## 2. Saturated-Unsaturated seepage Model
Its value is calculated by Equation (4) [33]:  
\[\theta_{w}=\theta_{r}+\frac{\theta_{s}-\theta_{r}}{\left[1+\left(\frac{\Psi}{ a}\right)^{n}\right]^{m}} \tag{4}\]  
where \(\theta_{s}\) and \(\theta_{r}\) indicate saturated and residual volume water content, respectively, in percentages; \(\Psi\) is the matric suction, in kPa; \(n\) and \(m\) are the fitting parameters of the soil-water characteristic curve, \(m=1-1/n\). When calculating the seepage of saturated-unsaturated soil, it is necessary to obtain the soil-water characteristic curve measured by the clay plate pressure test, and then use the van Genuchten model to fit the relationship curve between the water content of saturated-unsaturated soil, matric suction and permeability coefficient. The expression is [34]:  
\[k_{r}(\theta)=\frac{k_{w}}{k_{s}}=\Theta^{i}\frac{\int_{0}^{\theta_{w}}\frac{d \theta_{w}}{\Psi}}{\int_{0}^{\theta_{s}}\frac{d\theta_{w}}{\Psi}} \tag{5}\]  
where \(\Theta\) is the relative degree of saturation; \(i\) and \(j\) are constants, \(i=j=2\). Equation (6) can be obtained after Equation (5) is derived:  
\[k_{w}=k_{s}\frac{\left[1-\left(a\Psi^{n-1}\right)\left(1+\left(a\Psi^{n} \right)^{-m}\right)\right]^{2}}{\left(1+a\Psi\right)^{mm/2}} \tag{6}\]  
Compared with saturated soil, the permeability coefficient of unsaturated soil is no longer a constant but a function varying with soil saturation and is obtained from its soil-water characteristic curve function. Firstly, the relationship between the volume moisture content and the matric suction was obtained using the pressure test of the ceramic plate. Then, using the linear fitting function \"Newfunction\" in Origin 2018, the fitting parameters of the models of the soil-water characteristic curve were obtained. Figure 2a shows that, with decreasing water content, the matric suction of the loess soil increased gradually. The lower the soil moisture content, the greater the increase in matric suction. The reason for this is that, with decreasing water content and less free water in the soil pores, water remains only in the smaller pores between the soil particles. At this time, the transmission of pore water pressure depends mainly on the water-air interface in the smaller pores, namely the shrink film. Smaller pores provide shrink films with a smaller radius of curvature, thereby allowing the shrink films to withstand larger pore air pressure and transmit smaller pore water pressure. Therefore, the difference between pore air pressure and pore water pressure is greater when the soil-water content is lower, and the lower the water content, the greater the matric suction of unsaturated soil. The relationship curve between the permeability coefficient of saturated-unsaturated soil and the matric suction, as calculated by Equation (6), is shown in Figure 2a. As can be seen in Figure 2b, when the matric suction is less than the air inlet pressure, the soil permeability coefficient is close to the constant value; when the matric suction is close to the air inlet pressure, the soil permeability coefficient starts to change; when the matric suction is greater than the air inlet pressure, the soil permeability coefficient decreases rapidly. When the soil is saturated, the matric suction is equal to zero and the pore pressure is equal to the value of pore water pressure. The relationship curve between the permeability coefficient and volume water content exhibits a negative correlation. Saturated soil has high volume moisture content and low matric suction, and its permeability obeys [PERSON]'s law. The calculation formulas of head loss and hydraulic gradient in saturated soil follow [PERSON]'s law, and the permeability coefficient is a stable value, as shown in Equation (7). Calculation of the hydraulic gradient value was performed using Equation (8). As the volume water content of unsaturated soil decreases, the matric suction increases rapidly, and the pore water transport resistance between soil particles increases.