Bull Eng Geol Env (2000) 59 : 231

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231

Received: 5 July 1999 7 Accepted: 15 April 2000

M. Zairi (Y) 7 M. J. Rouis
Ecole Nationale d’Ingénieurs de Sfax, BP W, 3038 Sfax, Tunisia
e-mail: Moncef.Zairi6enis.rnu.tn
Tel.: c216-4-274088
Fax: c216-4-275595

Numerical and
experimental simulation
of pollutants migration
in porous media

M. Zairi 7 M. J. Rouis

Abstract Quantitative

prediction

of

pollutant

migration is now of great interest in the field of
waste

management

and

storage.

Numerical

modelling of contaminant transport through porous
media is an efficient tool to make this prediction
possible. Site selection and design of protective
liners for landfills and storage facilities are largely
related to the behaviour of contaminants in such
structures. This behaviour may be well defined by
numerical simulations. This paper proposes a two-
dimensional finite element model for pollutant
migration. The implementation of the advection-
dispersion equation in the numerical model and vali-
dation tests by comparison with analytical and semi-
analytical solutions are presented. Experimental
laboratory tests on cadmium and chloride migration
through liner samples are also discussed and
compared with the results from the numerical
model.

Résumé La prévision quantitative de la migration
des polluants présente un grand intérêt dans le
domaine de la gestion et du stockage des déchets. La
modélisation numérique du transport des polluants
à travers un milieu poreux est un outil efficace
rendant possible cette prévision. La sélection de sites
et la conception de barrières de protection pour des
installations de stockage dépendent fortement du

comportement des polluants dans de telles struc-
tures. Ce comportement peut être bien défini par des
simulations numériques. L’article présente un
modèle 2D en éléments finis pour la migration de
polluants. L’établissement des équations d’advec-
tion-dispersion dans le modèle numérique et les
tests de validation par comparaison avec des solu-
tions analytiques et semi-analytiques sont présentés.
Des expériences en laboratoire sur la migration
d’ions cadmium et chlorure à travers des barrières
de protection reconstituées sont également discutées
et les résultats comparés à ceux de la modélisation
numérique.

Keywords Environmental protection 7
Groundwater contamination 7 Laboratory studies 7
Models

Mots clés Protection de l’environnement 7
Pollution des eaux souterraines 7 Études en
laboratoire 7 Modèles numériques

Introduction

Numerical modelling of pollutant migration in porous
media has recently received a great deal of attention due to
an increased interest in the preservation of the quality of
the environment and particularly the protection of ground-
water from various pollutants. Mathematical modelling of
contaminant behaviour in soils is considered to be a
powerful tool for a wide range of problems concerned with
groundwater quality preservation or rehabilitation.
Enormous efforts have been made in recent years to
understand contaminant behaviour in porous media and
to represent it by mathematical models. Ogata (1970) was
the first to give a non-dimensional analytical solution to

M. Zairi 7 M. J. Rouis

232

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the mass transport equation in porous media. Others
models were then proposed, e.g. those of Guerghian et al.
(1980) and Rowe and Booker (1985). Although there are
now many numerical methods that can be used to solve the
mass transport equation, the finite element model is parti-
cularly appropriate for porous media with complex geom-
etry and flows. It also permits a good representation of
porous media heterogeneity.

Numerical model

Pollutant behaviour in the soil is subject to many pro-
cesses. The water–soil system is dynamic and many chem-
ical, biological and physical reactions occur which may
affect pollutant behaviour. Contaminant transport can be
reasonably approximated by one-dimensional solutions,
and the two-dimensional analysis will be adequate for the
vast majority of practical applications. In the case of a clay
layer acting as a liner below a landfill, for example, the
one-dimensional analysis may be adequate when the width
of the landfill is large compared to the thickness of the clay
liner and the direction of contaminant transport is
predominantly vertical. However, if the dimensions of the
landfill are comparable to the clay liner thickness, hori-
zontal and vertical transport may occur and the two-
dimensional analysis is more appropriate, especially when
concentration at points outside the landfill is of concern.
The governing equation of two-dimensional pollutant
migration in saturated porous media, also named the
advection-dispersion equation, is given by:

f C

f t

p

D*

x

f

2

C

f x

2

c

D*

y

f

2

C

f y

2

P

V*

x

f C

f x

P

V*

y

f C

f y

(1)

where C is the concentration of the considered contami-
nant at a point of coordinates (x, y) at time t; D*pD/R,
V*pV/R; R is the retardation factor, defined by Bear
(1972, in Auriault and Lewandowska 1993) which is given
by:

Rp1crK

d

/n

(1a)

where Vx and Vy are pore water flow velocities in X and Y
directions; Dx and Dy are hydrodynamic dispersion coeffi-
cients in X and Y directions; r is soil bulk density; K

d

is

distribution coefficient; and n is porosity.
In the case of conservative species which have no interac-
tion with porous medium particles, the retardation factor
is equal to 1. For species adsorbed by soil particles, the
value of this factor depends on the type of element studied
and the soil adsorption capacity expressed by the distribu-
tion or partitioning coefficient. It is appreciated that this is
a very simple description of the interaction between the
contaminant and the porous medium particles, but whilst
it is approximate, it has often been found to be an
adequate approximation.
K

d

, the distribution coefficient which reflects the degree of

retardation by reversible ion exchange, known as the equi-

librium sorption-desorption model of retardation, requires
particularly careful use in modelling. At high pH values,
removal of heavy metals from solutions is generally greater
than the exchange capacity of the soil, due to precipitation
as metal carbonate or hydroxide. To overcome this, an
apparent greater K

d

value is used to include precipitation

and fixation in modelling retardation of heavy metals.
In the finite element transformation of Eq. (1), the porous
media (or Eq. 1 domain) is discretized in a number (m) of
elements (e), which constitute the finite element mesh or
grid. The approximated solution for this equation may be
written for each element in the form:

C

e

(x, y)p

n

A

ip1

N

e

i

C

i

(2)

with

C

e

(x,y): approximate concentration of element (e)

N

i

e

: interpolation functions for each node of element (e)

C

i

: nodal concentration (unknown of the problem)

n: number of nodes of the element (e).

If C in Eq. (1) is substituted by the expression of Eq. (2), an
approximate solution for Eq. (1) is obtained.
The porous medium properties (e.g. hydrodynamic disper-
sion, flow velocities, porosity, distribution coefficient) are
considered constant in each element but may vary from
one element to another. The second derivatives of the
pollutant-migration-governing equation are generally
undefined for a variety of grid elements. They may be eval-
uated if written in the form of first derivatives, by using an
integration by parts of the diffusive terms of the equation
as given by Green’s theorem for integration in two dimen-
sions (Zienkiewicz 1977).
The integral on the boundary of the domain or contour
integral resulting from the integration by parts is intro-
duced in the form of a boundary condition. On the domain
boundary, with an imposed ingressing mass flux, the
global quantity QC

i

is substituted for the contour integral,

where Q is the water flux and C

i

its concentration with the

compound i. If the previous developments are written for
each element of the grid, the elementary equation may be
written in the matrix form as:

[K

e

]

3

C

1

C

n

4

c

[M

e

]

3

iC

1

it

iC

n

it

4

p

0

(3)

with

[K

e

]p[B]

T

[D][B]c[NN]

T

[V][B]

[M

e

]p[N]

T

[N]

where elements of matrices [B] and [B]

T

are composed of

the nodal values of interpolation function first derivatives.
The elements of matrices [N], [N]

T

and [NN]

T

are the

nodal values of interpolation functions and those of
matrices [D] and [V] are the elementary hydrodynamic
dispersion coefficients and flow velocities respectively. The

Pollutant migration in porous media

Bull Eng Geol Env (2000) 59 : 231

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233

elementary matrices [K

e

] and [M

e

] of Eq. (3) are assem-

bled for all domain elements to give the following global
system, in which m is the number of domain elements.

[K] Cc[M]

iC

it

p

0

(4)

with

[K]p

m

A

ip1

[K

e

]

[M]p

m

A

ip1

[M

e

]

The global system of Eq. (4) is a first-order differential
equation in time.

Definition of boundary conditions

Two types of boundary conditions may exist on the
domain limits. The first is a fixed concentration limit
which is a Dirchelet-type boundary condition. The second
is a fixed concentration flux limit and a Neumann-type
boundary condition. The first condition is introduced
when solving the global equations system (Eq. 4) by a
restructuring of the global matrix [K] to eliminate equa-
tions corresponding to an imposed concentration node
and replacing the corresponding concentration variables in
other equations by the specified values. The imposed flux
condition is represented by the quantity QCi which is
added to the second member of the elementary equation at
the node i. Each node affected by this condition is consid-
ered as a source term.

Solution implementation

The finite element spatial discretization of the governing
equation of pollutant migration in saturated homogenous
porous media has given a first-order differential equation
in time:

[K] Cc[M]

iC

it

p

F

for

t 1 t

0

(5)

C(t

o

)pC

o

In the previous equation, the right-hand side vector F
corresponds to the boundary conditions. The resolution of
Eq. (5) is in finding a C(t) function set that satisfies Eq. (5)
at each time t and the initial conditions for tpt

0

. Replacing

C and its time derivatives by their approximate finite
differences, writing the boundary conditions vector F for
times t and tcDt and using the concentration increment
D

C gives:

[M]caDt [K] DCpDta F

c

D

t

c

1Pa F

t

P

[K] C

t

(6)

with

D

CpC

tcDt

P

C

t

a

is the Euler constant. For a 1 0.5, Eq. (6) solution is

unconditionally stable. If 0~a~0.5, the solution is stable
only for a time step Dt~Dt

c

(critical time step) given by:

D

t

c

p

2

lP2a l

max

(7)

where l

max

is the greatest eigenvalue of the product of

matrices [M]

–1

and [K] (Dhatt and Tozot 1981). In order to

solve Eq. (6), the Newton-Raphson method (as described
by Dhatt and Tozot 1981) may solve linear or non-linear
non-stationary problems. This method was generally used
to solve non-linear problems. In the constructed algo-
rithm, for each iteration Eq. (6) is solved by the Gauss
elimination method.
The above developments were coded in a FORTRAN
program named MIGPO (Zairi 1996), using eight nodes
quadrilateral isoparametric elements for grid generation.
More details about these elements, interpolation functions
and coordinate transformations are given in Zienkiewicz
(1977).

Conditions for numerical solution stability

The discretization is the preliminary stage in transforming
a system of partial derivatives equations into an algebraic
one. In this process derivative approximation generates
truncation errors which cause the numerical dispersion of
the solution. Solution stability studies are generally carried
out by a Fourier series development of the amplitude and
the phase of the numerical solution; they may then be
compared to those of the analytical solution. Guerghian et
al. (1980) and Fletcher (1988) have demonstrated that the
stability of the advection-dispersion equation by a
Galerkin finite element method formulation is related to
two parameters: the Peclet number (Pe) and the Courant
number (Cr). These two parameters are expressed as:

PepVDL/D

(8)

CrpVDt/DL

(9)

with

V: flow velocity
D

L: characteristic length of the element (DX or DY)

D: hydrodynamic dispersion coefficient
D

t: time step.

The solution stability studies relative to these parameters
allow definition of the domain of applicability of a given
numerical scheme. These parameters also allow the deter-
mination of the optimum grid and time step for a stable
solution of the problem under consideration. In the litera-
ture, a value of less than or equal to 2 is considered as a
limit for the Peclet number; the Courant number generally
ranges between 0 and 1.6 for a Galerkin finite element
numerical scheme.
In order to determine the stability domain for the solution
scheme used in the MIGPO model construction relative to
the Peclet and Courant numbers, a 40-element, 147-node
grid is considered. Many simulations were executed for a
large range of these two factors in order to test the conver-
gence and the stability of the MIGPO solution scheme. The
simulation tests have shown that convergent and stable
solutions are observed for a Peclet number ranging from

M. Zairi 7 M. J. Rouis

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Fig. 1

Comparative concentration
breakthrough curves calcu-
lated from MIGPO and Ogata
solutions for time periods of
a 5 years and b 50 years

0.5 to 2 and a Courant number between 0.25 and 2 and for
a null value of these two numbers.
Non-convergent solutions are defined as those not
reaching the convergence criterion for high iteration
numbers. Numerical oscillation is expressed by negative or
very high concentrations relative to initial concentration
(contamination source). Practically, the stability domain is
reached by grid adjustment, choosing an element size (Dx
and Dy) that allows Peclet and Courant numbers in their
optimal ranges. It is also possible to vary the time step and
more accurate hydrodynamic dispersion coefficients in
order to obtain a stable solution.

Numerical model validation tests

The efficiency of a mathematical model depends on its
reliability. This is generally checked by comparisons with
experimental results or analytical solutions or with numer-
ical models whose efficiency is proved. For efficiency
testing of the “MIGPO”, three validations were undertaken:
(1) comparison with Ogata’s analytical solution (Ogata
1970); (2) comparison with Rowe’s semi-analytical model
(Rowe and Booker 1985); (3) experimental validation by
comparison with the results of laboratory tests.

Validation by comparison with Ogata’s analytical
solution

Ogata (1970) gave a one-dimensional analytical solution of
pollutant migration in porous media. In order to compare
the MIGPO results with those obtained by this analytical
method, a conservative solute with a concentration of
2 mg/l flowing through a 1 m thick clay layer having a
porosity of 0.43 was considered. The flow velocities were of
0.05 m/year in the flow direction and zero laterally. The
hydrodynamic dispersion coefficient was 1.25 and 0.46 m

2

/

year in the parallel and lateral flow directions respectively.
The grid used for this simulation incorporated four
elements with a time step of 5 years.
At xpo, a concentration-type boundary condition of the
form C(o,t)pCo is assumed for both Ogata and the finite
element model. For the lower boundary of the layer, at
xpL, the following condition is applied to the two solu-
tions:

f C

f x

L, tp0

(10)

The concentration evolution relative to the initial concen-
tration C/C

0

with depth is presented in Fig. 1 for time

periods of 5 and 50 years. Each of the curves plotted for the
MIGPO corresponds to the concentration evolution at grid
nodes which are lined in the flow direction. Figure 1 shows
that the three profiles obtained from the MIGPO results
have the same trend and evolution in time as that of the
analytical solution. Concentration values obtained by the
two methods are of the same magnitude, the greatest
difference between the two results being some 0.01 (for a
time of 5 years).

Validation by comparison with the Rowe semi-analytical
model

The model POLUTE, established by Rowe and Booker
(1985), was also used to verify the validity of the MIGPO
results. This method involves a transformation of all the
governing equations by introducing a Laplace transform;
the transformed equations are solved analytically and the
solutions are numerically inverted. The flow of a 2 mg/l
cadmium solute through a 1 m thick saturated layer having
a permeability of 10

–10

m/s and a porosity of 0.4 was

considered to compare the results of the two models. In
this theoretical example, only molecular diffusion was
considered as a transport mechanism. The dispersion coef-
ficient used was cadmium in an aqueous solution
(7.17!10

–8

m

2

/s; in Quigley et al. 1987) multiplied by a

tortuosity coefficient of 0.4. The cadmium adsorption by
soil particles was also modelled considering a retardation
factor of 3.
The evolution with depth of the ratio of the initial
cadmium concentration and that after 5 and 50 years is
presented in Fig. 2. The results from the two models differ
only slightly for the time period of 5 years (B0.02 mg/l)
with no difference discernible for the 50-year period.
Concentration profiles through the layer appear similar for
both models and show an increasing concentration with
time.

Pollutant migration in porous media

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235

Fig. 2a,b

Comparative concentration
breakthrough curves calcu-
lated from MIGPO and Rowe
solutions for time periods of
a 5 years and b 50 years

Table 1

Chemical composition of kiln dust and fly ash

Oxide (%)

CaO

MgO

Na

2

O

K

2

O

Fe

2

O

3

MnO

SiO

2

Al

2

O

3

SO

4

CO

3

Cl

Kiln dust

42.06

1.56

1.45

2.7

1.83

0.1

13.67

4.3

6.8

24.8

0.73

Fly ash

19.74

3.66

7.71

1.0

6.06

0.13

31.12

18.2

7.76

4.3

0.3

Table 3

Composition of liners 1 and 2

Liner

Kiln dust
(%)

Fly ash
(%)

Silica steam
(%)

Liner 1

95.24

0

4.76

Liner 2

76.19

19.05

4.76

Table 2

Mineral composition of kiln dust and fly ash

Mineral phase

Main phase

Secondary phase

Kiln dust

Calcite, lime, quartz,
anhydrite, arcanite, alite

Magnesite

Fly ash

Amorphous

Quartz, magnesite

Many simulation tests have demonstrated a good agree-
ment between MIGPO and POLUTE results both quantita-
tively and qualitatively. This is more evident for long time
periods and narrow grids.

Experimental study of pollutant migration through
liners

The experimental study of pollutant migration through
liner samples is part of a research programme on the use of
fly ash and stabilised kiln dust as major components for
the construction of liners. It is hoped that this may provide
a potential solution to the environmental problems caused
by this industrial refuse storage, particularly in five Tuni-
sian cities having cement works and a projected landfill in
their vicinity.
The aim of the experimental work was to outline the
mechanical and hydrogeological characteristics of the
liners being developed. Numerous experimental tests were
undertaken on a variety of liners, e.g. permeability,
leaching and longevity tests. In this paper the pollutant
migration tests for two liners are presented.
The mineralogical and chemical compositions of the kiln
dust and fly ash used are presented in Tables 1 and 2. As
can be seen, both the alkali (Na, K) and sulphate concen-
trations were high. The kiln dust is characterised by high
calcite and lime concentrations, while the fly ash was a C-
ASTM type with a moderate lime concentration compared
to that of kiln dust and particularly high concentrations of
alkalis, iron and alumina.
A number of liners were constructed using a variety of kiln
dust and fly ash mixtures. The compositions of liners 1 and
2 discussed in this work are presented in Table 3. Silica
steam was added to liners 1 and 2 to ensure a silica source

and to stabilise the mixtures (Rouis 1991). The prepared
mixtures were put into a PVC cylinder, 70 mm in diameter
and 20 mm in height. The mixture was slightly compacted
in three layers and the liners thus formed were matured for
1 week in a wet room prior to being left saturated in
distilled water for 3 weeks. Four weeks after preparation of
the mixture, the distilled water was replaced by a solution
with a known concentration of cadmium chloride (CdCl

2

,

2H

2

O). Samples of the floating solution were taken weekly

over the 67 days of the test. Each time a sample was taken,
it was replaced by an equivalent volume of water so that a
150-mm head was maintained above the liner. A sufficient
quantity of cadmium chloride was dissolved in this ad-
ditional water in order to maintain the chloride (Cl

–

)

M. Zairi 7 M. J. Rouis

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Fig. 3

Chloride concentration in liner interstitial water at end of diffu-
sion test

Fig. 4

Variation in cadmium concentration in solute over liner 1

Fig. 5

Cadmium concentration in liners 1 and 2 at end of diffusion
test

concentration at 1000 mg/l and the cadmium (Cd

2c

)

concentration at 1500 mg/l. At the end of the test, the 100-
mm liner sample was cut into five 20-mm-thick slices.
Each slice was dried for 24 h at 105 7C, then crushed and
the chloride concentrations determined on a solid sample
dissolved by acid.
During the 67 days of testing, no water flow took place in
the apparatus, which had a leachate collecting device. The
pH of the liner varied from 7 on the first day to 11.5 on the
seventh day and then stabilised at between 11.5 and 12.

Chloride migration

The chloride concentration of the interstitial water in the
liners was obtained by reducing the quantities given by the
chemical analyses to those of the water in the initial
sample prior to drying. This water content is considered to
equal effective porosity of the liner, measured by the
mercury porosimeter. In this way, chloride concentration
variation in the solute over the liner can be determined
together with the ion concentration profile in the liner.
The chloride concentration of the source solution
decreased with this ion migration into the liner, from
1000 mg/l at the beginning of each week (after the adjust-
ment to the initial concentration) to 750 mg/l at the end,
just before water sampling. A chloride concentration of
450 mg/l in the interstitial water at the liner base was
reached after 67 days of flow (Fig. 3). This is related to the
absence of any interaction between the chloride ion and
the liner constituents.

Cadmium migration

The variation in cadmium concentration of the source
solution with time for liner 1 is presented in Fig. 4. A sharp
decrease in cadmium concentration was recorded from the
first day of the experiment, from 1500 to 11.5 mg/l. It then
stabilised at about 0.05 mg/l, even though a cadmium-rich

solute was added after each weekly sampling. A similar
behaviour was recorded for all compositions of liner.
This decrease is related to the precipitation of cadmium
carbonate as a result of the increase in pH due to portlan-
dite dissolution from the liner into the solute. This is
confirmed by the cadmium concentration profiles through
liners 1 and 2 (Fig. 5) which show a sharp decrease in the
first 200 mm of sample. In their study of the influence of
pH on selectivity and retention of heavy metals in clay
soils, Yong and Phadungchewit (1993) found that the pres-
ence of carbonates in the soil contributes significantly to
their retention capability. At high pH values, precipitation
mechanisms, (for example as hydroxides and/or as carbon-
ates) dominate the cation exchange capacity.
It remains difficult to be confident of the real cadmium
concentration in the interstitial water in the liner. It is
likely that most of the analysed cadmium in the liners is
from the precipitates which cover their pore surfaces. This

Pollutant migration in porous media

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237

Fig. 6a,b

Comparison of liner chloride
concentration profiles from
MIGPO and experimental
tests for a 10 elements grid
and b 20 elements grid

leads to difficulties in calculating distribution and hydro-
dynamic dispersion coefficients.

Numerical simulation of experimental tests

The results of the pollutant migration tests on liners have
been used for the experimental validation of the MIGPO
model. For the numerical simulation of these tests the
following procedure was adopted:
1. The domain of the liner sample was divided into ten

elements.

2. Throughout the test, chloride concentration was main-

tained at 1000 mg/l at nodes on the limit between the
cadmium chloride solute and the liner.

3. Chloride

diffusion

coefficients

were

taken

as

DxpDyp0.02 m

2

/year.

4. The flow velocities Vx and Vy were taken as zero by

assuming that the transport mechanisms are limited to
molecular diffusion only.

5. Conservative values of chloride ions were considered –

the retardation factor equalling 1.

6. Simulations were carried out for ten time steps of

6.7 days each.

Simulation results

The simulated and measured interstitial water chloride
concentrations through the liner sample over a time period
of 67 days are shown in Fig. 6a. The curves obtained with
the MIGPO model represent the average concentration for
each 20-mm-thick slice of liner sample. As can be seen
from the figure, the simulated concentrations are close to
those obtained experimentally, with both profiles having a
similar appearance.
Experiments were also undertaken using the same simula-
tion conditions but with a finer grid of 20 elements and 79
nodes (Dxp1.75 cm and Dyp2 cm). The time interval in
this case was 2.68 days. Chloride concentration profiles for
the simulated liners are shown in Fig. 6b. As can be seen
from the figure, there is a closer correlation with the
experimental results than in the first tests, suggesting that
the grid refinement reduces the difference between the
MIGPO and experimental concentration profiles.

The inability of the model to reproduce the Cd data is
related to the fact that speciation of Cd with chloride is
important in evaluating its mobility. Results from theore-
tical and experimental studies (Yong and Sheremata 1991)
indicated that a better insight into the characteristics of
heavy metal adsorption modelling and prediction of
contaminant transport in soils can be obtained if the effect
of the other constituents in the solutions is taken into
account. Thermodynamic calculations may be used in
predicting the effect of the speciation and complexation by
inorganic and organic ligands. The model presented here
should be completed with a thermodynamic calculation
sequence to allow the prediction of such contaminant
transport.

Conclusions

This paper has outlined a two-dimensional finite element
solution for the advection-dispersion equation. The tech-
niques discussed have been implemented in the MIGPO
program. In order to optimise the use of this model, a
limited study of the numerical dispersion and oscillation
criteria was undertaken by an examination of a large range
of Peclet and Courant numbers.
Validation of the MIGPO results was carried out by
comparing the results with those obtained using two other
techniques: the Ogata analytical solution and Rowe semi-
analytical solution, together with experimental results. The
numerical simulation results were in good agreement with
those of analytical and semi-analytical solutions, while the
concentrations determined were of the same order of
magnitude and with a similar profile.
The experimental study of pollutant migration through
liners indicates that the behaviour of contaminants in such
structures depends on the type of species considered, the
composition of the liner and the physical and chemical
conditions and properties of the medium. The numerical
simulation test for chloride migration gave results in good
agreement with those measured during the laboratory

M. Zairi 7 M. J. Rouis

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tests, suggesting that this is an appropriate, conservative
model of contaminant migration. Laboratory measurement
of the distribution of cadmium in the liner material indi-
cated that more complex models are required to represent
the chemical and physical processes controlling its migra-
tion.

References

Auriault JL, Lewandowska J

(1993) Homogenisation analysis

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