# A Numerical Study of Chemical Compatibility of GCLs
## 3 The Numerical Results and Discussion
### An Equation for Calculating the Hydraulic Conductivity of GCLs under Chemical Solution  
[PERSON] [2], [PERSON] [42], [PERSON] [43], [PERSON] [44] investigated the chemical compatibility of GCLs experimentally. The experimental data are summarized in Table 3. The concentration of the chemical solution could be expressed as the ionic strength (\(I\)) [3].  
Substituting the experimental hydraulic conductivity data, the initial particle size, and the initial mobile porosity of [PERSON] [2], [PERSON] [42], [PERSON] [43], [PERSON] [44] into Formula (8), the final mobile porosity could be obtained. Therefore, the relationship between the final mobile porosity required to reach the experimental hydraulic conductivity and the ionic strength could be plotted as shown in Figure 6.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Literature**} & **Initial Particle** & **Initial Mobile** & **Chemical** & **Ionic Strength** \\  & **Size (\(d_{0}\))/mm** & **Porosity (\(n_{0}\))** & **Solution** & **(mol/L)** \\ \hline \multirow{8}{*}{Jo [2]} & & & 0.1 M NaCl & 0.1 \\  & & & 1.0 M NaCl & 1.0 \\  & & & 0.1 M KCl & 0.1 \\  & & & 1.0 M KCl & 1.0 \\  & & & 0.025 M MgCl\({}_{2}\) & 0.015 \\  & & & 0.1 M MgCl\({}_{2}\) & 0.3 \\  & & & 1 M MgCl\({}_{2}\) & 3 \\  & & & 0.1 M ZnCl\({}_{2}\) & 0.3 \\  & & & 1.0 M ZnCl\({}_{2}\) & 3 \\  & & & 0.01 M LaCl\({}_{3}\) & 0.06 \\  & & & 0.025 M LaCl\({}_{3}\) & 0.15 \\  & & & 0.1 M LaCl\({}_{3}\) & 0.6 \\ \hline \multirow{2}{*}{[PERSON] [42]} & & & 0.005 M CaCl\({}_{2}\) & 0.015 \\  & & & 0.01 M CaCl\({}_{2}\) & 0.03 \\  & & & 0.1 M CaCl\({}_{2}\) & 0.3 \\  & & & 0.5 M CaCl\({}_{2}\) & 1.5 \\ \hline \multirow{2}{*}{[PERSON] [43]} & & & 0.02 M CaCl\({}_{2}\) & 0.06 \\  & & & 0.05 M CaCl\({}_{2}\) & 0.15 \\  & & & 0.2 M CaCl\({}_{2}\) & 0.6 \\ \hline \multirow{2}{*}{[PERSON] [44]} & & 1 & 0.5 & - & 0.0395 \\  & & & - & 0.0968 \\ \multirow{2}{*}{[PERSON] [44]} & & & - & 0.048 \\  & & & 0.5 & - & 0.178 \\ \hline \hline \end{tabular}
\end{table}
Table 3: The experimental hydraulic conductivity of the GCLs [2; 42; 43; 44].  
It can be seen that with the increase in ionic strength, the hydraulic conductivity of GCLs showed an upward trend, indicating that the ionic strength and mobile porosity \(n_{\text{f}}\) have a very strong relationship (see symbols in Figure 6). Using the fitting method (see the solid curve in Figure 6), the relationship between the final mobile porosity and the ionic strength could be fitted, and the formula could be obtained as follows:  
\[n_{\text{f}}=\frac{0.0302I}{0.3568+I} \tag{11}\]  
Substituting Equation (11) into Equation (9), the hydraulic conductivity of GCLs in a chemical solution could be expressed as follows:  
\[k=\left(0.1114\left(\frac{0.0302I}{0.3568+I}\right)^{3}+0.4589\left(\frac{0.03 02I}{0.3568+I}\right)^{3}{n_{0}}^{3}\right){d_{0}}^{2} \tag{12}\]  
Figure 7 illustrates the relationship between the final mobile porosity with the hydraulic conductivity of GCLs, which was calculated from Equation (12). All the data came from [PERSON] [2], [PERSON] [42], [PERSON] [43], and [PERSON] [6]. It can be seen that the hydraulic conductivity decreased with the decrease in mobile porosity significantly.  
Figure 6: Relationship between final mobile porosity and Ionic strength.  
Therefore, using Equation (12), a series of the COMSOL model can be developed to investigate the mesoscopic mechanism of chemical compatibility of GCLs in detail. The next section will discuss this mesoscopic mechanism using [PERSON]'s data as an example [6].