## Appendix A  
As illustrated in Figure 11, the total circumferential stress around the borehole consists of three components, as follows: (1) the circumferential stress component (\(\sigma_{\theta}^{1}\)) resulting from the maximum and minimum horizontal stresses (\(\sigma_{H}\) and \(\sigma_{h}\)); (2) the circumferential stress component (\(\sigma_{\theta}^{2}\)) generated by the injection pressure of the fracturing fluid (\(P_{w}\)); and (3) the circumferential stress component (\(\sigma_{\theta}^{3}\)) induced by the pore pressure distribution (\(p\)) within the rock. The stress distribution in \(\sigma_{\theta}^{1}\) can be visualized as acting on the edges of a large rectangular plate containing a central circular hole. In \(\sigma_{\theta}^{2}\), the wellbore pressure rises from the original pore pressure (\(P_{0}\)) to the injection pressure (\(P_{w}\)). Finally, \(\sigma_{\theta}^{3}\) represents the radial fluid flow into the rock when there is a pressure differential between the injected fluid and the pore fluid, leading to an additional stress component.  
The in situ circumferential total stress field (\(\sigma_{\theta}^{1}\)), resulting from the maximum and minimum horizontal stresses, is calculated using Equation (A1) [28],  
\[\sigma_{\theta}^{1}=\left(\frac{\sigma_{H}+\sigma_{h}}{2}\right)\left(1+\frac {r_{w}^{2}}{r^{2}}\right)-\left(\frac{\sigma_{H}-\sigma_{h}}{2}\right)\left( 1+\frac{3r_{w}^{4}}{r^{4}}\right)cos2\theta+P_{0} \tag{11}\]  
where \(\sigma_{H}\) and \(\sigma_{h}\) represent the maximum and minimum horizontal stresses, respectively, with the condition \(\left|\,\sigma_{H}\,\right|>\left|\,\sigma_{h}\,\right|\). The angle \(\theta\) is measured counterclockwise from the direction of \(\sigma_{H}\), where \(\theta=0\) corresponds to the \(\sigma_{H}\) direction. The variable \(r_{w}\) denotes the borehole radius, while \(r\) represents the horizontal radial distance from the center of the hole.  
The circumferential total stress (\(\sigma_{\theta}^{2}\)) induced by the injection pressure of the fracturing fluid in the wellbore is calculated using Equation (A2),  
\[\sigma_{\theta}^{2}=\frac{r_{w}^{2}r_{e}^{2}p_{w}}{r^{2}(r_{e}^{2}-r_{w}^{2}) }-\frac{r_{w}^{2}p_{w}}{r_{e}^{2}-r_{w}^{2}} \tag{12}\]  
where \(r_{e}\) is the radius of the far-field boundary, and \(p_{w}\) is the pressure difference between the injected fluid pressure and the reservoir pore pressure, defined as \(p_{w}=P_{w}-P_{0}\).  
The circumferential total stress (\(\sigma_{\theta}^{3}\)), arising from the pore pressure distribution within the rock mass, is calculated using Equation (A3),  
\[\sigma_{\theta}^{3}=\frac{\alpha(1-2\
u)}{r^{2}(1-\
u)}\left[\left(\frac{r^{ 2}+r_{w}^{2}}{r_{e}^{2}-r_{w}^{2}}\right)\int_{r_{w}}^{r_{e}}p(r)rdr+\int_{r _{w}}^{r}p(r)rdr-p(r)r^{2}\right] \tag{13}\]  
where \(\
u\) is the [PERSON]'s ratio of the reservoir, \(\alpha\) is the Biot coefficient, ranging from 0 to 1, and \(p(r)\) is the pore pressure at a radial distance \(r\) from the center of the wellbore, exceeding the original pore pressure \(P_{0}\), such that \(p(r)=P_{r}-P_{0}\).  
The circumferential effective stress (\(\sigma_{\theta}\)) is the sum of these three stress components minus the original pore pressure \(P_{0}\), as shown in Equation (A4).  
\[\sigma_{\theta}=\sigma_{\theta}^{1}+\sigma_{\theta}^{2}+\sigma_{\theta}^{3}-P_{0}\] (A4)  
Assuming that the far-field boundary \(r_{e}\) is significantly larger than the wellbore radius \(r_{w}\), the circumferential stress components \(\sigma_{\theta}^{2}\) and \(\sigma_{\theta}^{3}\) can be expressed in simplified forms as follows:  
\[\sigma_{\theta}^{2}=-\frac{r_{w}^{2}p_{w}}{r^{2}}\] (A5)  
\[\sigma_{\theta}^{3}=\frac{\alpha(1-2v)}{r^{2}(1-v)}\bigg{[}\int_{r_{w}}^{r}p( r)rdr-p(r)r^{2}\bigg{]}\] (A6)