## 2 Erosion Theory
### Construction of the Erosion Model in PFC 3D  
The PFC particle flow method regards materials (e.g., rock, sand, and clay) as discontinuous, and regards the particulate matter composing the material as an independent basic unit. The interactions between particles reflect the macroscopic mechanical properties of the material. The particle flow numerical analysis method defines a material as a particle aggregate composed of finite particles, where a particle is a rigid body with mass; in three dimensions, a particle is a spherical particle with a unit mass. The basic idea of the discrete element method is to obtain the generalized unbalanced force of the particle through the force-displacement law, and then use [PERSON]'s second law to calculate the particle motion behavior under the action of the unbalanced force. In particular, the particle flow method uses [PERSON]'s second law alternately with the force-displacement law to perform cyclic calculations. In the discrete element simulation research, because the size of the actual soil particles is too small, it is almost difficult to use the actual soil particles for numerical calculation. Therefore, in the PFC simulation, the particle size is often enlarged, and the specific magnification factor is usually determined based on the needs of the research itself and the computing resources of the hardware. The dimensions of the numerical model were 130 m \(\times\) 66 m \(\times\) 66 m (length \(\times\) width \(\times\) height), considering the actual size. The particle size of the slope was correspondingly increased to avoid the problem of non-convergence and underestimation of the efficiency caused by the huge number of particles and the disparity of the particle size. Studies on the effect of the particle size [33; 34; 35; 36; 37; 38; 39] have shown that, for ratios between the model size and an average particle size \(>\) 30, appropriate changes in particle size will not affect the results of the simulation. The designed model was repeatedly checked and debugged. On the basis of improving the calculation efficiency and ensuring the rationality of the rainfall damage model, the average particle size of the particles was 1.9 m, the ratio of the minimum model size (66 m to 1.9 m) was 35, and the maximum model size was 35. The ratio of size (130 m to 1.9 m) was 68. Both data were much larger than the standard value of 30, so the particle size in this model was considered to be reasonable. At the same time, these ratio data are determined without affecting the simulation accuracy and taking into account the speed of numerical calculation. The particle size of the coal gangue soil triaxial test used in the meso-parameter calibration in this paper is consistent with the particle size of the final coal gangue slope numerical model. The total number of particles in the model was 12,786. The particles with different gradations in the model are shown with different colors in Figure 5, in order to visualize the gradation of different particles. Combined with an indoor gradation test of soil, the particle gradation in PFC is shown in Table 1. The meso-parameters of the model are summarized in Table 2, according to the debugging of the triaxial test (Figure 6). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Granules** & **Coarse Gravel** & **Medium Gravel** & **Fine Gravel** \\ \hline Particle size (m) & 2.2–2.4 & 1.4–1.8 & 1.2–1.4 \\ Amount (\%) & 15.22 & 69.43 & 15.35 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Particle gradation table. The data monitoring and output in the present study were mainly realized through measuring balls and monitoring points (Figure 5). We set up 60 measuring balls with a radius of 5.5 m, which were evenly distributed on the entire slope surface to monitor the real-time changes in soil porosity and coordination number over the entire slope surface, following the method described by [PERSON] [40]. The area comprising the measurement balls was subdivided into three areas (X1, X2, and X3) in the direction of the x-axis.