INDIAN GEOTECHNICAL CONFERENCE (DECEMBER 18

Finite Difference Method for Computation of Sodium and Chloride Migration in Porous Media
IGC 2009, Guntur, INDIA
FINITE DIFFERENCE METHOD FOR COMPUTATION OF SODIUM AND
CHLORIDE MIGRATION IN POROUS MEDIA
Ritwik Chakraborty
Lecturer, Dept. of Civil Engg., Ramakrishna Mission Shilpapitha, Belgharia, Kolkata–700 056, India.
E-mail: [email protected]
Ambarish Ghosh
Professor, Dept. of Civil Engg., Bengal Engrg. and Science University, Shibpur, Howrah–711 103, India.
E-mail: [email protected]
ABSTRACT: Finite difference method has been adopted herein to solve 1-D contaminant transport model to predict the
sodium and chloride migration through porous media in municipal waste landfill, considering both the short term and long
term effects. Providing an effective engineered barrier, which will separate the waste from ground water, can minimize the
potential contamination. Optimal design of waste disposal facilities requires an understanding of the fundamental mechanisms
and the material properties in the appropriate chemical and hydraulic environment, as well as the availability of mathematical
models. In the Finite difference technique, the velocity field is first determined within a hydrologic system, and these
velocities are then used to calculate the rate of contaminant migration by solving equation. This technique is well suited for
complex geometrics, complicated flow patterns heterogeneity and non-linearity. The computer program, CONTAMINATE
has been developed to solve the 1-D migration equation. The efficacy of the developed model has been studied comparing the
results available in literature.
1. INTRODUCTION
Geotechnical engineers are mostly involved in the analysis of
sodium and chloride migration from the leachates produced
in municipal waste landfill. This paper presents a technique
for the analysis of a single solute in a layer of finite thickness
of existing clay deposit or clay liner. This paper considers the
combined effects of advection, diffusion-dispersion, geochemical retardation and first order biological and chemical
decay under saturated condition in one model. Based on the
parametric values evaluated by earlier researchers have been
used for the execution of the programs and analysis of this
study. This paper neglects the three components of first order
decay due to radioactivity, biological decay and fluid
withdrawal for sodium migration; But 0.065 value is
assumed as first order decay constant due to biological and
chemical decay. The constant surface concentration Csur is
assumed for any time for the preparation of design charts.
The concentration profiles have been presented assuming the
value of height of leachate Hf as 1m simulating the field
condition such that the migration of the contaminant towards
the ground water beneath the clay deposit-liner from a finite
quantity of pollutant in the landfill. The next section presents
the governing equation of 1-D contaminant migration.
2. GOVERNING PARTIAL DIFFERENTIAL
EQUATION
The key factors governing contaminant migration are
advection, dispersion and geo-chemical and other chemical
reactions. The one-dimension form of the governing
equation applied to saturated homogeneous soil media
conditions (Rowe & Booker 1985), the ADRE (Advectiondispersion-reaction-equation) is:
R
c
 2c
c
 D 2 v
t
z
z
(1)
Again, the rate of reduction of concentration of radioactive
or biologically and chemically decaying solute species is
proportional to the available concentration of that solute,
considering first order decay such that
c
(2)
 c
t
Combining the effect of first order decay on ADRE, the
governing partial differential equation in this paper is
R
R
c
 2c
c
 D 2 v
 c
t
z
z
(3)
Where,
R
= Retardation Factor = 1 
K d
; [-].

ρ = Dry density of the subsurface medium; [ML -1].
Kd = Distribution co-efficient; [L3M-1].
c = solute concentration in the pore fluid of the liner;
[ML–3].
t
= time; [T].
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Finite Difference Method for Computation of Sodium and Chloride Migration in Porous Media
z
= Distance along the respective Cartesian coordinate
axis Z respectively; [L].
= dispersion coefficient; [L2T–1].
= discharge velocity calculated by Darcy’s law
= (Co-efficient of hydraulic conductivity k × hydraulic
gradient i); [LT –1].
= advective velocity = (discharge velocity, v) ×
(porosity,  ); [ LT –1].
D
v
va
= hydraulic gradient = (Difference of head, ∆h) /
(Length of medium, ∆l).
= the lumped first-order decay constant = sum of firstorder decay constant due to biological decay,
chemical decay, radioactivity, and fluid withdrawal
if any; [T–1].
i

Based on the above mentioned governing equation, finite
difference formulation has been made.
3. FINITE DIFFERENCE FORMULATION
Since the analytical solution technique is more complex,
simple Finite Difference Method (FDM) has been used to
solve this model. In this paper, a two dimensional domain
describing time t and spatial direction z as depth has been
considered with grid sizes k and h in the time and depth
directions respectively (Fig. 1).
where, j and i are the indices two dimensional concentration
array in the time and space directions respectively.
Therefore, finite difference form of 1- D contaminant
migration relation through soil is to be applied for computer
program as follows:
 1
c jIi     Qc j  1Ii  1  Sc j  1Ii   Tc j  1Ii  1 (5)
 P
Where, P 
R
;
k
S
2D
Q
D
h
2

v
;
2h
R
v
D
 ; T 
 2;
k
2h
h
(6–9)
Sum of fitting parameters = P  Q  S  T ;
(10)
h
2

A set of programs has been developed based on the above
mentioned finite difference formulation. The fitting
parameters in this study (equations 6–9) are used based on
the retardation factor, R, dispersion coefficient, D, discharge
velocity, v and the distance of the grid in the space and time
directions h and k respectively. The sum of the fitting
parameters P, Q, S and T (equation 10) showing zero
indicates consideration of non-decaying type of contaminant
solute in this study, otherwise decaying type of solute in this
study. This will also help to execute the programs. On the
basis of outputs of the above computer programs simulated
with the municipal waste landfill, the following results have
been drawn.
4. RESULTS AND DISCUSSION
Fig. 1: Grid Points in the Finite Difference Method for 1-D
Contaminant Migration Problem
Using the forward difference scheme for single order
derivative of concentration w.r.t. time and central difference
scheme both for single order derivative of concentration
w.r.t. space and second order derivative of concentration
w.r.t. space for better computation from the point of view of
accuracy and stability (Jain 1991), the general form of 1Dimensional model in this study can be replaced by finite
difference form as follows.
R
c[ j  1Ii ]  c[ jIi ]
c[ jIi  1]  2c[ jIi ]  c[ j  Ii  1]
D
k
h2
v
c[ jIi  1]  c[ jIi  1]
 c[ jIi ]
2h
(4)
The numerical model presented in this paper has been
simulated the sodium and chloride migration through the clay
liner from the municipal waste landfill. In this section, Figs.
2 to 4 present the concentration profiles of sodium, chloride
(neglecting decay) and chloride (considering decay) at
different depths of liner commencing from the bottom most
surface of the landfill. The parametric values have been taken
from Rowe et al. (2004). Whereas Figures 5 and 6 illustrate
the design charts of sodium and chloride migration through
compacted clay liner.
4.1 Concentration Profiles
Figure 2 presents concentration profile of non-decaying (first
order decay constant  or L as zero) reactive (R = 1.36) of
sodium. Figure 3 illustrates concentration profile of nonreactive (R = 1) solute chloride (neglecting decay, first order
decay constant  or L as zero), whereas Figure 4 shows
concentration profile of non- reactive (R = 1) solute chloride
(considering decay, first order decay constant  or L as
0.065.
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Finite Difference Method for Computation of Sodium and Chloride Migration in Porous Media
0.0
Rowe et al. (2004). From the above three graphs, it is noted
that as time passes the concentration in the liner increases;
chloride passes much faster than sodium species; since decay
indicates mass loss of a particular contaminant, height of
leachate (Hf) is essential for analysis of migration for
decaying solutes.
Depth below landfill surface (m)
0.5
0.0
Depth below landfill surface (m)
0.5
Concentration profile of sodium species
(non-decaying species)
Va= 0.01 m/a; D= 0.014 sq.m/a; RK=0.18;
Csur= 5100 mg/l; Cini=0; n=0.5; R=1.36;
1.0
Concentration profile of chloride species
(considering biological and chemical decay)
1.0
Va= 0.01 m/a; D= 0.018 sq.m/a; RK=0; Hf=1;
Csur= 2500 mg/l; Cini=0; n=0.5; R=1; L=0.065;
1.5
2.0
Time after peak source concentration (year)
6
12
2.5
1.5
Time after peak source concentration (year)
3.0
0
6
12
2.0
500
1000
1500
2000
Chloride concentration (mg/l)
2500
3000
Fig. 4: Concentration Profile of Chloride Migration at
Different Depths of Compacted Clay Liner for Various
Breakthrough Time (considering biological
and/or chemical decay)
2.5
3.0
4.2 Design Charts
0
1000
2000
3000
4000
Sodium concentration (mg/l)
5000
Fig. 2: Concentration Profile of Sodium Migration at
Different Depths of Compacted Clay Liner for Various
Breakthrough Time
0.0
Figures 5 and 6 present the design charts of liner thickness vs
breakthrough time for sodium and chloride migration
respectively for a given set of performance criteria. These
sets of curves are drawn based on fixed surface concentration
(Csur = 100 unit) and zero initial concentration (Cini =
0 unit) for the consideration of non-decaying solute nature
executing the set of programs CONTAMINATE.
1000
0.5
Design chart for Sodium migration
Va=0.01 m/a; n= 0.5; Csur= 100 unit; Cini= 0 unit;
D= 0.014 sq.m/a; RK=0.18; R=1.36;
Depth below landfill surface (m)
Concentration profile of chloride species
(neglecting decay)
1.0
Breakthrough time (year)
Va= 0.01 m/a; D= 0.018 sq.m/a; RK=0;
Csur= 2500 mg/l; Cini=0; n=0.5; R=1;
1.5
2.0
Time after peak source concentration (year)
6
12
100
C/Co (%)
90
65
50
40
20
5
10
2.5
3.0
0
500
1000
1500
2000
Chloride concentration (mg/l)
2500
3000
1
0.0
Fig. 3: Concentration Profile of Chloride Migration at
Different Depths of Compacted Clay Liner for Various
Breakthrough Time (neglecting decay)
0.5
1.0
1.5
2.0
Liner thickness (m)
2.5
Fig. 5: Design Charts for Compacted Clay Liner
for Sodium Migration
270
3.0
Finite Difference Method for Computation of Sodium and Chloride Migration in Porous Media
1000
5. CONCLUSIONS
Breakthrough time (year)
Design chart for chloride migration
Va=0.01 m/a; n= 0.5; Csur= 100 unit; Cini= 0 unit;
D= 0.018 sq.m/a; RK=0; R=1;
Based on the study in this paper as well as graphs of
concentration profiles and design charts of sodium and
chloride, the following may be concluded:
1. If the time increases the contaminant species passes
through a larger depth of the barrier.
2. Chloride species passes comparatively faster than that of
sodium species through the compacted clay liner and
this may be due to the higher molecular weight of
chloride ion than that of sodium ion.
3. Height of leachate consideration is essential for
analysing the migration of pollutant especially for
decaying contaminant species, since it includes the
implication of the mass available in the landfill leachate.
4. Service life of a liner may be increased upto a certain
limit by increasing the thickness of the liner.
100
10
1
0.0
C/Co (%)
90
65
50
40
20
5
0.5
1.0
1.5
2.0
Liner thickness (m)
2.5
3.0
Fig. 6: Design Charts for Compacted Clay Liner for Chloride
Migration
The curves have the similar trend and pattern with the design
charts of earthen barriers given by Acar and Haider (1990).
These set of programs CONTAMINATE may be used for
analyzing pollutant migration through different porous media
under varying hydrogeologic conditions of the landfill site
for the species of the contaminants under fully saturated as
well as partially saturated even multiple layered barrier
considerations.
REFERENCES
Acar, Y.B. and Haider, L. (1990). “Transport of Lowconcentration Contaminants in Saturated Earthen Barriers,”
J. Geotech. Engrg. ASCE, 116(7), 1031–1052.
Jain, M.K. (1991). “Numerical Solution of Differential
Equations”, Willey Eastern Limited, New Delhi.
Rowe, R.K. and Booker, J.R. (1985). “1-D Pollutant
Migration in Soils of Finite Depth,” J. Geotech. Engrg, ,
ASCE, 111(4), 479–499.
Rowe, R.K., Quigley, R.M., Brachman, R.W.I. and Booker,
J.R. (2004). “Barrier Systems for Waste Disposal”, Spon
Press, Taylor & Francis Group, London and New York.
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