## 2 Methods
### Mathematical Formulation
#### 2.2.1 Dams with a Low Groundwater Level
Leakage of the dam foundation includes leakage \(Q_{t_{1}}\) through the horizontal blanket and infiltration \(Q_{t_{2}}\) of the dam foundation through the riverbed surface at the front end of the blanket, as shown in Figure 1A. (1) Leakage \(Q_{t_{1}}\) through the horizontal blanket  
The blanket is divided into \(n\) calculation sections, so leakage \(q_{j}\) of the \(jth\) section can be expressed as:  
\[q_{j}=KAJ=K_{t}B_{i}dx\frac{H_{1}}{t_{0}+\frac{L_{0}+x}{L_{0}}(t_{1}-t_{0})}, \tag{4}\]  
where \(K_{t}\) is the permeability coefficient of the blanket, and \(B_{t}\) denotes the width of the blanket leakage area. Total leakage \(Q_{t_{1}}\) through the blanket can be calculated by integrating Equation (4) in the \((-L,0)\) interval:  
\[Q_{t_{1}}=\frac{K_{t}B_{t}H_{1}L_{0}}{t_{1}-t_{0}}\ln\left(\frac{t_{1}}{t_{0}}\right). \tag{5}\]When the blanket is of equal thickness \(t\), the total leakage \(Q_{t_{1}}\) through the blanket can be rewritten as  
\[Q_{t_{1}}=\frac{K_{s}B_{t}H_{1}L_{0}}{t}. \tag{6}\]  
(2) Infiltration \(Q_{t_{2}}\) of the foundation through the riverbed surface at the front of the blanket  
The Richards equation governing one-dimensional vertical flow in unsaturated soils can be written in the following form:  
\[\frac{\partial}{\partial z}\Big{[}K(\psi)\frac{\partial(\psi+z)}{\partial z} \Big{]}=\frac{\partial\theta}{\partial t}, \tag{7}\]  
where \(K(\psi)\) is the unsaturated hydraulic conductivity, which is a function of the pressure head \(\psi\) (negative for unsaturated flow), \(\theta\) is moisture content, \(t\) is the time, and \(z\) denotes the vertical coordinate pointing upwards. Since this is a steady-state seepage process, Equation (7) can be simplified as:  
\[\frac{\partial}{\partial z}\Big{[}K(\psi)\frac{\partial(\psi+z)}{\partial z} \Big{]}=0. \tag{8}\]  
To linearize Equation (8), the soil water characteristic curve and hydraulic conductivity function of unsaturated soil are described by the Gardner-Russo model as follows [40]:  
\[K(\psi)=K_{s}e^{\alpha\psi} \tag{9a}\] \[\theta(\psi)=\theta_{r}+(\theta_{s}-\theta_{r})e^{\alpha\psi}, \tag{9b}\]  
where \(K_{s}\) is the saturated hydraulic conductivity, \(\alpha\) is the soil parameter related to the pore size distribution, \(\theta_{r}\) is residual moisture content, and \(\theta_{s}\) denotes the saturated moisture content. Using the two constitutive relations above, Equation (8) can be transformed into the following linearized governing equation:  
\[\frac{\partial}{\partial z}\Big{[}K_{s}e^{\alpha\psi}\Big{(}\frac{\partial\psi }{\partial z}+1\Big{)}\Big{]}=0. \tag{10}\]  
A particular solution to Equation (10) requires two boundary conditions:  
* At the steady ground water table, \(\psi_{0}=0\), the boundary condition can be written as \[K(\psi_{0})\Big{|}_{z=0}=K_{s}.\] (11)
* The surface boundary is controlled by pore water pressure (which is the upstream water level \(H_{1}\) constantly): \[q_{j}=KAJ=K_{t}B_{i}dx\frac{H_{1}}{t_{0}+\frac{L_{0}+x}{L_{0}}(t_{1}-t_{0})}, \tag{4}\]0 (12) The solution for Equation (10) with the boundary conditions of Equation (11) and Equation (12) is: \[q_{j}=KAJ=K_{t}B_{i}dx\frac{H_{1}}{t_{0}+\frac{L_{0}+x}{L_{0}}(t_{1}-t_{0})}, \tag{4}\]1 (13)  
By substituting Equation (13) into Equation (9b), the total amount of soil infiltration \(Q_{t_{2}}\) can be calculated as:  
\[q_{j}=KAJ=K_{t}B_{i}dx\frac{H_{1}}{t_{0}+\frac{L_{0}+x}{L_{0}}(t_{1}-t_{0})}, \tag{4}\]2  
where \(\theta_{0}\) is the initial moisture content, and \(A_{s}\) is the leakage area of the dam foundation at the front of blanket. Consequently, the total leakage \(Q_{total}\) can be obtained by adding Equation (3), Equation (6), and Equation (14):  
\[q_{j}=KAJ=K_{t}B_{i}dx\frac{H_{1}}{t_{0}+\frac{L_{0}+x}{L_{0}}(t_{1}-t_{0})}, \tag{4}\]3