## 5 Conclusions  
The present study investigated the initiation of inhomogeneous shale fracture under the boosted pressure of an SC-CO\({}_{2}\) hydro-jet. Based on the field observations and experiments, the FSI model of SC-CO\({}_{2}\) hydro-jet fracturing is proposed using the method of user-defined code and two-step simulations. The mechanism of SC-CO\({}_{2}\) hydro-jet fracturing was explored, and the effects of various factors, such as the perforation pressure, the in situ stress, and the aperture ratio, were analyzed and discussed. Based on the results, the following conclusions were drawn.  
1. From the experiments and SEM analysis, shale rock exhibited distinct heterogeneity, including elastic modulus, tangent modulus, and Poisson's ratio. The numerical simulation results indicated that the Weibull probability distribution can be used to analyze fracture initiation. The parameters can explain the distribution of weak elements and can affect the stress field in the shale rock.
2. A high-velocity SC-CO\({}_{2}\) jet not only caused a hydraulic-fracturing ring, but also led to the increase in stress in the shale rock. The tensile stress in the direction parallel to the perforation (\(z\)-direction) led to crack initiation at the root and at the middle of the perforation. The tensile stress in the direction vertical to the perforation (\(y\)-direction) resulted in crack initiation at the position of the tip of the perforation.
3. It was found that, with the increase in perforation pressure, more cracks initiated at the tip. In contrast, the length of the cracks initiated at the root decreased. In SC-CO\({}_{2}\) hydro-jet fracturing, the in situ stress in the direction perpendicular to the perforation should be greater than that in the direction parallel to the perforation, which is favorable for fracture initiation at the tip and for the prevention of crack initiation at the root.
4. The structure of the nozzle outlet distinctly affected the pressurization of the SC-CO\({}_{2}\) jet. When the diameter of the nozzle outlet was increased, the perforation pressure became higher. These effects  
Figure 38: Number of failure elements under different pressure conditions. (**a**) The influence of \(\alpha\); (**b**) The influence of \(\beta\).  
were attributed to the non-linear distribution and different maximum values of the stress in shale rock.
* The aperture ratio between the nozzle outlet and the casing's perforation inlet significantly affected the SC-CO\({}_{2}\) jet fracturing, which resulted in the disappearance of the \"hydraulic isolation ring.\" With the increase in aperture ratio, the initiation position of the cracks' growth and the region of the maximum tensile stress extended, and the number of failure elements also significantly increased.  
The authors express their appreciations to the National Key Basic Research Program of China (No. 2014 CB239203), the National Natural Science Foundation of China (No. 51474158), the National Science Foundation of China (No. 51504166), the Fundamental Research Funds for the Central Universities (Grant No. 2042017 KF0227), Fund Project of Education Department of Sichuan (No. 17 ZB0198) and Fund project of LeShan Science and Technology Department (No. 17 GZD043) for the financial support of this work.  
[PERSON] and [PERSON] conceived and designed the schemes; [PERSON], [PERSON], and [PERSON] performed the experiments; [PERSON] and [PERSON] analyzed the data; [PERSON] contributed materials; [PERSON] and [PERSON] wrote the paper.  
The authors declare no conflict of interest.