Analytical Solution for Contaminant Transport through a Soil–Bentonite (SB)/Geosynthetic Clay Liner (GCL)/Soil–Bentonite (SB) Composite Cutoff Wall and an Aquifer

Abstract

This study develops a one-dimensional analytical solution for contaminant advection, diffusion and adsorption through a soil–bentonite (SB)/geosynthetic clay liner (GCL)/SB–aquifer composite cutoff wall (CCW) system. The solution agrees well with an existing double-layer model. Adopting toluene as a representative contaminant, using the present solution, the analysis systematically investigates the impact of hydraulic gradient (i) and the hydraulic conductivities of GCL (k_gcl) and SB (k_sb). The results show the following: (1) Increasing i from 0.1 to 1 reduces the concentration breakthrough time (t_cb) from 20 to 11 years and mass flux breakthrough time (t_fb) from infinite to 11 years, indicating lower i significantly extend both t_cb and t_fb, which is crucial for optimizing CCW barrier performance; (2) lowering k_gcl from 5.0 × 10⁻¹¹ m/s to 1 × 10⁻¹² m/s and reducing k_sb from 1.0 × 10⁻⁹ m/s to 1.0 × 10⁻¹¹ m/s, would increase the t_cb by 36% and 100%, respectively. It demonstrates that reducing k_gcl and k_sb could enhance barrier performance. (3) To achieve equivalent barrier performance, soil–bentonite cutoff wall (SBCW) requires greater thickness compared to SB/GCL/SB CCW, indicating that GCL reduces the required amount of bentonite; and (4) CCWs can use SB with lower adsorption capacity to achieve equivalent performance, further reducing bentonite requirements. The present solution can aid in the design and optimization of GCL-enhanced CCWs.

1. Introduction

The contaminants present in landfill leachate and industrial wastewater pose significant risks to human health and productivity (McGrath, 2000 [1]; Shackelford, 2014 [2]; Han et al., 2016 [3]; Kjeldsen et al., 2002 [4]; Islam and Rowe, 2008 [5]; Koda and Osinski, 2017 [6]). To cope with environmental contamination, vertical cutoff walls (CWs) characterized by highly adsorptive and low permeability were developed (Opdyke and Evans, 2005 [7]; Du et al., 2015 [8]; Katsumi et al., 2018 [9]). The high adsorption capacity ensures that pollutants are captured effectively, while their low permeability prevents any further spread of contaminants. For the quantitative design of the barrier performance of CWs, it is important to investigate the transport of typical contaminants in CWs.

Soil–bentonite cutoff walls (SBCWs) are a type of cutoff wall with a single vertical layer, i.e., soil–bentonite (SB) layer. Extensive studies have been conducted on the performance of SBCWs through experiments and theory analysis (Fattah and Al-Lami, 2016 [10]; Krol and Rowe, 2004 [11]; Li et al., 2017 [12]). Early proposed analytical models (Britton et al., 2004 [13]; Devlin and Parker, 1996 [14]; Neville and Andrews, 2006 [15]) assumed that the transport of contaminants in SBCWs is a steady-state process (i.e., a process that does not change over time). Obviously, it is insufficient to evaluate the barrier performance for the transient transport of contaminants within cutoff walls. Therefore, several one-dimensional transient transport models were developed to design CWs with an appropriate service life (Bharat, 2014 [16]; Chen et al., 2019 [17]; Li et al., 2017 [12]; Xie et al., 2020 [18]). It should be noted that although the one-dimensional model may not describe the actual migration process of special and complex conditions, it is possible to view various complex working conditions as one-dimensional transport problems using a limiting approach. This allows for the design of the barrier performance of CWs, whose strategy is feasible and practically effective for design purposes. However, these models fail to consider the influence of the aquifer layer. Consequently, a series of analytical models of CW accompanied by an aquifer was developed, revealing the significant impact of the aquifer on the barrier performance of SBCWs (Ding et al., 2022 [19]; Liu et al., 2023 [20]; Yan et al., 2021 [21]; Zhan et al., 2023 [22]).

For enhanced performance and reduced cost, inserting a vertical layer of geosynthetic material into a vertically oriented soil–bentonite (SB) layer forms composite cutoff walls (CCWs), which have been gaining increasing attention in engineering and theoretical fields recently (Zhan et al., 2014 [23]). CCWs mainly include GMB-enhanced CCWs and GCL-enhanced CCWs (GMB: geomembrane, GCL: Geosynthetic Clay Liner). With respect to GMB-enhanced CCWs (Qian et al., 2019 [24]; Thomas and Koerner, 1996 [25]), Zhan et al. (2014) [23] provided an analytical solution for the one-dimensional diffusion of organic pollutants in SB/GMB/SB CCWs. Considering the spatial distribution of the leachate contaminant and the existence of an aquifer adjacent to the outlet of a CW, Peng et al. (2020) [26] promoted the model from one dimensional to two dimensional. Due to the impenetrability of GMB for solute fluid, the advection process has not been considered in these models. Nevertheless, advection process should not be ignored in the model for contaminants migration in GCL-enhanced CCWs, which has been confirmed one of the dominant transport mechanism of solute in porous medium (Feng et al., 2019a [27], 2019b [28]; Liu et al., 2023 [20]; M.-Q. Peng et al., 2021 [29]; Peng et al., 2023 [30]; Xie et al., 2020 [18]; Zhan et al., 2023 [22]). However, analytical models for contaminant transport in GCL-enhanced CCWs are still relatively scarce compared to GMB-enhanced CCWs, even though they have been widely used in practices (Liu et al., 2023 [20]; Rowe and Jefferis, 2022 [31]; Zhan et al., 2023 [22]; Shan and Lai, 2002 [32]; Touze-Foltz and Barroso, 2006 [33]; Rowe and Jefferis, 2022 [31]; Thomas and Koerner, 1996 [25]). Only Liu et al. (2023) [20] provided an analytical solution for contaminant transport in SB/GCL CCW and an aquifer system. Unfortunately, the position of the GCL for the model can be only considered adjacent to the outlet vertical surface of the SB layer. In practice, it actually could be located anywhere within or adjacent to the SB layer (forming SB/GCL/SB CCWs). Nevertheless, to the best of the authors’ knowledge, an analytical model for contaminants diffusion and advection transport through SB/GCL/SB CCWs and an aquifer layer has not been available so far.

The primary aim of this study is to develop a one-dimensional analytical model for contaminant transport through SB/GCL/SB CCWs and an aquifer system that incorporates diffusion, advection and adsorption mechanisms. This study focuses on the influence of the placement of GCL on the barrier performance of the GCL-enhanced CCW. Additionally, the equivalence assessment on the barrier performance between GCL-enhanced CCW and SBCW is demonstrated.

2. Mathematical Model Development

2.1. Basic Assumptions

Figure 1 shows a schematic representation of contaminant transport through the SB/GCL/SB–aquifer CCW system, consisting of four layers, namely the first SB layer, the GCL, the second SB layer and the aquifer. Hydraulic advection, molecular diffusion and adsorption processes are considered in the four layers. The entrance boundary of the system serves as the coordinate origin for the one-dimensional Cartesian coordinate system, with the x-axis extending to the right. T_fsb [L], T_gcl [L], T_ssb [L] and T_a [L] stand for the thickness of the first SB layer, the GCL, the second SB layer and the aquifer, respectively. P_gcl [L] represents the relative position between the GCL and SB layers, specifically referring to the distance between the leftmost face of the first SB layer and the GCL. The mathematical model is facilitated by the following assumptions: (1) the contamination concentration at the source is thought to be constant and to stay at C₀, and the water head difference between the two sides of the CCW also is assumed to be constant, which are conservative for barrier performance design (Xie et al., 2020 [18]); (2) SB layers, GCL and aquifer were assumed to be uniform and fully saturated layers (Feng et al., 2019b [28]); (3) the adsorption of contaminants in the layers are thought to be a linear equilibrium process (Peng et al., 2021 [29]); (4) the diffusion process of contaminants within each layer is assumed to be molecular diffusion, satisfying Fick’s law; and (5) the advection process of contaminants within each layer is assumed to be Darcian advection, satisfying Darcy’s law.

2.2. Governing Equations, Initial, Continuity and Boundary Conditions

A one-dimensional advection–diffusion–adsorption equation was used to represent the transport of pollutants through the SB/GCL/SB–aquifer system based on the previously stated assumptions. The governing equation for the contaminant one-dimensional transient advection–diffusion–adsorption transport through the first SB layer is

$$R_{\mathrm{d},\mathrm{sb}}\frac{\partial C_{\mathrm{fsb}}\left(x, t\right)}{\partial t}=D_{\mathrm{sb}}\frac{{\partial }^2{C}_{\mathrm{fsb}}\left(x, t\right)}{\partial x^2}-v_{\mathrm{sb}}\frac{C_{\mathrm{fsb}}\left(x, t\right)}{\partial x}$$

(1)

The governing equation describes the transient transport of contaminants through the GCL as

$$R_{\mathrm{d},\mathrm{gcl}}\frac{\partial C_{\mathrm{gcl}}\left(x, t\right)}{\partial t}=D_{\mathrm{gcl}}\frac{{\partial }^2{C}_{\mathrm{gcl}}\left(x, t\right)}{\partial x^2}-v_{\mathrm{gcl}}\frac{C_{\mathrm{gcl}}\left(x, t\right)}{\partial x}$$

(2)

The governing equation describes the transient transport of contaminants through the second SB layer as

$$R_{\mathrm{d},\mathrm{sb}}\frac{\partial C_{\mathrm{ssb}}\left(x, t\right)}{\partial t}=D_{\mathrm{sb}}\frac{{\partial }^2{C}_{\mathrm{ssb}}\left(x, t\right)}{\partial x^2}-v_{\mathrm{sb}}\frac{C_{\mathrm{ssb}}\left(x, t\right)}{\partial x}$$

(3)

The governing equation describes the transient transport of pollutants through the aquifer as

$$R_{\mathrm{d},\mathrm{a}}\frac{\partial C_{\mathrm{a}}\left(x, t\right)}{\partial t}=D_{\mathrm{a}}\frac{{\partial }^2{C}_{\mathrm{a}}\left(x, t\right)}{\partial x^2}-v_{\mathrm{a}}\frac{C_{\mathrm{a}}\left(x, t\right)}{\partial x}$$

(4)

where C_fsb(x, t) [ML⁻³], C_gcl(x, t) [ML⁻³], C_ssb(x, t) [ML⁻³] and C_a(x, t) [ML⁻³], respectively, represent the contaminant mass concentration in the first SB layer, GCL, the second SB layer and the aquifer at the time t [T] and position x [L]; D_sb [L²T⁻¹], D_gcl [L²T⁻¹], D_a [L²T⁻¹] reflect the contaminants’ effective diffusion coefficient in the SBs, GCL and the aquifer, respectively; v_sb [LT⁻¹], v_gcl [LT⁻¹], v_a [LT⁻¹] represent the seepage velocity of the pore fluid in SB layers, GCL and aquifer, respectively; and R_d,sb [dimensionless], R_d,gcl [dimensionless], R_d,a [dimensionless] represent the retardation factor of contaminants in SB layers, GCL and aquifer, respectively.

The effective diffusion coefficient of contaminants in the SB layers as well as the aquifer can be expressed by

$$D_{\mathrm{sb}}={\tau }_{\mathrm{sb}}{D}_0, D_{\mathrm{a}}={\tau }_{\mathrm{a}}{D}_0$$

(5)

where τ_sb [dimensionless] and τ_a [dimensionless] indicate the tortuosity factors of the SB layers and aquifer, respectively; D₀ [L²T⁻¹] denotes the molecular diffusion coefficient of contamination in the free fluid.

The mass balance of porous fluid at each layer interface requires satisfaction of the Darcy velocity, where v_d = n_sbv_sb = n_gclv_gcl = n_av_a, and where n_sb [dimensionless], n_gcl [dimensionless], n_a [dimensionless] indicate the porosities of SB layers, GCL and aquifer, respectively.

The Darcy velocity, v_d [L T⁻¹], can be computed by multiplying the hydraulic gradient of the aquifer, i, by the average hydraulic conductivity of the SB/GCL/SB–aquifer system, k_s [L T⁻¹], i.e., v_d = k_si, where i = h/T_ccw (see Figure 1), and k_s is defined by (Feng et al., 2019b [28])

$$k_{\mathrm{s}}=\frac{T_{\mathrm{ccw}}+T_{\mathrm{a}}}{T_{\mathrm{fsb}}/k_{\mathrm{sb}}+T_{\mathrm{gcl}}/k_{\mathrm{gcl}}+T_{\mathrm{ssb}}/k_{\mathrm{sb}}+T_{\mathrm{a}}/k_{\mathrm{a}}}$$

(6)

where h [L] represents the difference of water head between the entrance and the outlet of the CCW, and T_ccw [L] represents the thickness of the CCW; and k_sb, k_gcl and k_a [L T⁻¹] represent the hydraulic conductivity of SB layers, GCL and aquifer, respectively.

Based on the linear equilibrium adsorption assumption in the CCW system, the retardation factor can be described by

$$R_{\mathrm{d},\mathrm{sb}}=1+\frac{{\rho }_{\mathrm{sb}}{K}_{\mathrm{d},\mathrm{sb}}}{n_{\mathrm{sb}}},R_{\mathrm{d},\mathrm{gcl}}=1+\frac{{\rho }_{\mathrm{gcl}}{K}_{\mathrm{d},\mathrm{gcl}}}{n_{\mathrm{gcl}}},R_{\mathrm{d},\mathrm{a}}=1+\frac{{\rho }_{\mathrm{a}}{K}_{\mathrm{d},\mathrm{a}}}{n_{\mathrm{a}}}$$

(7)

where ρ_sb, ρ_gcl and ρ_a [M L⁻³] are the dry densities of the SB layers, GCL and the aquifer, respectively; correspondingly, K_d,sb, K_d,gcl and K_d,a [M⁻¹ L³] are the distribution coefficients for them.

Assuming no contamination in the SB/GCL/SB–aquifer system at the initial time, t = 0:

$$C_{\mathrm{fsb}}\left(x,0\right)=C_{\mathrm{gcl}}\left(x,0\right)=C_{\mathrm{ssb}}\left(x,0\right)=C_{\mathrm{a}}\left(x,0\right)=0$$

(8)

A constant concentration boundary exists at the inlet boundary at the interface of the contaminated zone and the first SBCW.

$$C_{\mathrm{fsb}}\left(0,t\right)=C_0$$

(9)

Assuming the outlet boundary is a zero concentration boundary, meaning that the concentration at the interface of the aquifer and free water fluid is zero, namely

$$C_{\mathrm{a}}\left(T_{\mathrm{ccw}}+T_{\mathrm{a}},t\right)=0$$

(10)

Each layer’s interface mass flux and contamination concentration must meet the following requirements in order for continuity to exist.

For the interface between the first SB layer and the GCL

$$C_{\mathrm{fsb}}\left(L_{\mathrm{gcl}},t\right)=C_{\mathrm{gcl}}\left(L_{\mathrm{gcl}},t\right)$$

(11)

$$n_{\mathrm{sb}}{v}_{\mathrm{sb}}{C}_{\mathrm{fsb}}\left(L_{\mathrm{gcl}},t\right)-n_{\mathrm{sb}}{D}_{\mathrm{sb}}\frac{\partial C_{\mathrm{ssb}}\left(L_{\mathrm{gcl}},t\right)}{\partial x}=n_{\mathrm{gcl}}{v}_{\mathrm{gcl}}{C}_{\mathrm{gcl}}\left(L_{\mathrm{gcl}},t\right)-n_{\mathrm{gcl}}{D}_{\mathrm{gcl}}\frac{\partial C_{\mathrm{gcl}}\left(L_{\mathrm{gcl}},t\right)}{\partial x}$$

(12)

For the interface between the GCL and the second SB layer

$$C_{\mathrm{gcl}}\left(L_{\mathrm{gcl}}+T_{\mathrm{gcl}},t\right)=C_{\mathrm{ssb}}\left(L_{\mathrm{gcl}}+T_{\mathrm{gcl}},t\right)$$

(13)

$$\begin{array}{l}{n}_{\mathrm{gcl}}{v}_{\mathrm{gcl}}{C}_{\mathrm{gcl}}\left(L_{\mathrm{gcl}}+T_{\mathrm{gcl}},t\right)-n_{\mathrm{gcl}}{D}_{\mathrm{gcl}}\frac{\partial C_{\mathrm{gcl}}\left(L_{\mathrm{gcl}}+T_{\mathrm{gcl}},t\right)}{\partial x}=\\ n_{\mathrm{sb}}{v}_{\mathrm{sb}}{C}_{\mathrm{ssb}}\left(L_{\mathrm{gcl}}+T_{\mathrm{gcl}},t\right)-n_{\mathrm{sb}}{D}_{\mathrm{sb}}\frac{\partial C_{\mathrm{ssb}}\left(L_{\mathrm{gcl}}+T_{\mathrm{gcl}},t\right)}{\partial x}\end{array}$$

(14)

For the interface between the second SB layer and the aquifer,

$$C_{\mathrm{gcl}}\left(T_{\mathrm{ccw}},t\right)=C_{\mathrm{a}}\left(T_{\mathrm{ccw}},t\right)$$

(15)

$$n_{\mathrm{sb}}{v}_{\mathrm{sb}}{C}_{\mathrm{ssb}}\left(T_{\mathrm{ccw}},t\right)-n_{\mathrm{sb}}{D}_{\mathrm{sb}}\frac{\partial C_{\mathrm{ssb}}\left(T_{\mathrm{ccw}},t\right)}{\partial x}=n_{\mathrm{a}}{v}_{\mathrm{a}}{C}_{\mathrm{a}}\left(T_{\mathrm{ccw}},t\right)-n_{\mathrm{a}}{D}_{\mathrm{a}}\frac{\partial C_{\mathrm{a}}\left(T_{\mathrm{ccw}},t\right)}{\partial x}$$

(16)

2.3. Analytical Solution Derivation

The one-dimensional transient transport normalized governing equation for contaminants in each layer of the SB/GCL/SB–aquifer system can be expressed as follows in order to make calculations easier:

$$R_{\mathrm{d},i}\frac{\partial C_i\left(x, t\right)}{\partial t}=D_i\frac{{\partial }^2{C}_i\left(x, t\right)}{\partial x^2}-v_i\frac{\partial C_i\left(x, t\right)}{\partial x}, \left(i=1,2,3,4\right)$$

(17)

where the first SB layer, GCL, second SB layer and aquifer are represented, respectively, by the values i = 1, 2, 3 and 4. C_i [ML⁻³] is the contaminant concentration in layer i at time t [T]; D_i [L² T⁻¹] is the diffusion coefficient of the contaminant in layer i; v_i [L T⁻¹] is the seepage velocity in layer i; R_d,i [dimensionless] is the retardation factor of the contaminant in layer i.

Following normalization, the following updates are made to the initial and boundary conditions:

$$C_i(x, 0)=0,(i=1,2,3,4)$$

(18)

$$C_1\left(0, t\right)=C_0$$

(19)

$$C_4\left(L_4,t\right)=0$$

(20)

It is also possible to normalize the mass flux continuity condition and the concentration continuity condition between the layers.

The normalized concentration continuity is shown as

$$C_i(L_i,t)=C_{i+1}(L_i,t),\hspace{1em}(i=1,2,3)$$

(21)

The normalized mass flux continuity is

$$n_i{D}_i\frac{\partial C_i(L_i,t)}{\partial x}=n_{i+1}{D}_{i+1}\frac{\partial C_{\mathrm{i}+1}(L_i,t)}{\partial x},\hspace{1em}(i=1,2,3)$$

(22)

where L_i [L] is the distance between the SB/GCL/SB–aquifer system entrance interface and the layer i end interface.

The problem is split into two sub-functions in this study in order to derive the general solution for C_i (x, t). η_i (x) and δ_i (x, t), based on the superposition method. Equations (17)–(22) can be combined to decompose into the two sub-functions.

$$C_i\left(x,t\right)=C_0{\eta }_i\left(x\right)+{\delta }_i\left(x,t\right), (i=1,2,3,4)$$

(23)

The solution for the steady-state problem is represented by sub-function 1, η_i (x), and the solution for the transient-state problem is represented by sub-function 2, δ_i (x, t).

The general solution for the steady-state problem, η_i (x) is

$${\eta }_i\left(x\right)=j_{i,1}{e}^{r_{i,1}\cdot x}+j_{i,2}{e}^{r_{i,2}\cdot x}$$

(24)

The steady-state problem’s general solution has the eigenvalues j_i,1 = 0, and j_i,2 = v_i/D_i. The coefficients r_i,1 and r_i,2 can be obtained t using the method of separated variables and matrix transfer, as described in the solution procedure presented in Supplementary S1.

The general solution for the transient-state problem, δ_i (x, t) can be defined as

$${\delta }_i(x, t)=\sum _{\infty }^{m=1}{Q}_m{g}_{m,i}(x)e^{{\alpha }_{i}x-{\beta }_{m}t},\hspace{1em}(i=1,2,3,4)$$

(25)

The method of separated variables and matrix transfer can also yield the coefficients Q_m and α_i, eigenvalue β_m, and eigenfunction g_m,i (x). The solution procedure for obtaining these values is described in detail in Supplementary S2.

The general solution for C_i (x, t) can be obtained by integrating and substituting the solutions for the steady-state and transient-state problems into Equation (23), as follows:

$$C_i\left(x,t\right)=j_{i,1}{e}^{r_{i,1}\cdot x}+j_{i,2}{e}^{r_{i,2}\cdot x}+\sum _{\infty }^{m=1}{Q}_m{g}_{m,i}(x)e^{{\alpha }_{i}x-{\beta }_{m}t},\hspace{1em}(i=1,2,3,4)$$

(26)

The detailed description and derivation of the solution for the steady-state and transient problems are included in Supplementary S1 and Supplementary S2, respectively.

3. Model Validation

To confirm the accuracy of the derivation method, a one-dimensional model of contamination transport in a bilayer system of SB and aquifer reported by Xie et al. (2020) [18] is chosen. The contaminant’s constant concentration of 1 mg/L is what defines the inlet boundary. The relevant parameters for validation are shown in the diagram in Figure 2. The initial concentrations for the model were set to zero for both the SBCW and the aquifer at the initial time. The validation results, shown in Figure 2, show that the contaminant concentration distribution at t = 100 years, as reported by Xie et al. (2020) [18], closely matches the one-dimensional transient analytical solution proposed in this study.

4. Influence of Advection on the Barrier Performance of SB/GCL/SB CCW

In GMB-enhanced CCW models, the advection process is typically neglected due to the impermeability of GMB to solute fluid. However, the process should not be ignored in GCL-enhanced CCWs. Using the established analytical model, the influence of advection (i.e., the aquifer’s hydraulic gradient, the hydraulic conductivity of SB and GCL) on the barrier performance of the SB/GCL/SB CCWs is comprehensively examined.

Toluene (TOL) is chosen as the typical contaminant. The parameters used in these analyses are summarized as follows (Chen et al., 2018 [34]; Xie et al., 2011 [35]; Feng et al., 2019b [28]; Hong and Shackelford, 2017 [36];^. Liu et al., 2023 [20]; Xiao et al., 2023 [37]; Xu et al., 2016 [38]; Touze-Foltz et al., 2016 [39]; C.-H. Peng et al., 2021 [40]; Rowe et al., 2005 [41]): the thickness of CCW, GCL and aquifer layer (T_ccw, T_gcl and T_a) is 0.6138, 0.0138 and 5 m, respectively; the porosity of SB, GCL and aquifer (n_sb, n_gcl and n_a) is 0.4, 0.86 and 0.47, respectively; the dry density of them (ρ_sb, ρ_gcl and ρ_a) are 1.474, 0.790 and 1.431 g/cm³, respectively; they (k_sb, k_gcl and k_a) are 1 × 10⁻¹⁰, 1 × 10⁻¹¹ and 1 × 10⁻⁷ m/s, respectively, for their hydraulic conductivity; and for distribution coefficient, they (K_d,sb, K_d,gcl and K_d,a) are 0.54, 14.5 mL/g and 0; the effective diffusion coefficient in GCL and the molecular diffusion coefficient in free water (D_gcl and D₀) are 2 × 10⁻¹⁰ and 8.5 × 10⁻¹⁰ m²/s, respectively; the tortuosity factor of SB and aquifer (τ_sb and τ_a) are 0.4 and 0.5, respectively; the mass concentration of TOL at the source and the maximum allowable mass concentration in drinking water (C₀ and C_ma) is 10 and 1 mg/L, respectively; and the GCL is inserted in the midpoint of the two SB layers and the hydraulic gradient of CCW (i) is 0.5. The parameters will be adopted from the values provided above unless otherwise specified.

Two key performance indicators, i.e., the mass concentration breakthrough time (t_cb) and the mass flux breakthrough time (t_fb) are adopted in the following analysis. t_cb is defined as the time when the relative concentration at the exit interface (C_ssb/C₀) reaches the maximum allowable relative breakthrough concentration (C_ma/C₀). When C_ma/C₀ reaches 0.1, the CCW is considered broken through. t_fb refers to the duration required for the mass flux (F_ssb) at the exit surface of the second SB to reach the maximum allowable mass flux (F_ma), conveniently set at 40 mg/(m²·year) for the purpose of analysis. As for the mass flux at the exit surface of the second SB (F_ssb), it could be calculated by

$$F_{\mathrm{ssb}}\left(t\right)=n_3{v}_3{C}_3\left(T_{\mathrm{w}},t\right)-n_3{D}_3\frac{\partial C_3\left(T_{\mathrm{w}},t\right)}{\partial x}$$

(27)

4.1. Influence of the Hydraulic Gradient of the Aquifer

The influences of hydraulic gradient (i) of the GCL-enhanced CCW on the barrier performance of the CCW were studied. Three hydraulic gradients (i = 0.1, 0.5 and 1.0) were selected for the study. Figure 3a,b present the breakthrough curves of the SB/GCL/SB CCW under the three hydraulic gradients, using the concentration-based and flux-based breakthrough criteria, respectively. By adopting the respective breakthrough criteria, the corresponding breakthrough times can be obtained from the corresponding breakthrough curves.

Under the concentration-based breakthrough criterion (see Figure 3a), as the hydraulic gradient increased from 0.1 to 0.5 and further to 1.0, the mass concentration breakthrough time (t_cb) decreased from 20 years to 15 years and 11 years, respectively. The reduction in t_cb was 5 years and 9 years, corresponding to 25% and 45% decrease ratio, respectively. Under the flux-based breakthrough criterion (see Figure 3b), as the hydraulic gradient increased from 0.1 to 0.5 and further to 1.0, the mass flux breakthrough time (t_fb) decreased from infinite to 18 years and 11 years, respectively. The decrease magnitude of t_fb was infinite.

The results indicate that (1) maintaining a lower hydraulic gradient can greatly extend the contaminant breakthrough time for the SB/GCL/SB CCW system. Controlling the ratio of the hydraulic head difference across the barrier and the barrier thickness, i.e., the hydraulic gradient, is crucial for optimizing the contaminant isolation performance of the GCL-enhanced CCW; and (2) using the flux-based breakthrough criterion may inherently lead to the generation of anomalously high or even infinite breakthrough time estimates. Comparatively, the results based on the concentration-based breakthrough criterion are more stable and reliable.

4.2. Influence of the Hydraulic Conductivity of GCL

The influences of hydraulic conductivity of GCL (k_gcl) on the performance of the GCL-enhanced CCW were studied. Four values of the hydraulic conductivity for the GCL (k_gcl = 5.0, 1.0, 0.5 and 0.1 × 10⁻¹¹ m/s) were selected. The breakthrough curves of the SB/GCL/SB CCW with the four levels of hydraulic conductivity for the GCL are illustrated in Figure 4a and Figure 4b, respectively, using the concentration-based and flux-based breakthrough criteria.

Using concentration breakthrough criteria, when k_gcl = 5 × 10⁻¹¹ m/s, the mass concentration breakthrough time (t_cb) of the CCW was 14 years (see Figure 4a). Reducing the k_gcl by a factor of 5 to 1 × 10⁻¹¹ m/s, t_cb increased to 15 years, with an increase of 1 year and increase ratio of 7%. Further reducing the k_gcl by 10 times to 0.5 × 10⁻¹¹ m/s, increased t_cb to 16 years, with an increase of 2 years and an increase ratio of 14%. Reducing the k_gcl by 50 times to 0.1 × 10⁻¹¹ m/s increased t_cb to 19 years, an increase of 5 years, or 36%. Decreasing k_gcl significantly extends the t_cb, with larger reductions in k_gcl leading to more pronounced increases in t_cb.

A similar pattern was observed in mass flux breakthrough curves (see Figure 4b). The mass flux breakthrough time (t_fb) of the CCWs with k_gcl = 5.0, 1.0, 0.5 and 0.1 × 10⁻¹¹ m/s are 16, 18, 20 and 36 years. The increases in t_fb are 2 years, 4 years and 20 years, and the increase ratios are 13%, 25% and 125%, respectively, for k_gcl decreases from 5.0 × 10⁻¹¹ m/s to 1.0, 0.5 and 0.1 × 10⁻¹¹ m/s. Moreover, all the predicted t_fb are larger than the corresponding t_cb, and the magnitude of this difference increases as k_gcl is reduced. The results demonstrated that decreasing hydraulic conductivity of GCL (k_gcl) could significantly improve the barrier performance of GCL-enhanced CCW: reducing k_gcl from 5.0 × 10⁻¹¹ m/s to 1.0, 0.5 and 0.1 × 10⁻¹¹ m/s, t_cb would increase from 14 years to 15 (7%), 16 (14%) and 19 (36%) years; and t_fb would increase from 16 years to 18 (13%), 20 (25%) and 36 (125%) years.

4.3. Influence of the Hydraulic Conductivity of SB

The influences of hydraulic conductivity of SB (k_sb) on the performance of the GCL-enhanced CCW were investigated. Four values of the hydraulic conductivity for the SB (k_sb = 1.0, 0.5, 0.1 and 0.05 × 10⁻⁹ m/s) were adopted. The breakthrough curves of the SB/GCL/SB CCW with the four levels of hydraulic conductivity for the SB are shown in Figure 5a,b using the concentration-based and flux-based breakthrough criteria, respectively.

Using concentration breakthrough criteria, when k_sb = 1 × 10⁻⁹ m/s, the mass concentration breakthrough time (t_cb) of the CCW was 9 years (see Figure 5a). Reducing the k_sb by a factor of 2 to 0.5 × 10⁻⁹ m/s, t_cb increased to 10 years, with an increase of 1 year and increase ratio of 11%. Further reducing the k_sb by a factor of 10 to 0.1 × 10⁻⁹ m/s extended the t_cb to 15 years, resulting in a 6-year increase and a 67% rise. Reducing the k_sb by 20 times to 0.05 × 10⁻⁹ m/s increased t_cb to 18 years, a 9-year increase and a 100% rise. These results indicate that k_sb significantly extends t_cb, with larger reductions in k_sb leading to more pronounced increases in t_cb.

The mass flux breakthrough time (t_fb) of the CCWs with k_sb = 1.0, 0.5, 0.1 and 0.05 × 10⁻⁹ m/s are 7, 9, 18 and 27 years (see Figure 5b). The increases in t_fb for k_sb decreases from 1.0 × 10⁻⁹ m/s to 0.5, 0.1 and 0.01 × 10⁻⁹ m/s are 2 years (28%), 11 years (157%) and 20 years (286%), respectively (the values in the blankets are the corresponding increase ratios). The results demonstrated that decreasing hydraulic conductivity of SB (k_sb) could also significantly improve the barrier performance of GCL-enhanced CCW: reducing k_sb from 1.0 × 10⁻⁹ m/s to 0.5, 0.1 and 0.01 × 10⁻⁹ m/s, t_cb would increase from 9 years to 10 (11%), 15 (67%) and 18 (100%) years and t_fb would increase from 7 years to 9 (28%), 18 (157%) and 27 (286%) years.

Moreover, this study did not demonstrate the impact of aquifer hydraulic conductivity on the performance of the CCW barrier. This is because, as can be seen from Equation (6), the magnitude of k_s is primarily controlled by the smallest hydraulic conductivity among the components (k_gcl, k_sb and k_a). However, k_a is the largest among them. Therefore, its variation within a reasonable range (corresponding to various soil types) has a negligible impact on the value of k_s compared to k_sb and k_a, and, consequently, on the final migration results.

5. Equivalency Assessment between GCL-Enhanced CCW and SBCW

The excellent performance of GCL has made it a popular choice for creating a CCW barrier system. Before implementing GCL-enhanced CCW as an improved alternative to traditional SBCW, it is necessary and of great importance to evaluate the equivalency of their anti-pollution barrier performance. If they have the same breakthrough time, they are considered to have identical barrier performance. The anti-pollution barrier performance of cutoff walls is closely related to their geometric dimensions and material properties. Therefore, (1) the thicknesses required for SB/GCL/SB CCWs to achieve the same performance as traditional SBCWs are evaluated; and (2) the allowable adsorption ability of the SB layer within CCW are analyzed to achieve the same performance as SBCW.

Two different thicknesses of SBCW (T_sbcw = 0.60 and 1.20 m) are chosen to assess the equivalency of barrier performance between SBCW and SB/GCL/SB CCW. The breakthrough curves for both types of cutoff walls are shown in Figure 6. The concentration breakthrough time (t_cb) for SBCW with T_sbcw = 0.60 and 1.20 m is 12 and 45 years, respectively. The required thickness of SB/GCL/SB CCW to reach the corresponding breakthrough time is 0.53 and 1.12 m, respectively. The results show that SBCW needs a greater thickness to achieve an identical performance for GCL-enhanced CCW, indicating that GCL can reduce the amount of bentonite soil required.

As shown in Figure 7, for SBCWs with thicknesses of 0.6 and 1.2 m, the distribution coefficient of SB (K_d,sb) is set at 0.54 mL/g. For CCWs of equivalent thickness, K_d,sb only needs to reach 0.34 and 0.43 mL/g, respectively. The results demonstrate that for SBCW and CCW of the same thickness to achieve equivalent anti-pollution barrier performance, CCW can employ SB with lower adsorption capacity, thereby using less bentonite.

6. Conclusions

This paper presents the development of a one-dimensional analytical solution for contaminant advection, diffusion and adsorption through a SB/GCL/SB–aquifer composite cutoff wall (CCW) system. The proposed analytical solution was well validated with an existing model of double layers. Toluene is adopted as the representative contaminant in this study. Using the present solution, the influence of the advection processes (hydraulic gradient, hydraulic conductivity of GCL and hydraulic conductivity of SB) was systematically studied. Moreover, an equivalence evaluation was conducted between SB/GCL/SB CCW and SBCW. Some major conclusions of the study are drawn as follows:

(1)

The present solution can well accommodate contaminant advection–diffusion–adsorption transport through a SB/GCL/SB CCW and aquifer system. It is a practical tool for the design and optimization of the GCL-enhanced CCW by assessing the breakthrough processes of contaminant transport in the CCW system.

(2)

Maintaining a lower hydraulic gradient (i) can significantly extend both contaminant concentration and mass flux breakthrough times for the SB/GCL/SB CCW system, which is crucial for optimizing the CCW’s barrier performance. As i increases from 0.1 to 1, the mass concentration breakthrough time (t_cb) decreases from 20 to 11 years, and notably, the mass flux breakthrough time (t_fb) decreases from infinite to 11 years.

(3)

The results based on the concentration-based breakthrough criterion are more stable and reliable. Using the flux-based breakthrough criterion may inherently lead to the generation of anomalously high or even infinite breakthrough time estimates.

(4)

Exploring any strategies to reduce the hydraulic conductivity of GCL (k_gcl) and SB (k_sb) could significantly improve the barrier performance of GCL-enhanced CCW. For instance, reducing k_gcl from 5.0 × 10⁻¹¹ m/s to 1 × 10⁻¹² m/s would increase t_cb from 14 to 19 years, an increase ratio of 36%. Similarly, reducing k_sb from 1.0 × 10⁻⁹ m/s to 1.0 × 10⁻¹¹ m/s would increase t_cb from 9 to 18 years, an increase ratio of 100%.

(5)

To achieve equivalent contaminant barrier performance, SBCW requires a greater thickness compared to SB/GCL/SB CCW. Specifically, for t_cb of 12 years and 45 years, SBCW requires thicknesses of 0.60 m and 1.20 m, respectively, while the corresponding SB/GCL/SB CCW only requires 0.53 m and 1.12 m. This indicates that GCL can reduce the amount of bentonite soil needed.

(6)

For SBCW and CCW of the same thickness to achieve equivalent barrier performance, CCW can employ SB with a lower distribution coefficient (K_d,sb). For SBCWs with thicknesses of 0.6 m and 1.2 m, K_d,sb is set at 0.54 mL/g, whereas for CCWs of equivalent thickness, K_d,sb only needs to reach 0.34 mL/g and 0.43 mL/g, respectively. This demonstrates that CCW can use SB with lower adsorption capacity, thereby reducing the required amount of bentonite.