## 1 Introduction
The failure mechanism of rock and the changes in pore structure were also explored during the SC-CO\({}_{2}\) jet impinging [19]. There is still some degree of uncertainty in understanding the process of SC-CO\({}_{2}\) hydro-jet fracturing. Particularly, due to the absence of packers, the hydro-jet fracturing is regarded as a cost-effective technique, and since it isolates the annular from the reservoir, it is also considered not to damage the reservoir [20]. Obviously, it can offer the advantages of both the hydro-jet fracturing and the SC-CO\({}_{2}\) fluid used in the exploration and production of shale gas. In addition, the SC-CO\({}_{2}\) jet has significant mechanical effects and generates more extensive and complex fracture networks, which enhance the process of fracturing. Previously, the SC-CO\({}_{2}\) jet was investigated to comprehensively understand the process of SC-CO\({}_{2}\) jet fracturing. Some studies considered the influence of rock on the process of hydro-jet fracturing. Additionally, some experiments and numerical simulations have been carried out to investigate the pressurization process of SC-CO\({}_{2}\) hydro-jet fracturing in perforation tunnels. Various experimental setups and physical models have been proposed to determine the dynamic characteristics of high-pressure SC-CO\({}_{2}\) jets [21]. The flow field of SC-CO\({}_{2}\) jets was also studied to explore the properties of SC-CO\({}_{2}\) and analyze the sensitivity parameters using numerical simulations [14]. However, there are concerns about whether the use of SC-CO\({}_{2}\) produces the same pressurization effect in the perforation tunnel as that of the water jet [1; 19]. To address this concern, the pressurization effect of SC-CO\({}_{2}\) hydro-jet fracturing was investigated at higher temperatures and pressures. The results indicated that the hydro-jet fracturing parameters affected the pressurization in the perforation tunnel. However, due to the complicated interaction between the jet flow and the reservoir rock breaking, the studies focusing on SC-CO\({}_{2}\) jet fracturing did not consider the rock crack initiation and propagation [22]. Though the fracture initiation and propagation were considered, most of the turbulent jet flows were treated as laminar flows [23; 24]. In addition to the numerical simulation, for a given in situ stress condition and wellbore orientation parameters, a theoretical model was developed to predict the fracture initiation pressure and the orientation and location of fractures on the wellbore wall. The model has been applied in a series of in situ stress conditions to study the effect of wellbore orientation on the fracture initiation using non-dimensional parameters, which have enhanced the applicability of published results for any stress condition. Closed-form analytical solutions have also been obtained for the initiation of longitudinal, transversal, and complex multiple fractures from vertical and horizontal wellbores with/without perforation [25; 26]. A number of assumptions were made to simplify the theoretical model for the ease of calculations. In 1972, a Perkins-Kern-Nordgren type model was proposed, in which the created fracture was pressurized using the original laminar fluid flow that was constrained by various boundary conditions [27]. The previous studies focusing on SC-CO\({}_{2}\) hydro-jet fracture in perforation tunnels have two shortcomings. Firstly, most of the time, the study of CO\({}_{2}\) fracturing has only focused on the initiation and propagation of rock fracture, and the turbulent characteristics of SC-CO\({}_{2}\) jet flow were ignored [28; 29]. However, comparisons between the results and the parameters for laminar and turbulent hydraulic fractures have clearly shown that different models were suitable for different hydraulic fracture lengths and wellbore pressures [30].