Investigation on Releasing of a Stuck Drill String by Means of a Mechanical Jar
^{1}
Department of Higher Mathematics, IvanoFrankivsk National Technical University of Oil and Gas, IvanoFrankivsk,
15 Karpatska St.,
76019
IvanoFrankivsk, Ukraine
^{2}
Department of Oil and Gas Equipment, IvanoFrankivsk National Technical University of Oil and Gas, IvanoFrankivsk,
15 Karpatska St.,
76019
IvanoFrankivsk, Ukraine
^{*} Corresponding author email: kgl.imp.nan@gmail.com
Received:
9
October
2016
Accepted:
10
July
2017
Purpose. In this article the most important part is dedicated to the research of elimination of accident that is caused by drill string sticking during the process. That is why it is necessary to develop a mathematical model of the mechanic system: travelling system + drill string + mechanical jar + rock, to develop a computer model for numerical calculation of dynamic characteristics of firing gear. The aim is to use the results of the research and to work out recommendations for expediency of jar application. Methods. For description of the drill string we are using synthesis of the wave theory and theory of the local distortions. For mathematical modeling of firing device we are offering the use of the combined method that combines static solutions of the theory of elasticity for contact zone of drill string and method of a plain wave of SaintVenant. We solved systems of differential equations using the methods of mathematical physics. An algorithm of the numerical decision which mounted in the computing environment were used at simulation of the longitudinal impact to the stuck drill pipe. In this article we designed a wave chart of the equation system of the drill pipe and conducted stepbystep calculation of a collision momentum. We also designed a computer program for numerical modeling of the drill pipe mechanism with firing gear. We also designed a method of calculation of main dynamic characteristics of firing device that will help analyze and prove the performance of the mechanical jar. A wave diagram was built that shows the impact forces and speeds on the boundary surfaces of the sections of the drill string. There were calculated main dynamic characteristic of mechanical the jar. Originality. Authors also developed a dynamic mathematical model that combined elastic vibrations of continual system of loose part drill pipe, impact mechanisms and discrete movements of a given drill pipe. The process of a mechanical jar work that considers elastic deformations of contact areas was not published by researchers. Practical implications. Received results could be used for further research and perfection of existing engineering methods of modeling and calculations of drill pipes at the stage of their designing and their construction.
© V. Moisyshyn and K. Levchuk, published by IFP Energies nouvelles, 2017
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Introduction
The scientific and technical literature broadly covers the methods of elimination Drill String (DS) sticking using firing gear [1]. In general, mathematical modeling process is quite difficult, therefore to solve specific applied the impact theory we had to use some simplifications and assumptions that satisfy receive convenient engineering solution. Very important also is the right choice of mathematical description of the construction work which is related to the shock [2]. Methods of approximate calculation of impact include:

classic Newton method, which provides body and solid contact area;

Hertz method, which provides solid body and contact areas is elastic;

elasticity method, which provides elastic body, but the distribution of stress on bodies is immediate;

plane wave SaintVenant method;

combined method that combines lists.
The classical Newton’s method gives static elasticity solutions for contact zone and plane wave method to other bodies. Sufficiently accurate results are received if the percussion length of 3–5 times the periods of natural vibrations of the DS. But according to the studies [3] and experimental data [4] blow hammer and anvil last less than 0.1 s, and the greatest period of free oscillations of the drill pipe is 0.2–0.3 s. It should also be noted that the ratio of the classical theory of impact is impossible to determine the duration and force of percussion, tensions in DS, and accelerate their movement.
Despite a sufficient number of publications and developed advanced drilling technologies [5], potential risk of stuck DS is high. Thus, the task of developing new technological solutions and sophisticated technical means to release the drill string sticking remains relevant [6].
Modern methods of elimination stuck are based on the use of mechanical action stuck DS in the area. These methods are based on creating disturbances vibration, percussion or explosive loads applied in the area of the stuck DS. One of the promising areas of DS is releasing drill jars [7], which can eliminate the accident in the most difficult cases.
1 Mathematical model of elastic waves emitted by a drill pipe with a jar
Moving parts of the travelling system are presented by multiple mass m_{0} = 9855 kg and rigidity of travelling ropes c_{0} = 53 MN/m. Mechanical scheme of DS with a firing gear can be represented by discretecontinuous system comprising four sections (Fig. 1). For research of the selected DS layout we took the parameters that are presented in Table 1. The DS sections are composed of thin cylindrical steel pipes with outside diameter D, inside diameter d and total length l. Stuck part of the drill string will be divided into two sections: the third – a loose part, located above sticking zone and the fourth – stuck DS part. The density of the drilling fluid is q = 1200 kg/m^{3}. Young’s modulus (modulus of elasticity) for all three sections is E = 210 GPa. Coefficient of dry friction was taken f_{0} = 0.3 to the fourth section.
The mathematical model for the proposed scheme: travelling block + loose drill string + mechanical jar + sticking drill string (Fig. 1) describes dynamic processes in DS. According to the theory of elasticity [8], wave differential equations [9] for the crosscuts of each of the four sections can be written as: (1) where ; – multiple coefficient of viscous resistance (α – coefficient drilling fluid interaction with DS section, m = ρFl – mass of section); F – cross sectional area of drill pipes; – speed of elastic waves propagation (E – Young’s modulus); ; g – gravitation acceleration; f_{0} – coefficient of DS dry rod friction; x – current coordinate of DS section crosscut; t – current time.
Next, we have to attach boundary conditions at the ends and docking sections of DS to the dynamic equations of the drill string motion: (2) (3) (4) (5)
From static equilibrium equations of the mechanical system we got initial motion conditions u_{i} (x_{i}, 0): (6) (7) (8) (9) where P_{st} – stuck force of the drill string. Indexes in equations (2)–(9) indicate the number of section.
At the beginning of the motion in the position of static equilibrium speeds in the current DS crosscuts are .
Figure 1
Drill string with builtin mechanical jar. 
Geometric parameters of the DS layout.
2 Methods determining the boundaries of the sticking drill string
Before a decision on the choice of the liquidation of the accident, we should determine the type and location of stuck. The simplest way to determine the upper boundary sticking is an approximate calculation [10]. At the same time, it is believed that every 1000 m loose part of the stuck DS with force an effort that exceeds its own weight the drill string 200 kN is stretched in accordance with Table 2.
The upper boundary of stuck places is usually determined by the formula: (10) where 1.05 – coefficient takes into account hard locks; P_{1}, P_{2} – tensile forces; Δl – elongation the drill string with the subtraction load P_{1} − P_{2}. For the selected layout of the drill string: (11)
An oilman determines a place of the sticking DS more precisely using the identification stuck. This device is the electromagnet, the effect of which is based on the conversion of the sensor indications of magnetic waves in a stick impulses induction of electromotive force [11]. These indications are transmitted by cable to the drilling rig and recorded. Slight areas of pipes are magnetized using the direct current.
Acoustic cement bond log sonde is the most advanced method to determine the place of the sticking and allocation of plots with different degrees of pressing [12]. Amplitude and duration distribution refracted waves are measured using acoustic logging. Emitter generates sound pulses at a frequency of 10–30 Hz, which picks up the receiver. Elastic vibrations transmitted through the drilling fluid in DS, in which waves propagate at the speed of sound. Locations are stuck by a loss of energy that is not returned to the receiver. The intensity of the forces are measured at magnitude stuck energy that is absorbed by the rock. As a result of geophysical studies wells can get the curve that allows you to allocate areas of the sticking DS intensity and pressure forces distributed rocks in the pipe.
Lengthening drill pipes.
3 The sticking drill string
The drill pipe and the rock hole that surrounds it should be considered as a single structure in the case of simulation of wave characteristics of fourth section of the sticking DS. Given the density ρ_{r} = 3000 kg/m^{3} and Young’s modulus E_{r} = 129 GPa of rock we calculated multiple density ρ_{4}, multiple cross sectional area F_{4} and multiple Young’s modulus E_{4}. Thus, the sticking drill pipe consists of two bodies: steel pipe and rock stuck, unlimited nested in the hills. If we consider the section of described construction, the conditional parameters of rock stuck are completely determined by the stuck power pressure rock P_{st}, which is the sticking drill pipe (Fig. 2): (12) (13) where f(x_{4}, φ) – distributed power intensity rock stuck; ρ_{r}, F_{r} – rock density and rock strength equivalent crosscut of the rock sticking area.
Figure 2
Crosscut sticking drill pipe. 
4 Wave model of a mechanical jar
Let us review a percussion of two parts of the drill string. We consider the impact as being elastic: the possibility of residual deformation and dissipation energy is excluded. Therefore, during the formation of waves of an absolutely elastic percussion, we can observe the division of the energy into kinetic and potential. The process of partial transformation of kinetic energy into potential energy gradually spreads on adjacent layers of the drill pipe, whereas deformation spreads from one section to another section. When a percussion wave moves on rod of the drill string, starting from impact zone, dynamic characteristics change is explained by material properties and crosscuts of the drill pipes, as well as explained by the external environment. According to this, we will consider that the parameters of the wave, moving on rod of the drill string, will not change till it reaches the boundary surface. The boundary surfaces in DS are the following: end sections, adjusted to each other, and stuck surface of the drill string.
The demonstrated mechanic system (Fig. 1) has four boundary surfaces. As a result, a multiple reflection, refraction and imposition of shock waves are possible. It can cause a complication of general picture of wave process. This is why it is advisable to draw a wave diagram (Fig. 3), so as to demonstrate the drill pipes’ wave process. From Figure 3, we can see – the duration of the impact motion wave between the sections of the drill string; , , – efforts and speed on the section borders of the drill string before the percussion; – impact forces; – the speed of the waves between sections of the drill string.
Since all pipes are forged from the same material, and the distance between the impact zones and the drill string sticking l_{3} < l_{2} < l_{1}, thereafter t_{3} < t_{2} < t_{1}. Thus, the bouncing from the top of the drill string will take place times less than from the first section of the drill string and, consequently, times less than from the stuck boundary surface of the drill pipe. The percussion will be lasting as long as the impact forces will be gaining positive values and both components of the drill string will spring back.
The wave diagram is an important element in order to graph wave equation. At the beginning of a percussion, burdened drill pipes with end speed and ultimate tensile strength met with anvil – sticking component of the drill string, while its upper butt had a speed and a strike force : (14)
From the emplacement, percussion wave with force and speed spreads on the third section of the DS and then lapse of t_{3} reaches the stuck place of a static forth section of the drill string, under the force of : (15)
Similarly, in the following intervals 2t_{3}, impact forces on the drill pipes’ boundary sections DS (Fig. 3) can be calculated from the following equation system: (16) where i = 2, …, 2n_{2}. The lower part of the first section of the drill string, which had, before the percussion, speed and tensile strength will meet through the time interval t_{2} with the wave, which had spread with speed and force : (17)
Here , i = 1–4 – the impedance of the type of barrel drill hole. Impact forces and speeds are calculated stepbystep from these equation systems (14)–(17). Similarly for the upper part of the drilling string, which had, before the strike, speed and tensile strength will meet with a wave after a certain time t_{1}, that derived from the second section with the speed and force . Speeds , and forces , are calculated from the following equation systems: (18) where – air impedance. It is important to mention that impact force appears later in the sticking zone than in the zone of an actual percussion. This delay depends on the general distance between anvil, the place of the sticking of the DS and the type of borehole, so it means that it can be calculated due to duration of the wave’s passage till the place sticking t_{3}. The strike will be lasting till the moment when the force becomes negative and both parts of the drilling stick will spring back. Usually the wave never reaches the top of the drill stick, as till that moment t_{1} + t_{2} percussion ends up.
Figure 3
Wave diagram of the drill string. 
5 Findings
The research was conducted for the chosen component of the drill string with the modeling of different length sticking force with equally distributed load. The given wave dependencies are demonstrated in Figure 4.
The duration of the percussion was defined by the period of time from the moment of meeting t_{u} till the moment of separation of the hammer and the anvil. According to the wave diagram (Fig. 3) release of the sticking zone was considered as stepbystep summary displacement.
In case the integral impact forces occur by P_{st} and they exceed 100 kN, wave diagrams will have impulsive shape. And the collision momentum itself responds to the Hertz theory: at early convergence of hammer and anvil it quickly increases to the maximum value than it decreases till the complete stop of the drill string. The maximum value of impact force can reach tens of MN. Such stuck is characterized by length of the sticking zones more than 30 m or in places of strong monolithic formations. It is important to add that in practice the drillers meet with several sticking areas, and the sticking length does not exceed more than 20 m (Figs. 5 and 6).
From the researches we can see that with a decrease of pressure force of the rock on the drill pipe, the number of peaks of impact forces on wave diagrams and the speed of liberation of the drill string increase, and collision momentum takes the form of smooth curves.
Since every crosscut of the drill string's sticking gets negative speed, explained by elastic deformation, and some of the sections – characteristics of the material, therefore the kinetic energy of the drill string before percussion will have integral form: (19) where is the speed of crosscut of the section DS with coordinate x at the meeting moment of hammer and anvil t_{u}. The common energy that sticking area gets after the percussion can be calculated as such: (20)
With the help of elaborated computer program for a chosen composition of the drill string, we calculated time of meeting of hammer and anvil t_{u} = 28 ms, if the way of shock mechanism s = 3 m, rock pressure forces P_{st}, percussion duration τ, energy A, that was transmitted to the drill string sticking and the number of percussion Nud, needed to liberate the drill string. The results of calculations are summarized within Table 3.
Figure 4
Laws changing impact strength and speed of the stuck area of the drill string and for the pressure rocks P_{st} = 120 kN. 
Figure 5
Laws changing impact strength and speed of the stuck area of the drill string and for the pressure rocks P_{st} = 90 kN. 
Figure 6
Laws changing impact strength and speed of the stuck area of the drill string and for the pressure rocks P_{st} = 50 kN. 
Dynamic characteristics of the mechanical jar.
Conclusion
In this study we obtained mathematical and computer model of the firing gear, triggered with the help of longitudinal vibrations of the drill string. Those vibrations were generated by multiple harmonics spectrum. The given theory of dynamic modeling of the drill pipe with a mechanical jar gives us an opportunity to calculate the process of accident liquidation. One should take into account that mechanical jar is embedded into drill pipes section.
The researches have shown [1] that the rapidity of stuck liquidation depends on rock elasticity. The fastest method allows us to release the drill string from loam (E_{r} = 33 GPa), and the longest from crushed stone (E_{r} = 50 GPa) and sandy loam (E_{r} = 78 GPa).
The graphics show that after the meeting with a rock, the collision momentum of plane wave of the drill string (with an increase of rock pressure forces on the pipe) has lower duration and impulsive character. Therefore, while calculating we should take into account geographical data, which will enable us to define the number of stuck, its areas and length, as well as distributed force power on the pipe.
Acknowledgments
We express our sincere gratitude to Dr. Ja. Kuntsyak (Joint Stock Company « Scientific Design Bureau for Testing of Drilling Tools », Kyiv, Ukraine) and Prof. V. Vekeryk (IvanoFrankivsk National Technical University of Oil and Gas, Ukraine) for consultations on technical issues of borehole drilling.
Scientific advice was obtained from The National Academy of Sciences of Ukraine, Academic Society of Michal Baludyansky (Bratislava, Slovak Republic).
We greatly appreciate Prof. O. Ivasyshyn (G.V. Kurdyumov Institute for Metal Physics of the National Academy Sciences of Ukraine) for his valuable improvements in this manuscript.
References
 Dupriest F.E., Elks W.C., Ottesen S. (2011) Design methodology and operational practices eliminate differential sticking, SPE Drill. Complet. 26, 115123. [CrossRef] [Google Scholar]
 Hempkins W.B., Kingsborough R.H., Lohec W.E., Nini C.J. (1987) Multivariate statistical analysis of stuck drill pipe situations, SPE Drill. Eng. 2, 237244. [CrossRef] [Google Scholar]
 Levchuk K. (2015) Influence of location of the shock absorber on the performance of a shock pulse alarm layout of the drill string (Ukraine), Inf. Syst. Mech. Control 12, 7283. [CrossRef] [Google Scholar]
 Skeem M.R., Friedman M.B., Walker B.H. (1979) Drill string dynamics during jar operation, SPE J. Petrol. Technol. 31, 13811386. [CrossRef] [Google Scholar]
 Shook E.H., Lewis T.W. (2008) Cement bond evaluation, in: SPE Western Regional and Pacific Section AAPG Joint Meeting, 29th March to 4th April 2008, Bakersfield, California, USA. [Google Scholar]
 Shivers R.M., Domangue R.J. (1993) Operational decision making for stuck pipe incidents in the Gulf of Mexico: a risk economics approach, SPE Drill. Complet. 8, 125130. [CrossRef] [Google Scholar]
 Kyllingstad A., Halsey G.W. (1990) Performance testing of jars, in: SPE/IADC Drilling Conference, 27th February to 2nd March, 1990, Houston, TX. [Google Scholar]
 Shatskii I.P., Perepichka V.V. (2013) Shock wave propagation in an elastic rod with a viscoelastic external resistance, J. Appl. Mech. Tech. Phys. 8, 10161020. [CrossRef] [Google Scholar]
 Evans L. (1998) Partial differential equations, American Mathematical Society, Providence, p. 662. [EDP Sciences] [Google Scholar]
 Borwein J.M., Skerritt M.P. (2012) An introduction to modern mathematical computing with Maple, Springer Undergraduate Texts in Mathematics and Technology, p. 233. [Google Scholar]
 McCall K.R., Boudjema M., Santos I.B., Guyer R.A., Boitnott G.N. (2001) Nonlinear, hysteretic rock elasticity: deriving modulus surfaces, in: American Rock Mechanics Association DC Rocks, The 38th U.S. Symposium on Rock Mechanics (USRMS), 7–10 July, 2001, Washington. [Google Scholar]
 Siems G.L., Boudreaux S.P. (2007) Applying radial acoustic amplitude signals to predict intervals of sandstuck tubing, SPE Prod. Oper. (Society of Petroleum Engineers) 22, 254259. [Google Scholar]
All Tables
All Figures
Figure 1
Drill string with builtin mechanical jar. 

In the text 
Figure 2
Crosscut sticking drill pipe. 

In the text 
Figure 3
Wave diagram of the drill string. 

In the text 
Figure 4
Laws changing impact strength and speed of the stuck area of the drill string and for the pressure rocks P_{st} = 120 kN. 

In the text 
Figure 5
Laws changing impact strength and speed of the stuck area of the drill string and for the pressure rocks P_{st} = 90 kN. 

In the text 
Figure 6
Laws changing impact strength and speed of the stuck area of the drill string and for the pressure rocks P_{st} = 50 kN. 

In the text 