## 2 Methods
### Mathematical Formulation
#### 2.2.1 Dams with a Low Groundwater Level  
When the groundwater level is lower than the horizontal blanket, there is no direct hydraulic connection between the groundwater and reservoir water. Moreover, the permeability of the slope wall and horizontal blanket are usually much lower than those of the dam body and dam foundation. Under this situation, the slope wall and horizontal blanket are generally saturated, while the dam foundation under the horizontal blanket and the dam body behind the slope wall are unsaturated. Hence, it is assumed that the seepage directions are orthogonal to the slope wall and the horizontal blanket, as shown in Figure 1A. The total leakage of the dam is mainly divided into two parts for calculations: One is seepage through the slope wall; the other is seepage through the horizontal blanket. Figure 1: Layouts generalization of the slope-wall dam with a horizontal blanket (low groundwater level): (**A**) a typical cross section of the dam; (**B**) front view of the simplified slope wall. The coordinate system for a typical cross-section of the dam is established in Figure 1A, with the groundwater level as the x-axis whose positive direction extends from upstream to downstream, where the z-axis is a positive direction from a low elevation to a higher elevation, and point O is the origin. The symbols depicted in Figure 1A are defined as follows: the water level upstream is \(H_{1}\) (above the ground surface), the thicknesses of the slope wall at height \(H_{1}\) and the bottom are \(\delta_{0}\) and \(\delta_{1}\), respectively, the thicknesses of the upstream and downstream end of the blanket are \(t_{0}\) and \(t_{1}\), respectively, the blanket length is \(L_{0}\), the depth from the ground surface to the groundwater table is \(T_{1}\), and the angles of the dam slope upstream and downstream are \(\gamma\) and \(\beta\), respectively. Seepage through the Slope Wall  
The slope wall is divided into \(n\) sections for calculation, which can be seen in Figure 1B. Based on [PERSON]'s law, leakage \(q_{i}\) through the \(iith\) section of the slope wall can be computed as:  
\[q_{i}=KAJ=K_{c}B_{i}\frac{dx}{cos^{\gamma}}.\frac{H_{1}-xMany}{\delta_{1}- \frac{(\delta_{1}-\delta_{0})lux}{H_{1}}x} \tag{1}\]  
where \(K_{c}\) and \(B_{i}\) are the permeability coefficient and width of slope of the wall, respectively, and \(x\) denotes the horizontal distance between the ith section of the slope wall and point O. Under three-dimensional conditions, the front view of the slope wall is simplified as shown in Figure 1B. The leakage through the middle part is \(Q_{1}\), and the leakage through both sides is \(Q_{2}\) and \(Q_{3}\), respectively. The bottom width of the slope wall is \(B_{1}\), and the slope coefficients on both sides are \(m_{1}\) and \(m_{2}\), respectively. The total leakage \(Q_{c}\) of the dam body can be calculated by integrating Equation (1) in the \(\left(0,\frac{H_{1}}{lam\gamma}\right)\) interval:  
\[ \begin{array}{c}Q_{c}=Q_{1}+Q_{2}+Q_{3}\\ =\frac{K_{c}B_{1}H_{1}}{(\delta_{1}-\delta_{0})sin\gamma}\left[H_{1}+\frac{ \delta_{0}H_{1}}{\delta_{1}-\delta_{0}}Int(\frac{\delta_{1}}{\delta_{0}}) \right]\\ +\frac{(m_{1}+m_{2})K_{c}H_{1}}{(\delta_{1}-\delta_{0})sin^{2}\gamma}\left[ \frac{H_{1}^{2}}{2}+\frac{\delta_{0}H_{1}^{2}}{\delta_{1}-\delta_{0}}+\frac{ \delta_{0}\delta_{1}H_{1}^{2}}{(\delta_{1}-\delta_{0})^{2}}Int(\frac{\delta_{ 1}}{\delta_{0}})\right].\end{array}  \tag{2}\]  
When the slope wall is of equal thickness \(\delta\), the total leakage \(Q_{c}\) of the dam body can be rewritten as:  
\[Q_{c}=\frac{K_{c}B_{1}H_{1}^{2}}{2\delta sin\gamma}+\frac{(m_{1}+m_{2})K_{c} H_{1}^{3}}{6\delta sin^{2}\gamma}. \tag{3}\]  
When the groundwater level is too low to affect the seepage field of the foundation below, and the permeability of the unsaturated area is lower than that of the saturated area of the foundation, it is assumed that the water flows vertically through blanket and foundation below.