# From Climate Conditions to the Numerical Slope Stability Analysis of Surface Coal Mines
## 5 Numerical Analysis of Slope Stability of Lignite Mines under Extreme Rainfall Events  
### Numerical Model Overview  
The finite element method (FEM) is employed to investigate how rainfall affects pit mining slope stability. A fully coupled flow-deformation analysis is conducted using a geotechnical finite element software that allows for the simultaneous evaluation of the unsaturated transient seepage flow and soil mechanical response [47]. For slope stability analysis, the FEM needs to be combined with the strength reduction technique to calculate the safety factor (SF); the soil material's strength is gradually reduced until failure [48]. The simulation of rain infiltration requires the consideration of the unsaturated zone that lies above the groundwater table. The soil-water characteristic curve (SWCC) is usually employed for this type of analysis. It indicates the unsaturated soil's capacity to retain moisture at various stress levels and typically relates water content to soil suction. Furthermore, a relationship is needed to express the unsaturated soil permeability at different moisture levels, i.e., the unsaturated soil's drainage capacity. Finally, during rainfall, the precipitation does not completely infiltrate the soil. A possibly substantial part of it flows as runoff water on the soil's surface. The unsaturated soil permeability governs the part of the rainfall that infiltrates, and a small excess of water is considered to stagnate. The rest of the water is runoff water that does not affect stability. Figure 5: Mean and maximum monthly precipitation projections for the 2020–2099 period for the (**a**,**b**) RCP 4.5 and (**c**,**d**) the RCP 8.5 scenario. Through the SWCC and the permeability function, the assessment of soil behavior in the unsaturated zone is feasible for most geotechnical engineering applications [49]. In this study, unsaturated soil behavior is captured by a form of the widely known van Genuchten-Mualem model [50; 51]. The soil saturation, \(\mathrm{S_{w}}\), in the unsaturated zone, is described as a function of the matric suction, \(\mathrm{u_{w}}\) (negative pore-water pressure), according to:  
\[\mathrm{S_{w}}=\mathrm{S_{r}}+(\mathrm{S_{s}}-\mathrm{S_{r}})\cdot\left[1+ \left(\mathrm{g_{a}}\cdot\frac{|\mathrm{u_{w}}|}{\gamma_{w}}\right)^{\mathrm{g_{ a}}-1}\right]^{\frac{1-\mathrm{g_{a}}}{\mathrm{g_{a}}}} \tag{1}\]  
where \(\mathrm{S_{r}}\) denotes the soil's residual saturation; \(\mathrm{S_{s}}\) the soil saturation at the saturated state; \(\mathrm{g_{a}}\) and \(\mathrm{g_{n}}\) are material parameters governing the shape of the SWCC; and \(\gamma_{w}\) is the water's unit weight (9.81 kN/m\({}^{3}\)). The residual saturation degree \(\mathrm{S_{r}}\) and the material parameter \(\mathrm{g_{a}}\) are critical factors for the unsaturated soil's state [52]. In this work, the residual degree of saturation, \(\mathrm{S_{r}}\), was assumed to equal 0.2, a typical value for fine-grained soils, and the reference \(\mathrm{g_{a}}\) to equal 0.01 m\({}^{-1}\). However, the range of this parameter for fine-grained soil materials of 0.001-0.1 m\({}^{-1}\) was considered through a parametric analysis [16; 27; 30; 53; 54]. This range slightly differs from the literature, referring more to surface mines with fine-grained soils; thus, it has not been thoroughly examined, especially concerning the precipitation range investigated herein. Finally, the parameter \(\mathrm{g_{n}}\) is assumed to equal 1.3, representing fine-grained soils [55]. Figure 6 presents the SWCC for three different values of \(\mathrm{g_{a}}\). As \(\mathrm{g_{a}}\) decreases, the suction needed to initiate soil's desaturation (95% of the saturation degree) increases. The soil parameter \(\mathrm{g_{n}}\) is constant, leading to the same gradient of the linear part of the three SWCCs. Finally, the residual saturation \(\mathrm{S_{r}}=0.2\) is constant, and all three curves will eventually reach that value for large suction. Furthermore, unsaturated hydraulic conductivity is also a matric suction function [56].