# Hydraulic Fracture Propagation in Fractured Rock Mass
## 2 Crack Initiation and Propagation
### Crack Propagation  
With the initiation and propagation of the wing crack, the stress intensity factor at the tip of the wing crack will evolve as it propagates. Based on the fracture mechanics [35], the stress intensity factor at the tip of the wing crack under high hydraulic pressure can be simplified into the superposition of the two stress intensity factors, as shown in Figure 5:  
\[K_{I}=K_{I}^{(1)}+K_{I}^{(2)} \tag{8}\]  
The two wing cracks are linked up to form one isolation crack with the length of \(2l\), which under the combined action of the far field stress \(\sigma_{1}\), \(\sigma_{3}\), and hydraulic pressure \(P\), and this crack generates the stress intensity factor \(K_{I}^{(1)}\); simultaneously, the component \(K_{I}^{(2)}\) is due to effective shear stress induced by the presence of the main crack subjected to hydraulic pressure and far field stresses. The main crack length is \(2 al_{ty}\), where \(l_{ty}\) is the influence coefficient. According to the rock fracture mechanics criterion, the stress intensity factor \(K_{I}\) at the tip of the wing crack can be expressed as [36]:  
\[K_{I}^{(1)}=-\big{(}\sigma_{n}^{\prime}-P\big{)}\sqrt{\pi l}=-\frac{1}{2}[( \sigma_{1}+\sigma_{3})+(\sigma_{1}-\sigma_{3})\cos 2(\theta+\beta)-Pl]\sqrt{\pi l} \tag{9}\]  
\[K_{II}^{(2)}=3\tau_{tef}\sqrt{\frac{al_{ty}}{\pi}}\sin^{-1}\biggl{(}\frac{1}{ l_{ty}}\biggr{)}\sin\theta\cos\frac{\theta}{2} \tag{10}\]  
\[K_{I}=3\tau_{eff}\sqrt{\frac{al_{ty}}{\pi}}\sin^{-1}(\frac{1}{l_{ty}})\sin \theta\cos\frac{\theta}{2}-\big{(}\sigma_{n}^{\prime}-P\big{)}\sqrt{\pi l} \tag{11}\]  
\[l_{ty}=\biggl{[}1+\frac{9l}{4a}\cos^{2}\biggl{(}\frac{\theta}{2}\biggr{)} \biggr{]}\biggl{(}1-e^{-\frac{l}{a}}\biggr{)}+0.667\sec^{2}\biggl{(}\frac{ \theta}{2}\biggr{)}e^{-\frac{l}{a}} \tag{12}\]  
\[\sigma_{n}^{\prime}=\frac{1}{2}[(\sigma_{1}+\sigma_{3})+(\sigma_{1}-\sigma_ {3})\cos 2(\theta+\beta)] \tag{13}\]  
Figures 6a and 7a show the variation curves of normalized stress intensity factor \((K_{I}/\sigma_{1}\sqrt{\pi a})\) at the crack tip with the equivalent crack propagation length L = \(l/a\) under different hydraulic pressure \(P\), far-field stress \(\sigma_{3}\), and the crack angle \(\psi\). It is assumed that the maximum crack initiation angle \(\theta=70.5^{\circ}\), the crack length \(a=2.0\) m, the crack connectivity \(\beta=60\%\), the friction coefficient \(\mu=0.3\), \(\sigma_{3}=\lambda\sigma_{1}\), and the pressure transfer  
Figure 4: The crack initiation strength under different factors on confining pressure. (**a**) Theoretical data; (**b**) experimental data [33]. coefficient \(C_{n}\), shear transfer coefficient \(C_{v}\), and cohesion \(C\) are not considered. In order to study the evolution law of stress intensity factor of wing crack tip under different confining pressure and hydraulic pressure, it is assumed that far-field stress \(\sigma_{1}=40\) MPa, and \(\sigma_{3}=0\) (\(\lambda=0\)), 4 MPa (\(\lambda=0.1\)), and 8 MPa (\(\lambda=0.2\)). It shows that, under other given conditions, the stress intensity factor of wing crack decreases with the increase of remote field stress \(\sigma_{3}\), the wing crack tends to expand stably; the stress intensity factor of wing crack increases with the increase of hydraulic pressure \(P\), which indicates that under high hydraulic pressure, the rock crack will expand at a high speed as long as it cracks, and higher hydraulic pressure will lead to unstable expansion. As shown in Figures 6b and 7b, the effects of hydraulic pressure and confining pressure on the dimensionless stress intensity factor are consistent with the ANSYS numerical simulation data in [37]. Figure 5: Superposition of stress intensity factor of wing crack under high hydraulic pressure. (**a**) Wing crack stress intensity factor; (**b**) equivalent of crack stress intensity factor. Figure 6: The relationship curves between dimensional stress intensity factor at the crack tip wing stress and crack propagation length under different confining pressures.