## 1 Introduction  
Due to its clean nature, shale gas has become the focus of research and development in the field of energy sciences. Hydraulic fracture is an efficient method for commercial extraction of hydrocarbons in shale and tight reservoirs and has contributed to the boom of shale gas in the United States [1,2]. In order to reduce water consumption and CO2 emissions, SC-CO2 fracturing is regarded as a potential futuristic technology [3]. The SC-CO2 is a fluid state of carbon dioxide when it is held at or above its critical temperature (\(T_{c}\) = 31.10 \({}^{\circ}\)C) and critical pressure (\(P_{c}\) = 7.38 MPa). Compared to the conventional hydraulic fracturing technology, shale gas has several advantages, including low viscosity, small damage to the reservoir, outstanding permeability in fine pores and micro-cracks of shale. In particular, the SC-CO2 hydro-jet fracturing is regarded as a unique, cost-effective and efficient well-stimulation treatment owing to its capability to accomplish a multi-stage pin-point fracturing without the use of mechanical packers. The fracturing is accomplished due to the jet pressurization effect, while SC-COjet is pressurized in the perforation tunnel. Therefore, the pressurization process of SC-CO\({}_{2}\) jet and the crack initiation of shale rock have attracted significant research attention in recent years. Firstly, the SC-CO\({}_{2}\) jet was investigated for its structure, its expansion mechanism, and its application in industry. The rapid expansion of SC-CO\({}_{2}\) jet has been studied for nearly 20 years and is used to fabricate nanoparticles and grow thin films, and is used as an interface in various instruments including mass spectrometers and gas chromatographs [4]. Previously, the structure and the flow field of an SC-CO\({}_{2}\) free jet have been studied in different ambient environments [5; 6; 7]. Most of the previous studies on the simulations of rapid expansion of supercritical solutions (RESS) have employed a one-dimensional (1-D) method [8; 9]. Some scholars have used two-dimensional (2-D) and three-dimensional (3-D) simulations of RESS [10; 11]. To predict the fluid properties satisfactorily, even in the supercritical region, a series of studies have used cubic equations of state, such as the van der Waals and Peng-Robinson (PR) equations of state, to estimate some of the properties of the SC-CO\({}_{2}\) jet [12]. Furthermore, several researchers have used the Span-Wagner equations and the Bender equation of state to calculate the density and thermal conductivity coefficient of SC-CO\({}_{2}\) jets [13]. The rapid expansion of SC-CO\({}_{2}\) jets makes it one of the most commonly used supercritical solvents in the fields of chemistry and chemical engineering. Due to its liquid-like density, and gas-like diffusivity and viscosity, SC-CO\({}_{2}\) is used in a wide range of industrial applications. Due to its favorable properties, [PERSON] and [PERSON] reported that SC-CO\({}_{2}\) can be effectively used to enhance the fracturing and production of shale gas [3]. It was reported that a high rate of penetration was obtained when SC-CO\({}_{2}\) was used as the drilling fluid [14]. Moreover, rock breaking with SC-CO\({}_{2}\) exhibits advantages of a lower threshold pressure, a fast drilling rate, and a high rate of penetration [15]. In order to study the dynamics of wellbore, the effects of major influencing factors such as the nozzle diameter, the standoff distance, and the jet pressure of SC-CO\({}_{2}\) were investigated during well drilling [16]. In addition, many studies that focused on the rock-breaking performance of a high-pressure SC-CO\({}_{2}\) jet explored the influence of various factors on its rock-breaking performance. It was found that, under the same working conditions, the SC-CO\({}_{2}\) jet can efficiently erode rocks at a comparably low threshold pressure with a high rate of penetration [17]. Previously, researchers have investigated the structure of SC-CO\({}_{2}\) jets and measured the impinging pressure [18].