## 1 Introduction
The influence and mechanism of rainfall infiltration on the slope stability and possible rain erosion have been studied through theoretical analysis, numerical simulation, and modelling in previous studies. These studies have mainly focused on the influence of rainfall infiltration on the slope stability [6; 7; 8; 9], slope erosion characteristics [10; 11], and equally erosion generation mechanism [12; 13], as well as factors influencing erosion, such as rainfall [14; 15; 16; 17], slope length [18], and slope angle [19]. [PERSON] et al. [20] have obtained material parameters based on some closed-form expressions derived using physics-based models, and considered two typical triggering mechanisms to predict shallow landslidesunder expected rainfall scenarios. [PERSON] et al. [21] have identified a method to explain the different hazards associated with seasonality of hydrological processes within ash-fall pyroclastic mantle based on landslide physical model reconstruction, in situ hydrological monitoring, and hydrology and slope-stability-modeling conditions. [PERSON] et al. [22] have reconstructed the rainfall history that could have contributed to slope damage, determining the corresponding rainfall duration D (in hours) and cumulative event rainfall E (in mm). Then, using a power-law threshold model, the cumulative event rainfall-rainfall duration (ED) threshold was determined. Combined with FLAC2D, [PERSON] et al. [23] have combined Monte Carlo technology with an FDM (finite difference method) to study the stability of an actual slope, which provided reference for more accurate modeling and monitoring of rock and soil masses. Among the proposed numerical methods, the discrete element method combines the fluid module with the soil from the perspective of micro-dynamics, considers the interactions between water and soil, and can dynamically evolve the large deformation characteristics of the soil during rainfall. Therefore, discontinuous discrete element can be considered the most suitable method for deciphering the erosion mechanism of a coal gauge slope. To date, the specific formation mechanism of the erosion failure of coal gauge slope has not been studied using a three-dimensional discrete element method. [PERSON] et al. [24] have used three-dimensional particle flow (PFC3D) discrete element numerical software to determine that the hydrostatic pressure is closely related to the stability of the slope, and obtained threshold rainfall warning parameters. [PERSON] et al. [25] have used PFC2D to simulate the scouring and spalling process of a steep slope. [PERSON] et al. [26] have applied the PFC3D software to simulate the process of rainfall scouring on loess slopes with fluid-solid coupling, and compared the results with the experimental results from a laboratory test. [PERSON] [27] has conducted fluid-solid coupling simulation using PFC2D. The slope rate suitable for the stability of the roadbed against rain erosion was detected through simulation. However, the erosion mechanism of coal gauge slopes remains poorly understood. A review of the previous literature indicated that discrete element technology has not been applied to decipher the formation and evolution process of the main gullies. Moreover, the discrete element fluid-structure coupling technology has not yet been combined with shear erosion theory. In the present study, therefore, we use PFC3D as the key method, combined with optimized erosion shear failure theory, in order to determine the erosion failure mechanism of coal gangue slopes. The key contributions of this paper are as follows: 1.