Finite element modeling of the shaped charge jet and design of the reusable perforating gun

Finite element method is used to study the formations of the penetration jet, the bulge, and the burr in the designed reusable perforating gun. The attached layer of the soft metal on the perforator is studied for the controlling of the bulge height on the casing of the reusable perforating gun. Results indicate that the shaped charge jet is initially formed in the center of the shaped charge liner and then the material of the liner is driven to the centerline of the liner by the detonation wave. The attachment of the soft metal layer to the cartridge of the perforator can be beneficial to control the bulge height. The design on the blind holes on the casing can affect the burr height formed by the collision between the jet and the casing. With the increase in the liner angle, the penetration width on the cement wall of the wellbore is increased.


Introduction
Perforation is very important in the petroleum well completions. The perforation quality can directly affect the oil productivity. For perforation with high quality, different models have been proposed to predict the productivity of the perforated well or the permeability around the tunnel (Gambirasio et al. 2017;Hagoot 2007;Karacan and Halleck 2003;Du et al. 2017;Wang et al. 2010). Perforation by shooting with shaped charge is now commonly used in well completions due to its low cost and high efficiency. In the perforating process, a perforator is exploded to form a jet, which can penetrate the casing, the cement and the rock. The maximum velocity of the jet can reach 8000 m/s in some cases. Experiment shows the strength of the target can be treated as a factor for the penetration by the metallic jet in the perforating process (Eichelberger 1956). The perforating processes with shaped charge can be studied by both numerical simulation and experiment (Guo et al. 2011;Lee 2002;Molinari 2002;Murphy et al. 2003). A practical method was proposed to accurately predict the perforation depth and the size of the perforation hole (Ott et al. 1994). The shaped charge liner is one of the key components in the perforator, which serves as the basis for the jet formation (Xu et al. 2019;Guo et al. 2019;Eichelberger and Pugh 1952). The theory of the shaped charge formation, the steady-state and the quasi-steady-state theories of the jet formation and the target penetration were proposed (Eichelberger 1956;Eichelberger and Pugh 1952;Birkhoff et al. 1948). In jet formation, the liner material undergoes huge stretching deformations in the axial direction. When the liner material impacts on the charge axis, it ceases to move in the radial direction (Shvetsov et al. 2011). The penetration process and the destruction mechanism of concrete targets by the shaped charges and kinetic energy projectiles were analyzed (Wang et al. 2008). It was found that the shaped charge directly affects the penetration into the target and causes very large spalling area in comparison with the kinetic energy projectile. The liner collapsing and jet/slug formation of the double-layer shaped charge were analyzed for the improvement of the shaped charge performance (Zhang and Qiao 2011). It has been considered that the shaped charge with an axial channel can be beneficial to the composite shaped charge of the "base-head" type (Malygin et al. 2011).
The schematic of the perforating processes is shown in Fig. 1 Fig. 1 Schematic of the perforating process the cement and the rock oil. Due to the pressure in the rock oil, the oil can flow into the perforating tunnel and enter into the wellbore. At the same time, the cartridge can move in the opposite direction and collide on the casing of the perforating gun. Usually, there are blind holes on the casing to reduce the resistance to the jet. Thin wall tubulars with D/t = 7 ~ 15 are usually used for downhole perforating in well completion . When the jet penetrates the casing, burr is formed on the blind holes on the casing. Meanwhile, the cartridge moves in the opposite direction and collide the casing. Bulge can be formed in this collision, which can affect the reuse of the perforating gun.
The burr height can affect the reuse of the perforating gun. But till now, the relations between the burr height and the perforating gun have seldom been studied. To reuse the perforating gun, the collision between the cartridge and the casing of the perforating gun also needs to be studied to control the bulge height on the shell. The formations of penetration jet, the bulge and the burr need to be studied as well as the attachment of soft metal layer to reveal the controlling of the bulge and burr heights in the reusable perforating gun.

Model of shaped charge liner
The DP36RDX25-2 perforating gun is selected for simulation. The outer diameter of the casing is 154 mm and the inner diameter 132 mm. The material of the casing is 40CrNiMo. The yield stress of 40CrNiMo is taken as 888 MPa. The strength limit is 1043 MPa. The elongation is 12.7%. The shaped charge liner is made by copper. The density of the copper is 8932 kg/m 3 . The thickness of the shaped charge liner is 1.5 mm. The charge height is 16 mm. Johnson-Cook constitutive model (Mareau 2020) is used for the description of the mechanical behavior of shaped charge liner, where A, B, c, n and m are material constants, as shown in Table 1.
(1) y = A + B̄p ln (1 + c lṅ) 1 − (T * ) m T a and T m represent the ambient and the melting temperatures.

Description of explosive
The Jones-Wilkins-Lee (JWL) equation is used for the calculation of explosion (Yu et al. 1999;Yan et al. 2020), where V is the volume of the charge, E is the initial ratio of internal energy and A 1 , B 1 , ω, R 1 and R 2 are material constants. 32 g RDX is used as explosive for the perforating shaped charge in DP36RDX25-2 perforating gun. The parameters for RDX are indicated in Table 2.

Equation of state
Gruneisen state equation is used to describe the dynamic responses, where C is the intercept of v s -v p curve (shear and compressive wave speeds) (Meyers 1994). S 1 , S 2 and S 3 are the coefficients of the slope of the v s -v p curve. γ 0 is the Gruneisen gamma. a is the first-order volume correction to γ 0 and μ = ρ/ρ 0 − 1. The parameters for shaped charge liner and air are listed in Tables 3 and 4, respectively. (2)

ALE and adaptive re-meshing
To avoid mesh distortions and entanglements, arbitrary Lagrangian-Eulerian (ALE) mesh or adaptive re-meshing can be used Zhang and Tan 2019;Zhang et al. 2017;Zhang and Zhang 2014). In the procedure of ALE, one or more time-step computations are conducted.
Meshes are deforming with the flow of the material. When the deformation is completed, the meshes maintain the boundary condition of the object, and the internal meshes need to be updated. This step is called as smooth step. Then, the advection step is taken to transport the unit variables (density, energy, stress tensor, etc.) and  Figure 3 shows the perforator and the casing system. The outer diameter of the casing is 154 mm and the inner diameter 132 mm. The thickness of the casing is 11.0 mm. The length of the casing is 1213 mm. The material of the casing is 40CrNiMo. The thickness of the shaped charge liner made in copper is 1.5 mm. The charge height is 16 mm. The thickness of the attached soft metal is 8 mm. Dynamite and shaped charge liner exist in the shell as indicated in Fig. 1. Due to the characteristics of high ductility, aluminum and magnesium alloys are selected to be the attached soft metal.

Models of perforator and casing
The strain hardening constitutive model of the casing is shown as: where K is the tangent modulus and ̄p l is the equivalent plastic strain. When the equivalent stress reaches the strength limit, the casing is broken by the jet.
For the associated plastic flow, the flow rule can be expressed as: where g is the flow potential. dλ is the plastic state parameter.
The Cowper-Symonds model is used for the description of the material behavior of the cement,  where σ 0 is the initial yield stress. C and P are the Cowper-Symonds parameters. β is the hardening parameter. E p is the plastic hardening modulus, where E tan is the tangent modulus. The values of the above-mentioned parameters are listed in Table 5. 0.1 is taken to be the erosion strain for failure of cement in simulation.
According to the energy balance, the external work can be converted to kinematic energy, internal energy and frictional energy, Due to the short time for penetration, the integral in Eq. (11) must be very small and can be neglected. The frictional energy is small and nearly to be zero. The dissipated portions of internal energy E U are split off into the energy by dissipated viscous effect E V and the remaining energy E I , where σ c is the stress without viscous dissipation effect. σ v is the viscous stress. The internal energy includes the elastic strain energy, the energy dissipated by plasticity and the energy dissipated by time-dependent deformation, where ε e is the elastic strain, ε p plastic strain and ε c the creep strain.
LS-DYNA was used for the simulations of the jet formation, the collision between the cartridge and the casing and the casing and collision between the jet and the concrete. ABAQUS was used for the simulation of the collision between the jet and the casing. Figure 4 shows the velocity of the jet. The jet is initially formed in the center of the shaped charge liner, and then, the material of the liner is driven to the centerline of the liner by the detonation wave. The jet is divided into two parts: the head and the body. The velocity of the jet head is higher and can be increased to 4115 m/s in a short time when the liner angle is 80°. About 7 μs after the explosion, the jet velocity can reach the maximum value and then remains in this high speed until the jet penetrates the casing and the cement. The computational result is similar to the experimental and numerical figures on jet formation (Elshenawy and Li 2013).  The velocity of the cartridge can reach 150 m/s. Due to the fact that there is no restriction for the movements in the axial direction, the cartridge can collide the casing after the detonation. Then, a bulge can be formed on the surface of the casing. Figures 5, 6, 7, 8 and 9 reveal the collision between cartridge and the casing in different cases without attached layer, with attached layer of AA6061 and with attached layer of AZ80A-T5 magnesium alloy. It can be seen that the attachment of the layer of soft metal can provide good energy dissipations. The metal layer can dissipate the kinetic energy of the perforator by deformations, which can lead to the decrease in the bulge height. The provided relationship between the yield strength and the formed bulge height can be helpful for the selection of attached soft metal in the design of reusable perforating gun.

Results and discussions
The collision between the cartridge and the casing without soft metal is shown in Fig. 5. The annular and longitudinal stresses on the casing are similar. This phenomenon was also observed by experiment (Dou et al. 2019). The maximum effective strain on the casing is 0.67, and the maximum on the cartridge is 0.39. The dissipated plastic energy is dependent on the stress and the corresponding strain. So, the increase in the plastic deformation of the cartridge can be beneficial to the dissipation of energy on the cartridge. Meanwhile, the energy dissipated to the casing becomes smaller. This can protect the casing for the reuse of the perforating gun.
When the cartridge is attached by the layer of soft metal, the soft metal can be broken, which can be beneficial to the decreases of the bulge height. When a layer of AA6061 aluminum alloy is attached on the cartridge, as shown in Fig. 6, the effective strain of the shell is increased to 0.97. The increase in the plastic deformation on the cartridge leads to the decrease in the effective strain of the casing to 0.33. In comparison with the case in Fig. 5, the effective stain of the casing is obviously decreased. The bulge height on the casing is decreased. The effect of different attached Al alloys on the bulge height of casing is summarized in Table 6. The attachment of 7075 aluminum alloy can reduce the bulge height to 3.46 mm, i.e., 52.1% decrement compared to the case without attachment of soft metal. From Table 6, it can be seen that the increase in the yield stress of the attached soft metal can lead to the decrease in the bulge height. When a layer of AZ80A-T5 magnesium alloy is attached on the cartridge, as shown in Fig. 8, the effective strain of the shell is increased to 0.97. Meanwhile, the effective strain of the casing is 0.36. In comparison with the case in Fig. 5, the effective stain of the casing is obviously decreased. However, the effective strain is slightly increased from 0.33 to 0.36 in comparison with Fig. 6. The effect of different attached Mg alloys on the bulge height on casing is summarized in Table 7 and Fig. 9. 36.6% decrement can be found when the AZ61A-F is attached. In the current case, the bulge height on the casing is 4.58 mm.     Tables 6 and 7, the lowest bulge height can be obtained by use of 7075 aluminum alloy. The bulge height can be reduced by 52.1%. The effect of AZ31B-O on bulge height is the smallest. It can only reduce the bulge height by 21.6%. From the obtained results, it can be seen that the increase in the yield limit of the attached soft metal selected in current work can be beneficial to reduce the maximum bulge height. In similar yield stress, aluminum alloys show higher performance to reduce the bulge height on the casing in comparison with magnesium alloys.
When the jet penetrates the casing, the jet can be treated as a rigid body due to the high speed of the jet. Burr is formed on the casing in this collision. For the reuse of the perforating gun, the burr height must be small enough. The collision between the jet and the casing is completed within 8 μs. Moreover, the length size of jet is not a factor affecting the results in collision simulation because the jet in high velocity can be treated as rigid body (Xing et al. 2020). So, the state of tip jet at 8 s is used to be the rigid body in the simulation of the collision between the jet and the casing. The geometric parameters of the blind holes on the casing can be important factor to affect the burr heights. When the radius of the blind hole (R) on DP36RDX25-2 perforating gun is 15.875 mm and the thickness of the blind hole (t) is 9.7 mm, the average burr height is 1.70 mm as shown in Fig. 10, which is smaller than the threshold in the design, i.e., 3 mm. When the radius of the blind hole is increased to 20 mm as shown in Fig. 11, the average burr height is decreased to 1.204 mm. When the thickness of the blind hole is decreased to 7.7 mm as shown in Fig. 12, the average burr height is increased from 1.7 to 2.76 mm. This means that the increase in the radius or the decrease in the thickness of the blind hole is beneficial to control the burr height formed in the collision between the jet and the casing.
The effect of the liner angle on the penetration width is shown in Fig. 13. When the penetration hole is formed, the figure is sufficient to provide the studied penetration width. So, the initial stage of the penetration is used for comparison to reduce computational cost. With the increase in the liner angle, the penetration width is increased. More cases are summarized in Table 8. It reveals that smaller liner angle leads to narrower penetration width. The observation is fitted well with the experimental one (Shvetsov et al. 2011). The penetration width is a function of the maximum jet velocity, where W is the penetration width (mm) and v is the velocity of the jet (km/s).
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ig. 13 Collision of jet and concrete in different liner angles