## 3 Numerical Simulation Study
### Analysis of Numerical Simulation Results  
Initially, the model involved excavating the southern mining area and then underwent a simulation of 4000 steps under self-weight stress. As shown in Figure 7, this simulation generated a series of displacement contour maps, each representing the model's displacement status at 1000, 2000, 3000, and 4000 steps.  
Comparative analysis of the displacement contour maps at 1000 and 2000 steps revealed that, due to a lack of support and the influence of pre-existing subsidence, areas close to the subsidence zone experienced displacement earlier than other areas. Moreover, the magnitude of displacement in these regions was also the most significant.  
The displacement cloud in each direction is shown in Figure 8.  
Figure 6: (**a**) Model side constraints. (**b**) Velocity constraints to simulate the original collapse. (**c**) Model bottom constraints.  
Figure 7: Displacement cloud for each stage of numerical simulation. (**a**) 1000 steps. (**b**) 2000 steps. (**c**) 3000 steps. (**d**) 4000 step.  
Observing the displacement cloud maps at different stages, it is evident that, as mining-induced collapse progressed, surface subsidence gradually extended to the south. This non-uniform subsidence may lead to the formation of tension cracks consistent with the actual conditions in the study area. As shown in Figure 9, multiple cracks already existed on-site, further corroborating the results of the numerical simulations.  
As mining-induced subsidence developed, its effects gradually extended into the internal structural weak planes within the rock and soil mass. These weak planes, characterized by their lower strength, impeded the effective transmission of stress, leading to disparities in displacement in various directions. This process led to the evolution of weak surfaces in the geotechnical body into fissures, which, in turn, evolved into larger cracks. Based on the displacements and displacement differentials recorded at points A and B on both sides of the model interface, as depicted in Figure 10, it is evident that, with the progression of subsidence, both points A and B exhibited a certain degree of displacement. However, these two points had significant displacement disparities within the rock and soil mass. In particular, in the Y (north) direction, the displacement differential gradually increased, leading to the widening of cracks. In the Z direction, the increasing displacement differential resulted in an observable vertical offset between the two sides of the cracks, manifesting as distinct uneven subsidence. The results from the numerical simulation after 4000 steps revealed that the width of the cracks at the top reached 0.6 m, with a vertical offset of 1.8 m. Furthermore, from Figures 8 and 10, it can be observed that mining-induced surface subsidence led to displacement in the X direction on the south slope, potentially impacting the stability of the existing support structures in the drainage channel.  
Figure 8: Displacement cloud at 4000 steps in each direction. (**a**) X. (**b**) Y. (**c**) Z.  
Figure 9: Diagram of fine tension cracks on the south-side slope.  
Furthermore, a comparison was made between the displacement contour maps for the simulation of mining in the southern area and the displacement contour maps for the model only considering the initial state (as shown in Figure 6), both computed after 4000 steps. Figure 11 presents these displacement contour maps for both scenarios. The research results indicate that the displacement was significantly increased in the upper specific region within the southwest part of the subsidence crater, corresponding to the southern mining area. In contrast, the impact in other areas was relatively minor. This observation may be related to the simplified setup of the initial model. However, in practical production, it is still crucial to closely monitor the potential effects of new mining activities on the existing subsidence area.