1. Introduction
Escalating loads of municipal solid waste (MSW) end up in open dumps and landfills, producing continuous flows of landfill leachate. The risk of incorporating highly toxic landfill leachate into the environment is important to be evaluated and measured in order to facilitate decision making for landfill leachate management and treatment [
1]. In most countries, sanitary landfill is still the primary approach to the disposal of MSW [
2], which can generate pollution in soil, air, surface water, and groundwater because of the presence of leachate [
3]. The contamination of areas around solid urban waste dumps is a global challenge for the maintenance of environmental quality in large urban centers in developing countries [
4].
Derived from rainfall, surface drainage, and the decomposition of MSW, landfill leachate contains high concentrations of dissolved organic matter (DOM), inorganic salts, heavy metals, and xenobiotic organic compounds [
5]. If the landfill is not well lined with impermeable material or the lining fails due to geological conditions, landfill leachate will enter the groundwater as a contaminant front, which is one of the most significant environmental problems of landfills [
6].
One solution typically adopted to minimize and prevent groundwater contamination involves the construction of a liner composed of several materials [
7,
8]. The bottom cover layers (liners) in landfills are designed to make the leachate front in the groundwater compatible with potability standards. In all cases, the materials used must have certain technical characteristics, such as durability, adequate resistance, and low hydraulic conductivity. Hydraulic conductivity (k) describes the capacity of a porous medium to transmit a given liquid and is a function of both the medium and the liquid [
9].
One way to construct these liners involves low-permeability soil layers overlaid with a geomembrane (GM). The geomembrane in a single- or composite-liner barrier system inhibits the diffusive and advective migration of contaminants (in gas or liquid state) through the base liners and cover layer and to the surrounding environment [
10,
11,
12]. A geomembrane is a polymeric sheet material that is impervious to liquid as long as it maintains its integrity [
13].
Liners formed from other materials include soil liners (SLs), compacted clay liners (CCLs), and geosynthetic clay liners (GCLs). The choice of the most appropriate material depends on the characteristics of the waste, the operation of the landfill, and the cost of materials available in the region. A GCL is used in a landfill because of its relatively low cost compared with other materials, simple operational process, and low permeability [
14,
15]. CCLs have advantages such as good damping capacity and cheapness, as well as disadvantages such as high swelling and shrinkage [
16]. According to the United States Environmental Protection Agency (US EPA), the minimum thickness of CCLs should be 60 cm and the hydraulic conductivity of compacted liners should not be more than 1 × 10
−7 cm/s [
13].
A GCL is a relatively thin layer of processed clay (typically bentonite) either bonded to a geomembrane or fixed between two sheets of geotextile. A geotextile is a woven or nonwoven sheet material less impervious to liquid than a geomembrane, but more resistant to penetration damage [
17]. GCLs are 5 to 10 mm thick and their hydraulic conductivity typically ranges from 5 × 10
−12 to 5 × 10
−11 m/s [
18], and the reason for such low hydraulic conductivity is the bentonite [
19].
The liner materials should have an optimal cost/benefit ratio for each project. For this reason, detailed studies should be carried out for the retention of contaminants and thus for the prevention of contamination of the environment. Currently, analytical and numerical methods are essential for predicting the transport of leachate and contaminants in porous media. Analytical solutions, the basic tools for the initial design of liners, often involving simple assumptions [
20,
21], are preferred because they are continuous in space and time [
22].
In clay barriers, leachate contamination is mainly transported via a molecular diffusion mechanism rather than any advection mechanism because of the low hydraulic conductivity of the clays and the low leachate head (less than 30 cm) in sanitary landfills [
23]. However, the waste piles of landfills may generate a very high leachate head [
24]. Several solutions described in the literature disregard the advective term [
25,
26,
27,
28,
29], which is necessary in cases where the hydraulic head is significant. In such cases, the advective term should be considered in the transport process of the contaminant.
In addition to the advective term, the first-order term (decay, biodegradation) is also essential because of its effect on the mobility of organic contaminants [
30]. First-order kinetics is the most-used model because of its simple mathematical assumption in transport models [
31]. Similarly, considerations involving the generation or consumption of substances (zero-order term) have also interested researchers.
Analytical solutions are extensively used to simulate the transport of contaminants in a porous medium [
32,
33]. However, landfill cover layers commonly comprise multi-layered systems. Analytical solutions regarding multi-layered systems are described in the literature, but they include many simplifications [
34,
35]. For example, Carr [
22] proposed a semi-analytical solution to solving the advection–dispersion–reaction equation in a multi-layered system. Despite the solution’s novelty, the author did not consider the presence of thin layers (e.g., geomembrane), the partition coefficient, or mass flow calculation.
The semi-analytical model in this research may serve as a fundamental design tool that helps evaluate the potential contamination of groundwater and environmental damage caused to leachate in sanitary landfills. The model described in this paper is not only semi-analytical, but it also incorporates the complexity of multi-layered systems, recognizing the importance of thin layers as geomembranes, the influence of partition coefficients, and the criticality of mass flow calculations. As shown within the research, the model agrees well with other models in the literature, which includes those by Chen et al. [
36] and Rowe and Booker [
37].
In the simulations, a low-permeability material was overlaid with a geomembrane, because in the installation of the geomembrane, it is recommended to prepare the foundation area. The prepared foundation must be smooth and free of projections preventing damage to the liner; thus, even with the execution of the geomembrane, it is common to have a CCL or SL to prevent any damage. Moreover, the simulations considering the liner system with GM and molecular diffusion coefficient (DGM) greater than 10−13 m2 s−1 exhibited similar efficiency when compared with CCL (60 cm thick) for toluene contamination. Meanwhile, DGM with values less than 10−14 m2 s−1 proved to be an efficient liner system containing the toluene front. Hence, obtaining DGM with adequate precision is necessary to achieve the non-pollution of soil and groundwater. Considering a real case scenario in a sanitary landfill in Brazil, the simulations for the current liner system of the structure showed that benzene takes 50 years to reach the bottom of the liner. However, the life span of the landfill is just 13 years, showing the efficiency of the liner system and the applicability of the model.
Given the evolving complexities and challenges in environmental geotechnics, particularly concerning sanitary landfills, the need for more refined and versatile modeling techniques is urgent. Thus, the proposed model provides a more integrated understanding and prediction of any contaminant (dissolved organic matter, inorganic salts, heavy metals, and xenobiotic organic compounds) behavior as an essential tool for contemporary geotechnical engineers. This original perspective aims to elevate the standards of landfill liner design, making it more adaptable and effective in safeguarding our environment.
2. Mathematical Modeling
This study employed the advection–dispersion–reaction equation tailored for heterogeneous soil profiles commonly found in landfill settings. The model explicitly incorporates various barrier systems, including GMs, GCLs, CCLs, and SLs; it reflects the typical configurations observed in modern sanitary landfills.
The assumptions of the model proposed are (i) initial concentrations within layers may vary in each material; (ii) the top surface concentration is constant or an arbitrary function of time; (iii) interface concentrations and mass fluxes are continuous between layers; and (iv) the bottom boundary condition implies that the concentration derivative is zero (Neumann boundary condition).
The parameters considered in the model are as follows: the hydrodynamic dispersion coefficient
Di of each layer from
i = 1, 2, …,
m; the molecular diffusion of geomembrane (
DGM); the seepage velocity
v; the first-order decay constant
λ; the zero-order constant
γ; and the partition coefficients of the geomembrane
SL,GM and
S0,GM (
Figure 1).
The governing equations and boundary conditions used in this paper are as follows. The transport of contaminants in layer can be expressed using the advection–dispersion–reaction equation [
22]:
where
t is time (T);
z is the direction of the transport of the contaminant (L);
Dh,i is the hydrodynamic dispersion coefficient of layer
i (L
2 T
−1);
vi is the seepage velocity (LT
−1);
λi is the first-order decay constant (T
−1);
γi is the zero-order production/consumption constant (M L
−3T
−1); and
Ri is the retardation factor for linear sorption determined using Equation (2):
where
ρdi is the dry density of layer
I (ML
−3);
Kdi is the sorption distribution coefficient of layer
i (L
3M
−1); and
θi is the volumetric water content of layer
i (L
3L
−3). In geotechnical engineering, absorption and adsorption (referred to as sorption) can affect the concentration and mobility of solutes [
38].
The contaminant biodegradation may be expressed by the first-order decay constant [
7]. The decay constant
λi is described as a function of the half-life (
t1/2,i) of the contaminant [
36]:
where
t1/2 is the half-life of a contaminant of layer
i (T), defined as the time required for the quantity of the contaminant to decay to half of its initial value.
Seepage velocity in a multi-layer system considering the thickness of the contaminant above the
m layers is represented by the following Equation (3) [
23]:
where
hw is the contaminant head (L);
li is the thickness of each layer in the system (L); and
ke is the effective hydraulic conductivity of the system (LT
−1), which is equivalent to the harmonic mean of the hydraulic conductivities of the layers [
18]:
where
ki is the saturated hydraulic conductivity of layer
i (LT
−1).
The hydrodynamic dispersion coefficient can be expressed as the sum of the effective coefficient of molecular diffusion
Dm,I (L
2 T
−1) and the dispersion coefficient
DD,I (L
2 T
−1):
The solution to the multi-layer contaminant transport equation depends on the initial and boundary conditions. The initial condition can be expressed as follows [
18,
23]:
where
ci is the initial concentration of the contaminant in layer
i (M L
−3).
The top boundary condition of the imposed concentration can be approximated by a first-order rate equation [
3,
36,
37]:
where
c0 is the peak concentration of the contaminant (M L
−3) and
λ is the decay constant (T
−1) described in Equation (3). When the concentration of the contaminant in the overlying medium is not available, a constant concentration
c0 is often applied. Thus, Equation (8) becomes [
18,
23,
24]
In the transport of contaminants in multi-layers (
Figure 1), we need to account for the continuity between layers. Hence, the contaminant concentration and dispersive flux are considered continuous at the interfaces between adjacent layers [
22]:
When considering the seepage velocity, we obtain [
23]
If the first layer is composed of the GM (
Figure 1a), then the general transport equation, Equation (1), becomes [
18,
23,
24]
where
DGM is the molecular diffusion coefficient of GM (L
2 T
−1).
At the interface between the GM and contaminant, the inlet boundary condition is [
23]
where
S0,GM is the partition coefficient that indicates the ratio of the concentration of the contaminant to the concentration in GM.
The interface boundary condition between the GM and underlying SL is defined by the following equation [
21]:
where
SL,GM is the partition coefficient indicating the ratio of the concentration of the GM contaminant to that in the soil’s pore water at the interface of the two layers.
We now assume that the interface boundary conditions can be described as [
23]
The top boundary condition at the inlet (
z = 0) becomes
Equation (17) into Equations (15) and (16), we obtain
The GM is not at the top of the system but between two layers (
Figure 1b), then Equations (17), (19), and (20) are still valid.
The bottom boundary condition at the last layer is of the following type [
18,
23,
24]:
Based on the initial and boundary conditions and the continuity of the interfaces, the solution of the equation for a multi-layer system can be proposed. In
Appendix A, the solution is presented.
Based on the advection and dispersion mass balance, the mass flux
J of an arbitrary layer
i is [
37,
39] as follows:
4. Conclusions
Analytical models of contaminant transport are robust tools for parametric and numerical validation. Notably, both analytical and semi-analytical approaches maintain continuity across spatial and temporal domains. This research extends a semi-analytical model of the advection–dispersion–reaction equation to encapsulate these benefits.
Model validation, employing materials such as compacted clay liners (CCLs), SLs, GCLs, and geomembranes (GMs), effectively demonstrated the model’s capability to simulate multi-layer systems. The inherent heterogeneity of these materials indicates their critical importance when designing geostructures, especially landfill liners. The accurate comprehension of the transport of contaminants (like leachate) is pivotal for refining these geostructural designs.
The use of analytical and semi-analytical models eases parametric analysis greatly. Our research indicates the value of determining the geomembrane’s diffusion coefficient through lab analyses, facilitating an accurate prediction of contaminant transport and highlighting the trade-offs between opting for a geomembrane (GM) or augmenting the thickness of traditional soil liners (e.g., CCLs and SLs). Regardless, the geomembrane’s essential role as an advective flow barrier remains undebatable, necessitating rigorous checks to ensure its integrity. Moreover, incorporating first-order (decay/biodegradation) and zero-order (species source) terms is critical for a holistic prediction of contaminant transport dynamics.
The open-source model introduced here may be regarded as a versatile tool for simulating multi-layer systems to aid in optimizing landfill liner designs and thus contributing directly to minimizing soil contamination and mitigating groundwater pollution. Aiming for more comprehensive application, future research should investigate the applicability of the solution comparing with laboratory results analyzing multilayers of materials. Furthermore, for a holistic consideration of the contaminant transport phenomenon, the implementation of the material saturation (CCL and SL) and temperature in the model is recommended.