# A Numerical Study of Chemical Compatibility of GCLs
## 3 The Numerical Results and Discussion  
### An Equation for Calculating the Hydraulic Conductivity of GCLs under DIW  
12 conditions with different particle sizes and initial mobile porosity were considered. Twelve COMSOL models that represent the GCLs under DIW were developed and calculated. All the cases are summarized in Table 1.  
Figure 4 illustrates the relationship between hydraulic conductivity and mobile porosity (\(n_{\rm M}\)). It can be seen that the hydraulic conductivity decreased with the decrease in the mobile porosity. When the final mobile porosity was archived (\(n_{\rm f}\) = 0.025 here), selecting the final conditions of 12 cases (all the leftmost points in Figure 4), the data of 12 points corresponding to \(n_{\rm f}\) = 0.025 can be obtained. These relationships between the hydraulic conductivity and the final mobile porosity (\(n_{\rm f}\) = 0.025) were illustrated in Figure 5 (see three triangle symbols).  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{
\begin{tabular}{c} **The Initial** \\ **Mobile Porosity** \\ \end{tabular} } & \multicolumn{3}{c}{**The Particle Size (mm)**} \\ \cline{2-5}  & **0.02** & **0.08** & **0.2** & **1.0** \\ \hline
0.4 & \(1\) & \(1\) & \(1\) & \(1\) \\
0.5 & \(1\) & \(1\) & \(1\) & \(1\) \\
0.6 & \(1\) & \(1\) & \(1\) & \(1\) \\ \hline \end{tabular}
\end{table}
Table 1: Summary of Working Conditions in DIW.  
Figure 4: Relationship between hydraulic conductivity and the mobile porosity (\(n_{\rm f}\) = 0.025).  
The final mobile porosity was then changed to 0.01 and 0.0025, respectively. Following the same steps as above, the data of the other 24 points correspond to \(n_{\text{f}}\) = 0.01, and \(n_{\text{f}}\) = 0.0025, as shown in Figure 5, could be obtained.  
By fitting the data in Figure 5, the relationship between \(k\) and \(d_{0}\) as follows can be obtained:  
\[k=\beta{d_{0}}^{2} \tag{7}\]  
where \(\beta\) is a coefficient; \(d_{0}\) is the initial particle size (mm).  
As can be seen from Equation (7), the hydraulic conductivity was quadratic dependent on the particle size. This is consistent with those of [PERSON] and [PERSON] [37; 38; 39].  
Meanwhile, there was a relationship between the coefficient \(\beta\) and the final and the initial mobile porosity, as shown in Table 2 according to the COMSOL models.  
Therefore, using the fitting method, the coefficient \(\beta\) can be expressed by the flowing equation:  
\[\beta=0.1114{(n_{\text{f}})}^{3}+0.4589{(n_{\text{f}})}^{3}{n_{\text{f}}}^{3} \tag{8}\]  
where \(n_{\text{f}}\) is the final mobile porosity; \(n_{\text{f}}\) is the initial mobile porosity.  
\begin{table}
\begin{tabular}{c c c} \hline \(n_{\text{f}}\) & \(n_{\text{0}}\) & \(\beta\) \\ \hline
0.025 & 0.4 & \(2.2\times 10^{-6}\) \\
0.025 & 0.5 & \(2.6\times 10^{-6}\) \\
0.025 & 0.6 & \(3.3\times 10^{-6}\) \\
0.01 & 0.4 & \(1.4\times 10^{-7}\) \\
0.01 & 0.5 & \(1.7\times 10^{-7}\) \\
0.01 & 0.6 & \(2.1\times 10^{-7}\) \\
0.0025 & 0.4 & \(2.2\times 10^{-9}\) \\
0.0025 & 0.5 & \(2.6\times 10^{-9}\) \\
0.0025 & 0.6 & \(3.3\times 10^{-9}\) \\ \hline \end{tabular}
\end{table}
Table 2: \(\beta\) corresponds to the final and initial mobile porosity.  
Figure 5: The hydraulic conductivity vs. the mobile porosity.  
Substitute Equation (8) into Equation (7), the following equation can be obtained:  
\[k=\Big{(}0.1114(n_{\text{f}})^{3}+0.4589(n_{\text{f}})^{3}n_{0}^{3}\Big{)}{d_{0}} ^{2} \tag{9}\]  
where \(k\) is the hydraulic conductivity (m/s).  
When the final mobile porosity varies from 0.0025 to 0.025 [40; 41], considering the initial mobile porosity and the initial particle size, Formula (9) for calculating the hydraulic conductivity of GCLs in DIW could be obtained.  
Therefore, the final mobile porosity can be expressed as follows:  
\[n_{\text{f}}=\sqrt[3]{\frac{k}{(0.1114+0.4589n_{0}^{3}){d_{0}}^{2}}} \tag{10}\]