## 3 Results and Discussion
### Hcf
According to [PERSON]'s law, the permeability corresponding to the flow rate through a certain cross-section during a certain time period is equal to the ratio of flow velocity and mean hydraulic gradient at that cross-section and during that time period:  
\[k=-\frac{v}{i_{m,av}} \tag{13}\]  
The mean hydraulic gradient \(i_{m,av}\) is calculated by the forward difference method:  
\[i_{x,av}=\frac{1}{2}\bigg{(}\frac{h_{x_{i+1},t_{1}}-h_{x_{i},t_{1}}}{z_{x_{i+1} }-z_{x_{i}}}+\frac{h_{x_{i+1},t_{2}}-h_{x_{i},t_{2}}}{z_{x_{i+1}}-z_{x_{i}}} \bigg{)} \tag{14}\]  
where \(h_{xi+1,t1}\) and \(h_{xi,t1}\) represent the hydraulic heads at position \(x_{i+1}\) and position \(x_{i}\) at time \(t_{1}\), respectively; \(h_{xi+1,t2}\) and \(h_{xi,t2}\) represent the hydraulic heads at position \(x_{i+1}\) and position \(x_{i}\) at time \(t_{2}\), respectively; and \(z_{xi+1}\) and \(z_{xi}\) are the depths at positions \(x_{i+1}\) and \(x_{i}\), respectively. The permeabilities under different moisture contents, i.e., unsaturated infiltration curves, are obtained based on Equation (14), and combined with the SWCC, the unsaturated infiltration curve expressed as matrix suction-permeability can be obtained. There are many infiltration models for unsaturated soil, and statistical models can indirectly predict the unsaturated permeability based on the SWCC. The most commonly used models are the Gardner model, Van Genuchten (VG) model and Fredlund-Xing (FX) model [53; 54; 55]. In this study, two permeability models were used to predict the permeability of compacted loess. Gardner permeability model:  
\[k(\psi)=\frac{k_{s}}{1+a\psi^{n}} \tag{15}\]  
[PERSON] substituted Equation (15) into the statistical conductivity model proposed by [PERSON] and [PERSON] to obtain the unsaturated permeability model [56; 57], which can also be written in the form of effective moisture content \(\Theta\):  
Figure 13:0  
Figure 13:1  
Figure 13:2  
where \(a\), \(n\), and \(m\) are curve fitting parameters and \(n=1/(1-m)\). The vertical patterns volumetric moisture contents of the soil column are different at any time. Repeated calculations are performed according to Equations (8)-(14) to obtain the flow rates at different positions and in different time periods to obtain the HCF. Based on the average volumetric moisture content and average hydraulic gradient in two adjacent cross-sections, as well as the volume change and flow velocity of water for each cross-section during the time period of \(\Delta t\), a scatterplot of the permeability and volumetric moisture content (or matrix suction) was obtained. The complete curve was generated by fitting the data points using the Gardner model and VG model [53; 54], and the fitting parameters are shown in Figure 13:3 Figure 15 shows the soil column matric suction-permeability curve, \(k(\psi)\), and volumetric moisture content-permeability curve, \(k(\theta)\). This test involved a drying process, and the initial mass moisture content of the soil column was approximately 9.20%; therefore, the corresponding permeability when the mass moisture content is less than 9.20% was not obtained in this study. During the wetting process, the moisture content of each soil column increases from the initial value to the saturation moisture content, while the matric suction decreases from the initial value. The \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Parameters Model**} & \multicolumn{4}{c}{**Gardner**} & \multicolumn{4}{c}{**VG**} \\ \cline{2-7}  & **n** & **a** & **R\({}^{2}\)** & **n** & **a** & **R\({}^{2}\)** \\ \hline \multirow{3}{*}{\(k(\psi)\)} & C1 & 1.38377 & 0.54747 & 0.7536 & 1.64234 & 0.09035 & 0.8463 \\  & C2 & 3.18836 & 0.13933 & 0.81379 & 2.58966 & 0.06578 & 0.8167 \\  & C3 & 2.63563 & 0.1457 & 0.7698 & 1.50646 & 0.02109 & 0.7378 \\ \hline \multirow{3}{*}{\(k(\theta)\)} & C1 & & & & 2.32000 & & 0.8791 \\  & C2 & & & & 2.11924 & & 0.7871 \\  & C3 & & & & 2.16766 & & 0.7714 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Model parameters.