# Study on Composite Fracture Characteristics and Hydraulic Fracturing Behavior of Hard Rock
## 4 Characteristics of Composite Hydraulic Fracturing in Hard Rock
### Calculation of Critical Rupture Pressure  
Using Equation (3), the initiation angle \(\theta_{0}\) is determined without considering the T stress, and using the same procedure, the initiation angles \(\theta_{0}\) and \(\theta_{0,\rm T}\) are obtained when considering the T stress. The magnitudes of the initiation angles \(\theta_{0}\) and \(\theta_{0,\rm T}\) depend on the stress intensity factors \(K_{\rm I}\) and \(K_{\rm II}\). Therefore, the calculated values of \(K_{\rm I}\) and \(K_{\rm II}\) affect the prediction of the initiation path of hydraulic fractures.  
When hydraulic fractures initiate, the initiation angles of the hydraulic fractures are \(\theta_{0}\) and \(\theta_{0,\rm T}\). At this point, the hydraulic pressure inside the pre-existing fractures and inside the hydraulic fracture hole is equivalent to the formation breakdown pressure. Substituting \(\theta_{0}\) and \(\theta_{0,\rm T}\) into the tangential stress formula of the stress field at the crack tip in polar coordinates, we obtain:  
\[\sigma_{\theta_{c}}=\frac{1}{2\sqrt{2\pi r_{c}}}\cos\frac{\theta_{c}}{2}\! \left[K_{\rm I}^{\rm HF}(1+\cos\theta_{c})-3K_{\rm II}^{\rm HF}\sin\theta_{c} \right]+T^{\prime} \tag{21}\]  
\[\left\{\begin{array}{l}\theta_{c}=\theta_{0},T^{\prime}=0\qquad,\text{No T stress}\\ \theta_{c}=\theta_{0,T},T^{\prime}=T^{\rm HF}\sin^{2}\theta_{0,T}\,\text{T stress} \end{array}\right. \tag{22}\]  
where \(\sigma_{\theta_{c}}\sqrt{2\pi r_{c}}=K_{\rm IC}^{\rm HF}\), \(K_{\rm IC}^{\rm HF}\) is the pure Mode I fracture toughness of the rock under true triaxial hydraulic fracturing conditions. By substituting the stress intensity factors \(K_{\rm I}^{\rm HF}\) and \(K_{\rm II}^{\rm HF}\), given by Equations (19) and (20), respectively, into Formula (21), we obtain the breakdown pressure \(p_{\rm w}\)  
Figure 7: Principle of conformal mapping.  
\[\begin{array}{l}p_{w}\geq p_{b}=\frac{2a^{2}(K_{\rm{H}}-T^{\prime})}{\cos\frac{ \theta_{\rm{c}}}{2}(1+\cos\theta_{\rm{c}})\sqrt{\pi a\left(a^{4}-rs^{4}\right)} }\\ +\frac{1}{2}\Big{[}(\sigma_{H}+\sigma_{h})-(\sigma_{H}-\sigma_{h})\cos(2\beta) +3(\sigma_{H}-\sigma_{h})\frac{\sin(2\beta)\sin\theta_{\rm{c}}}{(1+\cos\theta _{\rm{c}})}\Big{]}\end{array} \tag{23}\]  
Here, according to the calculation formula for \(K_{\rm{II}}\), it is known that when the crack angle is \(0^{\circ}\) or \(90^{\circ}\), i.e., when the pre-existing fracture direction aligns with the direction of the principal stress, the value of \(K_{\rm{II}}\) is 0, resulting in a pure Mode I fracture. Therefore, Equation (23) should be an implicit function equation, requiring numerical iterative calculations. The solution process involves confirming parameters such as in situ stress, fracture hole radius, and pre-existing fracture length. Initially, set the injection pressure \(p_{\rm{w}}=0\). Subsequently, determine the values of stress intensity factors \(K_{\rm{I}}\) and \(K_{\rm{II}}\) using Equations (19) and (20), respectively, followed by calculating the initiation angle \(\theta_{0}\) or \(\theta_{0,\rm{T}}\). Using Equation (23), compute the breakdown pressure \(p_{\rm{b}}\) and check whether \(p_{\rm{b}}\) is equal to \(p_{\rm{w}}\). If the condition is not satisfied, increase the value of \(p_{\rm{w}}\) until \(p_{\rm{b}}=p_{\rm{w}}\). Since \(p_{\rm{w}}\) gradually increases from 0 to equal \(p_{\rm{b}},p_{\rm{b}}\) decreases with the increase in \(p_{\rm{w}}\) from a larger value in the calculation process, so \(p_{\rm{w}}\) gradually approaches the value of \(p_{\rm{b}}\) and is finally equal. Finally, output \(p_{\rm{b}}\). This solution process is illustrated in the flowchart in Figure 8.