# Analysis of Factors for Compacted Clay Liner Performance Considering Isothermal Adsorption
## 3 Scheme for Analyses and Discussion of Results
### Analysis of Influencing Factors of Advection-Dispersion Model for Linear Isotherm  
According to the expressions for the pore water concentration \(C_{\rm w}\) and effluent concentration \(C_{\rm e}\) recommended by [PERSON] et al. [18], the dimensionless expressions are shown in Equations (13) and (14). \[\frac{C_{\rm w}(x,t)-C_{\rm i}}{C_{\rm o}-C_{\rm i}}=\frac{1}{2}\varepsilon frc \left(\frac{\frac{t}{x}-\frac{T}{K_{\rm d}}}{2\sqrt{T/(K_{\rm d}R_{\rm i})}} \right)+\sqrt{\frac{TP_{\rm i}}{RT_{\rm d}}}\exp\left[-\frac{\left(\frac{T}{x }-\frac{T}{K_{\rm d}}\right)}{4T/(K_{\rm d}R_{\rm i})}\right]-\frac{1}{2} \left(1+P_{\rm L}\frac{x}{T}+\frac{TP_{\rm i}}{R_{\rm d}}\right)\exp\left(P_{ \rm L}\frac{x}{T}\right)erfc\left(\frac{\frac{t}{x}+\frac{T}{K_{\rm d}}}{2 \sqrt{T/(K_{\rm d}R_{\rm i})}}\right) \tag{13}\] \begin{table}
\begin{tabular}{c c} \hline \hline \(\rho_{\rm d}\) (g/cm\({}^{3}\)) & 1.04 \\ \(n\) & 0.62 \\ \(k\)(m/s) & 1 \(\times\) 10\({}^{-9}\) \\ \(L\)(m) & 2 \\ \(h\)(m) & 40 \\ \(C_{\rm 0}\) (mg/L) & 1000 \\ \(\alpha\)(m) & 0.047 \\ \(D_{\rm d}^{*}\)(m/s\({}^{2}\)) & 3.3 \(\times\) 10\({}^{-10}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Model calculation parameters. \[\frac{C_{\rm e}(L,t)}{C_{0}}=\frac{1}{2}erfc\left(\frac{1-\frac{T}{R_{\rm d}}}{2 \sqrt{T/(R_{\rm d}P_{\rm L})}}\right)+\frac{1}{2}\exp(P_{\rm L})erfc\left(\frac{1 +\frac{T}{R_{\rm d}}}{2\sqrt{T/(R_{\rm d}P_{\rm L})}}\right) \tag{14}\]  
where \(T=\frac{\gamma_{\rm e}t}{L}\), \(P_{\rm L}=\frac{\gamma_{\rm e}L}{D_{\rm h}}\), \(T\) is the number of pore volumes of the flow, and \(P_{\rm L}\) is the Peclet number of soil columns representing the relative effect of advection with respect to the dispersive/diffusive transport. The effluent concentration can be calculated according to Equation (16). In addition, the expression for the effluent concentration can be simplified when \(T_{\rm R}=T/R_{\rm d}\) is substituted into Equations (13) and (14). It is then found that the expressions in Equations (13) and (14) for adsorbing solutes and non-adsorbing solutes are exactly the same. Dimensionless parameters were used directly in the following discussion. Dry density, porosity, and water content were taken from Table 1.  
Figure 2a,b show the pore water and total concentration profiles in the clay liner corresponding to different retardation factors, respectively. Figure 2a shows that the larger the value of \(K_{\rm d}\) (i.e., \(R_{\rm d}\)), the shallower the contaminant front. Figure 2b shows that the larger the value of \(K_{\rm d}\) (i.e., \(R_{\rm d}\)), the shallower the total concentration profile curve in the soil and the bigger the peak value at the top. Figure 3 shows the breakthrough curves in terms of dimensionless parameters. As shown in Figure 3a, when \(T\) is taken as the abscissa, the greater the value of \(K_{\rm d}\), the later the effluent front appears and the wider the contaminant front in shape. When \(P_{\rm L}\) is the same, the concentration values corresponding to \(T=R_{\rm d}\) on the breakthrough curves for different \(K_{\rm d}\) values are equal: the greater the \(P_{\rm L}=\gamma_{\rm e}L/D_{\rm h}\) corresponding to the same \(K_{\rm d}\), the narrower the contaminant front. This is because a greater \(P_{\rm L}\) leads to a stronger advection effect; thus, its front is narrower. As shown in Figure 3b, when the abscissa is \(T_{\rm R}=\gamma_{\rm e}t/(LR_{\rm d})\), the outflow concentration curves for different \(R_{\rm d}\) values coincide, indicating that the effect of \(R_{\rm d}\) on advection and dispersion is as shown in the control equation, that is, both \(\gamma_{\rm e}\) and \(D_{\rm h}\) are \(1/R_{\rm d}\) of the past, so that the effect of \(R_{\rm d}\) on advection and dispersion migration is a linear superposition. Figure 4 shows the relative outflow concentration values for different  
The effects of the two diffusion parameters \({D_{\rm d}}^{*}(\tau)\) and \(\alpha\) are compared separately below.