## 2 Methods
### Mathematical Formulation
#### 2.2.2 Dams with a High Groundwater Level  
When the groundwater level is at/or higher than the horizontal blanket, the dam foundation and dam body below the phreatic line are in a saturated state. The coordinate system and head distribution for a typical cross-section of the dam is established in Figure 2A, with the impervious bed as the x-axis positive to the right and the z-axis in the positive direction from low to high elevation, with point O as the origin. The symbols are also depicted in Figure 2A and defined as follows: The water level downstream is \(H_{2}\) above the ground surface, the thicknesses of the slope wall and blanket are \(\delta\) and \(t\), respectively, the depth of the permeable foundation is T, the horizontal distance from the bottom of the slope wall to the beginning of the filter is \(L_{1}\), the head losses at the upstream and downstream ends of the blanket are \(\Delta h_{0}\) and \(\Delta h_{1}\), respectively, and the water level behind the slope wall is \(h_{1}\). The other symbols are the same as those described in Section 2.2.1. Considering the seepage of the layered soil, the equivalent permeability coefficients of the foundation are [41]:  
\[\overline{K_{y}}=\frac{\sum\limits_{i=1}^{n}l_{i}}{\sum\limits_{i=1}^{n}\frac {l_{i}}{K_{i}}} \tag{16a}\] \[\overline{K_{x}}=\frac{\sum\limits_{i=1}^{n}l_{i}K_{i}}{\sum\limits_{i=1}^{n}l_ {i}} \tag{16b}\]  
where \(l_{i}\) is the thickness of the _ith_ layer, and \(K_{i}\) is the permeability coefficient of the _i-th_ layer. The seepage flux of the slope-wall rock-fill dam with a horizontal blanket is calculated by the subsection method, taking the slope wall and blanket (including the foundation under the blanket) as section I and the dam body behind slope wall and the foundation under the dam body as section II, which can be seen in Figure 2A. Figure 2: Layout generalization of the slope-wall dam with a horizontal blanket (high groundwater level): (**A**) a typical cross section of the dam; (**B**) front view of the simplified slope wall. Seepage Flux \(q_{1}\) Per Unit Width through Section I  
1. Seepage flux \(q_{i}\) per unit width of the foundation passing through the vertical section at the end of the blanket If the ratio of the foundation permeability coefficient \(\overline{K_{x}}\) to blanket \(K_{t}\) is greater than 10, it is assumed that the water flow is vertical through the horizontal blanket and horizontal through the dam foundation [27]. Seepage flux through the blanket with \(dx\)-length is written as: \[dq_{t}=K_{t}\frac{\Delta h}{t}dx.\] (17) The increment of the \(dx\)-length foundation seepage flux is \[dq=\overline{K_{x}}T\frac{d^{2}(\Delta h)}{dx^{2}}dx.\] (18) The seepage differential equation for the blanket can be obtained as follows by equating Equation (17) with Equation (18): \[\frac{d^{2}(\Delta h)}{dx^{2}}=a_{1}^{2}\Delta h,\] (19) where \(a_{1}=\sqrt{\frac{K_{t}}{K_{x}T}}\). The boundary conditions are expressed as: \[\left\{ \begin{array}{ll}\Delta h=\Delta h_{0}&x=-L_{0}\\ \Delta h=\Delta h_{1}&x=0\end{array} \right..\] (20) The general solution of Equation (19) subjected to the boundary conditions given by Equation (20) is: \[\Delta h=\frac{1}{sh(a_{1}L_{0})}[\Delta h_{1}sh[a_{1}(L_{0}+x)]-\Delta h_{0} sh(a_{1}x)].\] (21) The seepage gradient of the foundation can be written as: \[J=\frac{a_{1}}{sh(a_{1}L_{0})}[\Delta h_{1}ch[a_{1}(L_{0}+x)]-\Delta h_{0}ch( a_{1}x)].\] (22) Therefore, the seepage flux at the front end of the blanket can be deduced: \[q_{0}=\overline{K_{x}}TJ_{x=-L_{0}}=\frac{a_{1}\overline{K_{x}}T}{sh(a_{1}L_{ 0})}[\Delta h_{1}-\Delta h_{0}ch(a_{1}L_{0})],\] (23) which can also be written as: \[q_{0}=\overline{K_{x}}\frac{\Delta h_{0}}{\zeta},\] (24) where \(\zeta=0.44\).