## 3 Data and Methods
### Numerical Simulation Software and Steps
#### 3.2.2 Numerical Simulation Steps  
To study the stability of single-step slopes in dump sites, the soil is assumed to be a homogenous material under ideal conditions. Based on this assumption, simplified numerical models for single-step slopes of varying heights and slope ratios were established using the finite element software GeoStudio. The models adopt an elastoplastic constitutive model and the Mohr-Coulomb strength criterion, with the grid discretization employing a combination of quadrilateral and triangular elements.  
To more accurately simulate rainfall infiltration, the SEEP/W module in GeoStudio was utilized to simulate and analyze the seepage lines representing groundwater level changes. By leveraging numerical simulation techniques, the variation of pore water pressure within the slope was determined through the analysis of infiltration lines. The analysis employed the seepage equation implemented in the SEEP/W module [22].  
\[\partial Q/\partial t=\partial/\partial x\ (k_{x}\ \partial h/\partial x)+ \partial/\partial y\ (k_{y}\ \partial h/\partial y)+q \tag{10}\]  
In the above equation, \(h\) represents the total head, \(k_{x_{-}}\) and \(k_{y}\) denote the permeability coefficients in the x and y directions, respectively, q is the boundary recharge rate, q also represents the unit storage capacity, and t is time.  
Van-Genuchten (VG) proposed a mathematical model for fitting the soil-water characteristic curve. The model can be expressed as [23]:  
\[\theta=S_{e}=1/[1+(a\psi)^{n}]^{m} \tag{11}\]  
The parameter relationships among a, n, and m significantly influence the predictive accuracy of the van Genuchten (VG) model. The inclusion of the maximum matrix suction point depends on the discreteness and range of a, m, and n. The VG model aims to optimize the parameter relationships to improve curve-fitting accuracy and enhance predictive precision. Compared to fine-grained soils, the VG model generally achieves better predictive accuracy for coarse-grained soils.  
In this simulation experiment, the soil's volumetric water content function and permeability coefficient function were established based on the VG model, derived from the basic physical parameters of the soil. Using rainfall data provided by the meteorological bureau, the unit rainfall intensity for this 10 h short-duration heavy rainfall was set at 0.05 m\({}^{3}\)/h/m\({}^{2}\). The height of the single-step slope varied between 5 and 30 m, with slope ratios selected as 1:1.5, 1:1.75, 1:2.0, and 1:2.25.  
Using the limit equilibrium theory and the Morgenstern-Price (M-P) method, the stability performance of each slope ratio under heights ranging from 5 to 30 m was calculated for the 10 h continuous rainfall event, with stability analyzed hourly. The modeling process utilized GeoStudio software. First, the SEEP/W module was employed to create a two-dimensional single-step slope model, applying seepage and boundary conditions, along with volumetric water content and permeability coefficient functions, to perform seepage calculations. Subsequently, the SLOPE/W module was used to incorporate geotechnical parameters for limit equilibrium analysis. Models were developed for slopes with heights of 5-30 m and slope ratios of 1:1.5, 1:1.75, 1:2.0, and 1:2.25. An example model of a slope with a height of 5 m and a slope ratio of 1:1.5 is shown in Figure 3.