## 4 The Inversion of HJC Constitutive Parameters for Hard Rock  
### Numerical Simulation of Simulated Reservoir Penetrating Hard Rock Target  
#### 4.1.1 Numerical Calculation Model  
The complete downhole perforation model is illustrated in Figure 13a. The black frame line represents an enlarged view of the local details, which depicts the schematic diagram of the numerical simulation conducted in this section. The model is also a 1/2 axisymmetric model, comprising a shell case, explosive, powder liner, gun holster, water layer, casing, cement ring, hard rock target, and surrounding air. The wellbore is filled with perforating fluid (represented by water in experiments and numerical simulations), while the cavity in the perforating gun contains air at standard atmospheric pressure. According to the actual situation, a blind hole is positioned in the central section of the gun holster, with its thickness constituting half of the overall thickness of the gun holster. The shaped charge utilized in the simulation remains consistent with the aforementioned description. Following experimental parameters, the distance (blast height) between the shaped charge and the gun holster is set at 2.3 cm, the water layer thickness is 2.5 cm, the casing thickness is 1 cm, the thickness of the cement ring is 1.2 cm, and the hard rock target is 45 cm thick. The air domain encompasses the entire model, as indicated by blue in Figure 13b. The primary distinction between this model and the penetrating steel target model lies in the incorporation of a gun holster, water layer, and cement ring. Addition  
Figure 12: Perforation results: (**a**) numerical simulation results based on known parameters; (**b**) numerical simulation results based on inversion parameters; (**c**) schematic diagram of experimental results. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Data Source** & **Perforation** & **Perforation** & **Depth Error** & **Depth Error** & **Aperture (cm)** & **Aperture Error** \\ \hline Experiment & 30.04 & 1.13 & - & - & - & - \\ \hline Numerical simulation & \multirow{2}{*}{19.85} & \multirow{2}{*}{1.31} & \multirow{2}{*}{10.19} & \multirow{2}{*}{33.92\%} & \multirow{2}{*}{0.18} & \multirow{2}{*}{15.93\%} \\ (known parameters) & & & & & &  
ally, subsequent numerical simulations consider the influence of wellbore pressure and confining pressure on the perforating outcomes. The numerical simulation still employs the Arbitrary Lagrange-Euler algorithm (ALE). The mesh division method remains consistent with the aforementioned approach; however, given the utilization of the HJC constitutive model for hard rock, an even distribution of meshes is employed in this case [27] We divided the rock target into uniform meshes and analyzed its grid sensitivity. The results are shown in Figure 14. When the grid is smaller than 0.08 cm, the simulation results tend to be stable, which indicates that it is correct for us to select 0.06 cm grid for the target. The mesh size of explosives, powder liners, and air is 0.03 cm, and the mesh size of shells, holsters, casings, cement rings, and hard rocks is 0.06 cm. The model generates a total of 116,230 nodes and 115,197 units. The model incorporates the downhole environment, so it is not only necessary to set axisymmetric calculation commands and flow-out boundary conditions (the green arrow in Figure 13b is shown) but also apply pressure to the water layer and target to simulate wellbore pressures and confining pressures. Specifically, the rock confining pressure is 30 MPa, and the wellbore pressure is 25 MPa, which is the same as the experiment, simulating the real underground environment. Confining pressure is added to the outer sides of the hard rock, casing, and cement rings by the \"LOAD_NODE_SET\" command, as shown by the blue arrows in Figure 13a. Using the \"INITIAL_HYDROSTATIC_ALE\" command, the wellbore pressure is added to the outside of the water formation, as shown by the red arrow in Figure 13a.