## 4 Results and Discussion
### Aperture Ratio (\(r_{0}/\)R)  
Unlike SC-CO\({}_{2}\) free jet, the SC-CO\({}_{2}\) will be affected by the casing and cement section of the tunnel. The results presented in previous sections have shown that the larger nozzle outlet radius will increase the number of failure elements. Therefore, the relationship between the nozzle outlet and the inlet of the perforation will affect the results of SC-CO\({}_{2}\) jet fracturing. In previous SC-CO\({}_{2}\) jet fracturing studies, this relationship was ignored. Therefore, in the present work, three cases with aperture ratios (\(r_{0}/\)R) of \(2/3\), \(1\), and \(4/3\) were studied to explore the influence of this relationship. The \"hydraulic isolating ring\" will disappear provided the nozzle outlet diameter is larger than the inlet of casing tunnel (Figure 31), which is due to the reason that the static pressure of annular is larger than that in the casing section. When the aperture ratio was less than \(1\), the perforation pressure was obviously reduced (Figure 30). It is necessary to maximize the aperture ratio as much as possible to ensure as large a pressurization as possible. However, the value cannot exceed unity. In comparison, Figure 27 shows that, when the nozzle outlet radius was increased, the perforation pressure obviously increased. Additionally, a large nozzle outlet will cover the inlet of the perforation to make sure that the CO\({}_{2}\) does not leak from the perforation. With the continuous SC-CO\({}_{2}\) jet, CO\({}_{2}\) was pressurized in the perforation, due to which a higher perforation pressure was obtained. Figure 30: Number of failure elements under different pressure conditions. Figure 29: Stress distribution under various in situ stress conditions. (a) Case for which \(L_{0}=2\), \(r_{0}=0.75\). (b) Case for which \(L_{0}=2\), \(r_{0}=1\). (c) Case for which \(L_{0}=2\), \(r_{0}=1.5\). Further analysis of the stress distribution (Figures 32 and 33) shows that the case for which \(r_{0}\)/R was unity resulted in the highest stress in the \(z\)-direction at the root of the perforation, which had a value of 26.06 MPa. For the case for which \(r_{0}\)/R was unity, the tensile strength exhibited a larger influencing region in the middle of the shale rock than the other two cases. At the tip, the three cases resulted in similar tensile stress values and had values of 19.58, 19.22, and 19.98 MPa. However, at the front of the tip, the case for which \(r_{0}\)/R was 1.33 resulted in higher stress due to higher perforation pressure. This means that the aperture ratio affects the stress at the tip and leads to different crack initiation situations. To accurately investigate the crack initiation at the tip, the stress distribution under different aperture ratios was compared. From Figure 34, it can be seen that, with the increase in aperture ratio, the number of cracks increased. Moreover, it was revealed that the cracks not only initiated at the front of the tip, but were also generated inside the shale rock. Interestingly, the region of maximum tensile stress extended as the nozzle radius increased. Significant differences in cracks were observed in the three studied cases. In the first case, there were only two short cracks initiated at the front of the tip. However, one longer crack grew in the middle of the shale rock (Figure 34a). It is clear that more cracks appeared in the second case than the first one, whereas the area influenced by the stress was the smallest (Figure 34b). The third case not only exhibited a large area influenced by the tensile stress, but also showed many cracks (Figure 34c). Additionally, the number of failure elements also increased significantly (Figure 35). These results were found to be in accordance with those obtained in previous sections. Figure 31: Distribution of the static pressure along the axial direction. Figure 32: \(z\)-directional stress distribution along the wall. Figure 33: \(y\)-directional stress distribution at the tip. Figure 34: Stress distribution under different aperture ratios.