A model of a multibody system is established to investigate the dynamic response of an oil tubeshock absorberperforating gun system in downhole perforationtest 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 perforationtest joint operation.
In the perforationtest joint operation of oilgas 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 Figure
Downhole tools in perforationtest joint operation.
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. [
A cross combination scheme of shock absorber and damping oil tube was proposed by Fan and Li [
A vibration model of perforated tubeshock absorber was established by Liu et al. [
Zhou [
A longitudinal vibration model coupling the testing tube and shock absorber was used by Huang [
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.
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.
(
(
(
(
(
On the basis of these assumptions, the mechanical model of downhole tools can be described in Figure
Mechanical model of oil tubeshock absorberperforating gun.
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 Figure
Mechanical analysis of an oil tube element.
The equilibrium equation of the element is given by
Based on the transformation of (
The local coordinates and the force analysis of the shock absorber are shown in Figure
Force analysis of a shock absorber.
The equilibrium equation of the shock absorber is given as
Equation (
Using Newton’s center difference formula to discretize (
Numerical form of the longitudinal vibration equation of oil tube and perforation gun:
By solving (
The force analysis of the lower end of the oil tube is shown in Figure
Force analysis of the lower end of oil tube.
Using Newton’s center difference formula, the numerical form of (
The oil tube is discretized by a finite number of elements. So, here
The force analysis of the upper end of a perforation gun is shown in Figure
Force analysis of the upper end of perforation gun.
Using Newton’s center difference formula, the numerical form of (
The performed gun is also discretized by a finite number of elements. So, here
Solving (
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 Table
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. 
Explosion test pool and positioning bracket.
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 Figure
Arrangement diagram of pressure sensors.
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 Table
Explosion overpressure values of perforation bullet measured in the experiment.
Serial number  Types of perforation bullet  Explosion overpressure (MPa)  





1  Type 73 (18 g HMX) 
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) 
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) 
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) 
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) 
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  18.105 
Based on the tested data in the experiment, the mass percentages (
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.
Fitting the peak pressure of each measuring point, an equation of peak pressure is obtained:
According to the different types of perforating bullet, the engineering calculation method of the pressure field of downhole explosion can be established.
It is worth noting that the application of (
A FORTRAN calculation code based on the dynamic model shown in Section
Basic perforation parameters.
Parameter  Value 

Liquid dynamic viscosity ( 
0.01 Pa⋅s 
Yield strength of tubing material ( 
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 ( 
206 GPa 
Density of tubing material ( 
7846 kg/m^{3} 
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 ( 
12~800 N/mm 
Damping coefficient of shock absorber ( 
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 
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.
Figures
Effect of shock absorber on the maximum dynamic axial force in oil tube string: (a) tensile force; (b) compressive force.
Effect of tube length on maximum dynamic axial force: (a) tensile force; (b) compressive force.
Effect of charge quantity on the maximum compressive force in oil tube string: (a) tensile force; (b) compressive force.
Figure
Figure
It is found in Figure
Helical buckling is a common failure form of tube string [
In order to distinguish the mechanical state of a tube string, here, a ratio
As
Figure
Effect of shock absorber on the buckling mechanical behavior oil tube string.
Effect of slenderness ratio of oil tube string on buckling mechanical behavior.
Effect of charge quantity on the buckling mechanical behavior of oil tube string.
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.
(
(
(
The authors declare that there are no conflicts of interest regarding the publication of this paper.