## 4 Results and Discussion
### Perforation Pressure  
To study the influence of perforation pressure, the FSI model and the method were simulated with the same in situ stress and material properties. In some previous studies, the perforation hole was simplified to be of a cylindrical shape. Additionally, the fracture was easily initiated from the tip of the perforation. Here, the perforation hole was simplified to be of a spindle shape. Under a load of nonlinear pressure, the spindle shape exhibited a more complex stress distribution. If the stress along the \(z\)-axis exceeded 0 MPa, the stress was considered the tensile stress. Furthermore, when it increased beyond the value of failure stress, fracture initiated. It was found that the maximum tensile stress (22.8 MPa) appeared at the root of the perforation (Figure 16). The tensile stress appeared on the perforation wall at various locations. The results demonstrated that, with the increase in perforation pressure in the perforation hole, more fracture initiation became apparent. When the perforation pressure was 3 MPa, which was smaller than the in situ stress, the in situ stress determined the initiation position. When the perforation pressure exceeded the in situ stress, the region of the tensile stress shrank, and more independent initiation positions appeared. It can be clearly seen that the tip and the root of the perforation were the most prone to fracture initiation during the SC-CO\({}_{2}\) hydro-jet fracturing process. These results indicate that the \(y\)-directional stress contributed to the tip fracture initiation, while the \(z\)-directional stress contributed to the root fracture initiation. When the perforation pressure increased from 3 to 10 MPa, the \(y\)-directional stress increased from 1.28 to 15.19 MPa (Figure 17). This means that the perforation pressure is proportional to the tensile stress at the tip. Higher perforation pressure is conducive to easy fracture initiation at the tip. At the root of the perforation, the \(z\)-directional stress decreased sharply with the increase in the \(y\)-position (Figure 18). On the perforation's inner wall, the \(z\)-directional stresses were 2.5, 10.81, 16.31, and 22.18 MPa for the perforation pressures of 3, 10, 15, and 20 MPa, respectively. The highest tensile stress was 10 times larger than the smallest tensile stress at the root of the perforation. However, with the increase in the \(y\)-position, the \(z\)-directional stress decreased below 0 MPa, which prevented fracture propagation and hence protected the oil well cement from getting damaged. Figure 15: Axial velocity and static pressure along the axial direction. (**a**) the results of this paper; (**b**) the results of He. The stress contours were obtained after the cracks initiated under the influence of different perforation pressures (Figure 19). The results showed that, with the increase in perforation pressure, more cracks initiated at the tip. In contrast, the length of the cracks initiating at the root decreased. Figure 19 shows that the number of elements supported the above conclusion (Figure 20). The shape  
Figure 16: \(z\)-directional stress distribution along the perforation wall. Figure 17: \(y\)-directional stress distribution at the tip. Figure 18: \(z\)-directional stress distribution at the root. and the path of initiating cracks were distinctly different at the tip of the perforation. When the perforation pressure was smaller than the in situ stress, small cracks appeared at the tip, while the longest cracks were observed at the root. When the perforation pressure rose to more than 10 MPa, more than one initiating crack appeared at the tip. However, when the perforation pressure increased significantly beyond 10 MPa (such as 20 MPa), the crack evolved into larger holes, while some small cracks were generated inside the shale rock. This is because the tensile stress increased inside the shale rock, while the perforation grew larger than the in situ stress.