An Inversion Study of Constitutive Parameters for Powder Liner and Hard Rock Based on Finite Element Simulation

Abstract

To acquire Johnson–Cook (J-C) constitutive parameters that accurately depict the mechanical behavior of powder liner under conditions of high pressure, elevated temperature, and large deformation, as well as Holmquist–Johnson–Cook (HJC) constitutive parameters that precisely describe the dynamic damage of hard rock and make them suitable for numerical simulations for hard rock perforation, the present study introduces a constitutive parameter inversion method based on finite element simulation. Firstly, based on the experiments of perforating steel targets and underground perforating hard rock targets, a dynamic simulation of the perforating process of a shaped charge perforating target was carried out using ANSYS/LS-DYNA, and the influence law of each constitutive parameter on perforating depth and perforating aperture was systematically analyzed. Subsequently, the key parameters of the J-C constitutive model for powder liner and the HJC constitutive model for hard rock were optimized and determined using a response surface method, multi-genetic algorithm, and experimental data. A numerical simulation of the perforating process was finally conducted using the retrieved constitutive parameters of powder liner and hard rock, which were then compared with the experimental results. The results demonstrated that the discrepancy between the experimental and simulated data was within 5%, indicating that the constitutive parameters obtained through this inversion method could more reliably reflect the mechanical behavior of the powder mold and hard rock used in this study during perforation.

1. Introduction

Well completion serves as a crucial final step preceding oil and gas exploitation, pivotal in the overall process. The quality of well completion directly impacts the productivity and lifespan of oil and gas wells. Currently, perforation completion technology is extensively employed both domestically and internationally [1]. This technique utilizes high-velocity metal jets generated by shaped charge during detonation to rapidly penetrate casing, cement ring, and formation rocks, thereby establishing efficient channels for oil and gas flow between reservoirs and wellbores. As the central component of the shaped charge, the liner plays a pivotal role in generating a metal jet to accomplish perforation and thus exerts a decisive influence on the perforating outcome [2].

Due to the volatile global geopolitical landscape and the escalating domestic energy demand, the exploration and production activities in the oil and gas industry have progressively extended towards deep–ultra-deep reservoir buried-hill hydrocarbon, as well as offshore fields. The Bohai Bay Basin is a prominent region for the development of buried-hill hydrocarbon reservoirs in China, and it holds significant potential for expanding oil and gas resources in the country. The reservoirs in this area exhibit characteristics such as multiple structures, origins, and forms, while the rocks are composed of complex lithology with compact hard textures (hard rock). Due to various factors, such as high pressure and tight reservoir conditions in the wellbore, conventional-shaped charges are unable to effectively release oil and gas from the reservoir. Many oil and gas wells encounter challenges in fluid production post-perforation or even experience no fluid production at all. In addition, it is difficult to ascertain whether the mud contamination zone has been penetrated.

In recent years, Li et al. [3] developed a novel powder liner by blending tungsten powder, aluminum powder, copper powder, and nickel powder in specific mass ratios based on previous studies on liners. The experimental results demonstrated excellent penetration capability in both ground-penetrating steel target tests and simulated, confining pressure, penetrating sandstone target experiments, with relatively clean perforation channels. However, due to the extremely short perforation time (on the order of microseconds), intricate processes (explosive detonation and high strain rate), and complex reservoir environment (high wellbore pressure and rock strength), the data collected by existing experimental observation methods often fragmented, rendering it arduous to comprehensively elucidate the complete physical process of perforation in hard rock [4]. In this case, investigating the perforating process in hard rock and enhancing the performance of shaped charges is hindered by difficulties and time constraints. The numerical simulation method not only effectively circumvents such issues but also significantly enhances research efficiency. However, the constitutive parameters of the powder liner and the hard rock suitable for perforating conditions must be determined first.

The selection of constitutive parameters for liner and rock materials in finite element simulations of perforation has long been a topic of significant interest and discussion within the academic community [5,6,7]. Currently, the majority of the key parameters in the material constitutive equation are acquired through experimental fitting. Liu [8] conducted Split-Hopkinson Pressure Bar (SHPB) tests and quasi-static tests to obtain stress–strain curves of the tungsten–potassium alloy used in the liner at different strain rates. Based on these results, key parameters of its J-C constitutive model were fitted. The J-C constitutive equation of an Al-Li alloy was calibrated by Li et al. [9] through quasi-static experiments and impact compression tests. Wen et al. [10] determined the parameters of the HJC constitutive model for granite porphyry under varying freezing and thawing durations by analyzing model parameters, combining SHPB experiments, and conducting numerical simulations. Ling et al. [11] established the parameters of the HJC constitutive model for sandstone through static experiments and SHPB laboratory tests.

However, there are inherent limitations in experimentally determining the constitutive parameters of powder liner (in shaped charge) and hard rock (in reservoir). First of all, in the preparation process of powder liner, usually using pressing technology, this process not only affects the microstructure of the material but also determines its macroscopic mechanical properties. Because of this, the traditional mechanical experimental methods, such as tensile tests, compression tests, etc., cannot effectively obtain its constitutive parameters. In addition, the experimental range of the strain rate cannot completely cover the actual range of the strain rate of the powder liner’s transformation from solid state to metal jet. Second, the properties of hard rock in the target reservoir and the actual downhole conditions are difficult to reproduce in these experiments. Finally, the experiment is time-consuming and laborious but also has a certain risk. These factors make the experimental method no longer suitable for fitting the constitutive parameters of powder coating and hard rock.

The present study proposes a parameter inversion method that integrates numerical simulation, multi-objective optimization, and experimental data to address the aforementioned issues. This approach aims to enhance reliability, simplicity, and universality in determining the constitutive parameters of powder liner and hard rock, thereby optimizing their applicability for hard rock perforation-related tasks.

2. Perforation Experiment

In the field of oil production engineering, perforation depth and aperture serve as crucial indicators for assessing the productivity and efficiency of oil and gas wells [12]. Moreover, during the process of conducting perforation experiments and numerical simulations, these parameters directly reflect the performance of shaped charges. Because of this, this paper first experimented on the ground using a shaped charge to hit a steel target, obtaining the depth and aperture of the metal jet penetrating the steel target. Subsequently, the constitutive parameters of the powder liner were inverted based on these data. We conducted further simulations of the underground environment to conduct an experiment using a shaped charge to hit a hard rock target to obtain the depth and aperture of the metal jet penetrating the hard rock, and then we used these data to inverse the constitutive parameters of the hard rock.

2.1. Experiment of Penetrating Steel Target

The experimental apparatus is depicted in Figure 1, comprising primarily of shaped charge, metal sheet, blast tube, and steel target. Among them, the primary structure of the shaped charge (with a body diameter of 5.2 cm) comprises a shell case, explosive, and powder liner. A metal sheet with a thickness of 0.25 cm is utilized to simulate the wall of the perforating gun (the blind hole region), while a blast height cylinder is employed to regulate the distance (blast height) between the shaped charge and the steel target. In this experiment, the blast height remains fixed at 7 cm. The steel target consists of steel 45 and takes on a cylindrical shape with dimensions measuring 10 cm in diameter and 50 cm in height.

At the onset of the experiment, detonation of the explosive is initiated through a detonating cord. After the explosion, rapid oxidation of the powder liner within the shaped charge occurs under high temperature and pressure generated by the explosive detonation, resulting in the formation of a metal jet possessing substantial impact force. Once formed, this jet effectively perforates both a metal sheet and subsequently penetrates through the steel target; ultimately, a perforating channel with a certain depth and aperture is formed on said target.

Based on statistical principles and economic considerations, we repeated seven sets of targeted charge shooting experiments with the above devices. During the implementation of the experiment, we strictly controlled the experimental environment (such as temperature, humidity, etc.) and ensured that the materials (shaped charge, sheet metal, and steel target) came from the same batch to reduce the variability of the experimental results. Key parameters (such as blast height, powder liner, sheet metal thickness, etc.) were consistent in all experiments to ensure standardization of the experimental process. In addition, we calibrated the detonating equipment to ensure the accuracy of the measurements. Upon completion of each experiment, a metal probe was inserted into the perforation channel to measure the maximum depth reached, referred to as the perforation depth. Additionally, the diameter of the steel target opening was measured and recorded as the perforation aperture. The experimental results are presented in Figure 2. The perforation depth and aperture of the seven groups exhibited relatively stable characteristics. The depth ranged from 28.90 cm to 31.10 cm, averaging 30.04 cm and displaying a standard deviation of 0.80 cm. The size of the apertures varied between 1.05 cm and 1.20 cm, with an average value of 1.13 cm and a standard deviation of 0.05 cm. The statistical results indicate a low degree of data dispersion, which can provide reliable support for subsequent numerical simulations of perforation and the inversion of constitutive parameters for the powder liner.

2.2. Experiment of Simulated Reservoir Penetrating Hard Rock Target

According to the Evaluation of Well Perforators issued by the American Petroleum Institute, a simulated reservoir perforation test device was designed and established according to the mechanical environment, rock properties, perforating gun parameters, and actual underground working conditions of the target reservoir. The device and test flow are illustrated in Figure 3. The device is mainly composed of a unit gun and hard rock in two parts and a shaped charge (the same model as above) assembled in the gun, and it is connected to a detonator, placed in the upper part of the device. The hard rock (a cylinder 20 cm in diameter and 45 cm in height) is vacuum-saturated and placed under the unit gun.

At the onset of the experiment, the shaped charge is fired through the detonator. The resultant metal jet sequentially penetrates the gun holster (containing blind holes), wellbore fluid (water), casing, and cement ring, and ultimately forms a perforation channel within the hard rock. Considering the actual reservoir conditions, wellbore pressure is regulated at 25 MPa, while confining pressure is maintained at 30 MPa.

Seven groups of experiments were repeated with the above device. After the experiment was completed, the experimental target was broken, the maximum depth of the hole was recorded as the perforation depth, and the diameter of the opening of the experimental target was measured as the perforation aperture. The experimental results are illustrated in Figure 4. The perforation depth and perforation aperture of the seven groups were relatively stable. The perforation depth fluctuated between 30.52 cm and 33.29 cm, the average depth was 32.00 cm, and the standard deviation was 1.05 cm. The perforation aperture fluctuated between 1.00 cm and 1.17 cm, with an average aperture of 1.08 cm and a standard deviation of 0.07 cm. The statistical results indicate a low degree of dispersion for each data point, thereby providing reliable support for subsequent numerical simulations of hard rock perforation and the inversion of constitutive parameters.

3. The Inversion of J-C Constitutive Parameters for Powder Liner

3.1. Numerical Simulation of Penetrating Steel Target

3.1.1. Numerical Calculation Model

The numerical calculation model of the penetrating steel target is illustrated in Figure 5. Considering the geometric characteristics of the perforating process, a 1/2 axisymmetric model is established, comprising shell case, explosive, powder liner, steel target, and air domain. Due to the relatively thin wall of the perforating gun, its impact on jet energy loss is negligible and thus disregarded in numerical simulations.

In numerical simulation, this paper employs the Arbitrary Lagrange–Euler (ALE) algorithm in numerical simulation, which integrates the merits of both Lagrangian and Euler algorithms, enabling faster and more accurate computation of explosion shock problems [13]. Among them, the Lagrange mesh is utilized for shell case and steel targets, while the Euler mesh is employed for explosives, powder liners, and air. Both grids are controlled during the calculation process to enable fluid–structure coupling of multiple substances. In addition, during the meshing process, it is imperative to utilize Euler mesh with co-nodes for explosives, powder liners, and air. Furthermore, a dense meshing of the powder liner and metal jet through the area is required [14]. Considering both computational cost and accuracy comprehensively, five tetrahedral meshes were created along the liner’s wall thickness direction with a mesh size of approximately 0.03 cm. The mesh size for both the explosive domain and the air domain is set to 0.03 cm. The shell employs a mesh size of 0.06 cm, while the steel target utilizes a gradient mesh, with the central region having a mesh size of 0.06 cm. This model resulted in a total of 27,211 nodes and 26,219 units, as shown in Figure 5b.

As the model is a 1/2 axisymmetric model, it is essential to define axisymmetric calculation commands separately for the Lagrange unit and Euler unit. Additionally, flow-out boundary conditions should be established at the periphery of the computational domain to simulate an infinite air domain environment, enabling unrestricted outflow of all substances from the model boundaries [15]. This approach prevents stress wave reflections during simulation and ensures maximum accuracy of the results.

3.1.2. Material Model and Parameters

(1)

Powder liner

Currently, the Johnson–Cook (J-C) constitutive model and Grunensin equation of state are commonly employed in numerical simulations of perforation to characterize the mechanical behavior of the liner during jet molding [16]. Specifically, the Grunensin equation of state is primarily utilized to describe the thermodynamic properties of a metal jet under high temperature and pressure conditions, encompassing relationships between physical quantities such as density, pressure, and temperature. The expression is presented below:

$$P={\displaystyle \frac{{\rho }_0{c}^2\mu \left[1+\left(1-\frac{{\gamma }_0}{2}\right)\mu -\frac{\alpha }{2}{\mu }^2\right]}{{\left[1-\left(S_1-1\right)\mu -S_2\frac{{\mu }^2}{\mu +1}-S_3\frac{{\mu }_3}{\mu +1}\right]}^2}}+\left({\gamma }_0+\alpha \mu \right)E$$

(1)

where ${\rho }_0$ is the initial density of the material, $C_0$ is the speed of sound, $S_1$, $S_2$, $S_3$, and $\alpha$ are material characteristic parameters, E is the initial internal energy, $\mu =\rho /{\rho }_0-1$ is the relative compression degree, ${\gamma }_0$ is the Gaussian coefficient, and $\rho$ is the density of the material. For the powder liner used in this paper, its initial density ${\rho }_0$ is 14.96 $\mathrm{g}\cdot {\mathrm{cm}}^{-3}$, and C₀ = 394 m/s, $S_1$ = 1.49, $S_2$ = $S_3$ = 0, ${\gamma }_0$ = 1.99, which are selected according to the literature [17].

The J-C constitutive model, proposed by Johnson and Cook in 1983, is designed to accurately capture the deformation flow behavior of metal materials under high temperatures, high strain rates, and large deformation conditions [18]. The mathematical expression is presented as follows:

$$\overline{\sigma }=\left[A+{\left(B{\overline{\epsilon }}^{\mathrm{p}}\right)}^n\right]\left[1+C\ln \left({\displaystyle \frac{{\overline{\overline{\epsilon }}}^{\mathrm{p}}}{{\overline{\epsilon }}_0}}\right)\right]\left[1-{\left({\displaystyle \frac{T-T_{\mathrm{r}}}{T_{\mathrm{m}}-T_{\mathrm{r}}}}\right)}^m\right]$$

(2)

where $\overline{\sigma }$ is the flow stress, A is the initial yield strength, B is the strain hardening coefficient, C is the strain rate sensitivity coefficient, m is the temperature coefficient, n is the strain hardening index, ${\overline{\epsilon }}^{\mathrm{p}}$ is equivalent plastic strain, ${\overline{\overline{\epsilon }}}^{\mathrm{p}}$ is equivalent plastic strain rate, ${\overline{\epsilon }}_0$ is reference strain rate, $T$ is temperature, $T_{\mathrm{r}}$ is the reference temperature (generally room temperature), and $T_m$ is the melting temperature of the material under normal conditions. Among them, A, B, C, m, and n are the key parameters of J-C constitutive, and also the potential, inversion parameters. The parameter values are shown in

Table 1. Key parameters of the constitutive equation of copper liner [19].

A/MPa	B/MPa	C	n	m
90	292	0.025	0.31	1.09

.

Since the constitutive parameters of the powder liner in this type of shaped charge are unknown, we utilize the constitutive parameters of the copper liner as reference values for conducting numerical simulation analysis. It should be pointed out that in current numerical simulation studies involving perforation, researchers mostly choose the constitutive parameters of copper as the constitutive parameters of the liner, regardless of the material of the liner, because copper liner was widely used in perforating operations in the early days, and its constitutive parameters have been well developed and clear (see Table 1). Therefore, in this paper, the mature constitutive parameters of copper liner are used as reference values to carry out numerical simulation analysis in order to evaluate the gap between the simulation results and the experimental results.

(2)

Explosive, air, steel target, and shell case

In related perforation simulations, the material models and parameters for explosive, air, steel target, and shell cases have been used extensively and are well established. Here, a brief overview is provided. The type of explosive in shaped charge is RDX, which is described by using the high-explosive material model in LS-DYNA software (16.0), the volume and pressure of the high explosive product are described by the JWL equation of state. The JWL state equation expressed in terms of the detonation product pressure P₀ is presented as follows:

$$P_0=A_0\left(1-{\displaystyle \frac{\omega }{R_{1}V}}\right)e^{-R_{1}V}+B_0\left(1-{\displaystyle \frac{\omega }{R_{2}V}}\right)e^{-R_{2}V}+{\displaystyle \frac{\omega E_0}{V}}$$

(3)

where V is the volume of the detonation products, E₀ is the internal energy of the detonation products, and A₀, B₀, R₁, R₂, and ω are constants related to the explosive; these parameters are determined experimentally, and the values of these parameters are listed in

Table 2. Material parameters of the explosive [20].

$\mathit{\rho }/\mathbf{g}\cdot {\mathbf{cm}}^{-3}$	E0/MPa	A0/MPa	B0/MPa	R1	R2	ω
1.82	9.2 × 103	8.5 × 105	1.8 × 104	4.6	1.3	0.38

.

The air is described jointly by the a Null model and a linear polynomial state equation, with the linear polynomial state equation represented by pressure value P₁, as follows:

$$P_1=C_0+C_1{\mu }_0+C_2{{\mu }_0}^2+C_3{{\mu }_0}^3+\left(C_4+C_5{\mu }_0+C_6{{\mu }_0}^2\right)E_1$$

(4)

where C₀, C₁, C₂, C₃, C₄, C₅, and C₆ are the coefficients of the equation, and ${\mu }_0=1/V_0-1$, V₀ is the relative volume. E₁ is the initial internal energy. The air material parameters are shown in

Table 3. Material parameters of air [21].

C0	C1	C2	C3	C4	C5	C6	V0	E1/MPa
0	0	0	0	0.4	0.4	0	1.0	0.253

.

The steel target and shell are both described by the plastic kinematic model, which is suitable for simulating isotropic hardening and kinematic hardening and can also take into account the effects of strain rate. This model is often used in numerical simulations of perforation to describe the deformation and flow behavior of the shell and target. The material parameters of the steel target and shell are shown in

Table 4. Material parameters of steel target and shell case.

Material	$\mathbf{Density}/\mathbf{g}\cdot {\mathbf{cm}}^{-3}$	Elasticity Modulus/MPa	Poisson Ratio	Yield Strength/MPa
Steel target [22]	7.83	2.06 × 105	0.269	500
Shell case [23]	7.83	2.07 × 105	0.3	400

.

3.1.3. Numerical Simulation Result

Based on the above model and known parameters, the numerical simulation of the penetrating steel target can obtain the perforation results shown in Figure 6, The green part is the metal jet. As can be seen from the figure, according to the constitutive parameters of copper, the continuity of the metal jet is poor, the final perforation depth in the steel target is only 19.85 cm, and the perforation aperture is 1.31 cm. However, the average perforation depth obtained in the penetrating steel target experiment of shaped charge is 30.46 cm, so the error between the simulated results and the experimental results using copper constitutive parameters is 34.83%. It can be seen that the constitutive parameters of copper cannot accurately describe the deformation and flow behavior of the new powder liner during perforation. Therefore, to improve the accuracy and reliability of the numerical simulation and make it better serve the perforation research of oil and gas wells, it is necessary to invert the constitutive parameters of the new powder liner under detonation.

Although parameters A, B, C, n, and m in the J-C constitutive model have the potential to be considered as design variables, during the inversion process, the computational workload increases exponentially with an increasing number of inversion parameters. To effectively conduct inversion design, this study initially focuses on analyzing the individual effects of five constitutive parameters on perforation simulation results (perforation depth and aperture). The parameters with more significant impacts are then selected as inversion variables. Subsequently, the expressions of the inversion variables of perforation depth and perforation aperture are fitted by the response surface method. Based on the expression, the distribution of the Pareto optimal solution is iterated by a multi-objective genetic algorithm. Finally, by considering the distribution of Pareto optimal solutions and selecting the solution closest to the experimental results, we determine corresponding parameter values that can be regarded as J-C constitutive parameters for powder liner in this paper.

3.2. Parameter Sensitivity Analysis

Based on the numerical calculation model and the constitutive parameters established in Section 3.1, the values of parameters A, B, C, n, and m were changed to perform numerical simulation, and then the depth and aperture of the penetration of the steel target by the jet flow under different parameter values were obtained. The values of parameters A, B, C, n and m are set in reference [24] as 10 MPa~900 MPa, 10 MPa~450 MPa, 0.1~1.0, 0.1~1.0, and 0.1~1.5, respectively. Figure 7 shows the variation in perforation depth and perforation aperture with initial yield strength A, strain hardening coefficient B, strain rate sensitivity coefficient C, strain hardening index n, and temperature correlation coefficient m. As can be seen from the figure, both perforation depth and aperture decrease with the increase in parameters A, B, C, n, and m, but the perforation depth and aperture are more sensitive to parameters A, B, and m, with a large change range, while the sensitivity to parameters C and n is small, with relatively minor change. Therefore, to improve the inversion efficiency, the influence of hardening index C and strain rate sensitivity coefficient n can be ignored in the inversion process, and the two can be fixed as C = 0.025 and n = 0.31, respectively.

3.3. Multi-Objective Optimization

The response surface method is an efficient approach for fitting polynomial functions to accurately simulate the real limit state. It enables the representation of the target variable as a display expression of the design variable, making it a commonly employed technique for solving problems involving multiple variables [25]. The specific form is presented below:

$$y(x)={\phi }_0+{\displaystyle \sum _{j=1}^N{\phi }_j{x}_j}+{\displaystyle \sum _{j=1}^N{\phi }_{jj}{x^2}_j}+{\displaystyle \sum _{i\ne j}^N{\phi }_{ij}{x}_j}$$

(5)

where $x_j$ is the design variable, N is the number of targets, the regression coefficient $\phi$ can be obtained by expression $\left(N+1\right)\left(N+2\right)/2$, ${\phi }_0$ stands for constant coefficient, and ${\phi }_j$, ${\phi }_{\mathit{jj}}$, ${\phi }_{\mathit{ij}}$ represent the primary coefficient, the pure quadratic coefficient, and the mixed quadratic coefficient, respectively.

The inversion variables were determined as parameters A, B, and m based on the results of sensitivity analysis in the previous section. Meanwhile, the target variables were identified as perforation depth L and perforation aperture D. Subsequently, a central composite design was employed to simulate various combinations of these parameters, enabling us to obtain a fitting surface for the target variables (L, D) and the inversion variables (A, B, m). The fitting surface of perforation depth and perforation aperture under the interaction of parameters B and m is demonstrated in Figure 8 as illustrative examples. It can be observed that the data points are accurately fitted onto the surface with small local fluctuations, thereby providing a reliable basis for subsequent display expressions.

Explicit expressions (6) and (7) of the target variable about the design variable can be further obtained from the fitting surface.

$$\begin{array}{ll}L=& 44.99+1.81\times {10}^{-4}A+0.032B-3.95m-1.6\times {10}^{-4}AB\\ & -2.79\times {10}^{-2}mA-0.015mB+1.2\times {10}^{-4}{A}^2-2\times {10}^{-4}{B}^2\\ & +2.23m^2+2.24\times {10}^{-4}mAB\end{array}$$

(6)

$$\begin{array}{ll}D=& 0.83+0.022A-1.13\times {10}^{-3}B+5.33\times {10}^{-2}m-9.2\times {10}^{-5}AB-\\ & 2.17\times {10}^{-3}mA-2.6\times {10}^{-4}{A}^2+2.9\times {10}^{-5}{B}^2-9.9\times {10}^{-4}{m}^2+\\ & 4.10\times {10}^{-7}{A}^{2}B+1.2\times {10}^{-5}m{A}^2+1.65\times {10}^{-7}A{B}^2+\\ & 9.58\times {10}^{-7}{A}^3-8.86\times {10}^{-8}{B}^3\end{array}$$

(7)

To assess the reliability of this display expression, additional determination coefficients R² are introduced for further evaluation, whose expression is as follows:

$$R^2={\displaystyle \frac{{\displaystyle \sum _{i=1}^Z{\left({\widehat{y}}_i-{\overline{y}}_i\right)}^2}}{{\displaystyle \sum _{i=1}^Z{\left(y_i-{\overline{y}}_i\right)}^2}}}$$

(8)

where Z represents the number of test points, $y_i$ denotes the true value of the test point, ${\widehat{y}}_i$ signifies the predicted value derived from the fitted curve, and ${\overline{y}}_i$ corresponds to the average of true values at each respective test point. After performing calculations, it is evident that the multiple determination coefficients of explicit expressions (6) and (7) are ${R^2}_L=0.956$ and ${R^2}_D=0.932$, respectively. Notably, both coefficients closely approach unity. This shows that the obtained display expression from fitting the surface demonstrates a higher level of reliability. The comparison between the simulated value and the predicted response value is further illustrated in Figure 9. The results demonstrate that the design points utilized in the simulation exhibit a small degree of dispersion around the regression line, indicating a strong agreement between the predicted and simulated target variable values. This signifies that the explicit expression derived from the response surface method effectively characterizes the relationship between the target variable and three J-C constitutive parameters.

After obtaining a reliable display expression through the response surface method, we further employ the genetic algorithm to iteratively optimize the solution distribution. To achieve quick and accurate identification of the optimal solution, this study adopts an improved genetic algorithm known as NSGA-II, specifically the fast non-dominated sorting genetic algorithm with elite strategy [26]. The algorithm under consideration is a multi-objective genetic algorithm that holds significant influence and boasts a wide range of applications. Its essence lies in effectively managing the interplay between diverse objective functions, thereby identifying the Pareto optimal solution set capable of maximizing (or minimizing) each function value. The process involved in NSGA-II is visually depicted in Figure 10. The distribution of Pareto optimal solutions tends to stabilize after approximately 200 iterations.

3.4. Inversion from Experimental Data and Evaluation of Results

Using NSGA-II for the iterative solution of the display expression, about two hundred Pareto optimal solutions can ultimately be obtained. To gain an intuitive understanding of these solutions, they are plotted in Figure 11 using blue dots, with each Pareto optimal solution corresponding to a set of J-C constitutive parameters. As can be seen from the figure, these Pareto optimal solutions exhibit a descending distribution trend, i.e., the deeper the perforating depth, the smaller the corresponding perforating aperture, which is consistent with the empirical rule of perforation hole formation. To obtain the J-C constitutive parameters applicable to the new powder liner, the results of the penetrating steel target experiment are also drawn in Figure 11 with a black triangle, the average value of the experimental results is taken, and the average value is marked as the green triangle; thus, the Pareto optimal solution (red dot) that is closest to the experimental average value can be found. The J-C constitutive parameters corresponding to the red dots are A = 90 MPa, B = 25 MPa, m = 0.1, C = 0.025, and n = 0.31, respectively, which can be regarded as the optimal J-C constitutive parameters of the new powder liner.

To verify the reliability of the retrieved J-C constitutive parameters of the powder liner, numerical simulation of the penetration of the steel target by the projectile flow of the shaped charge was carried out based on the retrieved constitutive parameters. The simulation results are shown in Figure 12b. The results were compared with the simulation results based on copper constitutive parameters (Figure 12a) and experimental results (Figure 12c). At the same time, to display the achievements of this research more intuitively, we present the experimental results in

Table 5. Comparison table between experimental data and numerical simulation results.

Data Source	Perforation Depth (cm)	Perforation Aperture (cm)	Depth Error (cm)	Depth Error	Aperture Error (cm)	Aperture Error
Experiment	30.04	1.13	-	-	-	-
Numerical simulation (known parameters)	19.85	1.31	10.19	33.92%	0.18	15.93%
Numerical simulation (inversion parameters)	31.50	1.1	1.46	4.86%	0.03	2.66%

. It was found that the perforation depth obtained based on the inversion parameters was 31.50 cm, and the perforation aperture was 1.10 cm. The perforation depth was 9.2 cm higher than that obtained based on the copper constitutive parameter, and the aperture was reduced by 0.21 cm; both the perforation depth and perforation aperture are closer to the experiment of the penetrating steel target. The error between the perforation depth and the experimental results obtained by the perforation simulation based on the constitutive parameters obtained in this paper is 4.86%, and the error of the perforation aperture is 2.66%—both of which are within 5%, which indicates that the constitutive parameters of the powder liner obtained by the inversion method in this paper are effective and reliable. Therefore, the constitutive parameters obtained from the inversion can be applied to the subsequent numerical simulation of hard rock perforation.

4. The Inversion of HJC Constitutive Parameters for Hard Rock

4.1. 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. Additionally, 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.

4.1.2. Material Model and Parameters

The material model used for the explosive, shell case, and air is the same as above, and the model parameters of the powder liner are selected after inversion. Furthermore, both the gun holster and casing are composed of steel 45, and its model and parameters are the same as above. Therefore, these material models and parameters will not be reiterated in this section. This section mainly introduces the constitutive models and parameters of hard rock, water, and cement ring materials not mentioned above.

(1)

Hard rock

Based on previous studies, the HJC model is utilized to characterize hard rock in this study [28]. The HJC model, proposed by Holmquist et al. [29], serves as a constitutive model for concrete under conditions of large strain and a high strain rate, enabling a more comprehensive depiction of its mechanical properties during explosion, impact, and penetration. This model effectively accounts for the accumulation of compressive damage while also capturing the influence of strain rate, compressive strength, and confining pressure on material behavior. Consequently, it provides a more accurate representation of dynamic damage occurring in materials subjected to significant deformation and high strain rates. In this paper, the HJC constitutive model considering strain rate and damage is described as follows:

Strength model equation:

$${\sigma }^{\ast }=[A_1(1-D)+B_1{P}^{\ast N_1}][1+C_7\ln {\epsilon }^{\ast }]$$

(9)

In the equation, ${\sigma }^{\ast }=\sigma /f_c$ is the characteristic equivalent stress, where σ is the actual equivalent stress, and f_c is the quasi-static uniaxial compressive strength of the material; $P^{\ast }=P/f_c$ is the characteristic hydrostatic pressure, where P is the actual pressure borne by the material; ${\epsilon }^{\ast }=\epsilon /{\epsilon }_0$ is the characteristic strain rate, where $\epsilon$ is the actual strain rate of the material, and ${\epsilon }_0$ is the reference strain rate; D is the damage parameter; and A₁, B₁, N₁, C₇, and S_max are material constants, Among them, A₁ is the standardized cohesion strength, B₁ is the standardized pressure hardening coefficient, N₁ is the pressure hardening index, C₇ is the strain rate coefficient, and S_max is the characteristic ultimate strength of the material.

Damage evolution equation:

$$\mathrm{D}=\sum {\displaystyle \frac{\Delta {\epsilon }_p+\Delta {\mu }_p}{{\epsilon }_p^f+{\mu }_p^f}}$$

(10)

In the equation, ${\epsilon }_p^f+{\mu }_p^f$ represents the plastic strain when the material breaks under normal pressure; ${\epsilon }_p^f+{\mu }_p^f=D_1{(p\ast +T\ast )}^{D_2}$, where $\Delta {\epsilon }_p$ and $\Delta {\mu }_p$ represent the equivalent plastic strain and plastic volume strain of the material in a calculation cycle; $T\ast$ represents the maximum characteristic hydrostatic tension that the material can withstand; and D₁ and D₂ are damage constants.

In the process of calculation with the LS-DYNA program, it is found that the FS failure parameter of the HJC constitutive model cannot effectively control the unit failure. The “*MAT_ADDE REROSION” erosion failure criterion is therefore introduced to regulate unit failure. This paper adopts the maximum principal strain failure criterion (${\epsilon }_{\max }$), which is currently the most widely employed criterion for rock materials [30]. Normalized parameters A₁, B₁, N₁, and C₇, damage parameter D₁, uniaxial tensile mild f_c, and maximum failure principal strain ${\epsilon }_{\max }$ were selected as design variables based on previous studies [31,32,33,34,35]. The values of key constitutive parameters of hard rock are shown in

Table 6. Key parameters of the constitutive model of hard rock [36].

A1	B1	N1	C7	D1	fc/MPa	εmax
0.55	1.23	0.89	0.0097	0.04	60	0.1

.

(2)

Water layer and cement ring

In this study, the wellbore pressure is considered in the numerical simulation of simulated reservoir penetrating hard rock targets. Therefore, it is necessary to utilize the Null model and the Gruneisen state equation to accurately describe the water layer. The material model is the same as the air domain, and the expression of the state equation is the same as that of the powder liner (Equation (1)). However, in this context, deviational stress should be disregarded, and its viscous stress equation can be expressed as follows:

$${\displaystyle \frac{d{\sigma }_{\mathit{ij}}}{dx}}=2\mu {\displaystyle \frac{d{\epsilon }_{\mathit{ij}}}{dx}}$$

(11)

where $\mu$ is the viscosity coefficient; ${\epsilon }_{\mathit{ij}}$ is the strain rate deviation. The key parameters are shown in

Table 7. Material parameters of the water layer [22].

S1	S2	S3	$\mathit{\gamma }$	Aw	E0	V0	$\mathit{\mu }$/mPa·s
1.92	−0.096	0	0.35	0	0	0	1.002

.

Similarly to hard rock, the HJC model is also employed to describe cement rings; however, since the model parameters for cement rings are widely adopted and relatively well established, it is unnecessary to reiterate them here, and the key parameters are shown in

Table 8. Key parameters of the constitutive model of cement ring [37].

A1	B1	N1	C7	D1	fc/MPa
0.79	1.6	0.61	0.007	0.03	80

.

4.1.3. Numerical Simulation Result

The numerical simulation of the simulated reservoir penetrating hard rock target was conducted based on the aforementioned model and parameters (the parameters obtained by inversion were adopted for the powder liner), and the corresponding results are presented in Figure 15. The simulation results indicate a perforation depth of 27.05 cm, while the experimental data show a penetration depth of 32 cm, resulting in an error of 15.47%. Furthermore, the simulated perforation aperture measures 1.76 cm compared to the experimental aperture of 1.08 cm, with a bigger discrepancy between them. This indicates that the existing HJC constitutive parameters make it difficult to accurately describe the mechanical behavior of hard rock under perforation detonation conditions. Therefore, it is imperative to invert the constitutive parameters of hard rock in a reservoir based on experimental data, following a similar approach as described above for inverting the constitutive parameters of powder liner.

4.2. Parameter Sensitivity Analysis

In this section, based on the numerical calculation model established in Section 4.1, normalized parameters A₁, B₁, N₁, and C₇, damage parameters D₁, uniaxial tensile mild f_c, and maximum failure principal strain ${\epsilon }_{\max }$ are changed, respectively, and the depth and aperture of jet flow penetration into hard rock targets under different parameter values are obtained. To ensure the comprehensiveness of subsequent fitting, we appropriately extended the value ranges of these parameters based on the relevant literature [38]; this is shown in Figure 16 and Figure 17.

Figure 15 illustrates the relationship between parameters A₁, B₁, C₇, and f_c for variations in perforation depth and aperture. With the increase in parameter A₁, there is an increase in perforation depth and a corresponding decrease in perforation aperture. With the increase in parameter B₁, both the perforation depth and aperture exhibit a gradual increment. With the increase in parameter C₇, there is a decrease in perforation depth and an increase in perforation aperture. With the increase in parameter f_c, there is a gradual decrease in both the perforation depth and aperture. Although the perforation depth and aperture exhibit variations based on parameters A₁, B₁, C₇, and f_c, they demonstrate consistent monotonic behavior.

The variation in perforation depth and perforation aperture with parameters N₁, D₁, and ${\epsilon }_{\max }$ is illustrated in Figure 16. With the increase in parameter N₁, the perforation depth initially exhibits an upward trend followed by a subsequent decline. But the perforation aperture demonstrates a positive correlation with parameter N₁. The perforation depth and aperture both exhibit an initial increase followed by a decrease with the increase in parameter D₁. The perforation aperture initially decreases and then increases with the increase in parameter ${\epsilon }_{\max }$. Obviously, the effects of parameters N₁, D₁, and ${\epsilon }_{\max }$ on the perforation depth and aperture are evidently more intricate and non-monotonic when compared to parameters A₁, B₁, C₇, and f_c.

4.3. Multi-Objective Optimization

Section 4.2 reveals that the influence of parameters N₁, D₁, and ${\epsilon }_{\max }$ on perforation depth and aperture exhibits a non-monotonic trend. Consequently, achieving convergence of these parameters in the subsequent fitting process becomes challenging. Therefore, the design variables in this study select parameters A₁, B₁, C₇, and f_c, while parameters N₁, D₁, and ${\epsilon }_{\max }$ are adopted based on the literature above. According to the previous literature research, a relationship between A₁ and C₇ can be observed when there is a change in uniaxial tensile strength, as demonstrated by Equation (12) [31], where c represents cohesion. Hence, parameters A₁ and C₇ are conjoined for investigation in this study.

$$A_1={\displaystyle \frac{c}{\left(1+C_7\ln {10}^{-4}\right)f_c}}$$

(12)

On this basis, using the central composite design to simulate the combination of different parameters, the fitting surface of the target variable (L, D) for the inversion variables (B₁, C₇, f_c) can be obtained. The fitting surface of perforation depth and perforation aperture under the interaction of parameters B₁ and C₇ is illustrated in Figure 18. The figure demonstrates a strong fit between the data points and the surface; therefore, the explicit expressions of the target variables obtained by fitting the surface for the design variables are Equations (13) and (14).

$$\begin{array}{ll}L=& 36+0.35B+20C+0.26f_c-0.11B^2-355.77C^2+8.23f_c^2\\ & -35.85BC-2.52B{f}_c-59.85C{f}_c+155.8BC{f}_c\\ & +13.16B^{2}C+0.98B^2{f}_c+676.32B{C}^2-0.20B{f}_c^2\\ & -210.02C^2{f}_c-0.012B^3-2501.33C^3\end{array}$$

(13)

$$\begin{array}{ll}D=& 1.3+\mathrm{0.0.016}B+1.2C-0.0023f_c-0.0039B^2-27.54C^2+0.39f_c^2\\ & -2.25BC-0.11B{f}_c-1.95C{f}_c+7.58BC{f}_c+0.83B^{2}C+0.045B^2{f}_c\\ & +54.43B{C}^2-0.015B{f}_c^2-30.36C^2{f}_c-0.011B^3-176.73C^3\end{array}$$

(14)

After performing the calculations, it is evident that the multiple determination coefficients of explicit expressions (13) and (14) are ${R^2}_L=0.941$ and ${R^2}_D=0.952$, which are also very close to 1. The comparison between the simulated values and the predicted response values is further illustrated in Figure 19. The results demonstrate strong agreement between the predicted and simulated target variable values, providing additional evidence that the explicit expression derived from the response surface method effectively characterizes the relationship between the target variable and the three HJC constitutive parameters.

4.4. Inversion from Experimental Data and Evaluation of Results

The distribution of Pareto optimal solutions of target variables, obtained through NSGA-II, is depicted by the blue dot in Figure 20. By comparing with the experimental value, the Pareto optimal solution (red dot) closest to the experimental mean value is found. The corresponding parameter values of this solution (perforation depth 32.01 cm, perforation aperture 1.05 cm) are A₁ = 0.9, B₁ = 0.8, C₇ = 0.1625 and f_c = 60 MPa, respectively, thus indicating that these parameters can be considered optimal HJC constitutive parameters for hard rock in reservoirs.

Similarly, to verify the reliability of the HJC constitutive parameters of the hard rock obtained by inversion, numerical simulations of the reservoir penetrating the hard rock target with the constitutive parameters obtained by inversion were conducted, and the simulation results are shown in Figure 21b. Compared with the simulation results based on known constitutive parameters (Figure 21a) and experimental results (Figure 21c), it can be found that the perforation depth obtained based on inversion parameters is 32.14 cm, and the perforation aperture is 1.13 cm. The perforation depth is 5.09 cm higher than that of 27.05 cm with known parameters, and the aperture is 0.60 cm smaller than the known parameter of 1.76 cm. Thus, the optimization effect is remarkable. In the experiment, the average perforation depth is 30.00 cm, and the average aperture is 1.08 cm. The error between the simulated perforation depth and the experimental results using the inversion parameters is 0.44%, and the error between the perforation aperture is 4.63%, both of which are also within the acceptable 5%. For ease of understanding, we present these results in their entirety in

Table 9. Comparison table between experimental data and numerical simulation results.

Data Source	Perforation Depth (cm)	Perforation Aperture (cm)	Depth Error (cm)	Depth Error	Aperture Error (cm)	Aperture Error
Experiment	32.00	1.08	-	-	-	-
Numerical simulation (known parameters)	27.05	1.76	4.95	15.47%	0.68	62.96%
Numerical simulation (inversion parameters)	32.14	1.13	0.14	0.44%	0.05	4.63%

. This indicates that the inverse parameters can more accurately describe the damage effects of the jet on hard rock than the known parameters in the numerical simulation of perforation, which also proves that the inverse calculation of the HJC constitutive parameters of hard rock is effective and reliable.

5. Conclusions

In this study, to acquire the parameters of the J-C constitutive model of powder liner in shaped charge and the parameters of the HJC constitutive model of hard rock in the reservoir, a penetrating steel target experiment and a simulated reservoir penetrating hard rock target experiment were first carried out, and the finite element software ANSYS/LS-DYNA (16.0)was used to reproduce the process of a jet penetrating a steel target and hard rock, respectively. Then, the expression of target variables (perforation depth L and perforation aperture D) regarding the parameters in the constitutive model was obtained through the utilization of the response surface method. Finally, a multi-objective genetic algorithm was employed to solve the display expression iteratively. We identified the Pareto optimal solution that closely approximated the experimental mean value by comparing it with experimental data. Then, we derived effective J-C constitutive parameters for powder liner and HJC constitutive parameters for hard rock.

The numerical simulation results, based on the reconstructed J-C constitutive parameters and HJC constitutive parameters, demonstrate that the perforation depth and aperture resemble the experimental results. This suggests that the constitutive parameters of the inversion offer a more accurate depiction of the mechanical behavior of powder liner and hard rock during actual perforation processes. Simultaneously, it was demonstrated that the perforation simulation based on the inversion of constitutive parameters could provide a more precise prediction of the damage induced by the metal jet formed through powder liner on hard rock. This finding establishes a fundamental basis for further investigation into both powdered liner and the process of hard rock perforation. At the same time, by combining numerical simulation and experimental data, the method overcomes the limitations of traditional experimental methods and reduces the dependence on experiments, thus reducing the time and economic cost that may be brought by experiments. The constitutive parameters of powder liner and hard rock can be obtained more efficiently in a short time, which also makes the results more applicable. In addition, this method not only provides a new theoretical framework for the current constitutive parameter inversion of powder liner and hard rock but also provides a new reference and inspiration for researchers in related fields in material behavior prediction, energy conversion, and transfer, etc., providing a new method for structural optimization in related engineering fields, which has great application potential. It further promotes the combination of basic research and engineering application.