## 4 Numerical Analysis
### Discussion on Lateral Pressure Coefficient  
Section 4.2 analyzed the effects of NIS distributed along the blasthole axis on the 3D damage distribution of rock caused by a lateral pressure coefficient of 1. However, the actual geological environment is often complex, and anisotropic in-situ stresses have meaningful effects on the crushed zone characteristics and the crack patterns around blasthole. Therefore, the vertical in-situ stress was defined as the side abutment pressure in the gob at a burial depth of 400 m using the numerical model established in Section 4.1. Five different groups of lateral pressure coefficients were established by varying the magnitude of the horizontal in-situ stress, namely \(\lambda\) = 0.25, \(\lambda\) = 0.5, \(\lambda\) = 1.0, \(\lambda\) = 1.5, and \(\lambda\) = 2.0. Similar to the investigation discussed in Section 4.2, cross sections \(S_{1}\), \(S_{2}\), \(S_{3}\), and \(S_{4}\) were taken in each of the five models with varying lateral pressure coefficients. Figures 9(b) and 12 illustrate the damage distribution around the blasthole for these cross sections. It can be deduced that when \(\lambda\
eq 1\), the primary crack tended to expand parallel to the maximum principal stress vector. To measure the degree of damage along the  
Figure 11: Time-dependent hoop stress profiles at A for \(S_{1}\), \(S_{2}\), \(S_{3}\), and \(S_{4}\) in Case 2. Figure 10: The extent of the cracked zones and crushed zones for \(S_{0}\), \(S_{1}\), \(S_{2}\), \(S_{3}\), and \(S_{4}\) under two working conditions: (a) the extent of the cracked zone for \(S_{0}\) in Case 1; (b) the extent of the cracked zones for \(S_{1}\), \(S_{2}\), \(S_{3}\), and \(S_{4}\) in Case 2.  
blashhole axis, the average lengths of primary cracks were determined in both horizontal and vertical directions for the four cross sections at different lateral pressure coefficients and used to generate variation curves (see Figure 13). When the action of all five lateral pressure coefficients, the gradual increase of damage ranges in both horizontal and vertical directions of each cross section was observed from the bottom of the blashhole to the orifice, which was attributed to the Influence of NIS distributed along the blashhole axis. The result corresponds to the investigation presented in Section 4.2. Therefore, the average length of the primary cracks should be considered according to the maximum confining pressure when determining the blashhole spacing. When the vertical loading remained constant and \(\lambda\) increased from 0.25 to 2, the changes of the damage ranges in the horizontal direction stayed within 10 cm, while the damage ranges in the vertical direction of \(S_{1}\), \(S_{2}\), \(S_{3}\), and \(S_{4}\) decreased by 64, 58, 51 and 49 cm respectively. In other words, as the lateral pressure coefficient increases, the damage in the vertical direction of the blashhole shows a significant attenuation trend. However, for non-directional blasting, the expansion of vertical blast-induced cracks may affect the stability of the coal pillar (Figure 14). If necessary, shallow-borehole crack filling should be performed on the coal pillar, and the specific parameters depend on the distribution range of blast-induced cracks within the coal pillar. Therefore, it is essential to guide the shallow-borehole crack-filling scheme based on the influence of NIS distributed along the blashhole axis and the lateral pressure coefficient on vertical cracks. Throughout the blasting simulations, changes in the hoop stress were monitored along both the vertical and horizontal directions of \(S_{1}\), \(S_{2}\), \(S_{3}\), and \(S_{4}\) because the lateral pressure coefficients varied. As indicated in Figure 15, the maximum hoop compressive stress was extracted at each monitoring point by the combined actions of blast loading and NIS distributed along the blashhole axis.