## 4 Discrete Element Numerical Model and Analysis Index
### Comparative Simulation of Narrow Pillar Width  
Firstly, the mining of seam No. 2 is completed in the model. According to the requirements of gob-side entry driving with a pillar, the width of the narrow pillar in lower seam is set to 4 m, 6 m, 8 m, 10 m, and 12 m, respectively, corresponding to five comparison models. Three types of analysis indexes are selected for comparison: (1) vertical stress is used to analyze the lateral pressure's distribution; (2) the maximum shear stress analysis of rock mass shear failure area and degree; (3) the second invariant of deviatoric stress (\(J_{2}\)) based on the maximum distortion energy criterion, which is applied to analyze the region and degree of plastic deformation [16; 38; 39]. The results of the comparative simulation are shown in Figure 10.  
Figure 10 illustrates the comparison of three indexes of five different width coal pillar types. The analysis is summarized as follows: (1) When the pillar's width is set to 4 m, the pillar has obvious stress concentration. Currently, the vertical stress peak reaches 34.8 MPa, and the shear stress peak is 11.4 MPa. When pillar's width is 6 m or 8 m, respectively, the peak vertical stress of the coal pillar is 36.3 MPa and 12.8 MPa, respectively, and the peak shear stress is 28.0 MPa and 10.5 MPa, respectively. Compared with the case of retaining an 8 m pillar, when a 6 m pillar is retained, the pillar stress concentration area is more extensive, which will make the deformation more difficult to control. (2) \(J_{2}\) represents a crucial metric for gauging plastic failure and the damage degree of rock. When the pillar's width is set at 4 m or 6 m, a pronounced \(J_{2}\) concentration manifests in the central part of the pillar. This clearly indicates that the pillar's plastic failure is more severe under such circumstances. Conversely, once the pillar's width exceeds 8 m, the \(J_{2}\) concentration degree is markedly attenuated.  
Figure 10: Model simulation results of different pillar widths.  
The monitoring line is located in the stope middle to further quantify the change in each index value under different pillar widths, as depicted in the illustration in Figure 11.  
From the perspective of stress monitoring, when the pillar's width is set at 4 m and 6 m, all three types of stress index exhibit a single-peak shape within the pillar, featuring significant stress concentration and large deformation. When the pillar's width is 8 m or above, the stress assumes the form of \"double-peaks and one-valley,\" and the stress indexes decrease significantly. The stress curve transforms from a single-peak parabolic type to a double-peak saddle type, indicating that there is an elastic area with favorable support performance in the pillars of 8 m or above. When the pillar's width is 10 m or 12 m, the differences between each index and 8 m are small and this would cause resource waste. Considering that a pressure-relief operation will be conducted, the support performance of the pillar will be further guaranteed. Furthermore, the pillar's width along the goaf is typically 3 m-8 m, as confirmed by the mines under numerous conditions. Hence, considering the geological parameters, model results, resource recovery rate, pressure relief measures, and engineering experience, the pillar's reasonable width is designed to be 8 m.  
Figure 11: Monitoring of stress indexes under different pillar widths: (**a**) vertical stress; (**b**) maximum shear stress; (**c**) the second invariant of deviatoric stress.