# Hydraulic Fracture Propagation in Fractured Rock Mass
## 2 Crack Initiation and Propagation
where \(\mu\) is the friction coefficient on the crack surface and \(C\) is the cohesion on the crack surface. \(C_{n}\) and \(C_{v}\) are, respectively, the compression transmitting factor and shearing transmitting factor,  
\[C_{n}=\frac{\pi a}{\pi a+\frac{E_{0}}{(1-v_{0}^{2})K_{n}}} \tag{3}\]  
\[C_{v}=\frac{\pi a}{\pi a+\frac{E_{0}}{(1-v_{0}^{2})K_{s}}} \tag{4}\]  
where \(a\) is half the length of the original inclined crack, \(v_{0}\) and \(E_{0}\) are, respectively, the Poison's ratio and elastic modulus of the rock, and \(K_{n}\) and \(K_{s}\) are, respectively, the normal stiffness and shear stiffness on the crack surface. According to the maximum circumferential normal stress criterion, the initial crack extends from the tip of the main crack in a direction \(\theta\) at which the transformed mode I stress intensity factor is at its maximum. \(\theta=70.5^{\circ}\) is obtained [32]. Additionally, stress intensity factor at the crack tip is [32]:  
\[K_{I}=\frac{2}{\sqrt{3}}\tau_{eff}\sqrt{\pi a} \tag{5}\]  
In Equation (5), \(K_{I}=K_{Ic}\), the initial crack strength criterion for the compression-shear rock crack is also obtained:  
\[\sigma_{1}=\frac{\frac{\sqrt{3}K_{Ic}}{\sqrt{\pi a}}+2C-2\mu\beta P+B_{1} \sigma_{3}}{A_{1}} \tag{6}\]  
\[A_{1}=-\mu(1-C_{n})(1-\text{cos}2\psi)+(1-C_{v})\text{sin}2\psi \tag{7}\]  
Referring to the shear fracture test data in the article of [33], a relationship curve between splitting strength values \(\sigma_{1}\) under different \(P\) is obtained (see Figure 2). It shows that the increase of the osmotic pressure results in an approximately linear decrease of the splitting strength value. This is because the hydraulic pressure reduces the normal stress and increases the shear driving force on the crack surface. Similarly, if the hydraulic pressure decreases, the initial crack strength of rock mass will be greater, and less likely to be damaged. At the same time, the confining pressure has a significant enhancement effect on the rupture strength. Figures 3a and 4b are, respectively, the initial crack strength curves under different confining pressure and different hydraulic pressure. It is assumed that crack length \(a=2.0\) m, crack connectivity \(\beta=60\%\), internal friction factor \(\mu=0.3\), the fracture toughness of rock compression shear failure crack \(K_{Ic}=11.2\) MPa m\({}^{1/2}\)[34], and the pressure transfer coefficient \(C_{n}\), shear transfer coefficient \(C_{v}\), and cohesion \(C\) are not considered. It shows that the initial crack strength is inversely proportional to the hydraulic pressure; with the increase of osmotic pressure, the initial crack strength of the fractured rock mass decreases; the crack angle \(\psi=30^{\circ}\) remains unchanged, when the hydraulic pressure increases from 0 to 5 MPa, the initial crack strength decreases from 18.16 to 15.65 MPa, which decreases by 13.8%; in Figure 3a, the crack angle is between 50 and \(60^{\circ}\), and the crack initiation strength changes faster than that of the smaller initial crack angle, indicating that the hydraulic pressure has a more obvious deterioration effect on the crack initiation strength of the larger crack angle. The initial crack strength is proportional to the confining pressure, the cracking strength of the fractured rock mass increase with the increase of the confining pressure; when the crack angle \(\psi=30^{\circ}\) remains unchanged, the confining pressure \(\sigma_{3}\) increases from 0 to 8 MPa, and the initial crack strength increases from 9.30 to 24.00 MPa, with an increase of 158.1%; and the initial crack strength is also relevant to the crack angle, which first decreases and then increases with the increase of the crack angle; when the crack angle \(\psi\) is approximately equal to \(40^{\circ}\), the initiation strength is minimum. Additionally, as shown  
in Figures 3b and 4b, the effects of hydraulic pressure, confining pressure, and the crack angle on the initial strength are consistent with the experimental data in [33].