## 4 Results and Discussion
### Nozzle Outlet Length-to-Diameter Ratio (L/\({}_{0}\))  
In order to estimate the influence of length-to-diameter ratio of the nozzle, three cases were simulated and their results compared. For SC-CO\({}_{2}\) hydro-jet fracturing, the perforation pressure, which was boosted by the SC-CO\({}_{2}\) jet, was the point of focus. As the radius of the nozzle outlet changed from 0.75 to 1.5 mm, the perforation pressure in the perforation tunnel sharply decreased with increases in \(z\) in the nozzle section, although the value remained steady throughout the annular (Figure 26). In the casing and cement section of the perforation, a significant difference among all cases was observed. When the radius was 1.5 mm, the total pressure sharply rose to its peak value (25.5 MPa) and remained steady in the casing and cement section. However, when the nozzle outlet was reduced to 1 mm, the total pressure gradually increased to its peak value (21.7 MPa) and then dropped a little to reach the value of its perforation pressure (21.5 MPa). In contrast to these two cases, when the radius was 0.75 mm, the total pressure first decreased slowly and then started to increase steadily after 7.4 mm. These results demonstrated that a larger nozzle outlet radius can reach a higher perforation pressure, while a smaller radius will contribute to generate the best \"hydraulic-fracturing ring\" in the perforation tunnel. On the other hand, a higher perforation pressure will contribute to crack initiation. Therefore, efforts should be made to decrease the value of the length-to-diameter ratio as much as possible, such that the energy loss remains within the acceptable range.  
Figure 25: Number of failure elements under different pressure values.  
Figure 26: Distribution of the total pressure on the perforation tunnel wall.  
The influence of length-to-diameter ratio on the distribution of stress in the shale reservoir was also investigated. The results showed that the higher value of length-to-diameter ratio (_L/r\({}_{0}\)_) caused a higher stress at the root and at the tip of the perforation. Figure 27 shows the distribution of \(z\)-directional stress along the wall of the perforation. The results reveal that the length-to-diameter ratio (_L/r\({}_{0}\)_) changed slightly with the \(z\)-directional stress at the middle of the perforation. However, at the root (Figure 28), the case for which \(L_{0}\) = 2 and \(r_{0}\) = 0.75 obtained the highest stress (26.33 MPa), while the other two cases resulted in stress values of 20.88 and 21.92 MPa. For the tip position, the case for which \(L_{0}\) = 2 and \(r_{0}\) = 0.75 also obtained the largest value of \(y\)-directional stress (20.74 MPa).  
It is observed again that the value of length-to-diameter ratio (_L/r\({}_{0}\)_) significantly affected the stress distribution (Figure 29). When the three cases were compared, it was found that, when \(r_{0}\) was equal to 1, the biggest region of maximum tensile stress obtained. However, the third case, for which \(r_{0}\) was 1.5, obtained the largest region of tensile stress and had higher potential for crack initiation during SC-CO\({}_{2}\) hydro-jet fracturing. It was noticed that the calculated stress profiles exhibited a non-linear distribution and different maximum values, which was attributed to an SC-CO\({}_{2}\) hydro-jet and inhomogeneous material properties. The failure elements were calculated by the software. Surprisingly, the trend of the curve indicated that a larger nozzle radius (\(r_{0}\) = 1.5) could result in a greater number of cracks (Figure 30).  
Figure 28: \(y\)-directional stress distribution at the tip.  
Figure 27: \(z\)-directional stress distribution along the wall.