# Analysis of Factors for Compacted Clay Liner Performance Considering Isothermal Adsorption
## 2 Introduction of Theory  
The mechanisms of pollutant migration include advection, hydrodynamic dispersion (including molecular diffusion and mechanical dispersion), and adsorption. The parameter representing advection is the pore flow velocity \(v_{\rm s}\), the parameters representing hydrodynamic dispersion are the effective molecular diffusion coefficient \(D_{\rm d}^{\ast}\) (bending factor \(\tau\)) and the diffusivity \(\alpha\), and the parameters representing the adsorption action are the retardation factor \(R_{\rm d}\) and the nonlinear adsorption parameters. The flow velocity determines the appearance of the pollution front, the hydrodynamic dispersion parameter determines the width of the pollution front, and the adsorption has a retardation effect on migration, which delays the appearance of the pollution front. Figure 1 shows the breakthrough curves with a continuous source \(C_{0}\) at the top of the soil column. As shown in Figure 1, when only advection exists, the pollution front is pushed vertically like a piston; this is called piston flow, and the occurrence time is \(t_{\rm c}=L/v_{\rm s}\), where \(L\) is the length of the column and \(v_{\rm s}\) is the seepage velocity. When there is both advection and hydrodynamic dispersion, the pollution front is a curve, and when there is also adsorption, the occurrence time of the pollution front is pushed backward, \(t_{\rm c}^{\ast}=t_{\rm c}/R_{\rm d}\). Whether the hysteresis phenomenon caused by adsorption with different adsorption models is the same has not been discussed. The one-dimensional transient solute transport through homogeneous soil can be described as  
\[\frac{\partial C_{\rm w}(x,t)}{\partial t}=\frac{D_{\rm h}}{R_{\rm d}}\cdot \frac{\partial^{2}C_{\rm w}(x,t)}{\partial x^{2}}-\frac{v_{\rm s}}{R_{\rm d} }\frac{\partial C_{\rm w}(x,t)}{\partial x} \tag{1}\]  
where \(D_{\rm h}\) is coefficient of hydrodynamic dispersion, \(D_{\rm h}=D_{\rm d}^{\ast}+D_{\rm m}\), \(D_{\rm d}^{\ast}=\tau D_{\rm d}\), \(D_{\rm m}=\alpha v_{\rm s}\), where \(D_{\rm d}\) is free diffusion coefficient, \(D_{\rm m}\) is mechanical dispersion. \(R_{\rm d}=1\) when it is nonabsorbable, \(R_{\rm d}\) is a constant greater than 1 under linear adsorption, and \(R_{\rm d}\) is a function of pore water concentration under nonlinear adsorption, with its value changing with pore water concentration. According to the adsorption model, a relationship between the total concentration in the soil and the pore water concentration can be established. The relationships for all three adsorption isotherms have been derived. The linear isotherm is the straight type, the Langmuir isotherm is the upper convex type, and the Freundlich isotherm (taking \(n_{\text{f}}\) greater than 1) represents the lower concave type. Linear isotherm (straight type). The linear isotherm is expressed mathematically as  
\[C_{\text{s}}=K_{\text{d}}C_{\text{w}} \tag{2}\]  
where \(C_{\text{s}}\) is the mass of contaminants adsorbed per unit dry mass of soil, \(K_{\text{d}}\) is the distribution coefficient, and \(C_{\text{w}}\) is the pore water concentration. The corresponding retardation factor expression is:  
\[R_{\text{d}}=1+\frac{\rho_{\text{d}}K_{\text{d}}}{n} \tag{3}\]  
where \(\rho_{\text{d}}\) is dry density, \(K_{\text{d}}\) is distribution coefficient, \(n\) is porosity.