## 2 Fracturing Tests
### Experimental Results  
The fracturing process is typically analyzed by monitoring the variation in injection pressure over time. The peak of the injection pressure curve, which corresponds to the first maximum fluid pressure, is identified as the breakdown pressure (denoted as \(P_{b}\) hereafter). Reaching this pressure signifies that the shale specimen is fractured [24; 25]. To illustrate the injection pressure curve, the results from the flow rate have been provided as an example in Figure 4. In addition, since the primary focus of this study is breakdown pressure, the breakdown pressure values for each scenario are listed in Figure 5.  
Figure 3: Fracturing experiment schemes.  
A comparison of breakdown pressures under different flow rates has revealed that higher flow rates resulted in higher breakdown pressures. As shown in Figure 4, the breakdown pressures at flow rates of 70 mL/min, 50 mL/min, and 30 mL/min were 37.70 MPa, 29.83 MPa, and 20.82 MPa, respectively. Correspondingly, the times to reach breakdown pressure were 30.0 s, 146.9 s, and 300.5 s, indicating that higher flow rates can raise CO\({}_{2}\) injection pressure, causing shale failure more quickly.  
As illustrated in Figure 5, the peak injection pressures for shale fracturing with water, L-CO\({}_{2}\), and SC-CO\({}_{2}\) were 33.16 MPa, 27.99 MPa, and 20.82 MPa, respectively. Compared to hydraulic fracturing with water, the breakdown pressure for SC-CO\({}_{2}\) decreased by 37.2%, and for L-CO\({}_{2}\), it decreased by 15.6%. This reduction may be attributed to the high diffusivity and superior penetrability of SC-CO\({}_{2}\), which enables it to infiltrate microfractures and natural crack tips, significantly enhancing its rock-breaking capability.  
Figure 5 shows that when the temperature increased from 35 \({}^{\circ}\)C to 45 \({}^{\circ}\)C and 55 \({}^{\circ}\)C, the breakdown pressure of the shale specimens decreased from 40.13 MPa to 37.85 MPa and 33.36 MPa, corresponding to reductions of 5.68% and 16.87%, respectively. This trend aligns with the findings of [PERSON] et al. [26]. Two potential explanations are available, as follows: First,  
Figure 4: Injection pressure versus time under various flow rates.  
Figure 5: Breakdown pressure under different fracturing schemes.  
higher temperatures reduce the viscosity and increase the diffusivity of SC-CO\({}_{2}\), resulting in lower breakdown pressures. Second, the thermal stress generated within the shale decreases the effective stress on the mineral matrix, further contributing to the reduction in breakdown pressure.  
Based on the theory of linear elastic tensile failure, the breakdown pressure models proposed by [PERSON] [6] (commonly referred to as the H-W model) and [PERSON] [7] (referred to as the H-F model) primarily account for the influences of horizontal stress and tensile strength. In this study, we varied the confining pressure to investigate the impact of in situ stress on breakdown pressure. As illustrated in Figure 4, higher confining pressures necessitated larger injection pressures to induce rock failure. Specifically, at a confining pressure of 5 MPa, the breakdown pressure was measured at 17.70 MPa. When the confining pressure was increased to 10 MPa and 15 MPa, the breakdown pressures rose by 43.85% and 54.95%, respectively.