## 4 Results and Discussion  
### Flow Field and Stress Filed  
#### 4.1.1 Flow Field and Stress Filed before Fracture Initiation  
After the solution, the flow field and the stress field before the initiation of fracture were obtained, and are shown in Figure 11. Three distinct features can be observed in Figure 11. Firstly, the SC-CO\({}_{2}\) jet velocity is shown, whereas the core of the jet velocity was at the end in the section of cement. At the end of the velocity core, discontinuity in the velocity occurred, which was due to the shock waves.  
\begin{table}
\begin{tabular}{c c c} \hline
**Item** & **Symbol** & **Value** \\ \hline pressure difference (MPa) & \(\Delta P\) & 10 \\ jet inlet pressure (MPa) & \(P_{in}\) & 20 \\ ambient pressure (MPa) & \(P_{am}\) & 10 \\ nozzle outlet length (mm) & \(L_{0}\) & 2 \\ perforation pressure (MPa) & \(P_{t}\) & 3/15/15/20 \\ in situ stress (MPa) & \((\sigma_{fz},\sigma_{fg})\) & \((5,10)/(10,10)/(12,10)/(15,10)\) \\ nozzle outlet radius (mm) & \(r_{0}\) & 0.75/1/1.5 \\ nozzle-to-casing inlet radius ratio & \(r_{0}/R\) & \(\frac{\delta}{3}/1/\frac{\delta}{3}\) \\ Weibull parameter & \(\alpha\) & \(2,3,4\) \\ Weibull parameter & \(\beta\) & \(1.2,1.6,2.0\) \\ \hline \end{tabular}
\end{table}
Table 2: Various parameters and their values used in the simulation.  
Secondly, the extreme tensile stress appeared when the SC-CO\({}_{2}\) jet was pressurized in the perforation tunnel. The discontinuity of the structure and the material resulted in extreme values of the tensile stress in the root of the perforation tunnel. Thirdly, the tensile stress concentration contributed to the fracture initiation and propagation. Many researchers have previously reported on the principle of tip fracture initiation.  
However, even the inhomogeneous properties of shale rock played an important role in the fracture initiation, the SC-CO\({}_{2}\) hydro-jet was the key in improving the inner pressure, which contributed to the rock's tensile stress field. The pressurization process affected the stress field of the casing, cement, and shale rock. As shown in Figure 12, this was due to the strongly turbulent velocity of SC-CO\({}_{2}\), which was different from that of the water jet. On the FSI boundary, pressure and velocity exhibited nonlinear changes. This nonlinear distribution on the FSI boundary influenced the stress field in shale rock, which led to fracture initiation at different positions.  
Figure 11: Flow field and stress field before fracture initiation.  
Figure 12: Flow field and stress field before fracture initiation.  
Figure 9 also shows the phenomenon of the \"hydraulic-fracturing ring\". Due to the high velocity of the SC-CO\({}_{2}\) jet core, the fluid in the annular was carried into the perforation tunnel. Closed to the inlet of the perforation, higher velocity led to a zone of negative pressure, which was beneficial in securing the reservoir from damage. This would also be better for casing, which will not bear a larger pressure during hydro-jet fracturing. Inside the perforation tunnel, the SC-CO\({}_{2}\) jet continuously expanded and boosted the perforation pressure. The strong turbulence occurred in the mixing layer of SC-CO\({}_{2}\) jet and led to non-uniform distribution of the perforation pressure on the FSI boundary.