## 2 Theoretical Analysis  
### Stress Distribution of Rock under NIS Distributed along the Blashhole Axis  
As can be observed from Figure 1, to sever the stress transmission path between the gob and the roadway roof, a blashhole is typically installed in the roof for a coal pillar by drilling from the roadway into the gob at a fixed angle. The blasting with NIS distributed along the blashhole axis can be simplified to a 3D model, as shown in Figure 2. To realize a convenient calculation, the angle of the blashhole relative to the horizontal direction was simplified to \(0^{\circ}\), and the orifice was positioned at the model boundary near the roadway. The boundary of the model near the gob was chosen as the location to establish the origin of a coordinate system. In this coordinate system, the \(x\) and \(y\) axes were positioned horizontally, while the \(z\) axis was positioned vertically. The \(y\) and \(z\) axes were perpendicular to the blashhole axis, while the x axis was aligned with it. The side abutment pressure in the gob \(\sigma_{v}\) was imposed on the model's upper and lower boundaries, and the model's front and rear boundaries were subjected to the application of horizontal stress \(\sigma_{h}\). Both stresses were non-uniformly distributed along the blashhole axis. Cross section I was perpendicular to the blashhole axis, while cross sections II and III were along the blashhole axis. The normal line of II was consistent with the \(z\) axis, and the normal line of III was consistent with the \(y\) axis. For the sake of computational simplicity, the influence of gob-side entry on the side abutment pressure in gob was ignored. Then, according to limit equilibrium theory, the side abutment pressure in the gob and horizontal stress can be expressed as follows [20]:  
\[\sigma_{v}\ =\ \begin{cases}\xi(P_{i}+Ccot\varphi)e^{\frac{2\gamma t \omega}{m}}-Ccot\varphi&0\leq x\leq x_{0}\\ (K-1)\gamma He^{b(x_{0}-x)}+\gamma H&x>x_{0}\end{cases}  \tag{1}\]  
\[\sigma_{h}\ =\ \lambda\sigma_{v} \tag{2}\]  
\(x_{0}\) can be defined as the distance between the maximum value of side abutment pressure in gob and the boundary of the model adjacent to gob [20]:  
\[x_{0}\ =\ \frac{m}{2\xi tan\varphi}ln\frac{K\gamma H+Ccot\varphi}{\xi(P_{i}+Ccot \varphi)} \tag{3}\]  
where \(\lambda\) is the lateral pressure coefficient; \(H\) is the burial depth; \(\gamma\) is the formation density; \(m\) is the height mining; \(P_{i}\) is the support intensity; \(K\) is the stress concentration factor;\(C\) is the cohesion between the layers; \(\varphi\) is the internal friction angle; \(\bar{\zeta}\) is the triaxial stress coefficient, \(\bar{\zeta}~{}=~{}(1+sin\varphi)/(1-sin\varphi)\); and \(b\) is the attenuation coefficient. The blasting of a single hole caused by static stress was reduced to a plane strain problem for ease of analysis, as presented in Figure 3. Assuming a circular blasthole of radius \(R\) exists in an infinite isotropic rock, and the vertical and horizontal stresses for cross section I are imposed on the boundaries of the plane. By using [PERSON]'s elastodynamic equations, the stress distribution surrounding the circular blasthole can be determined as shown below [21]:  
\[\left\{ \begin{array}{l}\sigma_{\rho}~{}=~{}\frac{\sigma_{v}+\sigma_{h}}{2}(1 -\frac{R^{2}}{\rho^{2}})+\frac{\sigma_{h}-\sigma_{v}}{2}(1-4\frac{R^{2}}{\rho^ {2}}+3\frac{R^{4}}{\rho^{4}})cos2\theta\\ \sigma_{\theta}~{}=~{}\frac{\sigma_{v}+\sigma_{h}}{2}(1+\frac{R^{2}}{\rho^{2} })-\frac{\sigma_{v}}{2}(1+3\frac{R^{4}}{\rho^{4}})cos2\theta\\ \tau_{\rho\theta}~{}=~{}\frac{\sigma_{h}-\sigma_{v}}{2}(1+2\frac{R^{2}}{\rho^ {2}}-3\frac{R^{4}}{\rho^{4}})sin2\theta\end{array} \right. \tag{4}\]  
where \(\sigma_{v}\) is the vertical stress; \(\sigma_{h}\) is the horizontal stress; \(\rho\) is the polar radius; \(\theta\) is the polar angle; \(\tau_{\rho\theta}\) is the shear stress; \(\sigma_{\theta}\) is the hoop stress; \(\sigma_{\rho}\) is the radial stress; and \(R\) is the circular borehole's radial.