## 2 Numerical Simulation Method
### Material Model Selection for Numerical Simulation  
To study the L-CO2 phase transition blasting permeability enhancement mechanism at an engineering scale, a large-scale numerical simulation model under in situ stress conditions is established. LS-DYNA R11.1.0, powerful dynamic simulation software, is widely used for dynamic impact simulations of rock and concrete materials. This study employs LS-DYNA R11.1.0 to numerically simulate the phase transition blasting of liquid carbon dioxide at an engineering scale, elucidating the mechanisms of L-CO2 phase transition fracturing and crack propagation under the combined effects of in situ stress and explosion load. Obtaining the load form for L-CO2 phase transition fracturing is critical for the reliability of the numerical simulation. The instantaneous heating and vaporization of L-CO2 produce high-pressure gaseous CO2, which applies explosive impact loads to the engineering rock mass. This process represents a typical physical explosion process without chemical reactions, relying primarily on gas impact generated by phase transition expansion to fracture the rock mass. This study selects the *MAT_HIGH_EXPLOSIVE_BURN material model built into LS-DYNA R11.1.0, paired with the *EOS_LINEAR_POLYNOMIAL equation of state, specifically designed to describe the physical processes of phase transition blasting. Through extensive trial calculations, model parameters that can produce different L-CO2 phase transition explosive loads are obtained. The LINEAR_POLYNOMIAL equation of state is shown in Equation (1):  
\[P=C_{0}+C_{1}\mu+C_{2}\mu^{2}+C_{3}\mu^{3}+\Big{(}C_{4}+C_{5}\mu+C_{6}\mu^{2} \Big{)}E \tag{1}\]  
The state equation is applicable for describing the pressure state of the gas as well as the compression and expansion processes. Therefore, this study selects this equation of state, which effectively simulates the adiabatic expansion in L-CO2 phase transition blasting. In the equations, \(P\) represents the pressure of the detonation products, in MPa;  
Figure 2: Pressure–temperature phase diagram of CO2. is the unit initial internal energy, in J; and C\({}_{0}\), C\({}_{1}\), C\({}_{2}\), C\({}_{3}\), C\({}_{4}\), C\({}_{5}\), and C\({}_{6}\) are the coefficients of the polynomial. \(\mu=\rho/\rho_{0}\) and \(\rho/\rho_{0}\) is the ratio of current density of reference density. Since the L-CO\({}_{2}\) phase transition blasting process does not involve chemical reactions, it is assumed that the gas follows the Gamma law, which models an ideal gas with a constant adiabatic index \(\gamma\). The relationship between pressure \(P\), density \(\rho\), and specific internal energy \(E\) under the Gamma law is given by \(P=(\gamma-1)\rho E\). By simplifying the LINEAR_POLYNOMIAL, a gas state equation consistent with the Gamma law can be derived. This simplification primarily employs the following equations, Equations (2)-(4):  
\[C_{0}=C_{1}=C_{2}=C_{3}=C_{6}=0 \tag{2}\]  
\[C_{4}=C_{5}=\gamma-1 \tag{3}\]  
\[\gamma=\frac{C_{p}}{C_{v}} \tag{4}\]  
where \(C_{p}\) is the specific heat capacity at constant pressure, in J/(kg\(\cdot\)K); and \(C_{v}\) is the specific heat capacity at constant volume, in J/(kg\(\cdot\)K). The validity of the mechanical model and parameters for coal rock materials is crucial for accurately reflecting the fracture distribution patterns during L-CO\({}_{2}\) phase transition blasting in coal seams. Unlike quasi-static loads, L-CO\({}_{2}\) phase transition fracturing, as a dynamic impact fracturing method, exhibits entirely different mechanical response characteristics under dynamic loads, primarily manifesting as significant strain rate effects and strengthening effects. Therefore, the constitutive equations for coal rock under quasi-static conditions cannot capture its impact dynamic behavior. The RHT constitutive model, modified from the HJC model by [PERSON], [PERSON], and [PERSON], effectively simulates the dynamic response of rock and concrete materials, as well as the tensile-shear damage process.