# Study on Dynamic Response of Downhole Tools under Perforation Impact Load

Authors: Liu Jun; Li Shide; Liu Qingyou; Liu Xian; Guo Xiaoqiang; He Yufa
Journal: Shock and Vibration
 (2017)

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## Abstract

A model of a multibody system is established to investigate the dynamic response of an oil tube-shock absorber-perforating gun system in downhole perforation-test joint operation. In the model, the oil tube and perforating gun are modeled as elastic rods and the shock absorber is modeled as single particle system with damping and a spring. Two force continuity conditions are used to simulate the interactions among the three components. The perforation impact load is determined by an experiment of underwater explosion of perforating bullets. Using the model, the effects of charge quantity of perforating bullet, the number of shock absorbers, and the length of oil tube on the dynamic response of oil tube and packer are investigated. On this basis, a basic principle of the combination design of shock absorber and oil tube is proposed to improve the mechanical state of downhole tools. The study results can provide theoretical support for the design of downhole perforation-test joint operation.


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## Body

## 1. Introduction

In the perforation-test joint operation of oil-gas development, in order to improve perforation efficiency, high perforation density and perforating bullet with high explosive charge are widely used, resulting in the strength increase of explosive load as well as worse mechanical state of the downhole tools, as shown in Figure1. In extreme conditions, the explosive load may cause overall buckling of tube string and damage of downhole tools [1, 2] including test equipment and packer, as shown in Figure 2. So, how to deal with the contradiction between the improvement of perforation efficiency and the safety of the tube string system becomes an urgent problem to be solved.Figure 1
Downhole tools in perforation-test joint operation.Figure 2
Tube buckling caused by perforation load.The study on the mechanics of tube string can be traced back to the helical buckling theory of packer pipe string, which was put forward by Lubinski et al. [3] in 1962. However, it is only in recent years that the research on the mechanics of the perforated tube string has been carried out [4–11]. Using the gap element theory of finite element method (FEM), a three-dimensional static analysis model of a perforated tube string was established by Zhang et al. [5]. Yin et al. [12] also used a static analysis model to examine the effect of high pressure and perforation gun on tube string and showed that the high pressure formed in the sealing section at the moment of perforation is one of the main factors that cause the vibration of the string and the damage of test equipment.A cross combination scheme of shock absorber and damping oil tube was proposed by Fan and Li [13] to minimize the effects of the shock wave energy on packer and test instrument. However, this method lacks theoretical basis. A ground simulation test system has been used by Lu et al. [14] to investigate the pressure and acceleration responses at the end of a perforation tube string. A preliminary understanding of the dynamic response characteristics of perforation string suffering explosion shock load is obtained. However, the dynamic response of under tube string and the effect of impact load on packer remained unclear.A vibration model of perforated tube-shock absorber was established by Liu et al. [15] to investigate the influences of system parameters on the vibration response of oil tube. However, in the model the effect of perforation gun is not considered and the perforating load was not verified by experimental data.Zhou [16], Teng [17], and Kang et al. [18] applied the software of LS-DYNA to simulate the process of perforation bullet explosion, jet formation, and the jet penetration of casing-cement- stratum. Zhang et al. [19] used ANSYS/AUTODYN software to investigate the effects of the length of perforation zone, the interval time of perforation bullet explosive, the charge quantity in a perforation bullet, the physical conditions of artificial bottom hole, “pocket” depth, and other parameters on the perforation impact load. Li et al. [20] presented a dimensionless expression of perforation impact loads as well as its engineering calculation formula. Based on the formula, the buckling behavior of the perforation tube string of an actual well was analyzed. In these studies, the focuses were put on explosive process of perforating bullet and the pressure distribution in the wellbore. The effects of perforation parameters on the dynamic response and stability of oil tube and the force acting on packer remained unclear.A longitudinal vibration model coupling the testing tube and shock absorber was used by Huang [21] to investigate the effects of perforation parameters on the dynamic behaviors of oil tube and packer. Another dynamic model for oil tubes in curved wells was also proposed by Cai et al. [22] to analyze the radial displacement, contact force, friction resistance, and axial force in oil tube. In the model, the nonlinear contact between oil tube and casing is simulated by the space beam element and spring element. In these studies, the mutual coupling effect of oil tube, shock absorber, and perforation gun and the effect of perforation parameters on the stability of oil tube and the force acting on packer were not considered. Effective experimental data are still lacking.The purpose of this paper is to find an effective engineering calculation formula of shock load based on experiment data and establish a dynamic model considering the mutual coupling effect of oil tube, shock absorber, and perforation gun. On this basis, the effects of main perforation parameters on the dynamic response of downhole tools are studied, focusing on the buckling of oil tube and the force acting on packer.


## 2. Vibration Differential Equation of Downhole Tools

### 2.1. Basic Assumptions and Mechanical Model

According to the structure of the perforation string system and the operation condition, the following assumptions are made in order to derive the dynamic governing equations of the system.(1) It is assumed that the material of oil tube and perforation gun is homogeneous and isotropic.(2) The shock absorber is equivalent to a mass-spring-damping system, ignoring the geometry of the shock absorber and the inhomogeneity of the mass distribution.(3) The packer is considered as a fixed support without considering the relative displacement between the packer and casing tube or oil tube.(4) The structural damping of the tube string is ignored.(5) Only the longitudinal vibration of the tube string is considered.On the basis of these assumptions, the mechanical model of downhole tools can be described in Figure3.Figure 3
Mechanical model of oil tube-shock absorber-perforating gun.


### 2.2. Longitudinal Vibration Differential Equation of a Downhole Tube

The oil tube is taken as an example to illustrate the process of establishing the vibration differential equation of a tube string including oil tube and perforation gun. The coordinate system and the mechanical analysis of an oil tube element are shown in Figure4. The forces on the element are composed of inertia force ρA∂2u1/∂t2dx, the damping force of the liquid in annular ν∂u1/∂tdx, gravity ρgAdx, and the elastic force in the tube EA∂u1/∂x, where E is elastic modulus, A is the cross section area of the oil tube, ρ is density, ν is the damping coefficient of the liquid in annulus, and g is gravity acceleration.Figure 4
Mechanical analysis of an oil tube element.The equilibrium equation of the element is given by(1)ρA∂2u1∂t2dx+EA∂u1∂x+ν∂u1∂tdx-EA∂u1∂x+∂∂xEA∂u1∂xdx-ρgAdx=0.Based on the transformation of (1), the partial differential equation of the vibration of an oil tube string can be expressed as(2)∂2u1∂t2-a2∂2u1∂x2+v0∂u1∂t=g,where a=E/ρ is the propagation speed of stress wave in perforation tube string; v0 is the damping coefficient of liquid inside and outside the oil tubing. A specific expression of v0 is given in [23] (3)v0=12πμρADrDti-Dr0.20+0.39DrDti+2.1970×10424DcDti-0.38102.57D2c-Dr2LDr,where μ is the dynamic viscosity of the fluid, Dc is the external diameter of oil tube, Dti is internal diameter, and Dr is the diameter of borehole.The local coordinates and the force analysis of the shock absorber are shown in Figure5, where fk1 and fc1 are, respectively, the spring and damping forces between the shock absorber and oil tube, fk2 and fc2 are correspondent forces between the shock absorber and perforation gun, and fI2 and m2g are, respectively, the inertia force and gravity of the shock absorber.Figure 5
Force analysis of a shock absorber.The equilibrium equation of the shock absorber is given as(4)fk1+fc1=m2g+fI2+fk2+fc2.Equation (4) can be further extended to(5)cddtu1dt-u2t+ku1dt-u2t=m2d2u2dt2+cddtu2t-u3ut+ku2t-u3ut+m2g,where u1dt, u2t, and u3ut are, respectively, the displacements of the lower end of the oil tube, shock absorber, and the upper end of perforating gun. m2, k, and c are, respectively, the mass, stiffness coefficient, and damping coefficient of the shock absorber.


### 2.3. Solving Scheme

Using Newton’s center difference formula to discretize (2) and (5), the following numerical forms are given, respectively, as (6) and (7).Numerical form of the longitudinal vibration equation of oil tube and perforation gun:(6)ui,j+1=a2Δt2/Δx2ui+1,j+ui-1,j-2a2Δt2/Δx2-2.0-v0Δtui,j-ui,j-1+gΔt21+vΔt1≤i≤N-1,1≤j≤K.Numerical form of the vibration equation of a shock absorber:(7)cu1,j+1-u1,jΔt-ui+2,j+1-ui+2,jΔt+ku1,j+1-ui+2,j+1=m2Δt2ui+2,j+1-2ui+2,j+ui+2,j-1+cui+2,j+1-ui+2,jΔt-ui+3,j+1-ui+3,jΔt+kui+2,j+1-ui+3,j+1+m2g.Let x2=c/Δt, x6=m2/Δt2; (7) can be written as (8)x2+ku1,j+1+-2x2-2k-x6ui+2,j+1+x2+kui+3,j+1=x2u1,j-x2ui+2,j-2x6ui+2,j+x6ui+2,j-1-x2ui+2,j+x2ui+3,j+m2g.By solving (6), the nodal displacements of a perforation tube string system at moment j+1 can be obtained, except for those of the lower end of oil tube, shock absorber, and the upper end of perforating gun. To determine the three unknown displacements, besides (8), at least two additional equations are needed. Here, two force continuity conditions are introduced.The force analysis of the lower end of the oil tube is shown in Figure6. According to the law of force continuity, the balance equation of these forces is given by(9)EA∂u1∂xx=L=fc1+fk1+fI1,where fI1 is the inertia force of the lower element of oil tube.Figure 6
Force analysis of the lower end of oil tube.Using Newton’s center difference formula, the numerical form of (9) can be expressed as(10)EOAOu2,j+1-u1,j+1loe=cu1,j+1-u1,j-ui+2,j+1+ui+2,jΔt+ku1,j+1-ui+2,j+1+moeu1,j+1-2u1,j+u1,j-1Δt2.The oil tube is discretized by a finite number of elements. So, heremoe denotes the mass of the bottom element of the oil tube and can be expressed as moe=1/4π(Dc2-Dti2)loeρ, where loe is the element length. Let x1=EoAo/loe, x3=moe/Δt2; (10) can be written as(11)-x1-x2-k-x3u1,j+1+x2+kui+2,j+1=-x1u2,j+1-x2u1,j+x2ui+2,j-2x3u1,j+x3u1,j-1.The force analysis of the upper end of a perforation gun is shown in Figure7. The balance equation of these equations is given by(12)fc2+fk2=EpAp∂u3∂xx=0+fI3.Figure 7
Force analysis of the upper end of perforation gun.Using Newton’s center difference formula, the numerical form of (12) can then be written as(13)kui+2,j+1-ui+3,j+1+cΔtui+2,j+1-ui+2,j-ui+3,j+1+ui+3,j=EpAplpeui+3,j+1-ui+4,j+1+mepΔt2ui+3,j+1-2ui+3,j+ui+3,j-1.The performed gun is also discretized by a finite number of elements. So, herempe denotes the mass of the top element of the performed gun and can be written as mpe=1/4π(Dpc2-Dpi2)lpeρ, where Dpc, Dpi, respectively, denote the external diameter and internal diameter of perforation gun; lpe is the element length. Let x4=EpAp/lpe, x5=mpe/Δt2; (13) can be written as(14)k+x2ui+2,j+1+-k-x2-x4-x5ui+3,j+1=x2ui+2,j-x2ui+3,j-x4ui+4,j+1-2x5ui+3,j+x5ui+3,j-1.Solving (8), (11), and (14) simultaneously, the displacements of the lower end of oil tube, shock absorber, and the upper end of perforating gun at moments j+1, u1,j+1, ui+2,j+1, and ui+3,j+1 can be determined.


## 2.1. Basic Assumptions and Mechanical Model

According to the structure of the perforation string system and the operation condition, the following assumptions are made in order to derive the dynamic governing equations of the system.(1) It is assumed that the material of oil tube and perforation gun is homogeneous and isotropic.(2) The shock absorber is equivalent to a mass-spring-damping system, ignoring the geometry of the shock absorber and the inhomogeneity of the mass distribution.(3) The packer is considered as a fixed support without considering the relative displacement between the packer and casing tube or oil tube.(4) The structural damping of the tube string is ignored.(5) Only the longitudinal vibration of the tube string is considered.On the basis of these assumptions, the mechanical model of downhole tools can be described in Figure3.Figure 3
Mechanical model of oil tube-shock absorber-perforating gun.


## 2.2. Longitudinal Vibration Differential Equation of a Downhole Tube

The oil tube is taken as an example to illustrate the process of establishing the vibration differential equation of a tube string including oil tube and perforation gun. The coordinate system and the mechanical analysis of an oil tube element are shown in Figure4. The forces on the element are composed of inertia force ρA∂2u1/∂t2dx, the damping force of the liquid in annular ν∂u1/∂tdx, gravity ρgAdx, and the elastic force in the tube EA∂u1/∂x, where E is elastic modulus, A is the cross section area of the oil tube, ρ is density, ν is the damping coefficient of the liquid in annulus, and g is gravity acceleration.Figure 4
Mechanical analysis of an oil tube element.The equilibrium equation of the element is given by(1)ρA∂2u1∂t2dx+EA∂u1∂x+ν∂u1∂tdx-EA∂u1∂x+∂∂xEA∂u1∂xdx-ρgAdx=0.Based on the transformation of (1), the partial differential equation of the vibration of an oil tube string can be expressed as(2)∂2u1∂t2-a2∂2u1∂x2+v0∂u1∂t=g,where a=E/ρ is the propagation speed of stress wave in perforation tube string; v0 is the damping coefficient of liquid inside and outside the oil tubing. A specific expression of v0 is given in [23] (3)v0=12πμρADrDti-Dr0.20+0.39DrDti+2.1970×10424DcDti-0.38102.57D2c-Dr2LDr,where μ is the dynamic viscosity of the fluid, Dc is the external diameter of oil tube, Dti is internal diameter, and Dr is the diameter of borehole.The local coordinates and the force analysis of the shock absorber are shown in Figure5, where fk1 and fc1 are, respectively, the spring and damping forces between the shock absorber and oil tube, fk2 and fc2 are correspondent forces between the shock absorber and perforation gun, and fI2 and m2g are, respectively, the inertia force and gravity of the shock absorber.Figure 5
Force analysis of a shock absorber.The equilibrium equation of the shock absorber is given as(4)fk1+fc1=m2g+fI2+fk2+fc2.Equation (4) can be further extended to(5)cddtu1dt-u2t+ku1dt-u2t=m2d2u2dt2+cddtu2t-u3ut+ku2t-u3ut+m2g,where u1dt, u2t, and u3ut are, respectively, the displacements of the lower end of the oil tube, shock absorber, and the upper end of perforating gun. m2, k, and c are, respectively, the mass, stiffness coefficient, and damping coefficient of the shock absorber.


## 2.3. Solving Scheme

Using Newton’s center difference formula to discretize (2) and (5), the following numerical forms are given, respectively, as (6) and (7).Numerical form of the longitudinal vibration equation of oil tube and perforation gun:(6)ui,j+1=a2Δt2/Δx2ui+1,j+ui-1,j-2a2Δt2/Δx2-2.0-v0Δtui,j-ui,j-1+gΔt21+vΔt1≤i≤N-1,1≤j≤K.Numerical form of the vibration equation of a shock absorber:(7)cu1,j+1-u1,jΔt-ui+2,j+1-ui+2,jΔt+ku1,j+1-ui+2,j+1=m2Δt2ui+2,j+1-2ui+2,j+ui+2,j-1+cui+2,j+1-ui+2,jΔt-ui+3,j+1-ui+3,jΔt+kui+2,j+1-ui+3,j+1+m2g.Let x2=c/Δt, x6=m2/Δt2; (7) can be written as (8)x2+ku1,j+1+-2x2-2k-x6ui+2,j+1+x2+kui+3,j+1=x2u1,j-x2ui+2,j-2x6ui+2,j+x6ui+2,j-1-x2ui+2,j+x2ui+3,j+m2g.By solving (6), the nodal displacements of a perforation tube string system at moment j+1 can be obtained, except for those of the lower end of oil tube, shock absorber, and the upper end of perforating gun. To determine the three unknown displacements, besides (8), at least two additional equations are needed. Here, two force continuity conditions are introduced.The force analysis of the lower end of the oil tube is shown in Figure6. According to the law of force continuity, the balance equation of these forces is given by(9)EA∂u1∂xx=L=fc1+fk1+fI1,where fI1 is the inertia force of the lower element of oil tube.Figure 6
Force analysis of the lower end of oil tube.Using Newton’s center difference formula, the numerical form of (9) can be expressed as(10)EOAOu2,j+1-u1,j+1loe=cu1,j+1-u1,j-ui+2,j+1+ui+2,jΔt+ku1,j+1-ui+2,j+1+moeu1,j+1-2u1,j+u1,j-1Δt2.The oil tube is discretized by a finite number of elements. So, heremoe denotes the mass of the bottom element of the oil tube and can be expressed as moe=1/4π(Dc2-Dti2)loeρ, where loe is the element length. Let x1=EoAo/loe, x3=moe/Δt2; (10) can be written as(11)-x1-x2-k-x3u1,j+1+x2+kui+2,j+1=-x1u2,j+1-x2u1,j+x2ui+2,j-2x3u1,j+x3u1,j-1.The force analysis of the upper end of a perforation gun is shown in Figure7. The balance equation of these equations is given by(12)fc2+fk2=EpAp∂u3∂xx=0+fI3.Figure 7
Force analysis of the upper end of perforation gun.Using Newton’s center difference formula, the numerical form of (12) can then be written as(13)kui+2,j+1-ui+3,j+1+cΔtui+2,j+1-ui+2,j-ui+3,j+1+ui+3,j=EpAplpeui+3,j+1-ui+4,j+1+mepΔt2ui+3,j+1-2ui+3,j+ui+3,j-1.The performed gun is also discretized by a finite number of elements. So, herempe denotes the mass of the top element of the performed gun and can be written as mpe=1/4π(Dpc2-Dpi2)lpeρ, where Dpc, Dpi, respectively, denote the external diameter and internal diameter of perforation gun; lpe is the element length. Let x4=EpAp/lpe, x5=mpe/Δt2; (13) can be written as(14)k+x2ui+2,j+1+-k-x2-x4-x5ui+3,j+1=x2ui+2,j-x2ui+3,j-x4ui+4,j+1-2x5ui+3,j+x5ui+3,j-1.Solving (8), (11), and (14) simultaneously, the displacements of the lower end of oil tube, shock absorber, and the upper end of perforating gun at moments j+1, u1,j+1, ui+2,j+1, and ui+3,j+1 can be determined.


## 3. Experimental Analysis of Downhole Perforation Load

An underwater explosion experiment of perforation bullet was carried out to investigate the pressure distribution of perforation explosion. The equipment needed for the experiment is mainly composed of a water tank, perforation guns, ammunition, and sensors which are shown in Table1 and Figures 8 and 9.Table 1
Experimental site and equipment.
Experimental site
A pool with diameter 2.0 m and water depth 2.7 m
Instruments
138A51 and 138A26 underwater explosion pressure sensor, F482A51 constant current source, DPO4034 Tektronix storage oscilloscope, computer, blasting line, data transmission line, detonator, trigger mutual inductor, positioning bracket.
Test material
Primer detonator, detonating cord, perforating bullet, perforating gun, contrast HMX bare drug column.Figure 8
Explosion test pool and positioning bracket.Figure 9
Perforating bullet for explosive test.In the experiment, the perforation gun with perforation bullet is placed in the pool, and the explosion pressure sensors are arranged in the position shown in Figure10. After the perforation bullet is detonated, the shock wave energy, namely, equivalent energy (as shown in Figure 11), can be collected by the pressure sensors, based on which the quality of equivalent explosive can be determined.Figure 10
Arrangement diagram of pressure sensors.Figure 11
Typical pressure signals collected in the test.More than twenty underwater explosion tests on 5 kinds of high temperature and high pressure oil perforation bullet were carried out to measure the underwater explosion overpressure which is shown in Table2.Table 2
Explosion overpressure values of perforation bullet measured in the experiment.
Serial number
Types of perforation bullet
Explosion overpressure (MPa)
1
# sensor (0.6 m)
2
# sensor (0.8 m)
3
# sensor (0.8 m)
1
Type 73 (18 g HMX)with 73 mm perforating gun
1 perforation bullet
15.181
15.460
12.102
11.747
12.013
11.871
2
16.017
10.774
11.73
3
15.181
12.364
—
4
3 perforation bullets
/
14.543
14.358
5
Type 89 (25 g HMX)with 89 mm perforating gun
1 perforation bullet
17.967
17.943
13.427
12.952
12.144
13.107
6
18.245
13.017
13.7
7
17.618
12.412
13.478
8
3 perforation bullets
22.284
20.373
17.256
9
Type 102 (32 g HMX)with 102 mm perforating gun
1 perforation bullet
19.359
21.588
15.688
15.786
—
15.862
10
22.981
15.458
16.216
11
22.423
16.212
15.508
12
3 perforation bullets
37.117
21.847
24.328
17
Type 89 (23g PYX)with 89 mm perforating gun
1 perforation bullet
16.365
16.481
11.102
11.462
12.144
12.40
18
15.808
—
11.366
19
17.27
11.822
13.7
20
3 perforation bullets
18.593
18.015
15.559
21
Type 102 (31 g PYX)with 102 mm perforating gun
1 perforation bullet
22.981
21.077
—
14.40
14.256
14.7
22
19.359
14.458
15.7
23
20.891
14.343
14.144
24
3 perforation bullets
22.423
18.212
18.105Based on the tested data in the experiment, the mass percentages (Q) of explosive mass to total charge mass (M) are determined. Here, explosive mass refers to those consumed by the shock wave produced by underwater explosion and the bubble pulsation of perforator tapes 73, 89, and 102 (main blasting charge is HMX). Figure 12 indicates that the mass percentage increases with increasing the charge quantity of perforation gun.Figure 12
Relation curve between mass percentage and total charge.With linear fitting the experiment data of types 73, 89, and 102, an equation describing the relation between the mass percentage and total charge mass is obtained.(15)Q=0.31624+0.00181M.Fitting the peak pressure of each measuring point, an equation of peak pressure is obtained:(16)pm=80.52995×e-X/4.96494+p0p0=1665.4×Q1/32.0608,where p0 is the pressure of stable stage, Q is explosive charge, pm is peak pressure, and X is the distance from explosion center of perforation gun.According to the different types of perforating bullet, the engineering calculation method of the pressure field of downhole explosion can be established.(17)p=pm×e-t/θ.It is worth noting that the application of (17) should be restricted to a certain extent. The first one is that the explosion peak pressure and the pressure of steady stage depend only on the equivalent explosive mass of the downhole explosion of a single perforation bullet and they have no direct relation with the number of perforation bullets detonated simultaneously and the time interval of detonation. The second one is that (17) is only suitable for perforation bullet with RDX type charge. As other charges (TNT, HMX, and PYX) are used in perforation bullet, it is needed to modify the peak pressure and the pressure of stable stage.


## 4. Dynamic Response Analysis of Downhole Tools

A FORTRAN calculation code based on the dynamic model shown in Section2 and the engineering calculation method of perforation pressure field shown in Section 3 are developed. Using the code, the dynamic response characteristics of the oil tube string and the packer in downhole perforation operation are investigated. The design, construction, and environmental parameters in calculation analysis are shown in Table 3.Table 3
Basic perforation parameters.
Parameter
Value
Liquid dynamic viscosity (μ)
0.01 Pa⋅s
Yield strength of tubing material (σs)
758 Mpa
Outer diameter of oil tube
88.9 mm
Inner diameter of oil tube
76 mm
Length of oil tube string
200 m
Element number of oil tube string
100
Elastic modulus of tubing material (E)
206 GPa
Density of tubing material (ρ)
7846 kg/m3
Total simulation time
80 s
Time step
0.001 s
Outer diameter of perforating gun
73 mm
Inner diameter of perforating gun
62 mm
Length of perforating gun
3.3 m
Charge quantity of a perforating bullet
16 g, 20 g, 32 g, 48 g, 64 g, 128 g
Spring stiffness coefficient of shock absorber (k)
12~800 N/mm
Damping coefficient of shock absorber (c)
15 N⋅s/mm
Element number of perforating gun
10
Distance between the upper end of the perforating gun and the explosion center
0.2 m


### 4.1. Dynamic Response of Oil Tube and Packer

In the calculation code, an array is used to record the maximum axial tensile and compressive force in the oil tube string. It is noted that the upper end of the oil tube string (its position in the coordinate system is 200 m) is connected to the packer. So, the axial force in the upper end of the tube string is equal to that acting on the packer.Figures13–15 show the distribution of the maximum dynamic axial force in oil tube string. It can be found that the axial tensile force decreases gradually along the well depth, with the maximum and minimum values appearing, respectively, at the upper and lower ends of the oil tube string. The distribution of maximum axial compressive force is just opposite to this phenomenon. The reason is that gravity is one of the main factors affecting axial forces.Figure 13
Effect of shock absorber on the maximum dynamic axial force in oil tube string: (a) tensile force; (b) compressive force.
(a)
(b)Figure 14
Effect of tube length on maximum dynamic axial force: (a) tensile force; (b) compressive force.
(a)
(b)Figure 15
Effect of charge quantity on the maximum compressive force in oil tube string: (a) tensile force; (b) compressive force.
(a)
(b)Figure13 shows that, compared with one-shock absorber scheme, the two-shock-absorber scheme can significantly reduce the axial forces in the tube string. However, excessive arrangements of shock absorbers in the same position have little effect on reducing axial forces and force acting packer.Figure14(a) shows that, except for the upper and lower ends, the maximum dynamic tensile force in the same position decreases with increasing tube length. However, increasing tube length has no significant effect on the distribution of axial tensile force, whereas it is found in Figure 14(b) that the tube length has a significant effect on the distribution and magnitude of axial compressive force. As the length is relatively small, there are two extreme points appearing on the curve of axial compressive force. With the length increasing, the axial compressive force in the lower part of the oil tube string increases and that in the upper part decreases.It is found in Figure15 that the axial forces including tensile force and compressive force in oil tube string increase with the increase of charge quantity. The greater the distance from the lower end of oil tube string, the greater the increment. As the explosive charge changes, the axial tensile force shown in Figure 15(a) increases linearly with the distance. Figure 15(b) shows that with the charge quantity changing from low to high, the axial compressive force in the mid-upper part of tube string, compared with that in lower part of tube string, changes from weak to strong. So the explosive charge has significant effect on the axial force in oil tube, especially for mid-upper part of the tube string. For the case of large charge, the packer may fail, since the force acting on it may exceed the releasing force.


### 4.2. Buckling Mechanical Behavior of Oil Tube String

Helical buckling is a common failure form of tube string [24]. To investigate the perforation impact load on this mechanical behavior, an equation for calculating critical helical buckling load of tube string is used as follows [25]:(18)Fcrh=5.55EIq23-poSo-piSi,where q=ρpVg is the effective line mass of oil tube, E is elastic modulus, ρp is density, V is line volume per unit length, I is inertia moment of cross section, pi and po are pressures of the fluid in oil tube and annulus, and Si and So are cross-sectional areas corresponding to inner and outer diameters.In order to distinguish the mechanical state of a tube string, here, a ratioPr (=Fcrh/Fc,max) is used, where Fc,max is the maximum dynamic compressive force.AsPr<1, helical buckling appears; otherwise, helical buckling does not appear.Figure16 shows that two-shock-absorber scheme is better than one-shock-absorber scheme in preventing helical buckling. Compared with two-shock-absorber scheme, three-shock-absorber scheme can only improve the capability of preventing helical buckling of the lower part of oil tube string; it has little effect on other parts. In fact, under the current conditions, one-shock-absorber scheme can meet the requirements of avoiding the helical buckling of the tube string. Figure 17 shows that with the tube length increasing the range of helical buckling in the lower part of tube string is expanded. However, in the other parts, the ability to resist buckling is enhanced. It is found in Figure 18 that with the charge of the perforating bullet increasing ratio Pr decreases. Especially for the mid-upper part, this trend is more significant. It is indicated that the charge has a greater effect on the mid-upper part of the tube than the lower part of the string.Figure 16
Effect of shock absorber on the buckling mechanical behavior oil tube string.Figure 17
Effect of slenderness ratio of oil tube string on buckling mechanical behavior.Figure 18
Effect of charge quantity on the buckling mechanical behavior of oil tube string.


## 4.1. Dynamic Response of Oil Tube and Packer

In the calculation code, an array is used to record the maximum axial tensile and compressive force in the oil tube string. It is noted that the upper end of the oil tube string (its position in the coordinate system is 200 m) is connected to the packer. So, the axial force in the upper end of the tube string is equal to that acting on the packer.Figures13–15 show the distribution of the maximum dynamic axial force in oil tube string. It can be found that the axial tensile force decreases gradually along the well depth, with the maximum and minimum values appearing, respectively, at the upper and lower ends of the oil tube string. The distribution of maximum axial compressive force is just opposite to this phenomenon. The reason is that gravity is one of the main factors affecting axial forces.Figure 13
Effect of shock absorber on the maximum dynamic axial force in oil tube string: (a) tensile force; (b) compressive force.
(a)
(b)Figure 14
Effect of tube length on maximum dynamic axial force: (a) tensile force; (b) compressive force.
(a)
(b)Figure 15
Effect of charge quantity on the maximum compressive force in oil tube string: (a) tensile force; (b) compressive force.
(a)
(b)Figure13 shows that, compared with one-shock absorber scheme, the two-shock-absorber scheme can significantly reduce the axial forces in the tube string. However, excessive arrangements of shock absorbers in the same position have little effect on reducing axial forces and force acting packer.Figure14(a) shows that, except for the upper and lower ends, the maximum dynamic tensile force in the same position decreases with increasing tube length. However, increasing tube length has no significant effect on the distribution of axial tensile force, whereas it is found in Figure 14(b) that the tube length has a significant effect on the distribution and magnitude of axial compressive force. As the length is relatively small, there are two extreme points appearing on the curve of axial compressive force. With the length increasing, the axial compressive force in the lower part of the oil tube string increases and that in the upper part decreases.It is found in Figure15 that the axial forces including tensile force and compressive force in oil tube string increase with the increase of charge quantity. The greater the distance from the lower end of oil tube string, the greater the increment. As the explosive charge changes, the axial tensile force shown in Figure 15(a) increases linearly with the distance. Figure 15(b) shows that with the charge quantity changing from low to high, the axial compressive force in the mid-upper part of tube string, compared with that in lower part of tube string, changes from weak to strong. So the explosive charge has significant effect on the axial force in oil tube, especially for mid-upper part of the tube string. For the case of large charge, the packer may fail, since the force acting on it may exceed the releasing force.


## 4.2. Buckling Mechanical Behavior of Oil Tube String

Helical buckling is a common failure form of tube string [24]. To investigate the perforation impact load on this mechanical behavior, an equation for calculating critical helical buckling load of tube string is used as follows [25]:(18)Fcrh=5.55EIq23-poSo-piSi,where q=ρpVg is the effective line mass of oil tube, E is elastic modulus, ρp is density, V is line volume per unit length, I is inertia moment of cross section, pi and po are pressures of the fluid in oil tube and annulus, and Si and So are cross-sectional areas corresponding to inner and outer diameters.In order to distinguish the mechanical state of a tube string, here, a ratioPr (=Fcrh/Fc,max) is used, where Fc,max is the maximum dynamic compressive force.AsPr<1, helical buckling appears; otherwise, helical buckling does not appear.Figure16 shows that two-shock-absorber scheme is better than one-shock-absorber scheme in preventing helical buckling. Compared with two-shock-absorber scheme, three-shock-absorber scheme can only improve the capability of preventing helical buckling of the lower part of oil tube string; it has little effect on other parts. In fact, under the current conditions, one-shock-absorber scheme can meet the requirements of avoiding the helical buckling of the tube string. Figure 17 shows that with the tube length increasing the range of helical buckling in the lower part of tube string is expanded. However, in the other parts, the ability to resist buckling is enhanced. It is found in Figure 18 that with the charge of the perforating bullet increasing ratio Pr decreases. Especially for the mid-upper part, this trend is more significant. It is indicated that the charge has a greater effect on the mid-upper part of the tube than the lower part of the string.Figure 16
Effect of shock absorber on the buckling mechanical behavior oil tube string.Figure 17
Effect of slenderness ratio of oil tube string on buckling mechanical behavior.Figure 18
Effect of charge quantity on the buckling mechanical behavior of oil tube string.


## 5. Discussion and Conclusion

The effects of three important parameters, charge quantity, shock absorber number, and tube length, on the dynamic behavior of downhole tools have been investigated in detail. Based on the results described, the following conclusions can be drawn.(1) The shock absorber between the perforating gun and the tube string can obviously improve the mechanical state of the oil tube and packer. However, it is not an economic option to set too many shock absorbers at the same location.(2) The charge has a significant effect on the mechanical state of tube string and packer. Especially for the mid-upper part of the tube, this effect is more evident. Thus, as the charge quantity is large, a subsection setting scheme of the shock absorber can be used in the mid-upper part of the tube string.(3) The length increase of the tube string will reduce the stability of the lower part but will improve the stability of the mid-upper part. Thus, as the charge quantity is large, the mechanical state of oil tube string and packer can be improved by extending the length of the tube string below the packer.



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*Source: 4894032-2017-08-24.xml*