## 2 Numerical Simulation Method
### Numerical Simulation Response Surface Design  
The response surface analysis method is advantageous due to fewer experimental groups, high prediction accuracy, and robust model validation capabilities, making it suitable for optimizing process parameters. Through the response surface method, not only can the relationship between the response target and a design variable be obtained, but also the interaction effects of different design variables on the response target can be analyzed. This study employs the Box-Behnken response surface method to conduct response surface design. This research selects the in situ stress, stress differential, and explosion load as  
Figure 3: Fluid–solid coupling model for coal L-CO\({}_{2}\) phase transition blasting. the dominant factors affecting the blasting effectiveness of a single coal seam. Therefore, based on the in situ stress conditions of common kilometer-deep coal layers in China and the actual conditions of L-CO\({}_{2}\) phase transition blasting technology, a three-factor, three-level response surface design scheme is developed, as shown in Table 4. The selection of parameters is based on the following considerations. Since coal seams in China are typically found at depths of less than 1000 m, where horizontal stress predominates, this study sets \(\sigma_{\rm y}\) to be smaller than \(\sigma_{\rm x}\). Based on the common relationship between vertical stress and burial depth, typical vertical stresses for depths of 200 m, 400 m, and 600 m are chosen, with values of 5 MPa, 10 MPa, and 15 MPa, respectively. To reflect the influence of stress difference on fracture propagation, three levels of differential stress--0 MPa, 5 MPa, and 10 MPa--are selected. The L-CO\({}_{2}\) phase transition blasting load is set to 150 MPa, 210 MPa, and 270 MPa, as previously calibrated. A total of 13 groups of numerical calculation model parameters are ultimately obtained, as shown in Table 5.  
Using the response surface analysis method requires first clarifying the response target. To comprehensively elucidate the effects of the vertical stress, stress differential, and explosion pressure on coal fracturing effects, this study extracts five fracturing parameters as response targets. To illustrate the determination method of response targets, for example, the damage cloud diagram is extracted after 3000 \(\upmu\)s following L-CO\({}_{2}\) phase transition blasting for the simulation group numbered 13, as shown in Figure 4. Based on the partition theory of borehole blasting, the coal layer sequentially forms a smash district, fracture zone, and vibration zone from the borehole to the far field after the blasting. The rapid pressure increase caused by the L-CO\({}_{2}\) phase transition leads to significant expansion at the borehole boundary, reflecting the power exerted during the rapid-pressure-rise phase of the blasting. Thus, the radius of the borehole after plastic deformation is defined as \(R_{\rm well}\), cm. In the near-borehole area, the dynamic compressive stress generated by the explosion load in the coal seam exceeds the dynamic compression strength of the coal, leading to compression-shear failure and forming a compression-shear smash district with shear cracks, as shown in the enlarged view in Figure 4. The area of the region with a damage degree of 1 was extracted and measured using ImageJ (FIJI win 64), denoted as \(A_{\rm crush}\), cm\({}^{2}\). The lengths of the major and minor axis of the outer ellipse enclosing the compression-shear smash district were extracted, and their ratio was defined as the shape factor of the smash district, \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Factor** & **Variable** & **Level 1** & **Level 2** & **Level 3** \\ \hline \(\sigma_{\rm y}\)/MPa & \(A_{1}\) & 5 & 10 & 15 \\ \(\sigma_{\rm x}-\sigma_{\rm y}\)/MPa & \(A_{2}\) & 0 & 5 & 10 \\ \(P_{\rm max}\)/MPa & \(A_{3}\) & 150 & 210 & 270 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Range table of values for the three factors in the response surface design.