# Study on Composite Fracture Characteristics and Hydraulic Fracturing Behavior of Hard Rock
## 4 Characteristics of Composite Hydraulic Fracturing in Hard Rock
### Influence of T Stress on \(p_{b}\) and \(\theta_{\rm{c}}\)  
The hydraulic fracture mechanics model under biaxial compression can be simplified as an inclined crack in an infinite plate under biaxial compression [30; 44; 45]. This mechanical model can directly provide a closed-form solution for the T stress parallel to the crack. Therefore, the T stress level is calculated according to the following equation:  
\[T=-0.5[(\sigma_{H}+\sigma_{h})-(\sigma_{H}-\sigma_{h})\cos 2\beta] \tag{24}\]  
As indicated in Section 3.2, T stress has a certain impact on the mixed-mode fracture characteristics of hard rocks, and this impact becomes more significant with the increase in the proportion of the Mode II stress intensity factor. To investigate the influence of T stress on the hydraulic fracture characteristics, based on Equations (3) and (23) and combined with the physical and mechanical parameters of the test material, the critical fracture pressure and fracture angle considering T stress and without considering T stress  
Figure 8: Fracture pressure calculation flowchart.  
are calculated. The results are shown in Figure 9. Here, considering the study focus on hard rocks, the critical fracture distance \(r_{\rm c}\) is relatively small, with values of 0.08 mm and 1 mm, respectively. The Poisson's ratio is \(v=0.235\); the maximum horizontal principal stress is \(\sigma_{\rm H}=9\) MPa; the minimum horizontal principal stress is \(K_{\rm IC}=0.84\); the rock tensile strength is \(\sigma_{\rm t}=13.5\) MPa; the fracture width is \(r_{\rm w}=0.006\) m; and the fracture length is \(a=0.009\) m.  
Figure 9 presents the critical hydraulic fracturing pressure and fracture angle calculated using Equation (23) considering and not considering T stress. The results show that when T stress is considered, the calculated critical fracturing pressure and fracture angle are relatively lower compared to the case without considering T stress. Particularly when the symmetric crack makes an angle \(\beta=45^{\circ}\) with the maximum horizontal principal stress, the impact of T stress on the critical fracturing pressure is most significant. Regarding the influence of T stress on the fracture angle, at \(\beta=45^{\circ}\), T stress has the most notable effect on the fracture angle \(\theta_{\rm c}\). However, at \(\beta=0^{\circ}\) and \(90^{\circ}\), \(\theta_{\rm c}\) is almost unaffected by T stress. By comparing the critical fracturing pressure and fracture angle calculated for different \(\beta\) values, it can be concluded that as \(\beta\) increases, the influence of T stress on the critical fracturing pressure gradually increases, but its impact on the fracture angle is relatively small. Furthermore, comparing Figure 9a,b, it can be observed that during the hydraulic fracturing process, when the critical crack size \(r_{\rm c}\) of the rock is larger, T stress has a more significant impact on both the critical hydraulic pressure and fracture angle. In contrast, for rock materials with a smaller \(r_{\rm c}\), T stress tends to affect the fracture angle more, with a relatively smaller impact on the critical fracturing pressure. The rock samples in this study exhibit obvious brittleness and belong to hard rocks with a very small critical crack size \(r_{\rm c}\). Therefore, the influence of T stress can be neglected in practical calculations.