## 3 Analysis of the Penetration Test Results
### Applicability Analysis and Parameter Determination of the Bottom Aquifer Permeability Model  
The permeability coefficient of clayey sand in the bottom aquifer of thick alluviums decreased nonlinearly with a decreasing void ratio during the consolidation process. In response to the nonlinear relationship between the permeability coefficient and void ratio of cohesive soil, scholars at home and abroad have conducted a vast amount of research, mainly including the following:  
Figure 13: Relationship between the permeability coefficient and hydraulic gradient of clayey sand. (1) The \(\lg[k_{v}(1+e)]-\lg e\) permeability model. [PERSON] [33] obtained the expression of the permeability coefficient of sand through a sand permeability test:  
\[k_{v}=\frac{Ce^{3}}{1+e}, \tag{2}\]  
[PERSON] [34] proposed the \(e-k_{v}\) relationship model for general normal consolidated clay according to Formula (2):  
\[k_{v}=\frac{Ce^{n}}{1+e}, \tag{3}\]  
That is, \(\lg[k_{v}(1+e)]=\lg C+n\lg e\), where \(C\) represents the reference permeability coefficient reflecting the properties of the soil and \(n\) represents the soil parameter. (2) The \(e-\lg k_{v}\) permeability model. \[e=C_{k}\lg(k_{v}/k_{v0})+e_{0}=C_{k}\lg k_{v}+e_{0}^{\prime}, \tag{4}\]  
where \(C_{k}\) is the permeability index; \(e_{0}\) and \(k_{v0}\) are the initial void ratio and the initial permeability coefficient, respectively; \(e^{\prime}{}_{0}\) is the pore ratio corresponding to the unit vertical permeability coefficient. Formula (4) is the most commonly used model to express nonlinear changes of the permeability coefficient with the void ratio. [PERSON] [35] pointed out that \(e-\lg k_{v}\) represented by Formula (4), has a linear relationship, which is applicable to common clays in engineering. (3) The \(\lg k_{v}-\lg e\) permeability model. [PERSON] [36] found, through research, that if the variation range of the clay pore ratio is too large, the applicability of Formula (4) is poor, and then put forward an improved \(\lg k_{v}-\lg e\) model:  
\[k_{v}=Be^{A}, \tag{5}\]  
That is, \(\lg k_{v}=Al\lg e+\lg B\), where \(A\) and \(B\) are the parameters of clay permeability. (4) The \(\lg k_{v}-\lg(1+e)\) permeability model. [PERSON] [37] once proposed the following nonlinear permeability model: \(k_{v}=k_{v0}\left(\frac{1+e}{1+e_{0}}\right)^{2}\). Then, the model was improved and a new model regarding the permeability coefficient and void ratio was put forward [38]:  
\[k_{v}=k_{v0}\left(\frac{1+e}{1+e_{0}}\right)^{\alpha}, \tag{6}\]  
That is, \(\lg k_{v}=\alpha\lg\left(\frac{1+e}{1+e_{0}}\right)+\lg k_{v0}=\alpha\lg(1+ e)+\beta\), where \(\alpha\) and \(\beta\) are the soil parameters, which can be obtained by analyzing test data and fitting curves. Through the regression fitting of the nonlinear relationship between four common pore ratios and permeability coefficients to clayey sand permeability test data, the statistical regression values and corresponding \(R^{2}\) parameters for the four nonlinear infiltration models' parameters were obtained, and the results are shown in Table 5. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Permeability Type**} & \multicolumn{8}{c}{**Permeability Model**} \\ \cline{2-13}  & \multicolumn{2}{c}{\(\lg[k_{v}(1+e)]-\lg e\)} & \multicolumn{8}{c}{\(e-\lg k_{v}\)} & \multicolumn{8}{c}{\(\lg k_{v}-\lg e\)} \\ \cline{2-13}  & \(n\) & \(C\) & \(R^{2}\) & \(C_{k}\) & \(e_{0}^{\prime}\) & \(R^{2}\) & \(A\) & \(B\) & \(R^{2}\) & \(\alpha\) & \(\beta\) & \(R^{2}\) \\ \hline Low-confined water & 14.24 & 0.034 & 0.9757 & 0.0656 & 0.8557 & 0.9595 & 13.951 & 0.0187 & 0.9746 & 47.899 & \(-\)14.374 & 0.9650 \\ High-confined water & 13.934 & 0.028 & 0.9888 & 0.0679 & 0.9004 & 0.9708 & 13.645 & 0.0154 & 0.9881 & 46.813 & \(-\)14.174 & 0.9769 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Four permeability model parameters and related parameters of clayy sand.