Keywords

1 Introduction

Perforation completion is a widely used completion method [1,2,3]. The energy of a perforating shaped charge can be divided into the kinetic energy of a jet, the residual energy of a wellbore, and energy dissipation after the perforating shaped charge explosion. During the perforating explosion, the liner gradually liquefies to form a perforating shaped charge jet. A perforating gun, casing, and formation are penetrated by the perforating shaped charge jet to make a channel that connects a reservoir to a wellbore [4, 5]. To increase the production of oil and gas wells and lower operating costs, various construction parameters should be set fairly according to different operating environments while constructing the perforation scheme on site. However, with various perforating conditions, the kinetic energy conversion of the perforating shaped charge jet change. Therefore, studying the conversion relationship and influence of the perforating shaped charge structure on the kinetic energy of the perforating shaped charge jet is critical to improve the conversion rate of the jet kinetic energy and increase the production.

The perforating shaped charge jet has attracted much attention among researchers, and some studies have been performed. In their studies, Brown et al. [6] demonstrated the derivative mechanism of a perforating shaped charge jet and offered suggestions for further research. Lee [7] used a simplified two-dimensional numerical calculation model combined with the Euler algorithm to simulate a perforating shaped charge jet process. Jin et al. [8] established a two-dimensional perforating shaped charge jet simulation model by using the ALE algorithm to study the formation mechanism, velocity evolution, and penetration characteristics of a perforating shaped charge jet. The obtained results indicated that the cone angle of the liner has a great influence on the velocity and shape of a perforating shaped charge jet, the mass of the jet head and the slug, and the penetration depth of a perforating shaped charge jet. Cao et al. [9] established a two-dimensional jet simulation model of two perforating shaped charges to analyze the influence of interference between perforating shaped charges on a perforating shaped charge jet and the factors causing interference between perforating shaped charges. Kang and Sheng [10] established a three-dimensional simulation model to study how a perforating shaped charge explosion crushes the liner and penetrates the casing. The change laws of the velocity and energy of a perforating shaped charge jet were carried. Suneson [11] used the LS-DYNA software to conduct a finite element simulation on the conversion of a solid metal liner covert into a high-speed metal fluid and the penetration of the high-speed metal fluid into a concrete medium. Liu et al. [12] established a two-dimensional simulation model of a perforating shaped charge jet to analyze the formation process of a perforating shaped charge jet and the proportion of the jet kinetic energy for different explosive types and densities. He [13] conducted an experiment on a new perforating shaped charge liner, and the results of the explosive test showed that the perforating shaped charge liner could achieve comparable or even a greater penetration performance. Elshenawy [14] and Du [15] studied the effect of the target strength on the shaped charge jet penetration. The findings demonstrate that when the target yield strength increases, The penetration depth of the shaped charge jet in the target significantly decreases.

Numerous studies on the generation of a perforating shaped charge jet, its penetrating properties, and the velocity have been conducted by academics. However, little research has been performed on the conversion relationship and influence of the perforating shaped charge structure on the jet kinetic energy and the improvement of the conversion rate of the jet kinetic energy. In this research, a jet penetration model is developed based on the ALE algorithm. The factors including the cone angle and thickness of the liner and the explosive mass affecting the energy conversion law of the jet kinetic energy are systematically studied.

2 Jet Kinetic Energy of a Perforating Shaped Charge

The liner is liquefied into a thinner jet in the front and a thicker slug in the back due to the explosive of a perforating shaped charge. A perforating gun, casing, and formation are penetrated by a perforating shaped charge jet created by the jet and the slug to create a channel that connects a reservoir to a wellbore. Meanwhile, the kinetic energy of the jet and slug gradually decreases during the penetration process. Figure 1 depicts a wellbore and a typical perforating shaped charge structure. The figure shows that a typical perforating shaped charge structure consists of a shell, explosive, and liner.

Fig. 1
A schematic depicts the intricate structures of a wellbore and a perforating-shaped charge in oil drilling. The wellbore includes elements like packer, casing, tubing, stratum, and perforating gun. The shaped charge consists of a shell, liner, and explosive.

Wellbore structure and typical perforating shaped charge structure

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Based on the kinetic energy theorem, the jet kinetic energy could be expressed as:

$$E_{J} = \frac{1}{2}(m_{j} v_{j}^{2} + m_{c} v_{c}^{2} )$$
(1)

where mj is the jet mass; vj is the jet velocity; mc is the slug mass; vc is the slug velocity.

In this research, a jet penetration model was established to study the proportion of the total energy of a perforating shaped charge converted into the jet kinetic energy and the influence of various factors on the kinetic energy of a perforating shaped charge jet.

3 Jet Penetration Model

3.1 ALE Algorithm and Geometric Model

The penetration simulation of a perforating shaped charge jet is a large deformation and fluid–structure coupling problem. The mesh of the jet penetration model will be deformed and the calculation will be simple to stop if the Lagrange algorithm is adopted. The Euler algorithm requires a high level of mesh precision, resulting in a high calculation cost. To effectively prevent grid distortion, The ALE algorithm can track the motion of a material's boundary and allow material to move freely within a space grid [16]. Therefore, the ALE algorithm was adopted to simulate the penetration process of a perforating shaped charge jet in this paper.

For the structure of a perforating shaped charge, the geometric model of the perforating shaped charge was established, as shown in Fig. 2. The cone angle of the liner was 60°. The thickness of the liner was 1.5 mm. The explosive mass was 25 g. The outer diameter of a casing was 127 mm. The wall thickness of a casing was10.36 mm. The outer diameter of a perforating gun was 86 mm. The wall thickness of a perforating gun was 6.45 mm. A blind hole diameter was 40 mm. The depth of a blind hole was 4 mm. In the meshing process of the jet penetration model, the solid mesh and the ALE mesh overlap. Therefore, a penalty function was used to realize the fluid–structure coupling of the jet penetration simulation. When the cell size of the mesh is 1 mm, the calculation time is 5 h, and the energy ratio is only 0.1% less than that of 0.8 mm. To improve computing efficiency, 1 mm was selected as the basic mesh size.

Fig. 2
A geometric model has rectangular, curved, and triangular components labeled explosive, shell, liner, perforating gun, blind hole, and casing.

Geometric model

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3.2 Material Constitutive and State Equation

The HNS explosive was used in the perforating shaped charge of this model. The specific parameters are displayed in Table 1. The High-Explosive-Burn material model was selected to define the HNS explosive properties. The JWL equation was conducted to calculate shockwave pressure due to the HNS explosive explosion, whose expression was as follows:

$$P_{1} = A\left( {1 - \frac{\omega }{{R_{1} V}}} \right)e^{{ - R_{1} V}} + B\left( {1 - \frac{\omega }{{R_{2} V}}} \right)e^{{ - R_{2} V}} + \frac{\omega E}{V}$$
(2)
Table 1 Explosive parameters
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where P1 is an isentropic pressure; V = ρ0/ρ = v/v0 is a relative specific volume; A, B, R1, R2, ω are undefined constants, respectively; E is the internal explosion energy per unit initial volume of the explosive.

Air parameters are shown in Table 2. The Mat-Null material model and The Eos-Linear-Polynomial state equation were conducted to defined air properties. The pressure linear polynomial of the state equation was expressed as:

$$P_{2} = C_{0} + C_{1} \mu + C_{2} \mu^{2} + C_{3} \mu^{3} + \left( {C_{4} + C_{5} \mu + C_{6} \mu^{2} } \right)E$$
(3)
Table 2 Air parameter
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where, P2 is air pressure; μ is air density ratio; If μ < 0, C2μ2 = C6μ2 = 0, C0 ~ C6 are the state equation coefficient.

Perforating fluid parameters are shown in Table 3. The Mat-Null material model and The Gruneisen state equation were conducted to defined perforating fluid properties. The Gruneisen state equation of a perforating fluid was expressed as:

$$p = \frac{{\rho_{0} C^{2} \mu_{k} \left[ {1 + \left( {1 - \gamma_{0} /2} \right)\mu_{k} - \frac{\alpha }{2}\mu_{k}^{2} } \right]}}{{\left[ {1 - \left( {S_{1} - 1} \right)\mu_{k} - S_{2} \frac{{\mu_{k} }}{{\mu_{k} + 1}} - S_{3} \frac{{\mu_{k}^{3} }}{{(\mu_{k} + 1)^{2} }}} \right]^{2} }} + \left( {\gamma_{0} + \alpha \mu_{k} } \right)E_{s}$$
(4)
Table 3 Perforating fluid parameters
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where, ρ0 is the initial density of perforating fluid; C is the intercept of the wave velocity curve; μk is the compressibility factor of perforating fluid; γ0 is the Grueisen constant; S1, S2, S3 are the slope coefficients of the wave velocity curve; α is the first order volume correction; Es is the internal energy.

The liner was made of a high-conductivity oxygen-free copper. The parameters are shown in Table 4. The Mat-Steinberg material model and the Gruneisen state equation were conducted to defined liner properties. The Mat-Steinberg material model was expressed as:

$$G = G_{0} \left[ {1 + b_{i} pV^{\frac{1}{3}} - h\left( {\frac{{E_{i} - E_{c} }}{{3R^{\prime } }} - 300} \right)} \right]\frac{{ - fE_{i} }}{{E_{m} - E_{i} }}$$
(5)
$$\sigma_{s} = \sigma_{0}^{\prime } \left[ {1 + b_{i} pV^{\frac{1}{3}} - h\left( {\frac{{E_{i} - E_{c} }}{{3R^{\prime } }} - 300} \right)} \right]\frac{{ - fE_{i} }}{{E_{m} - E_{i} }}$$
(6)
Table 4 Liner parameters
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where, p is pressure; V is the relative volume; Ec is the cold compression energy; Em is the melting energy; R′ = Rρ/A0, R is a gas constant; A0 is an atomic weight; As the material melts, σs and G are reduced to half of their initial values as the material melts.

Casing was made of 25CrMnMo. The parameters are shown in Table 5. The Johnson–Cook material model and the Gruneisen state equation were conducted to defined casing properties.

$$\sigma_{y} = \left( {A + B\overline{\varepsilon }^{{P^{n} }} } \right)\left( {1 + C\ln \dot{\varepsilon }^{*} } \right)\left( {1 - T^{*m} } \right)$$
(7)
$$\varepsilon^{f} = \left( {D_{1} + D_{2} \exp D_{3} \sigma^{*} } \right)\left( {1 + D_{4} \ln \dot{\varepsilon }^{*} } \right)\left( {1 + D_{5} T^{*} } \right)$$
(8)
Table 5 Casing parameters
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where, σy is a yield stress; A, B, C, n, m are the input constant, \(\overline{\varepsilon }^{{P^{n} }}\) is an equivalent plastic strain; \(\dot{\varepsilon }^{*} = \dot{\varepsilon }_{e}^{p} /\dot{\varepsilon }_{0}\) is a relative equivalent plastic strain rate; T* is a homologous temperature. σ* is a ratio of pressure to the Von-Mises equivalent stress; D1–D5 are damage coefficients.

The material 32CrMO4 was used to make a perforating gun and perforating bullet shell. The parameters are shown in Table 6. The Plastic-Kinematic material model was conducted to defined perforating gun and perforating bullet shell properties.

Table 6 Perforating gun and shell parameters
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4 Process of Perforating Shaped Charge Jet and Jet Kinetic Energy Conversion Analysis

4.1 Penetration Process of Perforating Shaped Charge Jet

Figure 3 depicts the forming and penetrating process of a perforating shaped charge jet a perforating. When t = 0 μs, the explosive started to explode from the top, producing a large amount of high temperature and high pressure gas that liquefied the liner and propagated the shockwave formed by the gas along the axis, as shown in Fig. 3a. When t = 10 μs, the shockwave swept along the liner and the liner was liquefied into a thinner jet in the front and a thicker slug in the back, as shown in Fig. 3b. When t = 40 μs, a perforating shaped charge jet still moved along the axis direction at a great speed and a violent collision occurred at a perforating gun and casing, which penetrated the perforating gun and casing and formed a channel connecting a reservoir and wellbore, as shown in Fig. 3c. In this process, due to the resistance of the above factors, the jet kinetic energy gradually decreased in this process until the energy was completely consumed and the jet penetration process also ended.

Fig. 3
Three illustrations of the gradual penetration process of a shaped charge jet into a material at 0, 10, and 40 microseconds. Initially, the jet appears pointed, but over time, it transforms, spreading out as it pierces deeper.

Penetration process of perforating shaped charge jet

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4.2 Kinetic Energy Conversion Analysis of Perforating Shaped Charge Jet

Figures 4 and 5 show the relationship between the total energy of the explosive and the kinetic energy of a perforation shaped charge jet over time. With the penetrating process of a perforating shaped charge jet, the total energy of the explosive gradually decreased from 110.92 kJ. The kinetic energy of a perforation shaped charge jet reached its maximum of 17.185 kJ at 12 μs and then decreased gradually due to air resistance and penetration influence. The kinetic energy conversion rate of a perforation shaped charge jet was 15.49%.

Fig. 4
A graph of total explosive energy versus time has some following values. (0, 110), (15, 60), (30, 30), (40, 20). Values are estimated.

Total explosive energy

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Fig. 5
A graph of the kinetic energy of jet versus time has some following values. (0, 0), (15, 18), (30, 10), (40, 8). Values are estimated.

Kinetic energy of perforation charge jet

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5 Analysis of Influencing Factors of Jet Kinetic Energy

5.1 Cone Angle of Liner

By changing the size of the cone angle of the liner and keeping all other conditions constant. The influence of the cone angle of the liner on the kinetic energy of a perforation shaped charge jet was studied. The cone angles were 45°, 50°, 55°, 60°, 65°, and 70°. Its energy conversion rate is shown in Fig. 6.

Fig. 6
A graph of the conversion rate of jet kinetic energy versus cone angle of the liner has some following values. (45, 14.6), (60, 15.5), (65, 15.7), (70, 16.1). Values are estimated.

Relationship between cone angle of liner and kinetic energy ratio of jet

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As shown in Fig. 6, with an increase in the cone angle of the liner, the kinetic energy ratio of a perforation shaped charge jet increase. The obtained results indicated that a perforation shaped charge jet is divided into the front jet and the rear slug. With an increase in the cone angle of the liner, the front jet becomes thicker. The mass proportion of the front jet gradually increases and the front jet velocity is increased. Therefore, according to the kinetic energy theorem, the kinetic energy conversion rate of a perforation shaped charge jet is increased with an increase in the cone angle of the liner.

5.2 Thickness Angle of Liner

By changing the thickness of the liner and keeping all other conditions constant. The influence of the thickness of the liner on the kinetic energy of a perforation shaped charge jet was studied. the thicknesses of the liner were 1 mm, 1.25 mm, 1.5 mm, 1.75 mm, 2 mm, and 2.25 mm. Its energy conversion rate is shown in Fig. 7.

Fig. 7
A graph of the conversion rate of jet kinetic energy versus thickness of liner has some following values. (1, 16.3), (1.6, 15.3), (1.9, 14.6), (2.2, 14.25). Values are estimated.

Relationship between thickness of liner and kinetic energy ratio of jet

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As shown in Fig. 7, with an increase in the thickness of the liner, the kinetic energy ratio of a perforation shaped charge jet decreases. The obtained results indicated that the energy derived from the explosive is used partly to melt the liner and partly to maintain the kinetic energy of a perforation shaped charge jet. With an increase in the thickness of the liner, the energy required to melt the liner is increased, the kinetic energy of a perforation shaped charge jet is decreased, and the kinetic energy conversion rate of a perforation shaped charge jet is decreased.

5.3 Explosive Mass

By changing the explosive mass and keeping all other conditions constant. The influence of the explosive mass on the kinetic energy of a perforation shaped charge jet was studied. Explosive masses were 25 g, 30 g, 35 g, 40 g, 45 g, and 50 g. Its energy conversion rate is shown in Fig. 8.

Fig. 8
A graph of the conversion rate of jet kinetic energy versus explosive has some following values. (25, 15.5), (35, 13.5), (45, 10.5), (50, 9.8). Values are estimated.

Relationship between explosive mass and kinetic energy ratio of jet

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As shown in Fig. 8, with an increase in the explosive mass, the kinetic energy ratio of a perforation shaped charge jet decreases. The obtained results indicated that the explosive total energy and the kinetic energy of a perforation shaped charge jet increase with an increase in explosive mass. But the increase rate in the explosive total energy is significantly greater than the kinetic energy of a perforation shaped charge jet. Therefore, the ratio of the two decreases. the kinetic energy conversion rate of a perforation shaped charge jet is decreased.

6 Conclusions

A jet penetration model was established based on the kinetic energy ratio of a perforation shaped charge jet. The typical perforating charge—perforating gun—casing structure, the penetration process of a perforation shaped charge jet, the kinetic energy conversion relationship of a perforation shaped charge jet, and its influencing factors were analyzed. The following are the conclusions:

In the angle range of 45° to 70°, the kinetic energy conversion of the 70° cone angle of the liner reached a maximum of 16.26%. In the thickness range of 1 mm to 2.25 mm, the kinetic energy conversion of the 1 mm thickness of the liner reached a maximum of 16.24%. In the mass range of 25 g to 50 g, the kinetic energy conversion of the 25 g explosive mass reached a maximum of 15.49%.

The cone angles and thicknesses of a liner and explosive mass all affected the kinetic energy of a perforation shaped charge jet. In the actual production operation, the effect of a perforating charge structure on the jet kinetic energy should be fully considered. It is helpful to improve the kinetic energy conversion rate of a perforation shaped charge jet and the use efficiency of explosives. Meanwhile, the ratio of the kinetic energy of a perforation shaped charge jet to its total energy is determined, providing pre-data support for future predictions of the transient pressure of perforating detonation.