INTRODUCTION
Leachate from Sanitary landfills is recognized as important groundwater
pollutants. The contaminants are released from the refuse to the passing
water by physical, chemical and microbial processes and percolate through
the unsaturated environment, polluting the groundwater with organic and
inorganic matter. If allowed to migrate freely from the landfill, leachates
may pose a serve pollution threat to groundwater (Demetracopoulos, 1986).
The modeling of leachate generation hinges on an understanding of the
mechanisms of mass release from the solid to the liquid phase and contaminant
decay. These mechanisms are influenced by such factors as climatic conditions,
type of waste, site geohydrologic conditions and chemical reactions as
well as microbial decomposition of organic matter (Demetracopoulos, 1986).
McCreanor and Reinhart (2000) developed a mathematical for the leachate
of landfill in the United States. The contaminant plume transport model
was presented by Aroa et al. (2007). Khire and Mukherjee (2007)
have done an extensive research on leachate injection using vertical wells
in bioreactor landfills.
The mathematical simulation of landfill leachate generation and transport
is addressed in the present study. The mechanisms controlling generation
and transport are incorporated in the appropriate governing equations
and a finite element solution is presented. Accuracy of Model has been
shown by comparison of results by experimental measurements.
EXPERIMENTAL WORK
Many empirical investigations have been made into the process of landfill
leaching and stabilization. The goals of leachate generation studies have
been to determine the volume and rate of leachate production, identify
the types and concentration histories of pollutants present in leachate
and to observe relationships among climate, age, refuse placement, leachate
production, gas production, temperature, settlement and overall landfill
stabilization. Several studies have utilized laboratoryscale columns
filled with compacted municipal refuse (Qasim and Burchinal, 1970; Fungaroli,
1971; Rovers and Farquhar, 1973; Walsh and Kinman, 1979; Wigh and Brunner,
1979; Scarpino and Donelly, 1979; Report To Congress, 1977). Others have
investigated the behavior of pilot or fieldscale landfills ranging from
0.94.7 m^{2} (Hentrich et al., 1979; Wigh, 1979; Leckie
et al., 1979). The behavior of experimental landfills operated
with leachate recalculation has been reported (Birbeck et al.,
1980; Leckie et al., 1979; Pohland et al., 1979; Pohland,
1975). A summery of reported empirical investigation has been prepared
(Straub, 1980a).
The methodologies incorporated in these studies have
been similar. An input/output approach is typical. Inputs to the experimental landfill, namely municipal solid waste, additives
and moisture were controlled or measured or both. Liquid and gaseous outputs
have been determined and analyzed. Refuse composition at various stages
of landfill stabilization has been determined (Chain and Dewalle, 1976;
Pohland, 1975). Little data have been gathered from the interior of experimental
landfills, except far temperature and transport in the landfill interior
have not been well documented.
Several similarities in the leachate behavior of the landfills may be
generalized. In virtually all the experimental landfills, significant
rates of leachate production were delayed from the initial application
of moisture, although small, intermittent volumes of leachate may be produced
prior to continuous leaching. Moisture balance on the landfill cells have
indicated that after the commence of significant leachate production,
the overall moisture content of the landfill remains essentially constant
and leachate is collected at roughly a onetoone proportion with net
moisture application. These observation have led to the concept of refuse
field capacity, defined fort porous medium as the average volumetric moisture
content above which continuous gravity drainage of water from the medium
will occur. Field capacities computed from input/output moisture data
from various studies indicate different values among several experimental
landfills, generally ranging between 0.3 and 0.4 cm/cm (Straub, 1980b).
High concentration of organic and inorganic contaminants are typically
associated with leachate. Peak concentrations of cod and total solids
above 50000 mg L^{1} are common. However, wide ranges of concentrations
of various contaminants have been observed for different landfills at
various ages. Chain and Dewalle investigated the characteristic of leachate
from 13 field and laboratoryscale landfills. While wide differences in
leachate composition were noted, a meaningful qualitative comparison was
made on the basis of landfill age, utilizing the ratios COD/TOC, BOD/COD,
VS/FS and percentage of free volatile fatty acid carbon to TOC. They suggest
a general decrease both in organic and inorganic leachate strength with
age and characterize young landfills as having high strength leachate,
while dilution and microbial utilization of organic reduce leachate strength
from older landfills. The pattern of high contaminant concentration near
the onset of leaching, followed by a general decrease, is typical among
other investigations (Walsh and Kinman, l979; Phelps, 1988; Raveh, 1979;
Scarpino and Donelly, 1979), although specific time rates of decrease
vary considerably.
Although basic methodological similarities exist among experimental landfill
studies, factors such as physical dimension of the landfills refuse composition,
refuse placement and density, additives to refuse, initial moisture content,
temperature and temperature control have varied greatly among the studies.
The rate and pattern of moisture application also have varied. Natural
moisture ranging from constant application to seasonal variation and average
rates ranging from less than 0.1 to over 1.0 cm day^{1}. The
time span of reported results ranges from several weeks to several year
and as the specific objectives of the studies differ, the measured variables
and the form in which results are reported have varied (Straub and Lynch,
1982).
LEACHATE MODELING
Fenn developed a method to estimate the time of first appearance of leachate
and subsequent rates of leachate production. This procedure applies a
water balance to estimate net moisture input to refuse from a soil cover
and utilizes field capacity assumption to calculate refuse moisture retention
and leachate production.
A semiempirical equation was developed by Wigh to describe the concentration
history of various contaminants in leachate generated from an experimental
landfill. The equation is based on two consecutive first order reactions
and expresses contaminant concentration as a function of cumulative leachate
volume, maximum concentration and two rate constants. Parameters of the
equation were evaluated to obtain good visual fit concentration histories
of several contaminants. The model captures the general decrease from
high initial concentration, which is typical of observed leachate behavior,
but its depth, etc. Similarly, Raveb observed declining concentrations
of various pollutants in leachate from experimental landfill column and
described the concentration histories of various pollutants with an exponential
function of time. Empirically fit parameters were evaluated for one set
of experiments.
Qasim and Burchinal operated three experimental landfill columns of varying
heights and applied column operation theory to describe the leaching of
chloride. The concentration histories of 14 other contaminants were related
to the chloride estimates, with reasonable agreement. The procedure relies
on empirically derived parameters, but is responsive to depth of refuse
and rate of moisture flow through the fill.
Phelps developed a model of sanitary landfill leaching utilizing mass
transfer equations based on flow through a moisture film the refuse particles.
The model is applied for assumptions of constant moisture infiltration
rate and constant moisture content above a wetting front. Model predictions
are compared with observed results from several experimental landfill
columns which were subjected to various moisture application rates and
contained refuse at different depths. Although the model requires the
empirical estimation of several parameters and makes limiting assumptions
about moisture flow, it is based on descriptions of fundamental mass transfer
and contaminant transport processes (Straub and Lynch, 1982).
Straub and Lynch (1982) and Demetracopoulos (1986) performed a finite
deference model for landfill leachate generation by using of convectiondispersion
equation in unsaturated medium of a landfill.
MATHEMATICAL MODEL
Water flow model: For solving convection dispersion equation,
water content and Darcy velocity is needed. These parameters are achieved
by solving moisture transport Equation. In this study water content and
Darcy velocity are obtain by using of LEACHW (Huston and Wagenet, 1989)
model. Equations that use in this model are as follows:
Continuity equation:
Darcy law in unsaturated soil:
Moisture transport equation:
Parameters are defined as:
θ 
: 
Water content [L^{3}/L^{3}] 
q 
: 
Darcy velocity [L/T] 
z 
: 
Elevation, computed from surface to bottom [L] 
K(θ) 
: 
Hydraulic conductivity [L/T] 

: 
:Suction head [L] 
t 
: 
Time [T] 
c 
: 
Specific moisture capacity equal to dθ/d
θ 
and Kθ. Equations are as follow: 
If 
Parameters are defined as:

: 
Suction head near saturation [L] 
K_{sat} 
: 
Saturation hydraulic Conductivity [L/T] 
b 
: 
An experimental constant 
θ_{s} 
: 
Water content [L^{3}/L^{3}] 
GOVERNING EQUATION
Contaminant transport through the landfill is controlled by the bulk
motion of the fluid and mechanical dispersion. Mixing due to molecular
diffusion is negligible compare to that caused by dispersion. Because
of discussing contaminant is inorganic total solids, decay not exists.
The mass transport equation is derived by applying continuity on an infinitesimal
control volume (Strub and Lynch, 1982):
Where:
C 
: 
Fluid phase concentration [M/l^{3}] 
D_{k} 
: 
Hydrodynamic longitudinal dispersion coefficient [L^{2}/T] 
λ 
: 
Dispersivity [L] 
K’ 
: 
A rate coefficient [1/T] 
S 
: 
Local mass per bulk volume of refuse available for transfer at time
t [M/T^{3}] 
S_{0} 
: 
Local mass per bulk volume of refuse available for transfer at time
t = 0 [M/T^{3}] 
C_{st} 
: 
Maximum contaminant concentration in the liquid phase [M/T^{3}]

FINITE ELEMENT SOLUTION
In this finite element solution two nodes one dimensional elements are
used (Fig. 1). The Shape functions are as follows:
The weak form of contaminant transport is:
where, Z_{A} and Z_{B} are coordinates of start and end
node of a typical element and w is trial function.
Approximation of solution is assumed as:
If w = N_{i} (ξ) then
Or

Fig. 1: 
One dimensional master element and shape functions 
In matrix form:
If master element coordinate is used then:
Where:
^ Symbol indicates average volume of variable in an element. With a CrankNicelson
time approximation scheme:
Where:
Δt 
= 
Time step 
K 
= 
Previous time 
K+1 
= 
Current time 
K+1/2 
= 
Interpolation between K and K+1 
After assembling boundary condition has been applied. The upper boundary
condition used here in is known concentration history.
The lower boundary is considered here in as:
Initial condition for this problem is:
Because of nonlinearity of equation, a Picard method is used to solve
system of equations.
MODEL VERIFICATION
For verifying model, two experimental measurements are used. The first
is an experimental landfill that history of TS concentration at the bottom
of landfill has been measured reported by Qasim and Burchinal (1970).
Characteristic of experiment are as follow:
Total time of simulation 
: 
163 days 
Length of landfill 
: 
195 cm 
Averaged pure precipitation 
: 
0.544 cm day^{1} 
The following parameters are assumed:
Cst 
: 
55000 mg L^{1} 
S_{0} 
: 
37000 mg L^{1} 
λ 
: 
5 cm 
b 
: 
7 

: 
100 cm 
K_{sat} 
: 
0.544 cm day^{1} 
A comparison between experimental measurement and model simulation is shown
in Fig 2.
The model is run for two value of initial concentration
but finally both of TS Cures are converges together and experimental result.
As one can see the difference between the experimental and simulated data shows
that initially is high, but as the time increase, the difference becomes minor.
The data from Qasim and Burchinal has used for the verification of the model
because of the simulation between their work and the present work which is Shiraz
Landfill.
The second is a long time experimental landfill. The leachate total solids
concentration from the Center Hill, Test Cell#4, as reported by Walsh
and Kimnan (1979), was simulated with the model. Characteristics of experiment
are as follows:
Total time of simulation 
: 
1000 days 
Length of landfill 
: 
240 cm 
Averaged pure precipitation 
: 
0.223 cm day^{1} 
The following parameters are assumed:
Cst 
: 
55000 mg L^{1} 
S_{0} 
: 
30000 mg L^{1} 
λ 
: 
5 cm 
b 
: 
7 

: 
100 cm 
K_{sat} 
: 
0.223 cm day^{1} 

Fig. 2: 
History of computed and measured bottom TS concentration 

Fig. 3: 
History of computed and measured bottom TS concentration 
A comparison between experimental measurement and model simulation is shown
in (Fig. 3). The model has been run for averaged rate
of daily precipitation, therefore no local picks bas been seen in the curve.
The peaks in the curve are due to the seasonal changes for irrigation application
rate.
Model has been run for three value of K’ that K’ have the best result.
A comparison between Demetracopoulos (1986) example and model has been
prepared. Characteristics of example are as follows:
Total time of simulation 
: 
1410 days 
Length of landfill 
: 
610 cm 
Averaged pure precipitation 
: 
0.213 cm day^{1} 

Fig. 4: 
History of Demetracopoulos (1986) and this model simulation bottom
TS concentration 
The following parameters are assumed:
Cst 
: 
45000 mg L^{1} 
S_{0} 
: 
45000 mg L^{1} 
λ 
: 
2.3 cm 
b 
: 
7 

: 
35 cm 
K_{sat} 
: 
0.0018 cm day^{1} 
Demetracopoulos (1986) results and model simulation is shown in Fig. 4. There are good agreements between Demetracopoulos simulation results
and present model ones. The main reason for the agreements between simulation
results by the present model with Demetracopoulos is that the chloride
is a conservative material with no sources and sinks in the model.
CONCLUSION
A mathematical model describing generation and transport contaminants
through a solid waste landfill was formulated and solved numerically by
the finite element method.
The result of model has been compared with two experimental works.
The
preceding results demonstrate the feasibility of analyzing compacted solid
waste as an unsaturated porous medium. The advantage of this analytical
approach lies in its process orientation leachate quantity and quality
results from the interaction of fundamental transport phenomena within
specific physical and environmental setting. At present, leachate model
is appropriate primarily for research purposes, insofar as they provide
organized analytical structures for the design and interpretation of experiments.
Simulation obtain with preliminary, aggregate representations of major
hydraulic and contaminant transport processes are overall agreement with
leachate data documented in separate studies. Additional experimental
work is required in order to identify and refine the description of basic
processes occurring in the landfill environment. Result of short time
experimental landfill works can’t be a good approximation of contaminant
transport terms and long term study is needed. Value of initial concentration
is an important parameter only in primary times and after a long time
period effect of initial concentration has been omitted.
K’ is a very important term in contaminant transport equation in a landfill
and a good value of K’ is obtain by several long times experimental works
on landfill samples. Ultimately, porous mediumbased landfill models could
be applied to fieldscale problems, although it must be recognized that
such applications would require extensive empirical support.