## 2 Methods
### A Simple Model for Estimating the Hydraulic Conductivity of Unsaturated Soil  
According to the soil-water interaction analysis, the hydraulic conductivity curve of unsaturated soil is divided into the boundary effect zone, transition zone and adsorptive dominant region by four critical points including the minimum suction at measurable saturation, air-entry value, residual suction and the highest suction. Specifically, the hydraulic conductivity in the boundary effect zone decreases slightly compared with the saturated one. Furthermore, the hydraulic conductivities of the transition zone and the adsorptive dominant region decrease approximately linearly on the log-log scale with different slopes. Consequently, the hydraulic conductivity curve can be simplified as three straight lines (Figure 2), and a simple piecewise model of the hydraulic conductivity is proposed as follows:  
\[ \begin{array}{l}\log k_{w}=\log k_{wa}+\frac{\log\psi-\log\psi_{a}}{\log\psi _{a}-\log\psi_{a}}\left(\log k_{s}-\log k_{wa}\right)\ \ \text{for}\ \psi\leq\psi_{a}\\ \log k_{w}=\log k_{wr}+\frac{\log\psi-\log\psi_{r}}{\log\psi_{a}-\log\psi_{r}} \left(\log k_{wa}-\log k_{wr}\right)\ \ \text{for}\ \psi_{a}<\psi\leq\psi_{r}\\ \log k_{w}=\log k_{wm}+\frac{\log\psi-\log\psi_{m}}{\log\psi_{r}-\log\psi_{m} }\left(\log k_{wr}-\log k_{wm}\right)\ \ \text{for}\ \psi>\psi_{r}\end{array}  \tag{3}\]  
where \(\Psi_{s}\), \(\Psi_{a}\), \(\Psi_{r}\) and \(\Psi_{m}\) are the minimum suction at measurable saturation, air-entry value, residual suction and the highest suction (i.e., \(10^{6}\) kPa), respectively. \(k_{s}\), \(k_{wa}\), \(k_{wr}\) and \(k_{wm}\) are the permeabilities corresponding to the auctions of \(\Psi_{s}\), \(\Psi_{a}\), \(\Psi_{r}\) and \(\Psi_{m}\). The saturated hydraulic conductivity is commonly obtained by direct measurements using constant head tests or falling head tests. The critical step is to determine the values of \(k_{au}\), \(k_{wr}\) and \(k_{wm}\). Figure 2: The proposed simple piecewise model of hydraulic conductivity. Parameters \(\Psi_{s}\), \(\Psi_{a}\), \(\Psi_{r}\) and \(\Psi_{m}\) are the minimum suction, air-entry value, residual suction and highest suction, respectively. \(k_{s}\), \(k_{tau}\), \(k_{wr}\) and \(k_{wm}\) are the hydraulic conductivities corresponding to the auctions of \(\Psi_{s}\), \(\Psi_{a}\), \(\Psi_{r}\) and \(\Psi_{m}\), respectively. Soil pores are almost water-filled in the boundary effect zone, and the hydraulic conductivity in this zone decreases slightly compared with the saturated one. For simplicity, it is suggested that the filled-water flow in this zone follows Darcy law. Thus, the hydraulic conductivity is proportional to the water cross-section area per unit area of the soil. Furthermore, the length of the water flow path is assumed to be invariable in the boundary effect zone, so the degree of saturation is equal to the ratio of the water cross-section area at an unsaturated state to that at a completely saturated state. Accordingly, the variation in the degree of saturation can be used to represent the variation in the water cross-section area, which can approximately characterize the change in the hydraulic conductivity in this zone. Hence, the hydraulic conductivity that corresponds to the air-entry value can be estimated by Equation (4):  
\[k_{wa}=S_{ra}k_{s} \tag{4}\]  
where \(k_{wa}\) is the hydraulic conductivity at \(\Psi_{a};k_{s}\) is the saturated hydraulic conductivity; \(S_{ra}\) represents the degree of saturation at \(\Psi_{a}\), whose expression is as follows:  
\[S_{ra}=\frac{1-\frac{\ln(1+\Phi_{a}/C_{r})}{\ln(1+10^{6}/C_{r})}}{\left\{\ln \left[e+\left(\psi_{a}/\alpha\right)^{n}\right]\right\}^{m}} \tag{5}\]  
where \(a\), \(m\), \(n\) and \(C_{r}\) are model parameters in [PERSON] and [PERSON]'s equation; and \(e\) is the natural constant. Furthermore, according to [PERSON] (2018) and [PERSON] et al.