## 4 Results and Discussion
### In-Situ Stress  
Many previous studies have shown that the in situ stress shows important effect on the crack initiation process. The difference between the \(y\)-directional in situ stress and the \(z\)-directional in situ stress would obviously affect the initiation and propagation of major fractures. In this regard, four cases with the same perforation pressure and rock material properties were investigated to study the influence of difference values of in situ stress. The perforation pressure was 10 MPa, and in all cases the material properties were those of the shale rock. The \(z\)-directional stress curve indicated that, when the in situ stress in the \(z\)-direction was smaller than that in the \(y\)-direction, the tensile stress exceeded the failure value on the perforation wall (Figure 21). With the decrease in difference between the \(y\)-direction in situ stress and the \(z\)-direction in situ stress, the \(z\)-directional stress obviously decreased. This means that the difference in in situ stress is favorable for the creation of cracks in the middle of the perforation region.  
For the tip region of the perforation, when the value of \(z\)-directional in situ stress varied from 5 to 15 MPa, the maximum tensile stress values were found to be 17.81, 18.87, 19.86 and 20.93 MPa (Figure 22). It means that the higher \(z\)-directional in situ stress created the cracks more easily at the tip of the perforation. However, at the root of the perforation (Figure 23), the smallest \(z\)-directional in situ stress resulted in the largest tensile stress in the \(z\)-direction. Therefore, during the SC-CO\({}_{2}\) hydro-jet fracturing, the in situ stress in the direction perpendicular to the perforation should be greater than that in the parallel direction, and will be favorable for the fracture initiation of the tip and prevention of the crack initiation at the root.  
The results presented in Figure 24 showed that the value of \(z\)-directional stress, which contributed to the crack initiation at the middle and at the root of the perforation, were found to be in accordance with the ones obtained above. With the increase in \(z\)-directional stress, the region that was under the influence of tensile stress became relatively smaller. On the other hand, the values of tensile stress at the tip and at the root of the perforation became larger. To accurately evaluate the fracture initiation, the number of failure elements provided the best results. The results showed that larger difference in the in situ stress benefited the crack initiation (Figure 25). When the casing safety and the cracks' initiation at the tip were considered, the better value of in situ stress in the \(z\)-direction became less than that in the \(y\)-direction. The results indicated that the perforation angle played an important role in the hydro-jet fracturing. Based upon the in situ stress distribution, the proposed method will effectively improve the fracture quality.  
Figure 21: \(z\)-directional stress distribution along the perforation wall.  
Figure 23: \(z\)-directional stress distribution at the root.  
Figure 24: Stress distribution under different in situ stress situations.  
Figure 22: \(y\)-directional stress distribution at the tip.