Numerical Analyses of Perforation and Formation Damage of Sandstone Gas Reservoirs

Abstract

Shaped charge perforation is an important technology for sandstone gas reservoirs. In the process of shaped charge perforating, the initial permeability and porosity of the formation are greatly reduced, directly affecting oil and gas production. This paper uses smooth particle hydrodynamics (SPH) and finite element methods (FEMs) to study the formation damage caused by the shaped charge perforation. In this perforation simulation, the impact characteristics of the metal jet formed by the liner are coupled with the damage characteristics of the sandstone. A new mathematical model is proposed to describe the damage of permeability and porosity based on the Morris and Xue models. The simulated results show that a large-scale abdominal section appears at the perforation site, and the axis of the hole tilts with the increase in the perforation depth. The porosity damage at the perforation site is the greatest, up to 60%, while the permeability recovers to 90% of its initial state after pressure relief. The degree of porosity damage decreases with the increase in negative pressure, and when the negative pressure is 30 MPa, it may cause a large amount of sand production.

1. Introduction

Shaped charge jet perforation is the key technology for a conventional oil and gas field [1,2,3]. Shaped charge perforation consists of four steps: detonating the explosive, the collapse of the liner, the formation and extension of the metal jet, and jet cratering and penetration. The high-pressure load produced by the high-speed metal jet hits the rock target and forms a hole. The plastic deformation of the rock is made near the hole, which creates the classical rock texture of the perforation damage area [4]. Therefore, the damage mechanism of the shaped charge jet perforation should be studied clearly, which will be helpful to improve the technology of shaped charge jet perforation and increase the production of oil and gas wells.

In recent decades, the numerical simulation of shaped charge jet perforation has been studied by many researchers. Zygmunt and Wilk [2] applied the X-ray pulse technique to research the process the jet stream formation and described the characteristics of shaped charges with metal powder liners. Karacan and Halleck [3] studied the damage of porosity and permeability for fractured zones caused by perforation in the gas-saturated and liquid-saturated Berea sandstone. Huang et al. [5] and Li et al. [6] investigated the mechanisms of and presented the results of laboratory experiments and the results of field tests on the abrasive water jet perforation for enhancing oil production. Li et al. [7] investigated the application of annular initiation to the explosively formed penetrator charge based on the LS-DYNA software (V6.5). Ayisit [8] performed numerical analyses to investigate the influence of shaped charge asymmetries on the jet characteristics.

In the above works, the researchers always used the Lagrange method, Euler method, and Arbitrary Lagrangian–Eulerian method (ALE) [9,10,11,12] to study the hole-forming process of shaped charge jet perforation. However, the Lagrange method can cause problems as the grids produce significant distortion and slip surfaces. With the Euler method, it is challenging to determine the position of the material interface, and the ALE method needs to build a more significant flow of space grids. These methods cannot sufficiently reflect the action process of metal particles on the rock. Therefore, it is better to choose the smoothed particle hydrodynamics (SPH) method [13,14,15] to simulate the shaped charge jet perforating metal particle flow.

In recent decades, considerable research has focused on simulating the shape of the shaped charge, the amount of ammunition, the dynamic form of the shaped charge jet, the jet velocity, and the hole-forming process through the metal target [16,17]. Gooch et al. [18] analyzed the target strength effect on penetration by shaped charge jets. Yin et al. [19] investigated the “white” etching layer on the perforation surface and adiabatic shear bands (ASBs) in the matrix of ultra-high-strength steel plates penetrated by shaped charge jets. Uhlig and Coppinger [20] used a means of the confined liquid to investigate the interplay of eroded target material and the remaining projectile to elucidate relevant penetration behaviors in solid materials. Zhu et al. [21] discussed the evolution of the response regions in front of the jet/target interface as the shaped charge jet moves into high- and ultra-high-strength concrete targets. Elshenawy and Li [22] studied the influences of concrete strength and its confinement pressure on the shaped charge jet penetration. Guo et al. [23] confirmed that the virtual origin theory could describe the interaction process between the target and the shaped jet based on the characteristic expansion hole and liquid return characteristics. However, little research has been conducted on the penetration of shaped charge jet through a sandstone target.

During the perforation process, the high-speed metal flow penetrates the formation while forming a compaction zone near the hole. The permeability and porosity of the sandstone near the hole are reduced. This extremely low permeability significantly reduces oil and gas wells’ production capacities. Therefore, it is necessary to analyze the distribution of porosity and permeability in the compaction zone and evaluate the damage degree. Morris et al. [24,25] proposed an evolution model of the porosity and permeability of sandstone with a low strain rate, which used plastic strain, effective stress, and damage variables to describe the evolution of porosity and permeability. Zhang and Shifeng [26] developed a mechanical model for evaluating the perforating damage of sandstone based on the sensitivity analysis of the deformation rate effect on porosity evolution. Kemmoukhe et al. [27] used AUTODYN-2D to study the effect of the main parameters (liner material, explosive charge, stand-off distance, and wave-shaper presence) on the jet formation, jet velocity, jet length, and penetration depth. Khamitov et al. [28] conducted a coupled 3D CFD–DEM analysis to simulate the perforation damage and sand production of soft sandstone materials. Yan et al. [29] set up a numerical model based on the LS-DYNA software (V6.5). A cylindrical target penetration test was used to verify the interface damage area caused by perforation. Fu et al. [30] studied the penetration behavior of convergent, divergent, and straight linear-shaped charges (LSCs) by numerical and physical methods.

Previously, most numerical simulation studies on shaped charges mainly used the LS-DYNA and AUTODYN software (V6.5). But the calculated results are not accurate, and the phenomena are usually different from the experimental results. Therefore, developing a new efficient method is necessary to provide more accurate results. Although the previous numerical simulation research about perforation compaction obtained excellent results, the compaction damage models in 2D and 3D in previous works were established by simplifying the perforating process, and the non-symmetrical and non-uniform shape of holes and the distribution of the compaction area caused by shaped charge jet were neglected.

This paper studies the pore-forming process of the shaped charge jet perforation and the influence of the perforation damage on the formation’s permeability and porosity numerically. The SPH algorithm is applied to establish the finite element model of the hole-forming process of sandstone by considerable high-speed jet particles impacting the sandstone. The ALE adaptive mesh technology is used to dynamically encrypt the mesh around the hole. The multi-node calculation is carried out by an HPC (high-performance computer) to ensure the accuracy of the results. The Hoffman damage model is used to assess the 3D hole model that is formed by the failure of sandstone. The hole-forming process and the 3D hole structure can be analyzed by the hole compaction zone, which is characterized by a custom field. Therefore, a set of mathematical modeling research methods are proposed, which has great significance in guiding the selection of perforation equipment and parameters of sandstone with low porosity and permeability, reducing the perforation damage and increasing the production of oil and gas wells. In addition, based on the model in Ref. [24], a novel mathematical model for evaluating the compaction damage in the perforation process of sandstone with low porosity and permeability is proposed to describe the permeability and porosity of the damaged formation. The distribution characteristics of compaction damage under different working conditions and the influencing factors are considered in this paper. The present work can provide a foundation for future research and has considerable referential significance.

2. Modeling Theory

The process of the shaped charge jet perforation is shown in Figure 1. The shaped charge is mainly composed of a liner and explosive, as shown in Figure 1a. When the explosion happens, the copper metal particle jet with an extremely high speed is formed, penetrating the perforating gun, well fluid, casing, and cement ring into the formation, as shown in Figure 1b. Amid the continuing explosion, the velocity of the jet becomes slower until a velocity gradient forms. A tremendous pressure is generated by the ultra-high velocity jet that creates perforation holes, as shown in Figure 1c.

2.1. Smoothed Particle Hydrodynamics Algorithm

The SPH method was applied to study the shaped charge jet perforation process in this paper. This method is a meshless fluid dynamic solution with pure Lagrange adaptive properties and is a clever method for discretizing continuous partial differential equations. It better reflects the process of metal particles acting on rock compared to the Lagrange method, Euler method, ALE method, and other numerical calculation methods.

The SPH method is used to solve hydrodynamics problems in two steps. The first step is the integral representation, which can be approximated by the integral over the influence region of the kernel function. The second step is the particle representation, which matches the function value of the discrete point by summing the importance of the nearest adjacent particles. SPH uses an evolutionary interpolation scheme to compare field variables at any point in the field. The kernel function is as shown:

$$〈f\left(x\right)〉={\displaystyle \underset{\mathrm{\Omega }}{\int }f\left(x^{\prime }\right)}W\left(x-x^{\prime },h\right)d{x}^{\prime }$$

(1)

where $W(x-x^{\prime },h)$ is a kernel function and is not zero; $x$ is the spatial coordinates of points; $x^{\prime }$ is the coordinate of the point in space that contributes to $x$; and $h$ is the smooth length that determines how many particles affect the interpolation of a particular point.

The kernel function $W$ plays an important role in the SPH approximation method, which determines the accuracy and computational efficiency of function expressions. It should satisfy three conditions. Namely, the compactness condition, the regularization condition, and Dirac function property as the smooth length approaches zero. The equations are expressed as follows:

$${\displaystyle \underset{\mathrm{\Omega }}{\int }W\left(x-x^{\prime },h\right)d{x}^{\prime }}=1,$$

(2)

$$\left|x-x^{\prime }\right|>kh, W\left(x-x^{\prime },h\right)=0,$$

(3)

$$\underset{h\to 0}{\lim }W\left(x-x^{\prime },h\right)=\delta \left(x-x^{\prime }\right), \delta \left(x-x^{\prime }\right)=\left\{\begin{array}{c}1,x=x^{\prime }\\ 0,x\ne x^{\prime }\end{array}\right.,$$

(4)

where $k$ is the constant that is related to the smooth function at $x$ and can determine the effective range (non-zero), which is called the integral in the support region of the smooth function at $x$. Therefore, the domain of integration $\mathrm{\Omega }$ is generally the support region.

According to the integration by parts and the divergence theorem, Equation (1) is transformed. The derivative of $f(x)$ is estimated as:

$$〈\nabla f\left(x\right)〉=-{\displaystyle \underset{\mathrm{\Omega }}{\int }f\left(x^{\prime }\right)\nabla W\left(x-x^{\prime },h\right)dx}.$$

(5)

From Equation (3), it is known that the SPH method transforms the spatial derivatives of functions into the spatial derivatives of smooth functions. Therefore, the spatial derivatives of arbitrary field functions can be obtained from the kernel functions.

After discretization, the integral expression of the function can be written into the particle approximation as follows:

$$〈f\left(x_i\right)〉={\displaystyle \sum _{j=1}^N\frac{m_j}{{\rho }_j}}f\left(x_j\right)W_{ij},$$

(6)

where $j$ and $i$ are particle numbers; $m_j$ and ${\rho }_j$ are the mass and density of the $j$ particles, respectively; and $N$ is the total number of particles.

It can be seen from Equation (7) that the value of the arbitrary function at a certain position can be expressed by applying the smooth function $W$ to the form of interpolation and summation of all particles in the compact support domain of smooth length h [31].

The SPH calculation equations of the shaped charge jet perforating metal flow simulation are [31,32]:

$$\left\{\begin{array}{ll}\frac{d{\rho }_i}{dt}& ={\displaystyle \sum _{j=1}^N}{m}_j\left(v_i^{\beta }-v_j^{\beta }\right)\frac{\partial W_{ij}}{\partial x_i^{\beta }},\\ \frac{d{v}_i^{\alpha }}{dt}& =-{\displaystyle \sum _{j=1}^N}{m}_j\left(\frac{P_i}{{\rho }_i^2}+\frac{P_i}{{\rho }_j^2}+{\mathrm{\Pi }}_{ij}\right)\frac{\partial W_{ij}}{\partial x_i^{\alpha }}+{\displaystyle \sum _{j=1}^N}{m}_j\left(\frac{S_i^{\alpha \beta }}{{\rho }_i^2}+\frac{S_i^{\alpha \beta }}{{\rho }_j^2}\right)\frac{\partial W_{ij}}{\partial x_i^{\beta }},\\ \frac{d{u}_i}{dt}& =\frac{1}{2}{\displaystyle \sum _{j=1}^N}{m}_j\left(\frac{P_i}{{\rho }_i^2}+\frac{P_i}{{\rho }_j^2}+{\mathrm{\Pi }}_{ij}\right)\left(v_i^{\beta }-v_j^{\beta }\right)\frac{\partial W_{ij}}{\partial x_i^{\beta }}+\frac{1}{{\rho }_i}{S}_i^{\alpha \beta }{\dot{\epsilon }}_i^{\alpha \beta },\\ \frac{dh}{dt}& =-\frac{1}{2}\frac{h}{p}\frac{d\rho }{dt}\end{array}\right. .$$

(7)

where $\rho$ is the density, $m$ is the mass, $P$ is the pressure, $u$ is the internal energy, $v$ is the velocity, $S$ is the deviatoric stress, ${\prod }_{ij}$ is the artificial viscosity, $\epsilon$ is the strain rate, and $\alpha$ and $\beta$ are the coordinate directions.

2.2. Damage Model

The local structure of the rock is deformed by high pressure, which is generated by the penetration of the metal particle flow of shaped charge jet perforation into the low-permeability sandstone target. The large deformation rate near the particle surface and the slow heat release rate on the sliding surface softens the material near the particle. The distance between the face and that of the local deformation surface is expressed by $\mathrm{\Delta }$, and the following inequalities can define the conditions for softening:

$$\frac{{\mathsf{\Delta }}^2}{\chi }>\frac{d}{{\upsilon }_p},$$

(8)

where ${\mathrm{\Delta }}^2/\chi$ is the relaxation time for the temperature caused by heat conduction, $d/{\upsilon }_p$ is the characteristic time of deformation, and $\chi$ is the temperature thermal conductivity coefficient. The distance $\mathrm{\Delta }$ between the sliding surfaces should be of about 0.1–1.0 μm order of magnitude.

The cavity will collapse in time $\tau$ as pressure $p$ is constantly applied on the outer boundary of the grid. Therefore, the ultra-deep penetration conditions of the particles are

$$\tau \le \frac{d}{{\upsilon }_p}\le \frac{{\mathsf{\Delta }}^2}{\chi }.$$

(9)

In the strongly excited state, the target material can be considered as a weakly interrelated particle whole. This weak complementarity can be regarded as being in a thermodynamic equilibrium state by comparing to the motion state of the penetration particle. Therefore, the theory of weakly interacting particles describes the interaction between the penetration particle and the target material particle. Assuming that only the force between the particle center is considered, the interaction between particles is

$$V_{jn}={\left(\frac{2\pi }{L}\right)}^3{\displaystyle \sum _l{V}_l\exp \left[il\left(x_j-x_n\right)\right]},$$

(10)

where $l$ is the distance between the particles, $V_l$ is a coefficient that only depends on the wave vector/absolute value, and $x_j$ and $x_n$ are the coordinates of the $j_{th}$ particle and the $n_{th}$ particle, respectively.

The force acting on the particle in the ultra-deep penetration state is expressed as

$$F_p=-\left[H^{\prime }+2.6\frac{{\rho }_s{\left({\overrightarrow{\upsilon }}_p-{\overrightarrow{\upsilon }}_1\right)}^2}{\sqrt{\mathrm{Re}}}\right]\frac{\pi d^2}{4}\frac{{\overrightarrow{\upsilon }}_p-\overrightarrow{\upsilon }}{\left|{\overrightarrow{\upsilon }}_p-\overrightarrow{\upsilon }\right|},$$

(11)

where ${\overrightarrow{\upsilon }}_1$ is the velocity of the target material, $H^{\prime }$ is the dynamic stiffness, $d$ is the particle diameter, ${\rho }_s$ is the density of the target material, and ${\overrightarrow{\upsilon }}_p$ is the particle velocity.

The friction generated by the contact between the moving particles and rock particles makes the energy and the velocity of the shaped charge perforating metal particle flow decrease rapidly. The friction coefficient between the particle and the target body is expressed as follows

$$\eta =\frac{16q\left(\upsilon \right)}{\upsilon },$$

(12)

with two limits

$$\left\{\begin{array}{c}\upsilon \to 0:q\left(\upsilon \right)~\upsilon , \eta =const,\\ \upsilon \to 0:q\left(\upsilon \right)=\frac{1}{2{\upsilon }^2}, \eta ~\frac{1}{{\upsilon }^3}.\end{array}\right.$$

(13)

The Hoffman criterion describes the penetration of jet particles into sandstone more comprehensively. In the numerical analysis, the dimensionless failure coefficient $F_d$ is introduced to represent the failure degree of the rock so that the Hoffman criterion can be written as

$$\begin{array}{l}{F}_d=C_1{\left({\sigma }_2-{\sigma }_3\right)}^2+C_2{\left({\sigma }_3-{\sigma }_1\right)}^2+C_3{\left({\sigma }_2-{\sigma }_1\right)}^2\\ +C_4{\sigma }_1+C_5{\sigma }_2+C_6{\sigma }_3+C_7{\sigma }_{23}^2+C_8{\sigma }_{13}^2+C_9{\sigma }_{12}^2,\end{array}$$

(14)

where

$$C_1=\frac{1}{2}\left(\frac{1}{Z_t{Z}_c}+\frac{1}{Y_t{Y}_c}+\frac{1}{X_t{X}_c}\right), C_2=\frac{1}{2}\left(\frac{1}{Z_t{Z}_c}-\frac{1}{Y_t{Y}_c}+\frac{1}{X_t{X}_c}\right),$$

(15)

$$C_3=\frac{1}{2}\left(\frac{1}{Y_t{Y}_c}+\frac{1}{X_t{X}_c}-\frac{1}{Z_t{Z}_c}\right), C_4=\frac{1}{X_t}-\frac{1}{X_c}, C_5=\frac{1}{Y_t}-\frac{1}{Y_c},$$

(16)

$$C_6=\frac{1}{Z_t}-\frac{1}{Z_c}, C_7=\frac{1}{S_{23}^2}, C_8=\frac{1}{S_{13}^2}, C_9=\frac{1}{S_{12}^2},$$

(17)

where $X_t$, $Y_{\mathrm{t}}$, and $Z_{\mathrm{t}}$ are the tensile strength values of the rock in the $X$, $Y$, and $Z$ directions, respectively. $X_c$, $Y_c$, and $Z_c$ are the compressive strength values of the rock in the $X$, $Y$, and $Z$ directions, respectively. $S_{12}^2$, $S_{23}^2$, and $S_{13}^2$ are on the 12, 23, and 13 surfaces’ in-plane shear strength, respectively.

Due to the complex mechanical properties of deep sandstone, establishing a damage plastic constitutive model is the key to accurately calculating the process of shaped charge jet perforation. Figure 2 plots the mechanical response curve of the damage plasticity of sandstone material. It can be seen that the unloading segment is weakened (the slope of the curve decreases) when the softened section is unloaded, which indicates that the material has been damaged (or weakened).

The damage variable $d_t$ is a function of the plastic strain, temperature, and field variable and can be written as

$$d_t=d_t\left({\tilde{\epsilon }}_t^{pl},\theta ,f_i\right) 0\le d_t\le 1,$$

(18)

where ${\tilde{\epsilon }}_t^{pl}$ is the tensile equivalent plastic strain, $\theta$ is the temperature, and $f_i\left(i=1,2,\dots \right)$ is another predefined field variable. $d_t=0$ corresponds to a non-damaged material, and a completely damaged material is $d_t=1$.

The softening section of the sandstone damage plastic constitutive model is controlled by the amount of damage and the energy released by the plastic deformation. The material damage plastic stiffness degradation model can be expressed as

$$E=\left(1-d\right)E_0, {\sigma }_t=\left(1-d_t\right)E_0\left({\epsilon }_t-{\tilde{\epsilon }}_t^{pl}\right),$$

(19)

where $E_0$ is the initial (damage-free) elastic stiffness of the material and ${\sigma }_t$ is the tensile strength.

2.3. Mathematical Model for the Perforation Compaction Damage

In this paper, based on the mechanical response characteristics of sandstone under perforation impact load, the modified evolution equation of sandstone porosity and permeability was employed, which was proposed in Refs. [30,31]. Therefore, a mathematical model of the compaction damage of perforation with low-porosity and permeability was established.

The porosity $\varphi$ of sandstone in the compaction zone consists of two parts: the initial porosity ${\varphi }_i$ and expanded porosity ${\varphi }_{dil}$. The initial porosity ${\varphi }_i$ is expressed as

$${\varphi }_i=\frac{{\varphi }_i^0}{1+D},$$

(20)

where ${\varphi }_i^0$ and $D$ are, respectively, the porosity of sandstone before damage and the damage variable of the rock mass element, which can be determined by Equation (16).

$${\varphi }_i^0=\left\{\begin{array}{c}{\varphi }_i-{\sigma }_{eff}\left({\varphi }_i-\left({\varphi }_{*}+C{P}_{*}\right)\right)/P_{*}\\ {\varphi }_{*}+C{\sigma }_{eff} {\sigma }_{eff}\ge P_{*},\end{array}\right.$$

(21)

$$D=\left\{\begin{array}{c}{C}_1{\epsilon }_p\min \left({\sigma }_{eff},P_s^{*}\right)-C_2{\epsilon }_v\\ 0\hspace{1em}\hspace{1em}\left(D<0\right),\end{array}\right.$$

(22)

where ${\varphi }_{\ast }$ is the fluid porosity intercept; $C$ is the porosity decay rate; $P_{\ast }$ is the critical effective stress; ${\sigma }_{eff}$ is the effective stress; $C_1$ is the plastic strain damage factor; $C_2$ is the hydrostatic pressure damage factor; $P_s^{\ast }$ is the maximum plastic strain damage stress; ${\epsilon }_p$ is the plastic strain of rock mass element; and ${\epsilon }_v$ is the volume strain of rock mass element.

The expanded porosity ${\varphi }_{dil}$ can be expressed as follows:

$${\varphi }_{dil}={\varphi }_{dil}^{\max }-\exp \left(\eta -\frac{m_d{\sigma }_{vm}{\epsilon }_p}{\max \left(P_m,P_{\min }\right)}\right),$$

(23)

where ${\varphi }_{dil}^{\max }$ is the maximum expanded porosity; $P_m$ is the average stress; $P_{\min }$ is the minimum expansion stress; $m_d$ is the coefficient of expansion; and ${\sigma }_{vm}$ is the von Mises stress.

According to the effective stress, volumetric strain, and plastic strain of the rock mass unit, the distribution of porosity and permeability in the compacted zone can be quantitatively analyzed through the damage evaluation mathematical model of the porosity and permeability in the sandstone perforated compacted zone. The expressions of the porosity $\varphi$ and permeability $k$ are

$$\varphi ={\varphi }_{dil}^{\max }-\exp \left(\eta -\frac{m_d{\sigma }_{vm}{\epsilon }_p}{\max \left(P_m,P_{\min }\right)}\right)+\frac{{\varphi }_i^0}{1+D},$$

(24)

$$k=k_0\exp \left(\frac{k_1{\varphi }_i^0\left({\sigma }_{eff},D\right)}{1+D}-B\min \left(D,D_{\max }\right)\right),$$

(25)

where $k_0$ is the initial permeability; $k_1$ is the permeability index; $B$ is the permeability damage attenuation factor; and $D_{\max }$ is the maximum permeability damage.

The evaluation index for the damage degree of the porosity and permeability can be expressed as follows:

$${\varphi }_{cd}=\frac{{\varphi }_i-\overline{\varphi }}{{\varphi }_i}\times 100\%, K_{cd}=\frac{K_0-\overline{K}}{K_0}\times 100\%,$$

(26)

where ${\varphi }_{cd}$ and $K_{cd}$ are, respectively, the damage degree of the porosity and permeability; and $\overline{\varphi }$ and $\overline{K}$ are the average porosity and permeability in the compacted zone, respectively.

In addition, the stress of, the strain of, and the changes in the porosity and permeability of any node under different negative pressure differences can be calculated by using the secondary development of ABAQUS. Stress and strain parameters can be automatically extracted through the secondary development of ABAQUS, thus obtaining the maximum principal strain ${\epsilon }_1$, intermediate principal strain ${\epsilon }_2$, and minimum principal strain ${\epsilon }_3$ of a node through numerical simulation. The volumetric strain ${\epsilon }_v$ is the sum of the three. Similarly, the maximum principal stress ${\sigma }_1$, intermediate principal stress ${\sigma }_2$, and minimum principal stress ${\sigma }_3$ of a node can be obtained. The average stress $P_m$ is the average of the three. The effective stress ${\sigma }_{eff}$ can be expressed as follows:

$${\sigma }_{eff}=P_m-P_c=\left({\sigma }_1+{\sigma }_2+{\sigma }_3\right)/3-P_c,$$

(27)

where $P_c$ is the confining pressure.

Finally, the plastic strain ${\epsilon }_p$ and von Mises stress ${\sigma }_{vm}$ of the same node are acquired. Substituting these parameters and other initial values into the formula for calculating the porosity and permeability yields the porosity and permeability of any node and the degree of damage.

3. Numerical Modeling and Analyses

This section firstly establishes the finite element model (FEM) of the hole-forming process and then compares the results of the sandstone target in penetration simulation with the experiment, thus verifying the reliability of the numerical method.

3.1. Material Parameters

The relevant material models and parameters were established and are briefly introduced in this section.

3.1.1. Material Parameters of the Sandstone Target

The hardness and tensile experiments of the sandstone required the following parameters: elasticity modulus of the rock was 30,217 MPa; the triaxial compressive strength was 249.1 MPa; the tensile strength was 2 MPa; the hardness was 1754 MPa; the uniaxial compressive strength was 70 MPa; and the Poisson’s ratio was 0.2216. The definitions of the model material parameters are shown in

Table 1. Model material parameter.

Material Name	Elastic Modulus/GPa	Density/kg·m−3	Yield Strength/MPa	Fracture Strain	Poisson’s Ratio
Casing (N80)	210	7800	551	0.0026	0.3
Cement ring	20	2500	50	0.0020	0.2
Stratigraphic rocks	15	2300	40	0.0020	0.25

.

3.1.2. Initial Values of the Parameters in the Mathematical Model

Combined with the values of the parameters in the mathematical model of perforation compaction damage in Refs. [30,31], the initial values of the parameters in this work are shown in

Table 2. Initial values of the parameters in the mathematical model of perforation compaction damage.

Parameters	Initial Values	Parameters	Initial Values
$\mathrm{Confining} \mathrm{pressure} P_c$/MPa	30	$\mathrm{Fluid} \mathrm{porosity} \mathrm{intercept} {\varphi }_{\ast }$	0.122
$\mathrm{Porosity} \mathrm{decay} \mathrm{rate} C$	−0.103	$\mathrm{Maximum} \mathrm{permeability} \mathrm{damage} D_{\max }$	0.15
$\mathrm{Coefficient} \mathrm{of} \mathrm{expansion} m_d$	1.3	$\mathrm{Maximum} \mathrm{expanded} \mathrm{porosity} {\varphi }_{dil}^{\max }$	0.15
$\mathrm{Maximum} \mathrm{pressure} \mathrm{for} \mathrm{hydrostatic} \mathrm{damage} P_{h2}$/GPa	0.42	$\mathrm{Minimum} \mathrm{permeability} K_0$/m2	1.00 × 10−17
$\mathrm{Constant} \eta$	−1.8971	$\mathrm{Hydrostatic} \mathrm{pressure} \mathrm{damage} \mathrm{factor} C_2$	27.8
$\mathrm{Minimum} \mathrm{expanded} \mathrm{stress} P_{\min }$/GPa	0.03	Permeability damage attenuation factor $B$	12.5
$\mathrm{Maximum} \mathrm{plastic} \mathrm{strain} \mathrm{damage} \mathrm{stress} P_s^{\ast }$/GPa	0.25	$\mathrm{Critical} \mathrm{effective} \mathrm{stress} P_{\ast }$/GPa	0.01
$\mathrm{Plastic} \mathrm{strain} \mathrm{damage} \mathrm{factor} C_1$	28.5	$\mathrm{Initial} \mathrm{permeability} K_k$/m2	2.00 × 10−15
$\mathrm{Permeability} \mathrm{index} K_1$	44.434	$\mathrm{Initial} \mathrm{porosity} {\varphi }_i$	0.12

.

3.2. FEM and Simulation Results

3.2.1. FEM of Sandstone Target Penetration

Based on the penetration mechanism of shaped charge perforating particles, the SPH method and Hoffman judgment criterion were used to establish the 3D FEM of the hole-forming process and sandstone compaction damage model. The VUSDFLD user subroutines created damage variables. The boundary conditions and load parameters were set by the target experiment of the perforation and the environmental parameters of temperature and pressure in the reservoir.

Primary hypothesis: The copper and tungsten of the perforating projectile are converted into particles; the velocity of the jet particles is much higher than the average thermomechanical velocity of the target body; the particles are not deformed or damaged and disappear when the velocity is lower than the critical value; the local high-temperature target material softens after impact; and the target material of the casing, cement ring, and sandstone are damaged.

Figure 3 plots the 3D FEM of the hole-forming process. The same mechanical parameters were used in the target shooting experiment. The thickness values of the casing, cement, sandstone, and shaped charge were 12 mm, 28 mm, 960 mm, and 250 mm, respectively. At the same time, the velocity, friction coefficient, confining pressure, and effective stress of the rock were set simulate the environment. In addition, the balance between the accuracy of the calculation result and the calculation time should be considered when selecting an appropriate grid size.

Figure 4 plots the initial velocity of the particles in different jet segments. The velocity can be divided into five stages: 7300 m/s → 7800 m/s → 7300 m/s → 1900 m/s → 800 m/s, according to the boundary conditions of the model. It can be seen from Figure 3 that the number of metal particles is different in different velocity segments.

Due to a large number of particles, the calculation efficiency of SPH is much lower than that of FEM. Therefore, FEM was applied to model the initial structure when the process of three-dimensional shaped charge perforation was analyzed. The FEM mesh units in the large deformation area were automatically converted into SPH particles in the calculation process. As shown in Figure 5, at a specific moment (7–10 s), before the jet beam touches the casing wall, all the mesh units are converted into particles that collide with the casing, cement ring, and formation rock.

3.2.2. Simulation of the Hole-Forming Process of the Sandstone Target

According to the mechanical parameters of sandstone and the shaped charge jet perforating metal particle flow when the confining pressure was zero, the 3D model of shaped charge jet perforating metal particle flow penetrating the casing–cement–sandstone target was established. After more than 100 h of calculation, the simulation results of the hole forming process were obtained. Figure 6 plots the sandstone stress distribution at 1.26 ms of the jet flow impact. It can be seen that the stress at the hole tip is larger than the residual stress of the rock around the perforation hole. However, the stress of the rock far away from the hole is minimal, which indicates that it is not disturbed by the perforation.

The tip of the jet flow stops moving at 1.26 ms, and a 3D hole with an unsmooth wall of about 56 cm in depth is formed. The residual stress distribution of the casing, cement stone, and rock near the hole is approximately 2–5 mm thick, and the axis is deflected with the depth of the hole.

3.3. Experimental Verification

According to the characteristics of the low porosity and low permeability of the target reservoir, the white sandstone, as shown in Figure 7, was selected to make the sandstone target for the perforation experiment. The porosity of the reservoir was 8.5–9.5%, and the permeability was 0.15–0.25 mD. The diameter of the cylindrical sandstone target was 200 mm, and the length was about 0.8 mm. The reservoir environment was simulated according to the API RP 19B standard [33]. The confining pressure, pore pressure, and wellbore pressure were applied, and the shaped charge was fired to complete the shaped charge jet perforation experiment.

Figure 7 plots the comparison diagram of the simulation and experiment. It can be seen from the profile that the perforating depth of the split target is greater than 50 cm, and the perforating depth obtained through simulation is 56 cm. Meanwhile, there is a silty loose fracture zone around the target hole with a thickness of about 10 mm. The particle size is significantly smaller than that of the original sandstone. Moreover, the black metal particles attached to the surface of the hole wall can be seen, which is denser than that of the original sandstone. The experiment results are in good agreement with the numerical simulation results. The error is less than 8%, which indicates that the established perforation compaction damage evaluation model is correct and reproduces the damage process of the shaped charge jet perforation.

4. The Analysis of the Compaction Damage

4.1. Instantaneous and Unloading High Pressure

In the perforation process, high pressure (1–2 GPa) is generated at the moment of shaped charge jet perforation. Therefore, an internal pressure of 1000 MPa was applied to the inner wall of the hole to simulate the immediate impact of the metal jet. Figure 8 plots the distribution nephograms of stress (a), density (b), and strain rate (c) at the moment of perforation. The result shows that the rock within 30 mm of the hole wall is significantly deformed by extrusion, and the deformation at the tip of the hole is the largest. The density increased from 2.3 g/cm³ to 2.5 g/cm³, which is caused by compaction. Then, the residual deformation and residual stress that appear due to the sandstone cannot be recovered completely after internal pressure is unloaded.

Figure 9 plots the distribution nephograms of the permeability (a) and porosity (b) at the moment of perforation. As shown in Figure 9, the permeability within 10 mm of the hole wall (blue part) is significantly reduced to 0.016 mD, and the maximum permeability is 1.9 mD. The porosity in the 5 mm range (blue region) is significantly reduced to 5%, and the tips are about 10.8%. The damage degree of the porosity and permeability can be calculated by the ratio of the variable of porosity and permeability near the hole to the initial porosity and permeability, which are shown in Figure 10.

Figure 11 shows the distribution nephograms of the permeability (a) and porosity (b) when the internal pressure is unloaded. After unloading the internal pressure in the hole, the partial elastic of the wall recovers. The permeability of the hole wall decreases to less than 0.2 mD and gradually increases in the radial direction to the original permeability. The porosity of the hole wall increases from 12% to 20%, which decreases first to 6.7% and then gradually increases in the radial direction to the original porosity.

The distribution nephogram of the damage degree is show in Figure 12. It shows that the damage degree of the permeability at hole wall exceeds 90%, while the porosity is 66%. The maximum porosity increases more than 60%, and the minimum damage degree of the porosity near the hole tip exceeds 44%. Thus, the distribution area of the porosity can be divided into the area of pore expansion and fracture, the area of pore damage and compaction, and the area of the nondestructive pore, which is similar to the experimental results.

4.2. Effects of the Different Negative Pressure

After the wellbore is formed, when the fluid pressure within the wellbore is lower than the formation pressure, negative pressure perforation occurs, causing the fluid to quickly flow back from the formation, thus flushing the borehole and compacting zone, removing perforation contamination. This results in a better communication between the reservoir and the wellbore, significantly increasing the near-wellbore area permeability. To study the effects of the different negative pressure on rock damage, different negative pressure values (10 MPa, 20 MPa, 30 MPa, 40 MPa, and 50 MPa) were applied to the hole wall according to the negative pressure difference after the instantaneous high pressure is loaded.

Figure 13 plots the distribution nephograms of the permeability (a) and porosity (b) with the differential pressure of 10 MPa. The maximum porosity of the hole wall increases from 20% to 22% because of the action of the tensile load. However, the minimum porosity and permeability distribution and values are basically the same as those in Figure 11. It shows that the negative pressure of 10 MPa has little effect on the porosity and permeability of sandstone with low porosity and permeability.

The corresponding distribution nephogram of the damage degree is shown in Figure 14. It can be seen that the damage degree of the permeability has no noticeable change compared to that in Figure 12a. Conversely, the damage degree of the porosity in the hole tip is less than that of Figure 12b. Therefore, the pore expansion increases under the pressure difference of 10 MPa.

As the negative pressure increases to 50 MPa, the distribution nephograms of the permeability and porosity and their damage degree are shown in Figure 15 and Figure 16, respectively. The local porosity of the hole wall increased continuously from 22% to 23.3%. Moreover, the distribution of the porosity compaction area within the range of 10–20 mm from the hole wall in the middle of the hole underwent significant changes. In addition, the damage degree of the permeability had a few changes, which are the same as those in Figure 14a. The pore in the hole wall continued to expand with a maximum increase of 94%.

4.3. The Distribution Characteristics of the Compaction Damage at Different Positions around the Hole

This section considers the stress distribution around the perforation under seven conditions, including perforation torque, perforation unloading, and a negative pressure of 10–50 MPa. In the absence of an external load, the stress distribution of the different cross-sections at various positions of the perforation is similar, as shown in Figure 17. Taking the middle A-A of the perforation as an example, it is located at a distance of 9–60 mm from the perforation axis.

Figure 18 plots the distribution curves of the sandstone porosity (a) and its damage (b) in the A-A path. It can be seen that the porosity changes significantly within 20 mm from the axis of the hole, and the porosity is less than 10% at the moment of perforation. In addition, with the increase in the negative pressure in the other six working conditions, the range of the pore expansion increases continuously.

Figure 19 plots the distribution curves of the sandstone permeability (a) and its damage (b) in the A-A path. It can be seen that the permeability is the same within 20 mm from the distribution curve, and the permeability of the perforation moment is significantly different from that of the other six working conditions within the range of 20–30 mm. As shown in Figure 19b, the damage degree of the permeability within 20 mm from the axis of the hole is the same, and there is no damage in the permeability of the outer 60 mm.

5. Conclusions

In the present study, the dynamic response of a sandstone target, the damage characteristics, and the influence factors of the hole parameters during perforation are investigated, and the sand production rule is explored. Based on the numerical results, the following conclusions are drawn:

• At the entrance of the perforation, the aperture is larger and appears at the extensive belly section. The axis of the hole is skewed with the perforation depth, and the hole wall is not smooth.
• The porosity damage is at its maximum at the moment of perforation, reaching 60%. The permeability reached a full 90% under unloading conditions.
• The porosity damage decreases with the increase in the negative pressure. However, a large amount of sand may be produced when the negative pressure is 30 MPa.

Although many studies have been carried out on the numerical simulation of the flow characteristics in the sandstone target perforation compacted zone at home and abroad, at present, a 3D damage mechanical model of the compaction zone considering the variation in the rock stress field and the distribution of the compaction degree under the action of non-uniform crustal stress has not been reported.

This paper aims to use the SPH method and the secondary development of ABAQUS to understand the numerical simulation of the perforation compaction process. Based on the FEM, a complete 3D impact damage-coupling dynamic perforation simulation method is established. A greater understanding of the damage, destruction, and hole-forming process after the collision of rocks and particles is obtained more realistically. This can be expanded to develop a more accurate numerical solution method for 3D flow to achieve the good simulation of the influence of the local micro-domains on multiple domains.