## 1 Introduction  
Rock-fill dams account for a large proportion of dams, due to their convenient supply of materials and few geological condition requirements. Considering the construction characteristics and operation mechanisms of rock-fill dams, their main failure modes can be divided into overtopping, slope stability, and seepage failures. Seepage failure is the main cause of rock-fill dam failure besides overtopping failures. Serious seepage failure not only affects the safe operation and economic benefits of reservoirs, but also threatens the safety of people's lives and property in the downstream area. Therefore, it is of great significance to calculate and analyze the seepage of earth rock dams. Theoretically speaking, the seepage calculation of a rock-fill dam involves solving a basic seepage equation with known definite conditions (initial conditions and boundary conditions) to obtain the seepage elements of dams, such as seepage flux, which is an important indicator for the state of dams during their operating period. Accordingly, many methods, such as numerical methods and analytical methods, have been employed to calculate the seepage flux of rock-fill dams. Numerical methods can be used to calculate the leakage of rock-fill dams with homogeneous, heterogeneous, isotropic, anisotropic, or complex boundary conditions. At present, the most accurate numerical methods for seepage analysis include the finite element method [1; 2; 3; 4], the finite difference method [5; 6; 7; 8], and the boundary element method [9; 10; 11; 12]. [PERSON] and [PERSON] [13] numerically studied the seepage characteristics of a face rock-fill dam with different geomembrane defects and reservoir water levels, and the infiltration line was obtained. [PERSON] [14] established a 2D finite element model of an embankment dam with an added vertical sheet pile, in order to study the effect of shapes and location of the sheet pile on the total seepage discharge and velocities. [PERSON], et al. [15] investigated the steady-state seepage through a homogeneous and simple zoned earthen dam using the finite-element method, subsequently researched the characteristics of blanket, chimney, and toe drains. Based on the generalized equivalent continuum model, [PERSON], et al. [16] established a finite element model of a high face rock-fill dam under complex hydraulic conditions and then analyzed the seepage characteristics in the dam area, including the seepage flux, seepage gradient, etc. Additionally, many studies have combined random field concepts with finite element methods to perform Monte Carlo simulations of seepage problem [17; 18; 19; 20; 21; 22; 23]. Analytical methods are also employed to solve seepage problems based on Darcy's law and some assumptions of rock-fill dams to ensure their simplification and intelligibility compared to numerical simulation. Slope-wall dams with horizontal blankets are usually divided into the following two sections during the derivation of seepage flux using an analytical method: One is leakage of the dam body--that is, seepage through the slope wall; the other is leakage of the dam foundation--that is, seepage through the horizontal blanket and the foundation below. Many studies on analytical methods for leakage calculation have been carried out by various researchers [24; 25; 26]. In the early stage of research, [PERSON] [27] pioneered a study of seepage flux through an equal thickness blanket, a curved blanket (approximate triangle), and a combination of natural and artificial blankets of slope-wall dams. After that, inspired by the segmentation method, [PERSON] [28] divided a slope-wall dam into three sections to calculate the seepage flow, but the head loss at the inlet was not considered. Subsequently, the head loss at the inlet was taken into consideration, and a complete formula for an equal thickness blanket was developed under the conditions that the blanket length is greater than the foundation thickness [29].