## 4 Numerical Analysis
### Discussion on the Side Abutment Pressure  
Two sets of working conditions were simulated using the numerical model developed in Section 4.1: in Case 1, without stress was applied to the model before blasting, i.e., \(\sigma_{h}=\sigma_{v}=0\); in Case 2, the curve of side abutment pressure in gob obtained by the theoretical calculation was simplified to Equation (12), in which \(x\) is the distance from the bottom of the blasthole, ranging from 0 m to 12 m. Then, the NIS distributed along the blasthole axis were applied to the boundary of the model. \[\sigma_{h}\ =\ \sigma_{v}\ =\ -1.25x+25 \tag{12}\]  
Following stress initialization, a local coordinate system was employed to obtain the distribution of hoop stress surrounding the blasthole in Case 2 prior to blasting. Four cross sections perpendicular to the blasthole axis were taken, and the distances between cross sections \(S_{1}\), \(S_{2}\), \(S_{3}\), and \(S_{4}\) and the bottom of the blasthole were 12, 8, 4, and 0 m, respectively, and the corresponding confining pressure was varied as 10, 15, 20, and 25 MPa. Figure 8 illustrates the hoop stress distribution around the blasthole for each cross section due to the influence of static stress, where a negative value presents compressive stress. Under isotropic in-situ stresses, the hoop stresses were all compressive with a circular distribution. Furthermore, the hoop compressive stress decreased sharply with increasing distance from the blasthole's center, tending to \(\sigma_{v}\). As the side abutment pressure in the gob increased, the hoop compressive stress gradually increased for locations equidistant from the blasthole's center. The hoop compressive stresses on the blasthole wall for \(S_{1}\), \(S_{2}\), \(S_{3}\), and \(S_{4}\) were 20, 30, 40, and 50 MPa, respectively, all of which were equivalent to 2 \(\sigma_{v}\); the hoop compressive stresses in the distant rock were 10, 15, 20, and 25 MPa, respectively, all of which were equivalent to \(\sigma_{v}\). The hoop stress distribution supported the theoretical calculations presented in Section 2.2 and confirmed the feasibility of the stress initialization used. Figure 8: The hoop stress distribution around the blasthole for \(S_{1}\), \(S_{2}\), \(S_{3}\), and \(S_{4}\) due to isotropic static stresses. The damage distribution of rock for the cross sections along the blasthole axis under the two considered working conditions is illustrated in Figure 9. In Case 1, the rock damage exhibited a uniform distribution within approximately 130 cm from the blasthole axis. In Case 2, the rock damage was unevenly distributed, and the extent of damage gradually declined with the increase in NIS distributed along the blasthole axis. The damage ranges at the orifice, and the bottom of the blasthole were 100 cm and 65 cm from the blasthole axis respectively, which was not conducive to the fragmentation of rock at the bottom of the blasthole. Furthermore, both damage ranges were smaller than those under the stress-free condition (Case 1). Therefore, it can be inferred that the NIS distributed along the blasthole axis inhibit the extension of rock damage, and the inhibiting effect rises with increasing stresses. For gob-side entry under the overhanging hard roof, the side abutment pressure affects the horizontal crack propagation of the blasthole. Therefore, to ensure optimal penetration, the spacing of blastholes should be estimated based on the extent of blast-induced cracks under the maximum confining pressure. To analyze the effects of NIS distributed along the blasthole axis on the damage induced by blasting, a cross section \(5_{0}\) perpendicular to the blasthole axis was taken at any location in Case 1, and four cross sections \(S_{1}\), \(S_{2}\), \(S_{3}\), and \(S_{4}\) perpendicular to the blasthole axis were taken in Case 2, consistent with the locations employed when calculating the hoop stress above.