 While a marked improvement isobserved by moving from 16 constant elements to 16 linear elements, 32 constant elements yieldaccurate results.

Case 5 and 6 (Deeply buried spherical repository). A special repository of 1 m radius is deeplyburied in a clayey soil (Figure 4(a)). Its centre has the co-ordinates (x

0, y

0, z

0). Two boundary

conditions are considered: case 5 with a constant concentration c."3000 mg/l and case 6 with

a constant-mass with an initial concentration cr.0

"3000 mg/l. Four di!erent mesh densities areused in the BEM simulations: 8, 32, 128 and 200 constant elements. The mesh with the highestelement density is shown in Figure 4(b).

Concentration pro"les along the positive y-axis after 10 years are shown in Figures 4(c) and4(d) where BEM predictions are compared to an analytical solution developed by Rahman andBooker.22 A relatively high number of elements is required to achieve good accuracy (200constant element). This is most likely due to the curvilinear geometry of the sphere beingmodelled with #at elements. It is highly probable that, using curvilinear elements, a smallernumber of elements would be required to achieve the same accuracy. On the other hand,curvilinear elements require additional components to be included in the algorithm, particularlyin evaluating singular integrals as the Jacobian of the transformation from local to globalco-ordinates is no longer constant over the surface of the element.



Figure 3(a). Deeply-buried cylindrical repository: longitudinal and cross-sectional views (cases 3 and 4).

Figure 3(b). Deeply buried cylindrical repository: 16-element mesh (cases 3 and 4)

Figure 3(c). Deeply buried cylindrical repository: 32-element mesh (cases 3 and 4)

Case 7 and 8 (Multi-layered soil). The case of non-homogeneous soils and non-equilibriumsorption is considered here. Two types of soil, consisting of two strata of 4 m depth each, restingon impermeable media, are analysed. The "rst soil (case 7, Figure 5(a)) consists of two layers ofclay with di!erent di!usion properties. The second soil (case 8, Figure 5(a)) is a sand layerunderlain by silt. In both cases, a "nite-mass source of contaminant with initial concentrationcr.0

"1000 mg/l and Hf"5 m is applied at the surface. This is a one-dimensional problem for

which analytical solutions have been developed by Rowe et al.40 As in cases 1 and 2, the analyticalsolution has been extended here to solve for non-equilibrium partitioning. The CONTAN 3Dmesh used in the simulation represents 2 zones and contains 44 elements (Figure 5(c)). Given thetime scale of migration in clay, sorption has been assumed to be instantaneous in case 7, whileboth instantaneous and non-equilibrium sorption have been considered in case 8.

Figure 5(d) shows concentration pro"les along the depth for case 7 after 100, 500, and 1500 and2000 years. There is clearly very good agreement between analytical and BEM results. The kink inthe curve at the interface between the two layers is well reproduced in the BEM pro"les.



Figure 3(d). Deeply buried cylindrical repository with constant concentration: c.

pro"le along the positive y-axis, BEMversus analytical results (case 3)

Figure 3(e). Deeply buried cylindrical repository with constant mass: c.

pro"le along the positive y-axis, BEM versusanalytical results (case 4)

Figures 5(e)}(g) show results for case 8 based on non-equilibrium sorption parameters includ-ing an instantaneous fraction F"0)3 and a slow sorption rate K

t"1./y. Figure 5(e) illustrates

the e!ect of non-equilibrium sorption by comparing the concentration pro"les at di!erent timesto pro"les obtained from instantaneous sorption (F"1.). In the sandy layer, the e!ect of non-equilibrium sorption gradually diminishes and disappears after "ve years. In the silt layer, thesame pattern is observed; however the e!ect of non-equilibrium is still discernable after 5 years.The shapes of the 1- and 5-y curves can be explained by the speci"ed-mass boundary condition atthe surface combined to the di!erential rate of transport between the two layers. Strong di!usion#uxes exist initially which transport contaminants from the surface into the soil. Contaminantsaccumulate at the interface because transport into the silt layer is slower. On the other hand, the



Figure 4(a). Deeply buried spherical repository: 3-D view (cases 5 and 6)

Figure 4(b). Deeply burried spherical repository: 200-element BEM mesh (cases 5 and 6)

Figure 4(c). Deeply buried spherical repository with constant concentration: c.

pro"le along the positive y-axis, BEMversus analytical results (case 5)



Figure 4(d). Deeply buried spherical repository with constant mass: : c.

pro"le along the positive y-axis, BEM versusanalytical results (case 6)

Figure 5(a). Cross-section view of a non-homogeneous clayey soil (case 7)

Figure 5(b). Cross-section view of a non-homogeneous sand/silt soil (case 8)

rate of reduction in surface concentration is higher than the rate of upward replenishment bydi!usion from the interface area. Consequently, concentration at the interface is temporarilygreater than elsewhere.

Figures 5(f ) and 5(g) compare analytical results to BEM predictions for the non-equilibriumsorption case. In Figure 5(f ), concentration pro"les after 3 months, 1 year and 5 years are shown,while in Figure 5(g), breakthrough curves at the interface between the two layers up to 100 yearsare reproduced. Figure 5(g) also shows the interface breakthrough curve obtained under assump-tions of instantaneous sorption. Both Figures 5(f ) and 5(g) demonstrate excellent agreementbetween analytical and BEM results.



Figure 5(c). BEM mesh for non-homogeneous soil: 2 zones, 44 elements (cases 7 and 8) (solid lines mark the interfacesurface)

Figure 5(d). Non-homogeneous clayey soil: c.

along depth at di!erent time stations up to 2000 years, BEM versusanalytical results (case 7)

Case 9 (Multi-layered soil polluted by an adjacent lake). A three-dimensional simulation ofa two-dimensional multi-layered soil is performed next. This problem features three zones and anorder of magnitude di!erence between longitudinal and lateral di!usion coe$cients. The soil ismade of a sand aquifer con"ned by two silt layers over an impermeable rock formation(Figure 6(a)). It is assumed to be semi-in"nite in the x direction, its total depth from surface torock base is 16 m and its width in the z direction is taken to be 2 m. The source of contaminationis an adjacent lake assumed to contain, over a short time scale of 10 years, a constantconcentration of pollutant at its interface with the soil. A soil length of 20 m was found tosimulate well semi-in"nite conditions over this time scale. An immobile fraction is included in thesandy layer while sorption in the silt layers is assumed to be governed by non-equilibriumequations. The soil is modelled with three zones, using 176 quadratic elements and 1066 degrees



Figure 5(e). Non-homogeneous sand/silt soil: c.

along depth at di!erent time stations, comparison of assumptions ofnon-equilibrium and instantaneous sorption, analytical solution (case 8)

Figure 5(f ). Non-homogeneous sand/silt soil: c.

along depth at di!erent times, BEM versus analytical results (case 8)

Figure 5(g). Non-homogeneous sand/silt soil: breakthrough curves at the interface between the two layers, BEM versusanalytical results (case 8)



Figure 6(a). Multi-layered soil polluted by a neighbouring lake: section view

Figure 6(b). Multi-layered soil polluted by a neighbouring lake. BEM mesh (the solid lines mark interface surfaces)

of freedom (Figure 6(b)). A conservative assumption of zero-#ux conditions is made at all surfacesexcept the interface with the lake. Simulation is run on supercomputer Fujitsu VPP 3000 at theAustralian National University.

Results in the form of concentration at 4 times stations are shown in Figure 6(c) and 6(d).Results from a two-dimensional boundary element solution of a similar problem have beenpresented by the authors earlier and the respective e!ects of non-linear sorption and immobilefraction analysed.39 Figure 6(c) shows pro"les along a central horizontal line running through theacquifer where two-dimensional and three-dimensional predictions are compared. There isclearly good agreement between the two solutions. Figure 6(d) depicts the advance of thecontamination front in a vertical pro"le at l m from the lake interface. Comparing the 6-monthand 10-y curves, it is evident that di!usion across layers is smoothing out the curves andeliminating the discrepancy between concentrations in the di!erent strata.



Figure 6(c). Multi-layered soil polluted by a neighbouring lake: concentration pro"le along a central horizontal line in theaquifer (y"2 m)

Figure 6(d). Multi-layered soil polluted by a neighbouring lake: concentration pro"le along a vertical line in the aquifer(x"1 m)

Multi-layered soil with anisotropic di+usion properties and surface load patch. To illustrate thescope of the method, a soil made of two sandy layers and penetrated by a concentration from twodi!erent sources (Figure 7(a)) is analysed with CONTAN 3D. The "rst source of contamination isa surface one in the form of a reservoir of 2]2 m2 surface area. The equivalent height H

fof the

reservoir is 5 m and the initial concentration of contaminant cr.0

is denoted by C0. The second

source of contamination is an adjacent river with a constant pollutant concentration of 0)03C0.

Each soil layer is 2 m in depth and the bottom layer is assumed to rest on an impermeablestratum. The origin of the axes is taken at the bottom of the soil in such a way that the soil/riverinterface lies at x"0, the lateral bounding surfaces of the soil lie at z"0 and 4 m, the top surfaceof the soil is at y"4 m and the impermeable bottom of the lower layer at y"0. The positivex-axis points away from the river and a negative groundwater #ow velocity therefore indicates



Figure 7(a). Multi-layered soil with two sources of pollutant and anisotropic di!usion: a 3D view

Figure 7(b). Multi-layered soil with two sources of pollutant and anisotropic di!usion: BEM mesh (dashed surfacerepresents reservoir location and solid lines mark the boundary of the interface between the two layers)

seepage from the soil into the river. The following parameters describe layer 1:

Dxx"2 m2/y, D

yy"D

zz"0)2 m2/y

<!x"!2 m/y, <

!y"<

!z"0

n."0)3

n*."0)1

o"1760 kg/m3

K$"0)000002 m3/kg

F"0)3

K5"1./y

w"1./y



Figure 7(c). Multi-layered soil with two sources of pollutant and anisotropic di!usion: longitudinal pro"le after 1 year, at1 m depth (y"3 m, z"2 m)

Parameters for layer 2 are:

Dxx"5 m2/y, D

yy"D

zz"0)5 m2/y

<!x"!3 m/y, <

!y"<

!z"0

n."0)25

n*."0)1

o"1720 kg/m3

K$"0)00001 m3/kg

F"0)3

K5"2./y

w"2./y

The BEM mesh used in the simulation consists of 112 quadratic elements and 790 degrees offreedom (Figure 7(b)).

Two sets of simulations are performed. The "rst type of simulations aim at validatingthree-dimensional BEM predictions by comparing them to BEM results from a two-dimensionalsimulation.39 The three-dimensional problem is "rst reduced to a two-dimensional one byremoving the lateral di!usion coe$cient D

zzand by using uniform contaminant sources in the

lateral z direction. Further simulations are performed in three-dimensional space in order toassess the e!ect of lateral di!usion and of the zero-#ux versus zero-concentration boundaryconditions. In this "rst instance, it is assumed that no sorption and no immobile fraction occur inthe soil and that the concentration of contaminant in the reservoir remains C

0throughout the

simulation. Results after 1 year are presented.Figures 7(c)}7(g) show results from three-dimensional solutions with and without lateral

di!usion, using zero-#ux boundary conditions at external surfaces where concentration is not



Figure 7(d). Multi-layered soil with two sources of pollutant and anisotropic di!usion: longitudinal pro"le after 1 year, at3 m depth (y"1 m, z"2 m)

Figure 7(e). Multi-layered soil with two sources of pollutant and anisotropic di!usion: vertical pro"le after 1 year(x"1 m, z"2 m)

Figure 7(f ). Multi-layered soil with two sources of pollutant and anisotropic di!usion: lateral pro"le after 1 year, at 1 mdepth (x"1 m, y"3 m)



Figure 7(g). Multi-layered soil with two sources of pollutant and anisotropic di!usion: lateral pro"le after 1 year, at 3 mdepth (x"1 m, y"1 m)

Figure 7(h). Multi-layered soil with two sources of pollutant, anisotropic di!usion, sorption, non-equilibrium processesand constant-mass boundary condition: longitudinal pro"les at 1 m depth (y"3 m, z"2 m)

known. Results based on zero-concentration boundary conditions with lateral di!usion are alsopresented (C

."0 at the surfaces z"0, z"4 m and y"4 m). A zero-concentration boundary

condition implies constant #ushing of pollutant by an external source, e.g. rainwater, whilezero-#ux leads to the accumulation of the contaminant. In the no-lateral-di!usion scenario,zero-#ux boundary conditions are used and the reservoir is assumed to extend over the wholewidth in the z direction. The problem therefore becomes two-dimensional, and two-dimensionalboundary element predictions are shown for comparison. Figures 7(c) and 7(d) are longitudinalpro"les along a centre-line at depths of 1 and 3 m, respectively (z"2 m). Figure 7(e) showsa vertical pro"le along a centre-line running at x"1 m and z"2 m. Figures 7(f ) and 7(g) containpro"les in the z direction at 1 and 3 m depths, respectively (x"1 m). Good agreement betweentwo-dimensional and three-dimensional predictions is obtained in the case of the no-lateral-di!usion scenario. While the e!ect of lateral di!usion is negligible over this time scale in thevertical and longitudinal directions, it is predictably far more pronounced in the lateral direction.



Figure 7(i). Multi-layered soil with two sources of pollutant, anisotropic di!usion, sorption, non-equilibrium processesand constant-mass boundary condition: longitudinal pro"les at 3 m depth (y"1 m, z"2m)

Figure 7( j). Multi-layered soil with two sources of pollutant, anisotropic di!usion, sorption, non-equilibrium processesand constant-mass boundary condition: longitudinal pro"les at 1 and 3 m depth (z"2 m)

The second type of simulations uses all parameters shown above including instantaneoussorption, non-equilibrium sorption, immobile fractions, constant concentration in the river,constant-mass condition in the reservoir, and zero-#ux boundary conditions elsewhere. Resultsare presented at di!erent time stations up to 20 years. Figures 7(h) and 7(i) show longitudinalpro"les along a centre-line (z"2 m) at di!erent time stations at depths of 1 and 3 m, respectively.The decreasing distance between the curves clearly re#ects the approach of a steady-state as thepenetration of pollutant from the river by di!usion is o!set by advective seepage. Figure 7( j)compares the longitudinal pro"les at the two di!erent depths. Figure 7(k) contains verticalpro"les at (x"1 m, z"2 m) at various time stations. In the lower layer, the contamination frontadvances due to the in#ux form the river and, consequently, concentration gradients between thelower and the upper layers of the soil give rise to strong vertical di!usion #uxes. The steady-stateconcentration in the lower layer is 0)0163C

0, at 54 per cent of the river pollution level. Over 87



Figure 7(k). Multi-layered soil with two sources of pollutant, anisotropic di!usion, sorption, non-equilibrium processesand constant-mass boundary condition: vertical pro"les at x"1 m, z"2 m

Figure 7(l). Multi-layered soil with two sources of pollutant, anisotropic di!usion, sorption, non-equilibrium processesand constant-mass boundary condition: breakthrough curves at di!erent points

and 98 per cent of this concentration has been reached after 1 and 5 years, respectively. Finally,breakthrough curves up to 10 years at di!erent points of the soil are shown in Figure 7(l).

This "nal simulation is a good demonstration of the method's capabilities: non-homogeneousmedia, three-dimensional transport, anisotropic di!usion properties, contamination source with"nite surface area, non-equilibrium sorption, immobile fraction in soil solution and "nite-massboundary condition. The opposing e!ects of groundwater seepage into the river and di!usion#uxes make qualitative predictions di$cult to obtain without numerical modelling. Togetherwith the validation cases, this example illustrates the ability of the algorithm to deal with complexphysicochemical conditions, di!erent geometries and multiple contamination sources.

COMPUTATIONAL PERFORMANCE

Figures 8(a)}8(c) show the computational time and memory requirements versus number ofboundary degrees of freedom for simulations performed on a DIGITAL ALPHA workstation



Figure 8(a). CPU time of simulation versus number of boundary degrees of freedom, for one time-station on a DIGITALALPHA workstation with a single 233 MHz processor operating as a shared resource

Figure 8(b). CPU time required for matrix-assembly and the solution of the algebraic system of equations as a percentageof total CPU time versus number of boundary degrees of freedom, on a DIGITAL ALPHA workstation with a single

233 MHz processor operating as a shared resource

with a single 233 MHz processor. The workstation is a shared resource at the Department of CivilEngineering of the University of Sydney and the "gures shown correspond to simulationsperformed under typical loading conditions rather than dedicated time. Times shown are CPUtimes for a full simulation yielding results at a single time station for problems made of 1, 2 or3 zones. Since the software requires little writing onto the hard disk and all arrays are stored inRandom Access Memory, CPU times were found to be almost identical to total elapsed times.The assessment of the computational performance of the proposed algorithm against othersimilar algorithms is beyond the scope of the paper. Such an assessment is complicated bya number of factors, not least of which are, the dependence of computational performance onparticular programming strategies independently of the algorithm itself (i.e. the degree ofmaturity of the software and the e!ort invested in its optimisation), and the importance of



Figure 8(c). Random access memory requirements versus number of boundary degree of freedom, on a DIGITALALPHA workstation with a single 233 MHz processor operating as a shared resource

including data preparation e!orts spent by the user in any assessment, if boundary elementsoftware is party to the comparison.

Figures 8(a) and 8(b), nevertheless, show that relatively large problems with around 1500boundary degrees of freedom can be analysed in a few hours of computing time with around 140Mbytes of memory using readily available computing power. Together, the matrix-assembly andsystem-solution components of the program account for 80}90 per cent of computing time(Figure 8(b)). Pseudo-singular integrals, arising from the proximity of the singularity point to theintegration element, require a large number of Gaussian integration points and account for thetime consumed by the matrix-assembly component of the program. No optimisation e!ort hasbeen applied to the software and a large scope for vector-optimisation is evident in Figure 8(b)where the percentage of total time taken by the solution of the algebraic system of equationsincreases with the number of boundary degrees of freedom at the expense of matrix-assembly. Inboundary elements, the system solution part of the algorithm lends itself to vector-optimizationfar more easily than the matrix-assembly part does. Most vector machines provide a readilyavailable library of optimised solution routines and large savings in computing time have beenachieved for boundary element software in the past.50 The di$culty of optimizing the matrix-assembly component stems from the variable number of Gauss-integration points required bypseudo-singular integrals and the special treatment reserved for singular integrals. Nevertheless,numerical schemes have been successfully used to improve the vector performance of theassembly part of a three-dimensional boundary element stress analysis program by more than50%.51 This is likely to be applicable to the present software. Memory requirements can also bereduced by avoiding the storage of blocks of zeros where the problem is made of more than onezone. Presently, global matrices for the entire problem are assembled directly resulting in thestorage of large numbers of o!-diagonal zeros. Further memory requirement reductions can beobtained by temporarily storing part of the program arrays in hard-disk space.

CONCLUSIONS

A three-dimensional boundary element solution of the di!usive}advective equations with non-equilibrium partitioning in saturated non-homogeneous media has been developed. The solution



is valid for any three-dimensional geometry of soil and contaminant source. It requires surfacemeshing only and no time stepping. A relatively small number of elements has yielded accuratepredictions when BEM results were compared to known solutions. A full-scale simulation ofa multi-layered soil polluted by a neighbouring river and a surface reservoir has shown that thealgorithm can be reliably used as an analysis tool.

REFERENCES

1. J. R. Matis, &Petroleum contamination of groundwater in Maryland', Ground=ater 9(6), 57}62 (1971).2. J. O. Osgood, &Hydrocarbon dispersion in groundwater: signi"cance and characteristics', Ground =ater, 12(6),

427}438 (1974).3. V. Pye and J. Kelly, &The extent of groundwater contamination in the United States', Groundwater Contamination,

National Academy Press, (1984).4. O. E. C. D. &Biotechnology for a clean environment. Prevention, detection, remediation', Organisation for Economic

Co-Operation and Development, 1994.5. R. Pickens, K. J. I. Jackson and W. F. Meritt, &Measurement of distribution coe$cients using a radial injection

dual-tracer test',=ater Resour. Res., 17(3), 529}544 (1981).6. M. N. Goltz and P. V. Roberts, &Interpreting organic solute transport data from a "eld experiment using physical

nonequilibrium models', J. Contaminant Hydrol., 1, 77}93 (1986a).7. R. C. Borden and M. D. Piwoni, &Hydrocarbon dissolution and transport: a comparison of equilibrium and kinetic

models', J. Contaminant Hydrol. 10, 309}323, 1992.8. R. S. Kookana, R. Naidu and K. G. Tiller, &Sorption non-equilibrium during Cadmium transport through soils', Aus.

J. Soil Res., 32, 635}651 (1994).9. L. A. Gaston and M. A. Locke, &Organic chemicals in the environment. Fluometuron sorption and transport in

Dundee soil', J. Environ. Qual., 24, 29}36 (1995).10. M. L. Brusseau, &Application of a multi-process nonequilibrium sorption model to solute transport in a strati"ed

porous medium',=ater Resour. Res., 27(4), 589}595 (1991).11. D. W. Smith, R. K. Rowe and J. R. Booker, &The analysis of pollutant migration through soil with linear hereditary

time-dependent sorption', Int. J. Numer. Anal. Methods Geomech., 17, 225}274 (1993).12. K. Hat"eld, J. Ziegler and D. R. Burris, &Transport in porous media containing residual hydrocarbon. II: Experi-

ments', J. Environ. Engng., 119(3), 559}575 (1993).13. M. L. Brusseau, R. E. Jessup and P. S. C. Rao, &Modeling the transport of solutes in#uenced by multiprocess

non-equilibrium',=ater Resour. Res., 25, 1971}1988 (1989).14. K. H. Coats and B. D. Smith, &Dead-end pore volume and dispersion in porous media', Soc. Pet. Engns. J., 4, 73}84

(1964).15. M. Th. van Genuchten, &A general approach for modeling solute transport in structured soils', Proc. Hydrogeology of

¸ocks with ¸ow Hydraulic Conductivity, Memoirs of the International Association of Hydrogeologists, Vol 17 (part 1),1985, pp. 513}526.

16. R. K. Rowe and J. R. Booker, &Two-dimensional migration in soils of "nite depth', Can. Geotech. J., 22, 429}436(1985b).

17. U. Y. Shamir and D.R. F. Harleman, &Numerical solutions for dispersion in porous mediums',=ater Resour. Res.,3(2), 557}581 (1967).

18. S. K. Gupta and K. K. Tanji, &A three-dimensional Galerkin "nite element solution of #ow through multiaquifersystem',=ater Resour. Res., 20(2), 553}563 (1976).

19. E. O. Frind and M. J. Verge, &Three-dimensional modeling of groundwater #ow systems',=ater Resour. Res., 14(5),844}856 (1978).

20. P. S. Huyakorn, B. G. Jones and P. F. Andersen, &Finite element algorithms for simulating three-dimensionalgroundwater #ow and solute transport in multilayer systems',=ater Resour. Res., 22(3), 361}374 (1986).

21. M. N. Goltz and P. V. Roberts, &Three-dimensional solutions for solute transport in in"nite medium with mobile andimmobile zones',=ater Resour. Res., 22, 1139}1148 (1986b).

22. M. S. Rahman and J. R. Booker, &Pollutant migration from deeply buried repositories', Int. J. Numer. Anal. MethodsGeomech., 13, 37}51 (1989).

23. F. J. Leij, N. Toride and M. Th. van Genchten, &Analytical solutions for non-equilibrium solute transport inthree-dimensional porous media', J. Hydrol. 151, 193}228 (1993).

24. S. Zou, J. Xia and A. D. Koussis, &Analytical solutions to non-Fickian surfaces dispersion in uniform groundwater#ow', J. Hydrol. 179(1}4), 237}258 (1996).

25. C. A. Brebbia, &¹he Boundary Element Method for Engineers1, Pentech Press, London, 1978, 189 p.



26. P. K. Banerjee and R. Butter"eld, 0Boundary Element Methods in Engineering Science1, McGraw-Hill, London, 1981,452 p.

27. F. Paris and J. Canas, &Boundary Element Method: Fundamentals and Applications1, Oxford University Press, Oxford,1997, 392 p.

28. J. A. Liggett and P. L.-F., &¹he Boundary Integral Equation Method for Porous Media Flow1, Georges Allen & UnwinLondon, 1983, 225 p.

29. A. J. Nowak, &Application of the multiple reciprocity BEM to nonlinear potential problems', Engng. Anal. BoundaryElements, 16(4), 323}332 (1995).

30. N. Kamiya, E. Andoh and K. Nogae, &New complex-valued formulation and eigenvalue analysis of the Helmholtzequation by boundary element method', Adv. Engng., Software, 26(3), 219}227 (1996).

31. T. A. Cruze and F. J. Rizzo, &A direct formulation and numerical solution in the general transient elasto-dynamicproblems I', J. Math. Analy. Appl., 22, 244}259 (1968).

32. D. W. Smith and J. R. Booker, &Boundary element analysis of transient thermoelasticity', Int. J. Numer. Anal. MethodsGeomech., 13, 283}302 (1988).

33. D. W. Smith, &Boundary integral methods for thermoelastic problems', Ph.D. ¹hesis, University of Sydney, 1990,292 p.

34. E. A. Sudicky, &The Laplace transform Galerkin technique: a time-continuous "nite element theory and application tomass transport in groundwater',=ater Resour. Res., 25(8), 1833}1846 (1989).

35. E. A. Sudicky, &The Laplace transform Galerkin technique for e$cient time-continuous solution of solute transport indouble porosity media', Geoderma 46, 209}232.

36. C. J. Leo and J. R. Booker, &A boundary element method for the analysis of contaminant transport in porous media I:homogeneous porous media', Int. J. Numer. Anal. Methods in Geomech., 23, 1681}1689 (1999).

37. C. J. Leo and J. R. Booker, &A boundary element method for the analysis of contaminant transport in porous media II:heterogeneous porous media', Int. J. Numer. Anal. Methods Geomech., 23, 1701}1715 (1999).

38. R. K. Rowe and J. R. Booker, &1-D pollutant migration in soils of "nite depth', J. Geotech. Engng., 111(4), 479}499(1985a).

39. A. Elzein and J. R. Booker, &Groundwater pollution by organic compounds: a two-dimensional analysis of con-taminant transport in strati"ed porous media with multiple sources of non-equilibrium partitioning', Int. J. Numer.Anal. Methods Geomech., 23, 1717}1732 (1999).

40. R. K. Rowe, R. M. Quigley and J. R. Booker, 0Clayey Barrier Systems for=aste Disposal Facilities1, F&FN SPON,1995, 390 p.

41. C. J. Leo, &Boundary element analysis of contaminant transport in porous media', Ph.D. ¹hesis, University of Sydney,1994, 353 p.

42. C. A. Brebbia and D.Nardini, &Dynamic analysis in solid mechanics by an alternative boundary element procedure',Int. J. Soil. Dyn. Earthquake Eng., 2(4), 228}233 (1983).

43. A. Elzein and S. Syngellakis, &Dual-reciprocity in boundary formulations of the plate buckling problem', Eng. Analysiswith Boundary Elements, vol. 9, No 2, Elsevier, Amserdam, (1992).

44. W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, 0Numerical Recipes in FOR¹RAN. ¹he Art ofScienti,c Computing1, 2nd eds, Combridge University Press, Cambridge, 1992.

45. A. Talbot, &The accurate numerical integration of Laplace transforms', J. Math. Appl., 23, 97}210 (1979).46. R. Piessens, &A bibliograpy on numerical inversion of Laplace transforms and its applications', J. Comput. Appl. Math.,

1, 115}128 (1975).47. B. Davies and B. Martin, &Numerical inversion of the Laplace transform: a survey and comparison of methods',

J. Comput. Phys., 33, 1}32 (1979).48. K. S. Crump, &Numerical inversion of Laplace transforms using a Fourier series approximation', J. Assoc. Comput.

Mach., 23(1), 89}96 (1976).49. A. Ogata and R. B. Banks, &A solution of the di!erential equation of longitudinal dispersion in porous media', ;.S.

Geol. Surey Prof. Paper, 411-A, 1961.50. R. A. Adey, &Computational aspects and applications of boundary elements on supercomputers', in Melli and Brebbia

(eds), Supercomputers in Engineering Structures, Springer, Berlin, 1989.51. A. Elzein, &Optimization of boundary element performance on super-computers', in Brebbia, Howard, Peters (eds),

Application of Supercomputers in Engineering II, (1991).



Documents

Groundwater pollution by organic compounds: a three-dimensional boundary element solution of contaminant transport equations in stratified porous media with multiple non-equilibrium