## 2. Saturated-Unsaturated seepage Model  
In recent years, the mathematical model based on saturated-unsaturated soil seepage provides a powerful means for studying transient unsaturated soil seepage. Due to the dynamic distribution of pore water pressure caused by the fluctuation of river water level, it is taken to be an important parameter to evaluate the induced seepage deformation of unsaturated soil. A large number of studies show that matric suction is the most important variable in unsaturated soil. Due to the complexity of its theory and application, its application scope is limited. Numerical analysis has proved to be an appropriate and effective tool to further consider the inherent variability of soil properties. In 1856, the French engineer [PERSON] proposed the linear seepage theory in saturated soil. In 1980, [PERSON] obtained the relationship between permeability and volume moisture content of unsaturated soil based on test statistics and established the van Genuchten model. This model expresses the characteristic parameters of water and soil through test data and incorporates the soil-water characteristic curve of unsaturated soil and seepage control differential equation into the mathematical model. In this paper, the medium is isothermal, and no chemical reactions were taken into account. It is assumed that the medium movement between the soil pores is single-phase flow, and the soil particles are filled with water and stagnant air. The motion of water in soil can be described using a flow control equation. When the water level changes, the water transfer of unsaturated soil is different from that of saturated soil, and the permeability coefficient is no longer a constant. It changes constantly with the change in soil-water content or matric suction. A generalized two-dimensional seepage differential equation based on [PERSON]'s law can describe the transient flow of water at any position in the soil under saturated-unsaturated conditions [30; 31]. The following equation was derived by [PERSON] for predicting the relative hydraulic conductivity from knowledge of the soil-water characteristic curve [32]. The general expression of this equation can be written as:  
\[ \begin{cases}\frac{\partial}{\partial x}\Big{(}k_{sx}\frac{\partial H}{ \partial x}\Big{)}+\frac{\partial}{\partial y}\Big{(}k_{sy}\frac{\partial H}{ \partial y}\Big{)}+Q_{w}=\frac{\partial\theta_{w}}{\partial t}&[(u_{a}-u_{w}) \leq(u_{a}-u_{w})_{b}]\\ k_{w}(\theta)=\left[\frac{(u_{a}-u_{w})_{w}\eta^{0}(e^{-\xi_{0}})}{u_{a}-u_{w}} \right]^{\alpha}&[(u_{a}-u_{w})>(u_{a}-u_{w})_{b}]\end{cases}  \tag{1}\]where \(k_{w}\) and \(k_{s}\) are the permeability coefficients of unsaturated and saturated soil, respectively, in cm/s; \(k_{sx}\) and \(k_{sy}\) are the permeability coefficients in \(x\) and \(y\) directions, respectively, in cm/s; \(H\) represents the total head, in m; \(Q_{w}\) is the unit volume flow applied on the boundary, represented as \(1/s\); \(t\) refers to time, in seconds; \((u_{a}-u_{w})_{b}\) indicates the generalized limit value of suction between saturated and unsaturated regions; \(e\) is the pore ratio, when \(e=e_{0}\), \((u_{a}-u_{w})_{b}=(u_{a}-u_{w})_{0}\); where a and \(\alpha\) are the parameters related to the intake value, in \(1/\)kPa, given here as Equations (2) and (3), respectively:  
\[a=\Delta\lg(u_{a}-u_{w})_{b0}/\Delta e \tag{2}\]  
\[\alpha=\eta_{0}+d(e-e_{0}) \tag{3}\]  
where \(u_{a}\), \(u_{w}\) are the pore pressure and pore water pressure, in kPa; when \(e=e_{0}\), \(\eta=\eta_{0}\), and \(d=\Delta\eta/\Delta e\); \(\theta_{w}\) represents the water content per unit volume of soil, in a percentage.