## 6 Conclusions  
Based on the improved erosion shear failure theory, we can calculate the erosion depth at any position of the soil as a function of time. The implemented shear stress of water flow can be applied to PFC simulation, thereby solving the problem associated with the particle velocity error. The erosion depth calculated using the improved equation was in good agreement with the erosion depth simulated by PFC, and both were basically consistent with the actual failure depth of the studied gully (No. 3) in the case study: a Guizhou coal gangue slope in Southwestern China. Therefore, the proposed calculation and modelling methods provide valuable new tools for engineering disaster early warning, as well as a simulation method for extreme precipitation events in geotechnical slopes (spatio-temporal connections, forecasting, generation, impact analysis, and vulnerability and risk assessment). The most serious erosion damages occurred at the mid-slope position, followed by those at the top of the slope. The accumulation of particles at the bottom of the slope was striking, and the degree of erosion on the slope surface was the lowest. The erosion damage type for the No. 3 gully was intermittent fragmentary damage. The following failure sequence was determined: initial fracture in the center of the slope, accumulation of continuous faults at the bottom of the slope, piping and failure at the top of the slope and, finally, the formation of a large-scale valley-type gully. The entire process of erosion damage could be divided into three stages: catchment-fracture, erosion-accumulation, and piping-penetration failure. The fracture of the force chain was the most severe in the first stage of erosion, where the coordination number decreased from 5.3 to 4.0, while the porosity increased from 0.42 to 0.45. In the third stage of erosion, the fracture of the force chain between the bottom and the top of the slope was striking, and the value of the coordination number decay reached 1.0-1.2. The peak porosity in the X2 region increased by up to 20%, forming an \"M\"-shaped course of the curve in the y-direction. Lateral erosion and expansion occurred at the mid slope, leading to a tendency to form tributaries, and also serving as the main condition for the formation of multiple ravines. This phenomenon should be further studied in the future. The unbalanced force value gradually increased from the top to the bottom of the slope. The degree of fluctuation was gradually severe, increasing to 16,000 N. The unbalanced force was generally lower in Section A than in Section C, and the direction of the water flow is more inclined in the x-direction. After ca. 37 min of water flow erosion, the unbalanced force reached 7500 N, inducing lateral erosion and expansion of the gully. The results indicated that, in engineering management, more attention should be paid to the lateral expansion and damage effects caused by rain erosion, in order to effectively avoid further debris flow and landslide disasters. The maximum kinetic energy of the erosion damage in No. 3 gully was concentrated in the first stage (25 MJ). The energy consumption of No. 3 gully during the water flow erosion was concentrated in the first stage of the central section of the slope (Section B). The other part was focused in the third stage, at the toe and top of the slope (Section C). Accordingly, we suggest adding a section of anti-erosion reinforcement treatment from the central section of the slope to the toe of the slope. Conceptualization, Y.T. ; Formal analysis, F.Z. ; Funding acquisition, X.L. ; Investigation, J.L., D.X. and Q.F. ; Project administration, Q.F. ; Software, Y.T. and J.L. ; Supervision, X.L. ; Validation, Y.T. and D.X. ; Writing--original draft, Y.T. ; Writing--review & editing, Y.W. All authors have read and agreed to the published version of the manuscript. This work was financially supported by Research on coordinated control technology of cadmium pollution in farmland soil and groundwater in Chengdu Plain (Project No. 2021 YFN0128).