## 2 Materials and Methods
### Mathematical Model  
A CGCW consisting of an upstream SBCW, an HDPE GMB and a downstream SBCW is installed to prevent the aquifer from being polluted by contaminants, as shown in Figure 1. A one-dimensional model considering the effect of advection, hydrodynamic dispersion and adsorption is established to evaluate the performance of the CGCW. The thicknesses of the upstream SBCW, HDPE GMB, downstream SBCW and aquifer are denoted as \(L_{1}\), \(L_{2}\), \(L_{3}\) and \(L_{4}\), respectively. The following assumptions are made to develop the mathematical model:  
1. The SBCWs, GMB and aquifer are saturated, homogeneous and isotropic. 2. The groundwater flows in a horizontal direction at a constant velocity. 3. The thickness of the aquifer is large enough to ignore the effect of contaminant outflow. 4. The adsorption of contaminants only occurs in SBCWs. The governing equations for contaminant transport in the CGCW and aquifer system can be given as follows [30]:  
\[R_{di}\frac{\partial C_{i}}{\partial t}=D_{i}\frac{\partial^{2}C_{i}}{\partial x ^{2}}-v_{i}\frac{\partial C_{i}}{\partial x}(i=1,\ 2,\ 3,\ 4) \tag{2}\]  
where, \(i=1\), \(2\), \(3\), \(4\) refer to the upstream SBCW, HDPE GMB, downstream SBCW and aquifer; \(C_{i}\) is the contaminant concentration; \(x\) is the contaminant migration direction; \(D_{i}\) is the diffusion coefficient; \(R_{di}\) is the retardation factor; \(v_{i}\) is the average linear velocity of contaminants. It should be noted that \(R_{d2}\) and \(R_{d4}\) are equal to \(1\)[31]. The retardation factors, diffusion coefficients and average linear velocities in the upstream SBCW and downstream SBCW are equal. It is assumed that there is no contaminant in the system initially [18]:  
\[C_{i}(x,t=0)=0 \tag{3}\]  
The constant contaminant concentration inlet boundary is assumed at \(x=0\)[19]:  
\[C_{1}(0,t)=C_{0} \tag{4}\]  
The outflow boundary is a semi-infinite boundary and can be expressed as [31]:  
\[\frac{\partial C_{4}(L,t)}{\partial x}=0 \tag{5}\]  
Figure 1: Mathematical model of contaminant transport in the CGCW and aquifer system. The concentration and mass flux continuity conditions at the upstream SBCW and the GMB interface, the GMB and the downstream SBCW interface, the downstream SBCW and the aquifer interface can be expressed as [12; 13]:  
\[C_{1}(x,t)\big{|}_{x=L_{1}} =\left.\frac{C_{2}(x,t)}{S_{gf}}\right|_{x=L_{1}} \tag{6a}\] \[D_{1}\frac{\partial C_{1}(x,t)}{\partial x}\bigg{|}_{x=L_{1}} =\left.S_{gf}D_{2}\frac{\partial C_{2}(x,t)}{\partial x}\right|_{x= L_{1}}\] (6b) \[\left.\frac{C_{2}(x,t)}{S_{gf}}\right|_{x=L_{1}+L_{2}} =\left.C_{3}(x,t)\right|_{x=L_{1}+L_{2}}\] (6c) \[S_{gf}D_{2}\frac{\partial C_{2}(x,t)}{\partial z}\bigg{|}_{x=L_{1}+L_{2}} =\left.D_{3}\frac{\partial C_{3}(x,t)}{\partial z}\right|_{x=L_{1}+L_{2}}\] (6d) \[R_{di}\frac{\partial C_{i}}{\partial t}=D_{i}\frac{\partial^{2}C_{i}}{\partial x ^{2}}-v_{i}\frac{\partial C_{i}}{\partial x}(i=1,\ 2,\ 3,\ 4) \tag{2}\]0 (6e) \[R_{di}\frac{\partial C_{i}}{\partial t}=D_{i}\frac{\partial^{2}C_{i}}{\partial x ^{2}}-v_{i}\frac{\partial C_{i}}{\partial x}(i=1,\ 2,\ 3,\ 4) \tag{2}\]1  
where, \(S_{gf}\) is the partition coefficient between the SBCW and the GMB.