Modelling Emergency Situations in the Drilling of Deep Boreholes [1 ed.] 1527540200, 9781527540200

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Modelling Emergency Situations in the Drilling of Deep Boreholes

Modelling Emergency Situations in the Drilling of Deep Boreholes By

Valery Gulyayev, Sergii Glazunov, Olga Glushakova, Elena Vashchilina, Lyudmyla Shevchuk, Nataliya Shlyun and Elena Andrusenko

Modelling Emergency Situations in the Drilling of Deep Boreholes By Valery Gulyayev, Sergii Glazunov, Olga Glushakova, Elena Vashchilina, Lyudmyla Shevchuk, Nataliya Shlyun and Elena Andrusenko This book first published 2019 Cambridge Scholars Publishing Lady Stephenson Library, Newcastle upon Tyne, NE6 2PA, UK British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Copyright © 2019 by Valery Gulyayev, Sergii Glazunov, Olga Glushakova, Elena Vashchilina, Lyudmyla Shevchuk, Nataliya Shlyun and Elena Andrusenko All rights for this book reserved. No part of this book may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner. ISBN (10): 1-5275-4020-0 ISBN (13): 978-1-5275-4020-0

CONTENTS PREFACE................................................................................................................ CHAPTER 1. PROBLEMS OF THE THEORETICAL MODELLING OF EMERGENCY SITUATIONS IN DEEP-WELL DRILLING ...................................................................................... 1.1. Technical aspects of deep-drilling problems .................................................. 1.2. Abnormal phenomena accompanying deep-drilling processes....................... 1.3. Mathematical aspects of deep-drilling string mechanic issues ....................... References to Chapter 1 .......................................................................................... CHAPTER 2. STABILITY OF THE COMPRESSED-STRETCHED, TWISTED, ROTATING STRINGS WITH INTERNAL FLUIDS IN VERTICAL WELLS .................................................. 2.1 Basic relations of quasi-static equilibrium and vibrations of rotating, twisted drill strings with internal fluid flows.................................... 2.2 Singularly perturbed equations in boundary value problems of the deep drilling string mechanics ........................................................................ 2.3 Procedure for constructing bifurcation solutions ............................................ 2.3.1 The method of initial parameters for construction of twopoint boundary value problem solutions ............................................. 2.3.2 The splicing technique and orthogonalisation of particular solutions .............................................................................................. 2.4 The bifurcation states of the vertical rotating strings ...................................... 2.4.1 Testing of the developed technique on examples of twopoint boundary value stability problems of long rods ........................ 2.4.2 Spiral wavelets in the modes of bending bifurcation buckling of drill strings....................................................................... 2.5 Drill string buckling according to the forms of spiral wavelets...................... 2.6 Two-point boundary value problems of drill strings buckling ....................... 2.7 Multipoint boundary value problems of bifurcation buckling of drill strings with centralisers ................................................................................... 2.7.1. The structure of multipoint boundary value problem solutions .............................................................................................. 2.7.2. Method for constructing particular solutions ...................................... 2.8. Analysis of quasi-static equilibrium stability of drill strings using integrated calculation schemes ...................................................................... References to Chapter 2 .......................................................................................... CHAPTER 3. BENDING VIBRATIONS OF THE DRILL STRING IN A VERTICAL WELL ..................................................................... 3.1 Technical and applied aspects of the problem of the prestressed rotating tubular rods with internal fluid flows ................................................ 3.2 Spiral progressive waves in infinite twisted tubular rotating rods with internal fluid flows .................................................................................. 3.2.1. Construction and analysis of the characteristic equation of

xi 1 1 3 6 10 12 12 17 21 21 27 33 33 36 39 41 51 51 53 57 63 66 66 68

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drill string vibrations ........................................................................... 3.2.2. Analysis of the properties of bending progressive waves in prestressed rotating rods (drill strings) ............................................... 3.2.3 Analysis of the effect of longitudinal force on the shape of free running waves .............................................................................. 3.2.4. Analysis of the influence of torque on the shapes of free running waves ..................................................................................... 3.2.5. Analysis of the influence of rotational motion inertia forces on the free vibration modes ................................................................. 3.2.6. Analysis of the effect of internal fluid flow on the free vibration shape of a tubular rod .......................................................... 3.2.7. Running bending waves in prestressed rotating tubular rods with fluid flow (general case) ............................................................. 3.3 Small free bending vibrations of rotating drill strings .................................... 3.3.1. Formulation of the problem of free bending vibrations of a drill string in a vertical well ................................................................ 3.3.2. Testing the developed method ............................................................ 3.3.3. Analysis of the influence of the angular velocity of drill string rotation on the frequencies and modes of its free vibrations ............................................................................................. References to Chapter 3 .......................................................................................... CHAPTER 4. EXCITATION OF TORSIONAL SELFOSCILLATIONS OF STRINGS IN DEEP WELLS ..................... 4.1 Self-oscillating systems and methods of their analysis................................... 4.2 Mathematical aspects of the problem of self-oscillation of deepdrilling strings.................................................................................................. 4.2.1 Hopf bifurcations in oscillatory and wave dynamic systems ............ 4.2.2 Typical non-linear models of resistance, friction and cutting forces in tasks for analysing self-oscillating processes ...................... 4.2.3 Regular and singularly perturbed equations with delay argument. Thomson and relaxation self- oscillations ......................... 4.3. Hopf bifurcations in problems of torsional self- oscillations of deep drilling string ................................................................................................... 4.3.1. Structural diagram of drill string torsion vibrations ........................... 4.3.2. Modelling drill string torsional self- oscillations on the basis of wave theory ............................................................................ 4.3.3. Modelling the drill bit cutting torque by nonlinear viscous forces and Coulomb friction moments................................................ 4.3.4. Wave model of drill string torsion self- oscillations........................... 4.3.5. Investigation of general regularities of the phenomena of limit cycle generation and loss in rotating strings .............................. 4.3.6. Dependence of the torsion self- oscillation nature of a drill string on its length ............................................................................... 4.3.7. Dependence of the drill string torsional self- oscillations on

68 75 78 81 83 86 88 91 91 94 95 104 106 107 111 111 115 117 121 121 126 129 132 134 135

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the nature of nonlinear frictional interaction of a bit with a rock ...................................................................................................... 4.3.8. Dependence of string torsion self-oscillations on the magnitude of the moment of the bit inertia......................................... 4.4. Torsional oscillations of composite drill strings ............................................. 4.4.1. Formulation of the composite drill string torsional selfoscillation problem .............................................................................. 4.4.2. Testing of the constitutive relations .................................................... 4.4.3. Analysis of the self- oscillations of composite drill strings using the viscous friction force moment model .................................. 4.4.4. Analysis of self-oscillations of a composite drill string using the model of a moment of the Coulomb friction force ............. 4.5. Self- oscillation of torsional vibrations of deep drill strings in a viscous liquid medium .................................................................................... 4.5.1. Self-oscillation models of a torsion pendulum with distributed parameters ......................................................................... 4.5.2. Model with one degree of freedom of elastic torsion pendulum in a dissipative medium ..................................................... References to Chapter 4 .......................................................................................... CHAPTER 5 SELF-EXCITATION OF DRILL STRING BIT WHIRLING .................................................................................... 5.1. Existing models of drill string whirling vibrations ......................................... 5.2. The main aspects of the whirling model based on the effects of friction and nonholonomic rolling of the bit ................................................... 5.3. The problem of Celtic stone mechanics and its analogy with bit rolling dynamics .............................................................................................. 5.4. Friction model of elastic oscillations of the system during rolling of a spherical bit on the spherical surface of the well bottom ............................. 5.4.1. Equations of elastic transverse oscillations of the drill string ............ 5.4.2. Boundary conditions at the ends of the drill string (friction model) ................................................................................................. 5.5. Analysis of frictional whirling vibrations of the system ................................. 5.6. Kinematic (nonholonomic) model of the spherical bit whirling on the spherical surface of the well bottom ......................................................... 5.7. Analysis of elastic vibrations of the system using the nonholonomic model of the bit rolling .................................................................................... 5.8. Friction and nonholonomic rolling of the spherical bit on the ellipsoidal surface of the well bottom ............................................................. 5.9. The frictional rolling of ellipsoid bit on the bottom surface of a well ............ 5.10 The analogy between the rolling dynamics of the ellipsoid bit and rattle back rotation ........................................................................................... 5.11 Dynamics of elastic bending of a drill string with the bit in the shape of an elongated ellipsoid ...................................................................... 5.12 Dynamics of elastic bending of a drill string with the bit in the

152 167 179 181 190 192 198 200 201 207 212 216 216 219 224 227 227 228 237 243 244 250 256 263 266

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shape of a flattened ellipsoid ........................................................................... 5.13.Elastic oscillations of the drill string for nonholonomic rolling of an ellipsoid bit on the surface of the well bottom ................................................ 5.13.1. Problem statement ............................................................................... 5.13.2. Dynamics of the drill string in nonholonomic rolling of the bit in the shape of a flattened ellipsoid ............................................... 5.13.3. Dynamics of the drill string at nonholonomic rolling of the bit in the shape of an elongated ellipsoid ............................................ References to Chapter 5 .......................................................................................... CHAPTER 6. MODELLING RESISTANCE FORCES AND DRILL STRING STICKING EFFECTS IN CURVILINEAR BOREHOLE CHANNELS ............................................................. 6.1. Features of modern technologies of oil and gas fuel extraction and problems of mechanics of elastic bending of drill strings in curvilinear boreholes ....................................................................................... 6.2. Mathematical aspects of the problem of forced elastic bending of the drill string in the channel of a curvilinear borehole .................................. 6.3. Stiff string drag and torque model of DS bending in curvilinear boreholes ......................................................................................................... 6.3.1. The features of the problem of elastic bending of DS in the channel of a curvilinear borehole during tripping in/out operations and drilling ........................................................................ 6.3.2. Theoretical modelling of the directional borehole trajectory with localised geometric imperfections .............................................. 6.3.3. Constitutive equations of the elastic bending of a drill string in the channel of a curvilinear borehole during the drilling process and tripping in/out operations ................................................ 6.3.4. Technique of computer determination of internal and external forces acting on the drill string ............................................. 6.4. Modelling resistance forces in a well with localised and harmonic imperfections ................................................................................................... 6.4.1 Results of calculations for a well with hyperbolic trajectory ............. 6.4.2 Calculations results for a borehole with an elliptical trajectory ............................................................................................. 6.5. Modelling resistance forces in a well with localised spiral imperfections ................................................................................................... 6.5.1. Model of spiral wavelets ..................................................................... 6.5.2. Results of calculations for a well with hyperbolic trajectory ............. 6.5.3. Study of the functions of resistance forces during the motion of the drill string in a well with breaks of the centreline ............................................................................................. 6.6. Simulation of energy-saving regimes of tripping in/out operations in a well with geometric imperfections ........................................................... 6.7. Method for minimising resistance force in the interfacing areas of

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the borehole curvilinear segments ................................................................... References to Chapter 6 .......................................................................................... CHAPTER 7. CRITICAL STATES AND BUCKLING OF DRILL STRINGS IN CHANNELS OF DEEP CURVILINEAR WELLS ........................................................................................... 7.1. State of the problem of drill string stability in a curvilinear borehole ........................................................................................................... 7.1.1. Technical aspects of the phenomenon of drill string buckling in deep curvilinear wells ...................................................... 7.1.2. Mechanical influences leading to drill string buckling in deep curvilinear wells ......................................................................... 7.1.3. Mathematical aspects of stability problems of drill strings in curvilinear wells .................................................................................. 7.1.3.1 Modified theory of flexible curvilinear rods and its features ........................................................................... 7.1.3.2. Singularly perturbed equations in boundary-value problems of strings buckling in channels of curvilinear wells .................................................................. 7.1.3.3. Stuck and invariant (dead) states of drill strings in curvilinear wells .................................................................. 7.1.3.4. The role of frictional effects in the overall balance of force impacts on strings in curvilinear wells .................................................................................... 7.1.3.5. The theory of bifurcations and the problem of string buckling .................................................................... 7.1.4. Review of the results of theoretical modelling of the bifurcational buckling phenomena of drill string in channels of curvilinear wells.............................................................................. 7.2. Buckling of a drill string in a directional channel of a rectilinear borehole ........................................................................................................... 7.2.1. Specifics of the problem of drill string bifurcational buckling in a directional channel of a curvilinear borehole ............... 7.2.2. General equations of elastic bending of a free curvilinear rod ....................................................................................................... 7.2.3. Geometrical prerequisites for analysis of drill string bending in a cylindrical cavity ............................................................ 7.2.4. Equations of non-linear bending and buckling of a straight drill string (frictionless model) ........................................................... 7.2.5. Equations of non-linear bending and buckling of a straight drill string (friction model) ................................................................. 7.2.6. Analysis of buckling of strings in directional rectilinear wells .................................................................................................... 7.3. Buckling of a drill string in a borehole channel with a circular outline ..............................................................................................................

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7.3.1. The main conceptual features of the problem of modelling bifurcations of a drill string in a curvilinear well ............................... 7.3.2. Geometry of circular centreline channel surface ................................ 7.3.3. Non-linear equations of bending of a drill string in the borehole with a circular centreline (frictionless model) ..................... 7.3.4. Results of modelling the buckling of a drill string in a circular well (a frictionless model) ..................................................... 7.3.5. Friction model of drill string bifurcational buckling in a circular borehole ................................................................................. 7.3.6. Results of buckling analysis of a drill string moving in a circular borehole channel 7.4. Global analysis of drill string buckling in a curvilinear well .......................... 7.4.1. Theoretical analysis assumptions ........................................................ 7.4.2. Analysis of the computer modelling results ....................................... References to Chapter 7 .......................................................................................... INDEX

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Preface

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Everyone can do whatever he would desire in his life. But he does desire only what he can do. PREFACE At the present time, we are all witnessing the sharp-plotted scenarios played in politics, business, and industry in connection with the problems of extraction and redistribution of the oil and gas resources. Additional freshness is contributed into this atmosphere by “shale revolution” broken out owing to development of new technologies of super-deep and long distance curvilinear drilling. Now, instances of vertical well boring to depths exceeding 10 km are not rare, and the record distance from the drilling rig of the horizontal well has reached more than 13.5 km. Such wells are drilled at the limit of current industrial capabilities at maximum velocities, hydrostatic pressures, and temperatures values as well as strength and wear parameters of drill string materials under the significant impact of violent vibration effects and entire system instability. These processes are often accompanied by emergency and failure situations, including: - Bifurcation buckling of vertical strings in the type of compressed-stretched, twisted, rotating tubular rods with internal fluid flows. - String resonance (bending) vibrations. - Self-excitation of torsional auto vibrations with slip-stick motions caused by nonlinear frictional forces between the bit and the rock being destroyed. - Whirling vibrations of the bit self-excited as a result of frictional and kinematic (nonholonomic) rolling of the bit over the surface of the hole bottom. - Deadlock states in wells with geometric imperfections caused by increasing contact friction forces between the string and the borehole wall. - Bifurcational buckling of the strings in curvilinear boreholes accompanied by a deterioration in the conductivity of the actuating cutting torque and axial force to the bit, increased wear of the string pipe, an increase in system power consumption, an increase in the intensities of stress fields and strains in the strings, and an increase in the probability of their destruction. These emergency situations are mainly caused by three factors. First, it is the long string length. In terms of geometric similarity, it is similar to human hair. Therefore, a phenomenon occurring at one end of the string can influence, have little or no effect on the phenomena occurring at the other end. In mathematics, the equations describing such phenomena are called singularly perturbed. They are characterised by poorly converging solutions, the forms of which have singularities in the shape of edge effects, internal harmonic wavelets or may contain irregularities.

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The second factor is related to the special character of frictional effects that appear in extended curvilinear boreholes. The fact is that the frictional force in them depends on the pressing force of contacting bodies. However, in curved sections, the force of pressing the string against the borehole wall is determined by the axial tension of the string. In this case, it is said in mechanics that frictional forces have a multiplicative (i.e., are multiplied) and not additive (i.e., not added) nature, as happens when the body slides over a rough flat surface. The third factor is that the issues of mathematical modelling of mechanical phenomena accompanying the drilling processes are multi-parametric since they depend on a large number of geometric and mechanical quantities and can hardly be solved in a general formulation. Therefore, it is very important to consider the implementation of these mechanical phenomena for specific values of the determining values and establish the general regularities of their behaviour. It should be noted that due to the great complexity of the tasks and the uncertainty of the initial data, the proposed drill string dynamic models reflect only the qualitative aspects of the phenomena considered, which are also of no small importance. Here, it is appropriate to recall Hemming who noted that in mechanics, when modelling, it is not the number but the understanding that is important. At the same time, it can be asserted that mathematical models of frictional phenomena and the processes of drill string buckling in curvilinear boreholes largely reflect also their quantitative aspects. In this regard, they can be directly used both in the design of wells and when drilling them. Ultimately, the practical application of methods for the computer simulation of abnormal situations arising in deep-well drilling can contribute to their prevention. The research given herein was discussed with specialists of the National Transport University and Smart Energy LLC (Kiev). The authors express their gratitude to all of them. Selected publications of the authors 1. Gulyayev V., Glazunov S. and Vashchilina O. Frequency analysis of drill bit whirlings on uneven bottoms of deep bore-holes. 2017. Journal of Mathematics and System Science, 7 , 14 – 24 (USA) 2. Gulyayev V.I., Gaydaychuk V.V., Andrusenko E.N., Shlyun N.V. Modelling the energy-saving regimes of curvilinear bore-hole drivage. 2015. Journal of Offshore Mechanics and Arctic Engineering. V.137, №1, 011402−1−011402−8. (USA) 3. Gulyayev V.I., Gaydaychuk V.V., Andrusenko E.N., Shlyun N.V. Critical buckling of drill strings in curvilinear channels of directed bore-holes. 2015. Journal of Petroleum Science and Engineering, 129, 168−177. (USA)

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4. Gulyayev V.I., Andrusenko E.N. Theoretical simulation of geometrical imperfections influence on drilling operations at drivage of curvilinear bore-holes. 2013. Journal of Petroleum Science and Engineering.V.112. 170 – 177. (USA) 5. Gulyayev V.I., Gaydaychuk V.V., Solovjov I.L., Gorbunovich I.V. The buckling of elongated rotating drill strings. 2009. Journal of Petroleum Science and Engineering.V.67, P.140 – 148. (USA) 6. Gulyayev V.I., Borshch O.I. Free vibrations of drill strings in hyper deep vertical bore-wells. 2011. Journal of Petroleum Science and Engineering. V. 78, P. 759 – 764. (USA). 7. Gulyayev V.I., Hudoly S.N., Glovach L.V. The computer simulation of drill string dragging in inclined bore-holes with geometrical imperfections. 2011. International Journal of Solids and Structures. V.48, P.110–118. (USA) 8. Gulyayev V.I., Glushakova O.V. and Glazunov S.N. Stationary and nonstationary self-induced vibrations in wave guiding systems. 2014. Journal of Mechanics Engineering and Automation. V. 4(3).P. 213-224. (USA) 9. Musa N., Gulyayev V., Shlun N., Aldabas H. Critical buckling of drill strings in cylindrical cavities of inclined bore-holes. 2016. Journal of Mechanics Engineering and Automation. V. 6. P. 25 – 38. (USA) 10. Gulyayev V.I., Shlun N.V. Influence of friction on buckling of a drill string in the circular channel of a bore hole. 2016. Petroleum Science. V. 13. P. 698 – 711.(China) 11. Gulyayev, V.I., Tolbatov E.Yu. Forced and self-excited vibrations of pipes containing mobile boiling fluid clots. Journal of Sound and Vibration 2002. – 257(3), 425–437. (Great Britain) 12. Gulyayev, V.I., Tolbatov E.Yu. Dynamics of spiral tubes containing internal moving masses of boiling liquid. Journal of Sound and Vibration 2004. – 274(2), P. 233–248. (Great Britain) 13. Gulyayev V.I., Hudoliy S.N., Glushakova O.V. Simulation of torsion relaxation auto-oscillations of drill string bit with viscous and Coulombic friction moment models. 2010. Journal of Multi-body Dynamics. V. 225. P. 139 – 152. (Great Britain) 14. Gulyayev V.I., Shevchuk L.V. Nonholonomic dynamics of drill string bit whirling in a deep bore-hole. 2013. Journal of Multi-body Dynamics. V. 227(3). P. 234 – 244. (Great Britain) 15. Gulyayev V.I., Shlyun N.V. Global analysis of drill string buckling in the channel of a curvilinear bore-hole. 2017. Journal of Natural Gas Science and Engineering. V.40. P.168-178. (Netherlands)

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16. Gulyayev V., Andrusenko E., Glazunov S. Computer simulation of dragging with rotation of a drill string in 3D inclined tortuous bore-hole. 2019. SN Applied Sciences. V. 126(1). (Switzerland) 17. V.I. Gulyayev, O.V. Glushakova, S.N. Khudoliy. Innovational attractors in wave models of torsional vibrations of deep drilling strings. 2010. Proceedings of the Russian Academy of Sciences. – Mechanics of Rigid Body No. 2. 134-147. (RF) 18. Gulyayev V.I., Andrusenko E.N., Shlyun N.V. Theoretical modelling of post buckling contact interaction of a drill string with inclined bore-hole surface. 2014. Structural Engineering and Mechanics, V. 49(4), 427-448. (South Korea) 19. Gulyayev V.I., Khudoliy S.N., Andrusenko E.N. Sensitivity of resistance forces to localised geometrical imperfections in movement of drill strings in inclined bore-holes. 2011. Interaction and Multiscale Mechanics. V.4(1), P.1–16. (Taiwan) 20. Gulyayev V.I., Glushakova O.V. Large-scale and small-scale self-excited torsional vibrations of homogeneous and sectional drill strings. 2011. Interaction and Multiscale Mechanics. V. 4(4), P. 291 – 311. (Taiwan) 21. Musa N.W., Gulyayev V.I., Shevchuk L.V., Hasan A. Whirl interaction of a drill bit with the bore-hole bottom. Modern Mechanical Engineering.-2015.-V.5.P.41-61. (Hong Kong)

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CHAPTER 1. PROBLEMS OF THE THEORETICAL MODELLING OF EMERGENCY SITUATIONS IN DEEP-WELL DRILLING 1.1. Technical aspects of deep-drilling problems Energy problems, which are becoming ever more acute in the 21st century, are caused by the approaching exhaustion of oil and gas resources and the fact that their production has become more complex. As a result of prolonged and inefficient extraction and consumption using low-cost technologies, the time of light oil and gas ended in the 20th century [5]. Therefore, deposits found in shale rocks and at depths of up to 10,000 m are now very promising. For example, in the United States, the possibility of extracting fuel from a depth of 30,000 feet (9,150 m) is being studied, and goals are being set for developing inclined and horizontal offshore wells with a distance of up to 15 km from the drilling platform [4]. Taking into account the increase in the depth and range of drilling, the cost of these wells already exceeds $50 million [3], and every third well is an emergency one; but reliable methods for the theoretical modelling of their functioning have not yet been developed, the conclusion can be drawn on the importance of theoretical forecasting of critical states of drill strings (DS) and the price of the forecast error. The main global energy consumers are [2, 6] industry, transport, agriculture, the residential sector, and commercial and public services. The most energy-intensive is transport, followed by industry and the residential sector. The energy type to be used by these consumers is selected based on six main factors: 1) fuel and energy feedstock calorific value; 2) ease of production; 3) ease of transportation; 4) ease of energy production and use; 5) associated hazard to health and life of employees; 6) presence of waste and potential environmental contamination. Given these factors, we can conclude that oil and gas will remain the most attractive energy sources in the coming decades. An important circumstance contributing to the complication of the situation in the oil and gas industry is that under normal conditions usually only 40% of hydrocarbon fuels filling the cracks and pores of underground reservoirs can be extracted using conventional production techniques. One of the ways to increase the volume of fuels extracted from underground reservoirs is associated with the drilling of curvilinear boreholes penetrating the oil-bearing and gas-bearing beds along their laminated structure and, therefore, covering large areas of fuel intake [7]. Since, when using this technique, the total number of wells drilled is reduced, and the flow rate of curvilinear boreholes turns out to be significantly higher than that of vertical wells, in the near future drilling directed holes will become the main technique in much of the world. The development of curvilinear drilling techniques is also facilitated by the need to extract hydrocarbons from shale formations.

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G i G j

G k

X

y

Z

Drilling fluid flow

L

Centring devices (additional intermediate supports)

ln 2

ln 1 ln

Z,z Fig. 1-1 Geometrical layout of a drill string in a deep well

2

In accordance with economic requirements, the geological conditions of the deposit and the process capabilities of oil and gas companies, vertical, directional, horizontal, and multilateral oil and gas wells of different depths are now being drilled. However, the practical introduction of drilling techniques for deep wells of different spatial orientation is associated with the need for the theoretical modelling of mechanical phenomena occurring in the drilling equipment structures to prevent critical modes of operation. In addition, one of the most important aspects of this area is the theoretical modelling of quasi-static and dynamic behaviour of deep-drilling strings. However, this task is significantly complicated by the fact that under conditions of geometric similarity the drill string is similar to human hair, and at the same time it is subject to intensive loads and a large number of different factors complicating the methods of theoretical modelling. The most popular oil and gas well-drilling technique is the rotary method when the rock is cut using a bit attached to the bottom end of the drill string suspended in the borehole at the top end. In this case, the bit rotates because of the entire drill string rotation caused by the effect of actuating torque on its top end (Fig. 1-1). A drill string is assembled from pipes 12–15 m long using threaded connections. Drilling efficiency and quality are determined mainly by its regime and the bottom hole assembly

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(BHA) structure with reamers, centralisers, and stabilisers. The generation of high axial loads ensuring stabilisation and controllability of the wellbore trajectory is associated with the use of multi-bearing BHA, where the number of centralising components does not usually exceed five. The drill string bottom structure also includes weighting agents, calibrators, and other elements. To remove rock particles chopped as a result of being cut with a bit from the borehole, drilling fluid is supplied inside the string by a special pumping system that—rising in the outer space between the string and the borehole wall—entrains and carries away these particles. The drilling fluid also plays other important functional roles. It is known that rocks in the interior of the earth experience significant threedimensional stresses caused by a comprehensive compression of upstream solids by gravity forces. Due to the continuity of the rock, these stresses counterbalance each other, much as hydrostatic forces in the liquid. However, rock discontinuity during well drilling leads to a redistribution of these stresses in the vicinity of the well, rock imbalance, and destruction. Should the well be filled with liquid with a specific gravity equal to that of the rock, its hydrostatic pressure on the walls will balance the imbalance of forces in the rock, and it will remain stable (using the language of drillers). It is also important to maintain a balance between the average density of the liquid and rock (especially for deep wells). In addition, the drilling liquid plays an important role in the formation of frictional interaction between the string and the borehole wall. Finally, the drilling liquid serves to cool the drill tools. 1.2. Abnormal phenomena accompanying deep-drilling processes When suspended, the drill string is exposed to distributed gravity forces. They generate a tensile axial force therein, which reaches a maximum at the suspension point and decreases to zero at its bottom end. When operating, the string is resting against the bottom of the well, it is exposed to the vertical reaction compression force; therefore, the entire string is in a stretched-compressed state of stress. To impart a rotary motion to the string, torque is applied to its top end. For shallow wells, it can be assumed that in the stationary state this moment is equal to the cutting moment applied to the bit, and then the torque in the string itself remains constant along its length. However, if the string is deep, then to calculate the actuating torque at the suspension point, the distributed moments of frictional forces between the walls of the borehole well and the string shall be added to the cutting torque. Then, the internal torque in the string becomes variable. And it turns out to be significantly variable at longitudinal and twisting vibrations of the string, when the bit approaches and terminates to contact with the rock at the borehole bottom, and the cutting process becomes intermittent.

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The angular velocity value has a significant influence on the cutting moment (rotation moment) value. As the speed change causes non-linear alteration of the torque, self-excitation of torsional vibrations leading to emergency situations is possible in the drill string. An important factor affecting the quasi-static and dynamic behaviour of the drill string is its rotation. For strings with geometric imperfections and with mass imbalances, this leads to the appearance of centrifugal forces of inertia that significantly affect the stability of the drill string's straight shape. For bending vibration excitation, the rotation is the generating source of the gyroscopic (Coriolis) forces of inertia. These forces connect different types of movements (rotational and linear) and lead to the disturbance of the equiphase condition of vibrations. Very complex effects in drill strings are generated by external and internal drilling fluid flows. Their motion is first associated with additional friction forces that affect the drill string dynamics. Second, for DS bending vibrations, internal flows (similar to rotational motion) also generate centrifugal and gyroscopic forces of inertia destabilising the straight shape of the string and changing its own bending vibration spectrum. The features of these forces have been studied in detail in the pipelines theory. Their manifestation in the DS has not been sufficiently investigated. It should be noted that, in essence, all the above forces and effects can occur simultaneously with different combinations of their intensities and lead—depending on the DS length—to various unacceptable modes.

vert horiz

(a)

(b) (c) (d) Fig. 1-2 Vibration modes of the drill string structure bottom part: a = bit axial beating; b = transverse string beating; c = torsional self-excited vibrations of the bit and the string; d = string bottom whirling motion

Chapter 1. Problems of the theoretical modelling of emergency situations…

5

Therefore, when extracting fuels from great depths, the increased drilling efficiency of vertical wells using a rotary system is associated with the issue of identifying critical drill string operation modes and developing measures aimed at reducing their negative impact on the drilling process. Such phenomena negatively affecting the drilling process include: x helical buckling failure of the straight shape of the DS in its bottom part in the form of an extra-long compressed-stretched, twisted rotating rod, x excitation of DS longitudinal vibrations when exposed to various process disturbances (Fig. 1-2, a), x excitation of DS resonant bending vibrations due to geometric imperfections and imbalance of the whole system and its individual parts (Fig. 1-2, b), x parametric build-up of DS slip-stick vibrations caused by non-linear frictional forces between the cutting tool (bit) and the rock being processed (Fig. 1-2, c), x self-excitation of the bit whirling motion associated with its rolling around the system axis under the conditions of frictional or nonholonomic interaction of the bit with the well bottom surface (Fig. 1-2, d), x sidewall sticking (loss of mobility) in long curvilinear boreholes with geometrical imperfections (dead lock states) as a result of a sharp increase in the forces of contact and frictional interaction, x bifurcation buckling of the drill string in the curvilinear borehole channel with non-predictable zones of buckling localisation. These phenomena can lead to emergencies accompanied by breakage of the DS pipe, sticking of the cutter tool in the rock cutting zone and mashing of the DS sections into the rock, unscrewing of the DS pipes, vertical deviation of the borehole axis and its unplanned distortion, as well as the loss of stability of the borehole walls and their destruction. The drilling process parameters when critical states occur can be determined using mathematical model methods. However, attempts to conduct practical mathematical experiments on the prediction of the DS critical states are associated with significant computational challenges. Above all, this is because of the features of the relations between the DS geometric parameters. So, for example, as the diameter of a long string is equal to a 10 -5 part of its length, theoretically it appears to have negligible bending and torsional stiffness. Therefore, mechanical models of strings or absolutely flexible cables are often used for their theoretical investigation. At the same time, to correctly describe the edge and local effects of the DS bending deformation, they shall be calculated based on the beam theory; therefore, using this theory for lengths of several kilometres leads to the appearance of a so-called ‘computational rigidity accompanied by a significant

Modelling emergency situations in drilling deep boreholes

6

deterioration in the convergence of computational algorithms. In mathematics, the equations modelling these effects are called singularly perturbed [1]. The second complication of the task of the DS quasi-static and dynamic behaviour modelling is associated with a complex combination of forces and influences that affect their quasi-statics and dynamics. Therefore, the problems under consideration are significantly multiparametric. In a general formulation, such problems are unsolvable. So, one of the most rational approaches to their solution is to separate the bending, longitudinal, and torsional movements of the string, consider them separately, and establish the most common patterns of the processes with the determination of their critical states. 1.3. Mathematical aspects of deep-drilling string mechanic issues The issues of mathematical modelling of static and dynamic mechanical phenomena and critical states arising in deep-drilling strings are associated with considerable theoretical difficulties. Above all, these difficulties are due to the complex nature of the static and dynamic impacts on the drill string and the complexity of the mechanical processes generated by them. Second, the factor of the large drill string length, which leads to a virtual loss of its bending stiffness, has a significant (and determining) effect on the specifics of the processes, the formulation of tasks, and the methods for their solution. Therefore, resolving DS bending equation by integration methods on large integration segments turns out to be difficult to implement. Most noticeably, these difficulties occur while attempting to solve tasks concerning DS bifurcational buckling and free vibrations. So, the problems regarding DS bending stability are given in two formulations. The first statement is based on the Sturm-Liouville problem formulation on large length L of the string, where the so-called ‘computational rigidity’ phenomenon is very noticeable. This is caused by the fact that the resolving functions of transverse displacements u  x , v  x of non-trivial solutions vary rapidly with large derivatives on a small segment adjacent to the DS bottom end and have small values with small derivatives on the remaining part. The extraordinary complexity in the study of the combined equations, determining functions u  x , v  x , is due to the fact that the points of the onset of rapid changes in the resolving functions are not known beforehand. In this case, the effects of ‘computational rigidity’ are caused by the high degree of the differential equations and small factors implicitly presenting before the high order derivatives. The smallness of these coefficients appears when scaling integration length L of the equilibrium and vibration equations to a unit segment. Then, the terms with the fourth derivatives shall be divided by L4 and the role thereof in the overall balance of internal forces and moments is significantly reduced. As a result, the solution acquires areas of fast (such as boundary layer) and slow (regular)

Chapter 1. Problems of the theoretical modelling of emergency situations…

7

changes. Should this solution be combined in the form of a superposition of particular solutions that increase and decrease exponentially, then on large integration intervals the first group solution values tend to infinity, the second group solutions go to zero, and the problem of constructing the required solutions of the initial equations becomes impractical even for the two-point boundary value problems. In mathematics, such systems are called singularly perturbed [1]. Due to the above difficulties, the issues of long DS bending stability and natural vibration study have been virtually unexplored. This book offers a technique for their solution based on the application of the initial parameters method in conjunction with the Godunov orthogonalisation procedure (for vertical wells) and the finite-difference procedure with very small step (for curvilinear boreholes). The formulation of the Sturm-Liouville problem used in the investigation of DS stability makes it possible to determine the beginning of the bifurcation bending process initiation. When it is implemented, the string protrudes and comes into contact with the borehole wall. At this stage, the second DS stability loss step is implemented, at which its supercritical state is examined, and the DS element equilibrium beyond the range of stability is studied for a given (usually, a regular spiral) deformation geometry. Problem formulations based on this approach are widely used in the world literature. They are based on the application of the flexible curved rods theory and are usually related to a number of simplifying assumptions on the nature of the supercritical DS behaviour, which substantially reduce the value of the results obtained on its basis. This paper considers the initial stage of DS bifurcation buckling (first formulation), for the analysis of which the corresponding Sturm-Liouville problems are formulated along the entire length of the DS in a vertical well. Particular attention is paid to the DS stability calculation, based on singularly perturbed equations using the so-called integrated design schemes, which leads to multipoint boundary-value problems for continuous strings (Chapter 2). It is shown that the loss of stability of strings usually occurs in the shape of spiral wavelets. Even greater computational difficulties are associated with the problems of DS free vibrations in vertical wells. In Chapter 3, dispersion analysis methods are used to establish that - in contrast to the vibrations of ordinary rods of infinite length that allow for solutions in the mode of standing plane waves - the vibrations of infinite rotating twisted tubes with fluid flows can only occur in the modes of progressive spiral waves (waves with circular or cylindrical polarisation). In addition, each progressive wave length corresponds to four phase-velocity values, different for the left and right spirals and depending on the wave direction. This circumstance indicates that free bending vibrations of a long (albeit finite) DS can only be implemented in spatial modes.

Modelling emergency situations in drilling deep boreholes

8

Due to the large length of the DSs, their torsion self-excited vibrations can also assume a special shape (Fig. 1-2, c). They are caused by a significantly non-linear dependence (with extremum points) of the bit rock cutting moment on its angular velocity and are generated through the cycle birth bifurcation. In mathematics, Henri Poincare was among the first to pay attention to the bifurcation nature of self-excited vibrations. A. Andronov considered the likelihood of these effects appearance in mechanical oscillatory systems with nonlinear friction. Later, Hopf provided a mathematical justification for this theory. Therefore, the effects of the build-up of vibrations in non-linear frictional systems were named the Poincare-Andronov-Hopf bifurcations. Applied mathematics and physics distinguish two types of self-excited vibrations - that is, Thomson and relaxation. Thomson self-excited vibrations proceed in modes close to harmonic ones; relaxation self-excited vibrations are described by periodic or quasi-periodic broken-line functions with almost discontinuous derivatives (velocities). As our analysis shows, torsional self-excited vibrations of drill strings can be classified as relaxational ones. In this regard, their modelling involves much greater computational difficulties. Chapter 4 offers three models of torsional self-excited vibrations in drill strings: wave, vibration model with distributed parameters, and vibration model with a single degree of freedom. The calculations showed that, although the wave model involves quantum nature solutions, in the integral sense the solutions virtually coincided in all cases. However, in the case of self-excited vibrations in extended inclined wells with significant friction forces, the model with distributed parameters turns out to be more accurate. Apparently, a special type of mechanical vibration called ‘whirling’ is only observed in drilling practice. When they build-up, the bit deviates from its balanced state and starts rolling—generally with slip and torsion—around the system axial line (Fig. 1-2, d). In these cases, the constraints imposed on the bit movement may be nonholonomic. As a result, the bit torsion can be stable and unstable, occurring in forward and backward rotation directions, or the bit centre can move along the most intricate trajectories. Nonholonomic mechanics and geometry methods should be applied to describe these types of vibrations (Chapter 5). Drilling curvilinear super long boreholes is associated with heavy technical and theoretical difficulties. When drilling, the string can be in rather severe conditions caused by contact and frictional forces. During drilling or tripping operations (for example, to change the bit), these forces reach very high values, especially in places of geometric irregularities of the well centreline in the shapes of dog legs and harmonic or spiral wavelets (Fig. 1-3). They are often the main cause of drilling technique disturbance and lead to the DS sticking. To model these effects, the theory of curvilinear flexible rods, differential geometry, and computational mathematic

Chapter 1. Problems of the theoretical modelling of emergency situations…

9

methods (Chapter 6) shall be applied. Of particular interest is the matter of interfacing two well sections with different curvatures. This book shows that the use of the minimal curvature method, based on the soft string drag torque model, for this operation is irrational, but it is necessary to match the well sections by introducing small sections in the shape of the Cornu spiral (clothoid) or a cubic parabola. To model this effect, it is advisable to use the stiff string drag torque model developed by the authors.

Fig. 1-3 Axial dog leg path (a), dog leg shape (b)

Finally, in terms of mechanics and mathematics, the matter of the DS Eulerian (bifurcational) buckling in a curvilinear borehole channel is of great interest. Above all, it must be considered on the basis of the Sturm-Liouville problem formulation for a curved elastic rod. In this case, it is broken down into two separate problems. First, using the stiff string drag torque model, axial force and torque functions shall be built, and then, using them as coefficients, eigen value equations for the string along the entire length shall be formulated. As the problem formulated is also singularly perturbed, the buckling modes implemented based on the solution have the shape of localised harmonic wavelets, whose locations are not known in advance. In this regard, to solve the problem, an integrated (global) approach should be applied, and

Modelling emergency situations in drilling deep boreholes

10

localised bulges should be searched for along the entire DS length. In addition, as the bulged string movements are limited by the borehole channel walls, the formulation of the problem requires consideration of non-linear constraints. We eliminate the constraint equations using differential geometry methods, the channel surface theory and using a special mobile trihedron. The solution of this problem makes it possible to find the critical loads for the string in a curvilinear borehole channel and indicate the location of its buckling zone (Fig. 1-4).

bоrehole wall

DS buckled

Fig. 1-4 Diagram of drill string local buckling a curvilinear borehole channel

Mathematical features of the formulated problems set forth in this subsection make them both rather time-consuming and appealing for a scientist. Some of these problems are solved in the papers of authors (see References [1–20] of Preface). References to Chapter 1 1. Chang K., Howes F. Nonlinear Singular Perturbation Phenomena: Theory and Applications М., Mir. – 1988. – 247 p. 2. Editorial. Avoiding an oil crunch // Science. – 1999. – V.286, № 5437. – P. 47. 3. Iyoho A.W., Meize R.A., Millheim K.K., Crumrine M.J. Lessons from integrated analysis of GOM drilling performance // SPE Drilling & Completion. – March 2005. – P. 6–16. 4. Jonggeun Choe, Jerome J.Schubert, Hans C. Juvkam-Wold. Well-control analyses on extended-reach and multilateral trajectories // SPE Drilling & Completion. – June 2005. – P. 101–108. 5. Kerr R.A. Bumpy road ahead for world’s oil. // Science, 18 Nov.2005, Vol. 310.– P. 1106–1108.

Chapter 1. Problems of the theoretical modelling of emergency situations…

11

6. Leonardo Maugeri. Oil: never cry wolf- why the petroleum age is far from over// Science. – May 2004. – V.304.– P. 1114–1115. 7. Myslyuk M.A., Rybchich I.I., Yaremiychuk R.S. Well Drilling. Vol. 3, Vertical and directed drilling – Kyiv: “Interpress LTD”. – 2004. – 294 p. (in Ukrainian).

Modelling emergency situations in drilling deep boreholes

CHAPTER 2.

12

STABILITY OF THE COMPRESSED-STRETCHED, TWISTED, ROTATING STRINGS WITH INTERNAL FLUIDS IN VERTICAL WELLS

2.1 Basic relations of quasi-static equilibrium and vibrations of rotating, twisted drill strings with internal fluid flows The bifurcation buckling of a rotating DS is described by the equations of its neutral equilibrium in a perturbed state, which are composed accounting for the presence of internal longitudinal tensile and compression forces, torques, and forces of inertia due to rotation and motion of the internal fluid flow. Let us formulate a problem of the dynamic equilibrium of a DS in operating conditions, considering the effect of its quasi-static buckling as a special case of its motion. Let the drill string rotates at angular velocity Z . To formulate the equations of its motion, let us introduce inertial coordinate system OXYZ with the origin at the point of suspension and coordinate system Oxyz with unit vectors i , j , k associated with the string and rotating together with it (Fig. 1-1). In the initial undeformed state, axes OZ and Oz coincide with the longitudinal axis of the string. We will investigate the stability of the straight shape of the string in rotating coordinate system Oxyz . Assume that the elastic displacements of its elements along axes Ox and Oy are equal to u and v , respectively, displacements along Oz axis will be ignored. Consider that the drill string (Fig. 1-1) is an elastic tubular rod loaded by longitudinal force T and torque M z , which rotates at constant angular velocity Z about its longitudinal axis. In the pipe channel, fluid with density U f flows at velocity V . We will investigate the rod vibration in rotating coordinate system Oxyz with axis Oz directed along the longitudinal axis of the undeformed rod. To derive the dynamics equations, separate a pipe element dz long and consider the equilibrium of internal moments relative Oy , Ox axes system. For the considered combination of forces, these moments include [10, 11, 14, 17] the increments of elastic moments dM y , dM x ; moments  Qx dz ,  Q y dz of shear elastic forces Qx , Q y with arms dz ; moments  Tdu ,  Tdv of internal axial force of prestress T generated by increments du , dv in interval dz of transverse displacements u , v along axes Ox , Oy ; bending moments  M z d dv dz , M z d du dz caused by the alteration of torque M z orientation due to increments d dv dz , d du dz of angles of rotation dv dz , du dz in interval dz . Let us sum up these moments

Chapter 2. Stability of the compressed-stretched, twisted, rotating strings…

§ dv · dM y  Qx dz  Tdu  M z d ¨ ¸ 0, © dz ¹ § du · dM x  Q y dz  Tdv  M z d ¨ ¸ 0. © dz ¹

13

(2.1)

The bending moments of elastic forces, being a part of this system, are calculated based on beam theory formulas My

 EI

d 2u , dz 2

Mx

 EI

d 2v . dz 2

(2.2)

The equilibrium of forces applied to the element in the direction of axes Ox , Oy is described using equations

dQx  qx dz

dQ y  q y dz 0 ,

0,

(2.3)

where q x , q y are the internal distributed forces directed along the corresponding axes. Let us change system (2.2), (2.3) to the following form: Qx

dM y dz

T

dQx  qx dz

du d 2v M z 2 , dz dz

Qy dQ y

0,

dz

dM x dv d 2u T M z 2 , dz dz dz  qy

(2.4)

0.

With equations (2.2), system (2.4) may be simplified to the system of two bending equilibrium equations of a rod prestressed by longitudinal force T and torque M z , EI

wv · w 4u w § wu · w 2 §  ¨T ¸  2 ¨ M z ¸ qx , 4 z z wz ¹ w w wz © ¹ wz ©

wu · w 4 v w § wv · w 2 § EI 4  ¨ T ¸  2 ¨ M z ¸ q y . wz ¹ wz © wz ¹ wz © wz

(2.5)

The right-hand sides of these equalities contain distributed forces q x , q y . Taking into account that the DS is not exposed to active forces, transverse load q based on d'Alembert's principle shall be taken to be equal to the forces of inertia caused by rod motion qr and liquid flow q f , that is q

qr  q f .

For the rod element, distributed force of inertia q r shall be calculated as follows:

Modelling emergency situations in drilling deep boreholes

14

 Ur Far ,

qr

where U r is the linear density of the rod; F is its cross section area; ar is the absolute acceleration of the element. In rotating coordinate system Oxyz , absolute acceleration ar shall be calculated from the Coriolis formula [8, 21]

are  arr  arc ,

ar

where are , arr , arc are the bulk, relative, and Coriolis acceleration vectors, respectively. Bulk acceleration vector are shall be calculated from the formula

are

ω u (ω u r ) ,

(2.6)

ui  vj  zk is the radius-vector of a beam element in coordinate

where r system Oxyz .

Having performed the corresponding vector operations, we obtain Z 2u ,

are, x

Z 2v ,

are, y

are, z

0.

(2.7)

The components of the relative acceleration vector in the directions of the coordinate system Oxyz axes are determined by equalities arr, x

d 2u , dt 2

arr, y

d 2v , dt 2

arr, z

0.

(2.8)

Coriolis acceleration a c vector of the rod element shall be calculated from the formula

ar

c

r 2ω uVr ,

(2.9)

where Vr r is the relative velocity vector of the element with components

Vxr

du , dt

V yr

dv , dt

Vzr

0.

(2.10)

Accounting for equalities (2.9) and (2.10), we have arc, x

2Z

dv , dt

arc, y

2Z

du , dt

arc, z

0.

(2.11)

By adding the obtained component values of accelerations (2.7), (2.8), (2.11), we obtain the components of the rod element rotational motion inertial force vector

Chapter 2. Stability of the compressed-stretched, twisted, rotating strings…

qrZ, x Z

qr , y

dv d 2u  ), dt dt 2 du d 2v  U r F (Z 2v  2Z  ). dt dt 2

15

 Ur F (Z 2u  2Z

(2.12)

The distributed force of inertia acting on the moving fluid element shall be calculated as follows: qf

U f Ff a f ,

(2.13)

where U f is the linear density of the fluid; F f is the cross section area of the pipe channel; a f is the absolute acceleration of the fluid element. It consists of rotary acceleration together with the rod and acceleration due to self-motion in the pipe channel. The first component is calculated according to the pattern of formulas (2.6)– (2.12). When calculating the second component, let us take into account that the element occupies a new position on the beam at each instant; therefore, its velocity, for example, along Ox axis is determined not only by the velocity of the rod point, where the element is located, but also by the fact that the fluid moves to an adjacent point on the rod with a different z coordinate. Then, we can write dx dt

wx wx wz  ˜ wt wz wt

x  x cV f .

(2.14)

Here, V f is the fluid velocity along axis Oz . The dot is used to indicate the differentiation with respect to t ; the prime mark, with respect to z . The presentation of velocity dx dt as (2.14) has an analog in field theory, where operator d dt is called the substantial time derivative; w wt , the local derivative operator, and the expression wx wz ˜ wz wt is referred to the convective component of the field value alteration. By differentiating again both parts of expression (2.14) with respect to time, we can find the transverse component of the fluid element absolute acceleration d 2x dt 2

w2 x w2x wx wV f w 2 x 2 wx wV f 2  V  ˜  2 Vf  f wzwt wz wt wz wt wt 2 wz dV f x  2 xcV f  V f xc  xccV f2  xcVf dz

(2.15)

Modelling emergency situations in drilling deep boreholes

16

This formula can be compared with the Coriolis theorem formula for the absolute acceleration of a material point, where x constitutes the relative acceleration; 2 xcV f , the Coriolis acceleration; xccV f2 , the centrifugal acceleration; xc , the angular rotation velocity of the rod element. Using our notations u , v for displacements along axes Ox , Oy and assuming that the fluid flows inside the tubular rod at constant speed V f , we obtain expressions for the accelerations due to its motion in the oscillating (not rotating) tube d 2u f dt 2 d 2v f dt 2

2 w 2u r w 2u r 2 w ur 2 ,  V  V f f wzwt wt 2 wz 2

w 2vr w 2vr w 2v  2V f  V f2 2r . 2 wzwt wt wz

(2.16)

Here, u f , v f are the transverse displacements of the fluid element; ur , vr are the transverse displacements of the rod element. As the fluid also participates in the rotational motion along with the tube, it is also exposed to distributed forces of form (2.12); the complete components of the distributed forces of inertia applied thereto are determined by the following relations: q f ,x

ª§ wv w 2u · § w 2u w 2u · º  U f F f «¨¨  Z 2u  2Z  2 ¸¸  ¨¨ 2V f  V f2 2 ¸¸», wt wt ¹ © wzwt wz ¹¼ ¬©

qf ,y

ª§ wu w 2v · § w 2v w 2v ·º  U f F f «¨¨  Z 2v  2Z  2 ¸¸  ¨¨ 2V f  V f2 2 ¸¸». wt wt ¹ © wzwt wz ¹¼ ¬©

(2.17)

Substituting the right-hand sides of these equalities into equation (2.5), we obtain the vibration equations of a rotating tubular rod prestressed by force T , torque M z and containing fluid flows EI

wv · w 4u w § wu · w 2 § 2  ¨T ¸  ¨ M z ¸  U r F r  U f F f Z u  wz ¹ wz 4 wz © wz ¹ wz 2 ©

 2U r F r  U f F f Z

EI

wv w 2u w 2u w 2u  V 2 U f F f 2  2VU f F f  U r F r  U f F f 2 wt wzwt wz wt

0,

wu · w 4v w § wv · w 2 § 2  ¨T ¸  ¨ M z ¸  U r F r  U f F f Z v  wz ¹ wz 4 wz © wz ¹ wz 2 ©

w 2v w 2v w 2v wu  U r F r  U f F f 2  V 2 U f F f 2  2VU f F f  2U r F r  U f F f Z wzwt wt wt wz

(2.18)

0.

Chapter 2. Stability of the compressed-stretched, twisted, rotating strings…

17

Equations (2.18) can be used to study free vibrations of the DS and to simulate transient regimes of acceleration and deceleration of its rotation. The presence of terms with coefficients M z and UF  U f F f makes this system coupled; this excludes the possibility of DS deformation and vibrations in a plane [12, 31]. All this leads to the probability of effects in the rotating DS, specific to compressed-bent rods [7, 32], rotating shafts [8], and pipelines with internal fluid flows [16, 17]. Should the terms containing the derivatives with respect to t be excluded from (2.8), we obtain the following reduced system: EI

d 4u d § du · d 3v d 2u  ¨ T ¸  M z 3  ( UF  U f F f )Z 2u  V 2 U f F f 2 4 dz © dz ¹ dz dz dz

0,

EI

d 4v d § dv · d 3u d 2v  ¨ T ¸  M z 3  ( UF  U f F f )Z 2v  V 2 U f F f 2 4 dz © dz ¹ dz dz dz

0,

(2.19)

which can be used to analyse the spatial stability of the DS stationary rotation equilibrium. Using it, various problems concerning DS stability can be formulated. First of all, for M z , Z and V values equal zero, one can determine the critical value of arbitrary force T, for which homogeneous system (2.19) has a non-zero solution. To do this, a problem shall be formulated for eigen values (or eigen functions), which is then solved numerically. Similarly, it is possible to determine the critical values of independent variables M z , Z and V or various combinations thereof. However, as noted above, this problem has a significant difference for long drill strings caused by a change in the role of the various terms being part of system (2.19). Below, we discuss the influence of the structure of equations (2.19) on the form of the solutions thereto for long DS of length L. 2.2 Singularly perturbed equations in boundary value problems of the deep drilling string mechanics Many mathematical models that adequately describe physical processes in terms of differential equations include (explicitly or implicitly) various parameters, and in a typical situation their values are known only approximately, with a certain degree of accuracy. Therefore, the matter of the nature of the behaviour of the differential equation solutions for a small change in the value of the parameter being a part of the equation is of fundamental interest. Starting with the conventional papers by H. Poincare and A. M. Lyapunov, the so-called regular case has been studied in great detail when the right-hand side of the second-order type equation

Modelling emergency situations in drilling deep boreholes

18

xcc F (t , x, xc, H ) (0 d t d 1) ,

(2.20)

regularly (continuously, smoothly, analytically) depends on parameter H in the vicinity of H 0 value, and the solutions to the equation are considered on a finite interval 0 d t d 1 of the independent variable t variation. In this case, to solve this equation for any small H , a solution close to it, which corresponds to the value of parameter H 0 , exists. The singular case is much more diverse and complex [4] when the solutions depend on small parameter H before the second derivative (here, above all, the Krylov-Bogolyubov-Mitropolsky averaging theory should be mentioned), and when the assumption of the regularity of the parameter inclusion in the equation is not met. Clearly, the last of these situations occurs, for example, for equation

H xcc

f (t , x, xc, H ) (0 d t d 1) ,

(2.21)

where H is the parameter that adopts arbitrarily small positive values (often written as 0< H

greatest

@

value

At first, the problem of determining critical rotation velocity Zcr of the DS with a length of L 2 km that does not contain centring devices was solved. Calculated values Zcr for different R and M z are shown in Table 2.9. Modes of DS stability loss without centring devices are shown in Fig. 2-18, positions a–f correspond to positions a–f in Table 2.9.

a

b

c

d

e

f

Fig. 2-18 Shapes of incipient buckling of a rotating drill string without centring devices (L=2,000 m)

It is interesting to trace the dependence of the critical value of angular rotation velocity Zcr of the DS on value M z . As it follows from Table 2-9, loading the

Chapter 2. Stability of the compressed-stretched, twisted, rotating strings…

59

rotating DS by torque at R 0 can lead to both an increase and decrease of Zcr . This phenomenon, contradictory at a first glance, can be explained by the fact that when M z 0 (case a) the shape of buckling of the rotating rod is flat, the centrifugal forces of inertia lie in the same plane, they act in the same direction, and their destabilising effect is added. However, if the rotating rod is also impacted by torque M z (cases b– d), then, when it is buckled in its lower least stressed part, bending occurs along a spiral line; the inertia centrifugal forces acting on the spiral elements do not lie in the same plane, they can neutralise each other, and their destabilising effect can be reduced. If the DS has a segment compressed by reaction R at the bottom, the critical value of angular velocity Zcr can also fall (case e) or increase (case f). The force of gravity of the considered DS turned out to be equal to G2 9.81 ˜ ( U  U f ) FL 2296183 N. As the DS does not experience axial forces at the selected boundary conditions at its lower point (positions a–d) or prestressed by compressive force T ( L) R (positions e, f), it has a lower bending stiffness in this zone, and, as shown by preliminary calculations, DS bifurcation buckling occurs right at this spot. To increase its stiffness in the specified zone at points z1 L  l1 , and z4 L  (l1  l2  l3  l4 ) , z 2 L  (l1  l2 ) , z3 L  (l1  l2  l3 ) , l 13 . 5 where m, m, z5 L  (l1  l2  l3  l4  l5 ) l1 l2 9 l4 l5 18 m, 3 additional supports are introduced, and the supporting conditions of type (2.71) are used. Fig. 2-19 shows the shapes of the initial stability loss of DS in two projections. As it follows from these figures and the table, the buckling of the rotating DS in the absence of torque M z and reaction R occurs in a planar form, while in the zone where centralisers with a length of l1 + l2 + l3 + l4 + l5 = 67.5 m are installed deflections u (z ) , v (z ) practically equal zero, and at the exit from this zone they increase sharply. If the rotating DS is additionally prestressed by torque M z (position b–d in Fig. 2-19), the mode of stability loss becomes spatial and more complex, with its greatest complication occurring in the bottom part. Comparison of the results of the stability calculation of a 2 km long DS allows us to conclude that the shape of their buckling represents a superposition of a regular function throughout length L of the string and the function of the boundary effect adjacent to its lower end. As the zone of the boundary effect does not have a clear upper limit, and its length varies significantly with a change in the conditions of the DS operation, it is impossible to distinguish it in its pure form, and the problem has to be solved entirely. Due to the very small relative bending stiffness of the DS, the addition of five supporting devices in the zone of the boundary effect only made it

Modelling emergency situations in drilling deep boreholes

60

possible to locally make the structure rigid in the rock cutting zone (see Figs. 2-18 and 2-19), while critical values Zcr of the angular velocity changed insignificantly (Table 2-9). y

O

z

y

x

x

x

x

x

x

a

b

c

d

e

x

f

Fig. 2-19 Shapes of incipient buckling of a rotating drill string with five centring devices at the bottom (L=2,000 m)

During computer modelling to verify the convergence of calculations in the building of the Cauchy matrix Y (z ) using the Everhart method, the integration step was chosen equal to 'z L 80000 and 'z L 160000 , the number of points of orthogonalisation was brought to 40 and 80. As the results of the calculation for these values of the calculating parameters coincided with high accuracy, numerical studies were carried out at value 'z L 80000 with a selection of 40 points of orthogonalisation. Similar patterns of bifurcation behaviour of DS also occur at L 7 km (Fig. 220), L 9 km (Fig. 2-21), and L 10 km (Fig. 2-22). So, in the transition from length L 2 km to length L 7 km, values Zcr almost halved, but the nature of dependence Zcr on M z and R has not changed qualitatively. Note that the convergence of the solution of these problems was achieved at the integration step 'z L 640000 and the choice of 320 orthogonalisation points.

Chapter 2. Stability of the compressed-stretched, twisted, rotating strings…

a

b

c

d

e

61

f

Fig. 2-20 Shapes of the incipient buckling of a rotating drill string with five centring devices at the bottom (L=7,000 m)

a

b

c

d

e

f

Fig. 2-21 Shapes of the incipient buckling of a rotating drill string with five centring devices at the bottom (L=9,000 m)

Modelling emergency situations in drilling deep boreholes

x

a

x

b

x

x

c

d

62

x

x

e

f

Fig. 2-22 Shapes of the incipient buckling of a rotating drill string with five centring devices at the bottom (L=10,000 m)

Figure 2.23. The shape of stability loss of the multi-supported bottom part of the rotating drill string in enlarged scale ( M z 0, V 0 , L=2,000 m, case ‘a’ in Table 2.9, ωcr = 0.13452 s–1)

Chapter 2. Stability of the compressed-stretched, twisted, rotating strings…

63

Bringing the length of DS to 9 km and 10 km (Table 2-9, Figs. 2-21 and 2-22) did not qualitatively affect critical values of ωcr and shapes of stability loss. In these cases, the relative stabilisation zones of DS centring devices become quite small, and centring devices are appropriate only to provide a given orientation of the drill bit. Gravity forces for the considered drill strings were G9 10332823 N and 11480914 N, respectively. Fig. 2-23 shows a scale-up of the lower part of the DS for case ‘a’ in Table 2-9. It can be seen that the addition of intermediate supports leads to a significant decrease in the deformation of the DS only in its lower part. The obtained results lead to the conclusion that DS stability during its operation is determined by a number of factors, the destabilising or stabilising effects of which depend on their combination and the intensity of each individual demonstration. When these effects are superimposed, they can enhance or weaken their actions. As the length of the drill string increases, the critical values of the angular velocity of its rotation fall, the stabilising role of the centring devices decreases, and their efficiency is limited only to providing a given orientation of the bit. This effect testifies that boundary conditions at the DS bottom end do not essentially influence on the overall buckling phenomenon.

G10

References to Chapter 2 1. Alfutov N. A. Basics of Calculation on Stability of Elastic Systems. – M.: Mashinostroyenie, 1978. – 312 p. (in Russian). 2. Barakat Elie R., Miska Stefan, Takach N. The effect of hydraulic vibrations on initiation of buckling and axial force transfer for helically buckled pipes at simulated horizontal wellbore conditions // SPE / IADC Drilling Conference, 20-22 February 2007, Amsterdam, Netherlands. 3. Beederman V. L. Mechanics of Thin-Walled Structures. – Moscow, Mashinostroyenie. – 1977. – 488 p. (in Russian). 4. Chang K., Howes F. Nonlinear Singular Perturbation Phenomena. M.: Mir. – 1988. – 247 p.(in Russian). 5. Choe Jonggeun, Schubert Jerome J., Hans C. Juvkam-Wold. Well-control analyses on extended-reach and multilateral trajectories // SPE Drilling&Completion. – June 2005. – P. 101–108.(26) 6. Chui C. An Introduction to Wavelets. M.: Mir. – 2001. – 412 p.(in Russian). 7. Feodosiev V.I. Selected Problems and Questions on Strength of Materials. M.: Nauka. – 1967. – 237 p. (in Russian). 8. Filippov A.P. Oscillations of Deformable Systems. Kiev: Naukova Dumka. – 1970. – 673 p. (in Russian).

Modelling emergency situations in drilling deep boreholes

64

9. Gudkov V. V., Klokov Yu. A., Lepin V. D., Ponomarev A. Ya. Two-point Boundary Value Problems for Ordinary Differential Equations. – Riga: Zinatne. – 1973. (in Russian). 10. Gulyayev V. I., Gaydaychuk V. V., Koshkin V. L. Elastic Deformation, Stability and Vibrations of Flexible Curvilinear Rods. – Kiev: Naukova dumka. – 1992. – 344 p. (in Russian). 11. Gulyayev V. I., Gaydaychuk V. V., Soloviev I. L., Gorbunovich I. V. Quasistatic critical states of drill strings for deep drilling. Problemy Prochnosti. – 2006. - No 5. – pp.109-119. (in Russian). 12. Gulyayev V. I., Gaydaychuk V.V., Abdullayev F. Ya. Self-excitation of unstable oscillations in tubular systems with moving masses. Prikladnaya Mekhanika, – 1997. – No. 3 – pp. 84–90. (in Russian). 13. Gulyayev V.I., Gaydaychuk V.V., Khudoley S.N., Golovach L.V. Computer simulation of static states of drill strings in directional wells with geometric imperfections. Neftegazopromyslovy Inzhiniring. – M.: 2006, 4. – pp. 26–29. 14. Gulyayev V.I., Gaydaychuk V.V., Solovjov I. L., Gorbunovich I. V. The buckling of elongated rotating drill strings// Journal of Petroleum Science and Engineering, 2009.-67, P. 140-148. 15. Gulyayev, V.I., Shlyun N. Influence of friction on buckling of a drill string in the circular channel of a bore-hole// Petroleum Science, 2016, 13. –P.698–711. 16. Gulyayev, V.I., Tolbatov E.Yu. Dynamics of spiral tubes containing internal moving masses of boiling liquid. Journal of Sound and Vibration 2004. – 274(2), P. 233–248. 17. Gulyayev, V.I., Tolbatov E.Yu. Forced and self-excited vibrations of pipes containing mobile boiling fluid clots. Journal of Sound and Vibration 2002. – 257(3), 425–437. 18. Huang N.C., Pattilo P.D. Helical buckling of a tube in an inclined well bore // International Journal of Non-Linear Mechanics. – 2000. – 35. – P. 911–923. 19. Iyoho A.W., Meize R.A., Millheim K.K., Crumrine M.J. Lessons from integrated analysis of GOM drilling performance // SPE Drilling&Completion. – March 2005. – P. 6–16. 20. Kerr R.A. Bumpy road ahead for world’s oil. // Science, 18 Nov.2005, Vol. 310.– P. 1106–1108. 21. Lurie A.I. Analytical mechanics. M.: Fizmatgiz, 1961. – 834 p. (in Russian). 22. Maugeri Leonardo. Oil: never cry wolf- why the petroleum age is far from over// Science. – May 2004. – V.304.– P. 1114–1115. 23. Myslyuk M.A., Ribchich I.I., Yaremiychuk R.S. Drilling of wells. Vol. 3, Vertical and Directional Drilling. – Kiev. “Interpress LTD”. – 2004. – p.294. (in Ukrainian).

Chapter 2. Stability of the compressed-stretched, twisted, rotating strings…

65

24. Newland D. E. Wavelet analysis of vibration – part I: Theory // Department of Engineering, University of Cambridge, Cambridge CB2 1PZ, UK. – P. 1–23. 25. Newland D. E. Wavelet analysis of vibration – part II: Wavelet maps // Department of Engineering University of Cambridge, Cambridge CB2 1PZ, UK. – P. 1–19. 26. O’Donnell M.A. Boundary and corner layer behaviour in singularly perturbed semi-linear systems of boundary value problems // SIAM J. Math. Anal. – 1984. 27. Radchenko V.N.. To a determination of the natural frequencies of longitudinal oscillations of the drill staged string. Masiny i Neftyanoe Oborudovanie. – 1976. – No 5. – pp.13-16. (in Russian). 28. Skrypnik S.G., Golovin A.A., Ryabikhina S.M. et al. Some issues of complex research of dynamic loads in the system of the lifting mechanism of the drilling rig - drill string. Mashiny i Neftyanoe Oborudovanie. – 1977. – No 10. – pp.2124. (in Russian). 29. Vasilyeva A. B., Butuzov V.F. Singularly Perturbed Equations in Critical Cases. – M.: Ed. Moscow State University. - 1978. (in Russian).(3) 30. Walker J.S. A primer on wavelets and their scientific applications. Eau Claire, University of Wisconsin, USA. – 2007. – 320 p. 31. Woods G., Lubinsky A. Curving of Wells During Drilling. M.: Gostoptekhizdat. – 1960. – 161 p. (in Russian). 32. Zigler G. Fundamentals of the Theory of Structural Stability. М., Mir, 1971.– 192 p. (in Russian).

Modelling emergency situations in drilling deep boreholes

CHAPTER 3. 3.1

66

BENDING VIBRATIONS OF THE DRILL STRING IN A VERTICAL WELL

Technical and applied aspects of the problem of the prestressed rotating tubular rods with internal fluid flows

The problems of studying the dynamic behaviour of rotating tubular systems prestressed by longitudinal forces and torques and containing internal moving masses and mobile fields of internal pressure are widely encountered in engineering. The first results of their solutions seem to be associated with engineering and, first and foremost, with the power engineering industry. They relate to the analysis of the stability loss of rotating shafts [3, 12] mainly due to the action of centrifugal forces of inertia without taking into account shaft pretension by longitudinal force and torque. However, since the main purpose of the turbine shaft is associated with the transmission of longitudinal forces and torques, later attention was drawn to the formulation of problems that took into account the additional effects of prestresses. Quite complex processes in tubular systems can be caused by internal flows of moving fluid. Such systems emerged in the last century, but there already exist many examples. In the oil and gas industry, these are main pipelines [17], in the tanker fleet, in aviation, and space technology they are hoses for pumping liquid fractions of oil and fuel when refuelling aircrafts in flight, in engineering they are hydraulic drive pipes [2], in nuclear and thermal energy they are tubular contours of heat exchangers with straight and curved elements [19, 20]. There are numerous projects that use vertical pipelines up to 6 km long to supply minerals (concretions) from the depths of the world ocean using the effect of air-water mixture lifting forces. Examples of such systems can also be pneumatic tube and pipe transport pipelines as well as gun barrels, inside which the shells and the mobile field of increased internal pressure move. The functioning of these mechanical systems can be accompanied by complex dynamic effects due to the possibility that the system bodies participate in several types of motion simultaneously, the complex interaction of these movements and gyroscopic connection between them, the possibility of static (divergent) loss of stability of stationary motion [3, 11], the appearance of unstable oscillatory movements of the flutter type, and parametric resonances. However, the most complex combinations of power effects are felt by tubular deep drilling strings used in the rotary method for creating oil and gas drilling wells. In these technologies, rock destruction at the bottom of the drill hole is performed by a drill bit attached to the lower end of the rotating string, as a result of applying torque to its upper end. The drilling fluid is passed through the string's pipe cavity and serves to remove rock particles destroyed by the drill bit through the outer cavity, reinforce the rock medium by hydrostatic pressure of the rock media weakened by the cavity, implement the lubricating effect between the string and the wall of the drill hole, and remove the heat released during drilling [25, 26].

Chapter 3. Bending vibrations of the drill string in a vertical well

67

In the process of drilling operations, the string is subject to the influence of a number of static and dynamic force factors, among which the most significant are the uneven tension force along the drill string, torque caused by the movement of the internal flow of the drilling fluid, centrifugal and Coriolis’ forces of inertia [16], the frictional interaction forces of the string with the wall of the drilling well, etc. The listed factors initiate the occurrence of longitudinal, torsional [15], and bending vibrations in the string and promote its bending buckling [13, 27]. As noted above, as a result of this, it is possible that the DS pipe will stick, the walls of the drilling well will collapse, and there will be an overall loss of system stability. The problem of vibrations and propagation of bending waves in long prestressed rotating pipes with internal flow of liquid has a direct application in the dynamics of deep-well drilling (Fig. 1-2), whose working lengths are close to 10 km and under operating conditions are exposed to a complex combination of forces [1, 4 8–10, 14, 16, 21–23]. The action of these factors leads to the fact that standing waves and planar progressive waves cannot be formed in such rods, and only spiral progressive waves can occur, whose speed depends on the orientation of the spiral (left or right) and the direction of propagation (along or against the positive direction of the longitudinal axis). Due to the long length of the DS in the first approximation, it can be considered as an open (without boundaries and boundary conditions) infinite system. However, since in the lower part it has a system of additional supports (centralisers), it is also interesting to consider it as a beam with supports (closed system). The equations of transverse string vibrations are built in Chapter 2. They have the form: EI

wv · w 4u w § wu · w 2 §  ¨ T ¸  2 ¨ M z ¸  U r F r  U f F f Z 2u  4 wz ¹ wz © wz ¹ wz © wz

 2U r F r  U f F f Z

EI

w 2u w 2u w 2u wv  U r F r  U f F f 2  V 2 U f F f 2  2VU f F f wzwt wt wt wz

0,

wu · w 4v w § wv · w 2 §  ¨ T ¸  2 ¨ M z ¸  U r F r  U f F f Z 2v  4 wz ¹ wz © wz ¹ wz © wz

w 2v w 2v w 2v wu  U r F r  U f F f 2  V 2 U f F f 2  2VU f F f  2U r F r  U f F f Z wzwt wt wt wz

(3.1) 0.

The notation system is also provided there. As these equations are related, it can be concluded that the drill string cannot perform planar vibrations, and its shape of motion can only be spatial. An important factor is the presence of odd derivatives with respect to z in this system. To further evaluate the effects caused by this feature, let's simplify system (3.1). To do this, we consider the statics of the rod loaded only by constant force T , constant torque M z , and constant uniformly distributed load q x , q y 0 without taking into account the internal flow. Then, system (3.1) takes the form:

Modelling emergency situations in drilling deep boreholes

d 4u d 2u d 3v T 2  Mz 3 4 dz dz dz d 4v d 2v d 3u EI 4  T 2  M z 3 dz dz dz EI

68

qx , 0.

Consider the case of the beam of length L pivotally supported at points z r L 2 . Analysis shows that even in the simplest case, when qx const , q y 0 , reduced equations cannot have a solution symmetric relatively to z 0 ˗ that is, solutions in the form of even functions. Indeed, let u  z and vz be symmetric (even) functions. Then, even derivatives d 2u dz 2 , d 4u dz 4 , d 2v dz 2 , d 4v dz 4 will also be even, while odd derivatives d 3u dz 3 , d 3v dz 3 represent odd functions. But in this case, the sum of even and odd functions cannot be equal to an even function. It follows that even in such a simple case, when the rod is loaded with uniformly distributed loads q x , q y , the solution acquires a complex asymmetric spatial form. Of course, it becomes even more complicated when the general dynamic case for system (3.1) is considered. 3.2 Spiral progressive waves in infinite twisted tubular rotating rods with internal fluid flows In the analysis of the dynamics of technical objects, designers and estimators, as a rule, have to deal with deformable bodies of limited dimensions and conduct research on the basis of the formulation of boundary value problems for the equations of their motion. However, in these cases, most often it is necessary to use the mechanical and physical characteristics established in the study of the most common laws of the dynamic behaviour of unlimited media and bodies. Such characteristics include, for example, the types and speeds of wave propagation and the possibility of their propagation without distortion (non-dispersive media) and with distortion (dispersive media). Therefore, this section aims to settle the task of investigating periodic movements that can be carried out in infinite rotating rods prestressed by longitudinal forces and torques and study the effect of taking into account additional more precise fixed moment factors on the shapes and velocities of the waves as well as the frequency and modes of the vibrations. 3.2.1. Construction and analysis of the characteristic equation of drill-string vibrations Let the infinite elastic rod stressed by longitudinal force T and torque M z rotate at constant angular velocity ω around its longitudinal axis. To derive the equations of its bending motion, we introduce inertial coordinate system OXYZ with the origin at some point of the rod and coordinate system Oxyz with unit vectors

Chapter 3. Bending vibrations of the drill string in a vertical well

69

i , j , k rotating with it. Axes OZ and Oz of these systems coincide with the longitudinal axis of the rod. The frequency spectra and the modes of its free vibrations depend on the selection of the reference system (coordinate system). Since from the point of view of assessing the deformable state of the rod it is more convenient to consider its dynamic behaviour in rotating coordinate system Oxyz , let's choose elastic displacements u and v of its elements along axes Ox and Oy as the required variables, and we will ignore the displacements along axis Oz . With rod vibrations, internal elastic forces and inertia forces act on each element, so do the external distributed forces caused by the flow of liquid with the local forced bending turning of the axis of rotation. The vibration equations of such a rod are written in the form of (3.1). Due to the fact that, in general, this system has components with variable coefficients T (z ) and M z (z ) , it does not have solutions in the form of harmonic waves, and the issue of using it to study natural vibrations of the tubular rod is associated with certain difficulties. To simplify it, we will accept T const , M z const . Then, the second additive components Tw 2u / wz 2 , Tw 2 v / wz 2 and the





sixth  V U f F f w u / wz , V U f F f w v / wz additive components in these equations will have the same form, differing only in signs and constant coefficients  T and V 2 U f F f . Therefore, it can be concluded that the movement of a liquid with linear density U f F f and velocity V is equivalent to the action of longitudinal compressive force  T on the rod. Furthermore, if we consider that in additive components 2VU f F f w 2u / wzwt , 2VU f Ff w 2v / wzwt velocity V is included in the first degree, and in 2

2

2

2

2

2

additive components V 2 U f F f w 2u / wz 2 , V 2 U f F f w 2v / wz 2 , in the second degree, then to simplify at V !! 1 , the first additive components at this stage can be ignored. In this case, considering that the effect of the destabilising action of the mobile fluid can be simulated by providing force T of the corresponding value, we exclude from system (3.1) the members containing multiplier U f F f . We should emphasise that the generality of this approach will not be broken, and the transformations will become more visible. Then, system (3.1) will become more compact w 4u w 2u w 3v wv w 2u  T 2  M z 3  UFZ 2u  2 UFZ  UF 2 4 wt wz wz wz wt 4 2 3 wv wv wu wu w 2v EI 4  T 2  M z 3  UFZ 2v  2 UFZ  UF 2 wt wz wz wz wt

EI

0,

(3.2) 0.

With the help of this system, the harmonic oscillations that may take place in the rod can be investigated. Note that despite its linearity it has a rather complex structure due to the presence of summands M z w 3v / wz 3 ,  M z w 3u / wz 3 and

Modelling emergency situations in drilling deep boreholes

70

 2UFZ wv wt ,  2UFZ wu wt in the first and second equations. So, the presence of the first two summands leads to the fact that the system does not allow solutions in the form of a planar curve, and the elastic line of the curved rod can only be a threedimensional curve (in this case, a spiral). The presence of the second two summands leads to a more complex law of changing the shape of the vibration in both spatial and temporal coordinates, thus eliminating the possibility of moving the elements of the rod with one common phase. The specified complication of system (3.2) by these members is due to the fact that they contain odd derivatives with respect to z and t , and the coefficients before these summands form skew-symmetric matrices. The problem of the harmonic vibrations of tubular twisted rotating rod prestressed with an axial force is multi-parametric. By direct substitutions u ( z, t )

Acoskz ˜ sinct, v( z , t )

Bsinkz ˜ sinct.

(3.3)

u ( z, t )

Acoskz ˜ sinct, v( z , t )

Bcoskz ˜ cosct ,

(3.4)

we can verify that system (3.2) does not allow solutions in the form of standing waves or progressive waves with nodal points. Therefore, we will build its solutions in the form of progressive cylindrical spiral waves

u ( z, t )

Acos (kz  ct ), v( z, t ) Bsin(kz  ct ) ,

(3.5)

where k is the wave number; c is the cyclic frequency. By substituting (3.5) into (3.2), we obtain a homogeneous system of algebraic equations

EIk  Tk  UFZ  UFc A  M k  2UFZc B M k  2UFZc A  EIk  Tk  UFZ  UFc B 4

z

2

3

2

2

z

4

2

3

2

2

0, 0.

It has a non-trivial solution when the determinant of the coefficient matrix is equal to zero. From this condition, the characteristic (dispersion) equation follows:

EIk

 2

 Tk 2  UFZ 2  UFc 2  M z k 3  2 UFZc It decomposes into two lower-degree equations 4



2

EIk 4  Tk 2  UFZ 2  UFc 2  M z k 3  2 UFZc 0, EIk 4  Tk 2  UFZ 2  UFc 2  M z k 3  2 UFZc 0,

0.

(3.6)

which connect wave number k and cyclic frequency c . This system has two roots for cyclic frequencies c1, 2 determined by the dispersion equality

Chapter 3. Bending vibrations of the drill string in a vertical well

c1, 2

Zr

71

k EIk 2  M z k  T , UF

(3.7)

which corresponds to a wave in the shape of a right spiral  A / B ! 0 , and two roots

c3, 4

Z r

k

UF

EIk 2  M z k  T ,

(3.8)

corresponding to the wave in the shape of the left spiral  A / B  0 . Let's make sure of this by substituting frequencies ci in (3.6) and counting the ratio A / B . For frequency c1 , we find

M z k 3  2Zk UF ˜ EIk 2  M z k  T  2 UFZ 2

A1 B1

M z k 3  2Zk UF ˜ EIk 2  M z k  T  2 UFZ 2

1.

(3.9)

From relations (3.5), (3.7), (3.9), it follows that because A1 B1 , with vibrations at this frequency, all elements of the rod move clockwise along circular trajectories with phases shifted by k ˜ 'z counter clockwise, looking from the end of axis Oz . The wave formed as a result of such movement is a cylindrical right-screw spiral of a circular profile, which shifts in the positive direction of axis Oz (Fig. 3-1, a). As the equality

M z k 3  2Zk UF ˜ EIk 2  M z k  T  2 UFZ 2

A2 B2

M z k 3  2Zk UF ˜ EIk 2  M z k  T  2 UFZ 2

1,

(3.10)

Is valid for frequency c2 , the corresponding shape of vibrations is also a circular right-screw spiral, which, however, is shifted in the negative direction of axis Oz at c2  0 . In cases where the vibration frequency equals с3 and с4 , we obtain A3 B3



M z k 3  2Zk UF ˜ EIk 2  M z k  T  2 UFZ 2 M z k 3  2Zk UF ˜ EIk 2  M z k  T  2 UFZ 2

1,

(3.11) A4 B4

M z k  2Zk UF ˜ EIk  M z k  T  2 UFZ 3



2

2

M z k 3  2Zk UF ˜ EIk 2  M z k  T  2 UFZ 2

1.

Modelling emergency situations in drilling deep boreholes

x

v1

Z

v2

72

z 2A

y a

x

v3

Z

v4 z 2A

y b Fig. 3-1 Progressive bending waves in the shapes of right (a) and left (b) orientation spirals

Since the gained dependencies are negative, at these frequencies the rod elements move clockwise with the phases displaced along circular trajectories. The waves that correspond to such movements represent left-screw cylindrical spirals of circular profile, moving in the positive direction of axis Oz at c3 ! 0 and in the negative direction for frequency c4 (Fig. 3-1, b), because it is always negative.





Having determined four values (3.7), (3.8) of circular frequencies ci i 1,4 , it is possible to calculate the corresponding phase velocities vi ci k in rotating coordinate system Oxyz . As a result, we have

v1, 2 v3, 4

c1, 2

Z

k

k

c3, 4 k

r

Z

1 UF

1  r k UF

EIk 2  M z k  T , (3.12) 2

EIk  M z k  T .

These equations show that at Z 0 the velocity of propagation of each of the considered waves in the positive and negative directions of axis Oz will be the same,

Chapter 3. Bending vibrations of the drill string in a vertical well

73

although with positive M z the speed of the right-screw wave turns out to be greater than the speed of the left-screw wave. If the rod begins to rotate at angular velocity Z ! 0 , the velocity of the rightscrew spiral in the positive direction of axis Oz becomes greater than its velocity in the negative direction, and this velocity exceeds the velocity of propagation of all other steady-state spiral waves of this length. Upon reaching the value k

Z

EIk 2  M z k  T

UF

the right-screw spiral, moving in the positive direction of axis Oz , doubles its speed, and the wave propagating in the opposite direction becomes stationary. As a result of further increase in Z , both right-screw waves move in the positive direction of axis Oz but at different speeds. Changing the direction of rod rotation ( Z  0 ) leads to an increase in the phase velocity of the right-screw wave in the negative direction and a decrease in its





2 velocity in the positive direction, and at Z   k EIk  M z k  T / UF the waves of this polarisation move only in the negative direction of axis Oz . The nature of spiral wave propagation also depends significantly on parameters EI and T . For positive M z with an increase in these values, the phase velocity of propagation of the right-screw spiral in the positive direction increases, and the provision of negative values T   EIk 2  M z k leads to the appearance of nontransmission zones for these waves at the selected k . If we analyse the wave processes in relation to inertial reference system OXYZ , it is necessary to impose rotation with angular velocity Z on the considered motion. As for the right-screw spiral waves, their movements with frequencies (3.7) correspond to the clockwise rotation of the elements of the rod with angular velocity ci , and with positive Z rotation of the entire system is counter clockwise movement at velocity Z , then, as a result of superposition of these two movements, the elements will move with angular velocities

c1, 2  Z

r

k

UF

EIk 2  M z k  T ,

(3.13)

independent of Z , and the axis of the circular directrix of the cylindrical surface of the spiral will rotate at angular speed Z . At the same time, the effects of wave propagation along axis OZ fail to depend on angular velocity Z v1, 2 

Z k

r

1 UF

EIk 2  M z k  T .

(3.14)

The type of relations (3.5)–(3.14) shows that if the sign of torque M z is changed to the opposite, the wave shapes of right and left spirals interchange their properties.

Modelling emergency situations in drilling deep boreholes

74

Let's analyse some more properties of the revealed waves.





For the right spiral at k1, 2  M z r M z2  4EIT 2EI , circular frequencies c1, 2 go to zero, and at k2 d k d k1 , a radical expression in (3.7) becomes negative. Therefore, this range of k values composes the zone of non-transmission of the rightscrew waves. But this conclusion has no significant applied value because it refers to the negative values of wave number k . Of great interest is the positive range of the wave number change k4 d k d k3 for the left-screw spiral, where





k3,4 M z r M z2  4EIT 2EI because in this range the wave of the polarisation under consideration cannot propagate. These considerations allow us to make important conclusions. At first, only spiral waves can propagate in the twisted rotating rods. Second, at the same step of the spiral and at the selected directions of rotation and torque in the positive and negative directions of the Oz axis, waves propagate in the shapes of left and right spirals with four different cyclic frequencies ci (and, therefore, with different values of phase velocities). For each of these waves, the character of the dispersion curves ci ci k is determined by the relations between bending stiffness EI , the value of torque M z , and the value and sign of longitudinal force T . However, the cylindrical spiral of the progressive wave fails to be circular if the axial moments of inertia I x , I y of the cross section of the rod are not the same. Then, in relations (3.1), one needs to make the appropriate changes, and they will obtain the form: wv · w 4u w § wu · w 2 § EI y 4  ¨ T ¸  2 ¨ M z ¸  U r F r  U f F f Z 2u  wz ¹ wz © wz ¹ wz © wz w 2u w 2u w 2u wv  U r F r  U f F f 2 0,  2U r F r  U f F f Z  V 2 U f F f 2  2VU f F f wzwt wt wt wz wu · w 4v w § wv · w 2 § EI x 4  ¨ T ¸  2 ¨ M z ¸  U r F r  U f F f Z 2v  wz ¹ wz © wz ¹ wz © wz w 2v w 2v w 2v wu  U r F r  U f F f 2 0.  V 2 U f F f 2  2VU f F f  2U r F r  U f F f Z wzwt wt wt wz System (3.6) in this case is reduced to the form:

EI k M k y

z



B

4

 Tk 2  UFZ 2  UFc 2 A  M z k 3  2 UFZc B 0,

3

 2 UFZc A  EI x k 4  Tk 2  UFZ 2  UFc 2



As a result, the characteristic equation takes a more complex form EI y k 4  Tk 2  UFZ 2  UFc 2 EI x k 4  Tk 2  UFZ 2  UFc 2  M z k 3  2 UFZc







(3.15)

0.



2

0,

Chapter 3. Bending vibrations of the drill string in a vertical well

75

and expressions for frequencies ci become quite cumbersome and are not given here. However, by substituting them in system (3.15), it can be established that in this case A z B , and the progressive wave takes the shape of an elliptical cylindrical spiral. The third important feature of the considered dynamic processes is that unlike the harmonic waves in the infinite elastic medium and the harmonic longitudinal waves and torsion waves in the infinite rods, which are non-dispersible, the bending waves of the considered type are dispersible. This is due to the fact that wave number k and circular frequency c are connected by non-linear dependencies (3.7) and (3.8). Therefore, waves with different lengths propagate at different speeds (3.12). This leads to the fact that under the considered conditions only cylindrical spiral waves can be stationary eigen waves. In an infinite elastic medium, a stationary wave can be any longitudinal or transverse wave of an arbitrary profile. The same properties have arbitrary longitudinal waves and torsion waves in infinite ordinary rods. 3.2.2

Analysis of the properties of bending progressive waves in prestressed rotating rods (drill strings)

In this paragraph, the problems of the propagation of bending periodic waves in an elastic rod are considered; it is shown that in the case of prestressing the rod by torque the application of the concept of wave polarisation is principal. It is widely used in the electromagnetic wave theory of light propagation. Transversality of electromagnetic waves deprives each wave of axial symmetry in respect of the direction of propagation through the presence of selected directions in a plane perpendicular to the direction of propagation. It is known that the light emitted by any single elementary emitter (atom, molecule) in each act of radiation is always polarised. But macroscopic light sources are composed of a large number of such emitters, each of which has its own polarisation. Therefore, the total radiation, the so-called natural light, is nonpolarised. Light is called fully polarised if two mutually perpendicular components of electric field intensity vector E of the whole beam vibrate with a time-constant phase difference. As is usual, the state of polarisation of light is represented by an ellipse of polarisation—the projection of the trajectory of the end of vector E on the plane perpendicular to the beam. The projection picture of the fully polarised light in a general case has the shape of an ellipse with the right or left directions of rotation of vector E in time (Fig. 3-2, b, d, and f). This light is called elliptically polarised. The limiting cases of elliptic polarisation are linear (Fig. 3-2, a, e) and cylindrical spiral (or circular) when the polarisation ellipse represents a circle (Fig. 3-2, c). As shown in paragraph 3.1, for a rod with an axial-symmetric cross-section, prestressed by torque, the propagation of only the cylindrical spiral bending waves with circular polarisation is possible. Taking into account the clear visual visibility of such waves, which lead to the bending of the rod in the mode of a cylindrical spiral, such waves are called spirals. Polarisation of the wave in the rod can be elliptical

Modelling emergency situations in drilling deep boreholes

76

only in cases where the cross section of the rod has no cyclic symmetry, and, therefore, moments of inertia I x , I y have different values ˗ that is, I x z I y . x

A

B

y Phase difference

G

0 a

S 6

S

b

2

5 S 6

S

7 S 8

c

d

e

f

Fig. 3-2 Examples of polarisation of a light beam (axis Oz of propagation is directed deep into the plane)

It should be noted that in the optical medium the generation of waves of any polarisation (presented in Fig. 3-2) can be implemented, and such waves can be both stationary and progressive. This becomes possible due to the existence of a simple plane harmonic progressive wave in the optical medium, which propagates, for example, in plane xOz (Fig. 3-3, a)

U1 z, t Acoskz  ct .

(3.16)

But in this medium, the same harmonic wave of the same polarisation can propagate at the same velocity

U 2 z, t Acoskz  ct ,

(3.17)

which, however, moves in the opposite direction. Due to this, by the superposition of two harmonic waves (3.16), (3.17) running in opposite directions, a wave can be formed U 3  z , t U 1  z , t  U 2  z , t Acos kz  ct 

 Acos kz  ct Acoskz ˜ cosct  Asinkz ˜ sinct   Acoskz ˜ cosct  Asinkz ˜ sinct

(3.18)

2 Acoskz ˜ cosct ,

which will be stationary. So, in optical media, stationary planar harmonic waves can exist.

Chapter 3. Bending vibrations of the drill string in a vertical well

x

y

y

2A

y

x

x 2A

2A

x

77

y

2A 2B

a

b

c

2A

d

Fig. 3-3 To the problem of superposition of stationary and progressive waves in optical and elastic media

Similarly, we can show the possibility of generating plane-polarised stationary and progressive harmonic waves

V1 z, t Bcoskz  ct

or

V2 z, t Bsinkz  ct

(3.19)

in plane yOz (Fig. 3-3, b). Then, the sum (or superposition)

U1 z, t  V2 z, t Acoskz  ct  Bsinkz  ct

(3.20)

gives a propagating spatial wave with elliptic polarisation (Fig. 3-3, c), and at A B , we get a propagating spatial wave with circular polarisation (Fig. 3-3, d). The reverse procedure can also be performed by decomposing progressive wave (3.20) into several stationary waves

Acoskz  ct  Bsinkz  ct Acoskz ˜ cosct   Asinkz ˜ sinct  Bsinkz ˜ cosct  Bcoskz ˜ sinct

(3.21)

Each of the summands in the right part of this equality is a solution of the wave equation. The reflections stated above can be repeated for elastic infinite isotropic media in which both stationary and running waves can exist, with both planar and elliptic and circular polarisations. And, in the end, the same properties have waves in elastic non-rotating rods that are free from prestressed torque. Then, in the case of circular polarisation of a progressive wave (Fig. 3-3, d), the axial line of the rod takes the shape of a circular cylindrical spiral (left or right), which moves in all cases at the same velocity for a given length, regardless of the direction of propagation. We must also emphasise that in rods of circular cross-section prestressed by torques, as shown in our book, free transverse vibrations can be carried out only in the shape of spiral progressive waves of the form:

u ( z, t )

Acoskz  ct ,

v( z, t )

Asinkz  ct .

(3.22)

Modelling emergency situations in drilling deep boreholes

78

Moreover, spiral wave (3.22) can also be represented as a superposition of stationary waves

u ( z, t )

Acoskz ˜ cosct  Asinkz ˜ sinct ,

v( z , t )

Asinkz ˜ cosct  Acoskz ˜ sinct ,

(3.23)

but here (unlike case (3.21)) each of the summands in right part (3.23) no longer satisfies the wave equation, which can now only be satisfied by the total sum in (3.23) but not separately by each summand. The last property is the explanation for the fact that in the rod prestressed by torque only running spiral waves can propagate. To trace the mechanism of this pattern, we shall consider the effect of each of the prestress factors as well as the rotation of the tubular rod and the motion of the liquid in it on the shape of periodic waves (stationary and progressive). 3.2.3. Analysis of the effect of longitudinal force on the shape of free running waves The task of investigating the wave vibration motions of the rotating rod prestressed by the considered system of forces is multi-parametric and, therefore, difficult to analyse. In this regard, it is advisable to analyse separately the effect of each of the factors on the possible forms of periodic movements of the rod. Let's first consider the simplest case when the rod does not rotate and is free from the influence of prestressing. Then, system (3.1) is reduced to the simplest form of decoupled equations EI

w 2u w 4u  U r Fr 2 4 wt wz

0,

EI

w 2v w 4v  U r Fr 2 4 wt wz

0.

(3.24)

They allow solutions in the form of the simplest functions forms of stationary waves (Fig. 3-4) u z, t

Acoskz ˜ cosct ,

uz, t 0 ,

v z , t 0 , v z , t Bcoskz ˜ cosct ,

(3.25)

with the help of combinations of which one can form any stationary and progressive waves with linear (Fig. 3-5) or elliptical and circular polarisations shown in Figs. 3-2 and 3-3. The lengths of such waves are l 2S k . In this case, the characteristic equations are also simplified

EIk 4  Ur Fr c2

0,

EIk 4  Ur Fr c2

0,

(3.26)

Chapter 3. Bending vibrations of the drill string in a vertical well

x

79

A 2S /k

y

Stationary nodal po int

Stationary nodal po int

z

Stationary nodal po int

Stationary nodal po int

x ci

y

ci

Fig. 3-4 Stationary plane polarised wave

x

Moving nodal point

A i 2S /ki

y

Stationary nodal po int

Stationary nodal po int

Stationary nodal po int

Stationary nodal po int

vi ci /ki

x y

z

ci ci

Fig. 3-5 Progressive plane polarised wave

and the frequencies will be multiple and acquire known values (Fig. 3-6) c1, 2

r k 2 EI U r Fr ,

r k 2 EI U r Fr .

c3, 4

(3.27)

The presented transformations are trivial. However, they can be used to make an important conclusion, according to which the considered harmonic waves are dispersing since dependencies (3.27) between c and k are non-linear, and the propagation velocities of the waves depend on their length. To illustrate the established effects, we shall consider examples for the case





S r14  r24 / 4 3.07 ˜105 m4; T

E

2.1 ˜1011 Pa; I

Uc

7.8 ˜103 kg/m3; Fc



S r12  r22



r2 ˜106 N; M z

4.2 ˜107 N·m;

5.34 ˜103 m2; r1 0.1775 m; r2

0.99 ˜ r1 .

Here, r1 , r2 are the outer and inner radii of the DS pipe. The values of the given mechanical parameters are typical of deep-drilling strings. As frequencies c2 and c4 differ from c1 , c3 only in terms of signs, they are not shown in the figures.

Modelling emergency situations in drilling deep boreholes

ci , s 1

1000

vi , m s

1000

800

800

600

600

c1,3

400

80

400

v1,3

200

200

k , m 1

k , m 1 0

0 0

0.4

0.8

1.2

0

1.6

0.4

a

0.8

1.2

1.6

b

Fig. 3-6 Dispersion curves for frequencies c1 , c3 (a) and dependence of the phase velocities v1 , v3 on the wavelength (b) (T 0 , Mz 0, Z 0 , V 0)

The dynamics of an infinite rod prestressed by longitudinal constant force T is described by an uncoupled system of equations EI

w 2u w 2u w 4u U   F T r r wt 2 wz 2 wz 4

0, EI

w 2v w 2v w 4v U   F T r r wt 2 wz 2 wz 4

0,

(3.28)

from which the frequency dependencies follow (Fig. 3-7): k

r

EIk 2  T  0

EIk 2  T , c3, 4

k

(3.29) EIk 2  T . Ur Fr Ur Fr They indicate that the movements in planes xOz and yOz are also not coupled in this case, but there may be cases when no bending waves can propagate in the rod. This situation occurs under condition c1, 2

r

or

T d EIk 2 .

(3.30)

This means that for arbitrary compressive longitudinal force T  0 the following values can always be found k d  T EI ,

(3.31)

under which vibration and wave periodic movements are not possible. Therefore, dependence (3.31) can be called a condition of non-transmission of progressive bending periodic waves in the rod, and value kcr

 T EI is critical.

Chapter 3. Bending vibrations of the drill string in a vertical well 1

ci , sc-1-1

1000

vi , m s

1000

800

800

600

600

c1,3

400

81

400

v1,3

200

200

1

k , m1

k, m м -1 0

kkcr Wave non-- кр transmission zone 0

0.4

0

0.8

1.2

kkcr Wave non-- кр transmission zone

1.6

0

a

0.4

0.8

1.2

1.6

b

Fig. 3-7 Dispersion curves for frequencies c1 , c3 (a) and dependence of phase velocities v1 , v3 on the wavelength (b) ( T

2 ˜106 N, M z

0, Z

0,V

0)

If the rod is prestressed by longitudinal tensile force T ! 0 , the radical expressions in (3.29) are always positive, and with such T , the distribution of periodic progressive waves of any length is possible. 3.2.4. Analysis of the influence of torque on the shapes of free running waves Spectral analysis of the vibration equations of an infinite rod prestressed by torque M z const is sharply complicated since the equations of motion that describes them take the form: EI

w 2u w 4u w 3v  M  U F z r r wz 4 wz 3 wt 2

0, EI

w 4v w 3u w 2v  M  U F z r r wz 4 wz 3 wt 2

0

(3.32)

and become coupled. They no longer allow solutions of form (3.25) and can only satisfy functions in the shape of spatial circular cylindrical spirals (Fig. 3-8)

uz, t Acoskz  ct ,

vz, t Bsinkz  ct .

(3.33)

Modelling emergency situations in drilling deep boreholes

82

A i 2S /ki

X,x

Z,z

Y, y

vi 2S /ci

X

ci

Вузлові точки Nodal points

Y

відсутні are absent

Fig. 3-8 The shape of vibrations of an infinite non-rotating rod prestressed by torque M z

This feature also leads to a significant change in vibration frequencies ci , which are split and acquire values (Fig. 3-9) c1, 2

r

k

Ur Fr

EIk 2  M z k , c3, 4

k

r

Ur Fr

EIk 2  M z k .

(3.34)

By using calculations similar to the transformations given in subsection 3.1, it can be shown that for frequencies c1, 2 there are ratios

A1 B1

A2 B2

1,

whereas for c3, 4 A3 A4 1. B3 B4 and c3, 4 correspond to the modes of progressive

Therefore, frequencies c1, 2 waves in the shape of right and left spirals, respectively. They propagate at phase velocities v1, 2 v3, 4

c1, 2 k c3, 4 k

r

EIk

r

EIk

2

2



 M z k / U r Fr ,



(3.35)

 M z k / U r Fr .

The graphs of ci k and vi k dependencies for the values of the rod parameters given in subsection 3.2.3 are shown in Fig. 3-9, a, b. It follows from them that the addition of torque M z led not only to the addition of frequencies c3 , c4 but also to a fundamental change in their dependence on M z .

Chapter 3. Bending vibrations of the drill string in a vertical well

1

ci , sc-1

30000

3000

20000

83

vi , m / s

v1

2000

c1

c3

10000

v3

1000

k , m 1

k , m 1 0 0

2

4

6

Non-transmission zone of the left spiral

k cr

0 8

10

0

2

4

6

Non-transmission zone of the left spiral

a

k cr

8

10

b

Fig. 3-9 Dispersion curves for frequencies c1 , c3 (a) and phase velocity dependences v1 , v3 on wave number k (b)

(T

0 , M z = 4.2·10 7 N·m, Z

0)

Indeed, it can be seen that the right and left spirals propagate at different velocities r v1 , r v3 , and when M z EIk the radical expression in second dependence (3.34) turns to zero, and c3 , c4 acquire zero values, and at M z ! EIk frequencies c3 , c4 are lost. Values kcr M z / EI are critical because at k  kcr one of the spiral waves is lost. Therefore, the interval 0  k  kcr is the wave non-transmission zone for waves in the shape of the left spiral. It should also be noted that the velocities of the spiral waves of both types do not depend on the direction of propagation. 3.2.5. Analysis of the influence of rotational motion inertia forces on the free vibration modes In the phenomenon of bending dynamics of the drill string, additional specificity is introduced by its rotation around the longitudinal axis. To study it, it is convenient to introduce two coordinate systems: inertial OXYZ and moving Oxyz rotating together with the rod. It is convenient to consider the movement of each point of the axis line of the rod in rotating coordinate system Oxyz and consider it compound. Then, the equations of spatial vibrations of the rod will take the wellknown form [3, 11, 12, 18]:

Modelling emergency situations in drilling deep boreholes

84

w 2u wv w 4u  U r FrZ 2u  2 U r FrZ  U r Fr 2 4 wt wt wz 4 w v wu w 2v  U r Fr 2 EI 4  U r FrZ 2v  2 U r FrZ wt wz wt

EI

0,

(3.36) 0,

which also follows from system (3.1). As can be seen, these equations are connected, so, in general, the oscillations in this system can only be spatial. For the circular cross-section, the simplest mode of free vibrations is described by relations

uz, t Acoskz ˜ sinct,

vz, t Acoskz ˜ cosct ,

(3.37)

in which frequencies ci are expressed in terms of wave numbers k by relations c1, 2

Z r k 2 EI U r Fr ,

x

c3, 4

Z r k 2 EI Ur Fr .

(3.38)

A i 2S /ki Неподвижна Stationary я узловая point ynodal

Неподвижная Stationary узловая nodal point

точка

ci x y

точка

Неподвижна Stationary я узловая nodal point точка

z

Неподвижна Stationary я узловая nodal point точка

ci

Fig. 3-10 The mode of vibrations of an infinite rod in a rotating coordinate system

X

y

xZ

A i 2S /ki

x

Z

Неподвижная Stationary узлова я nodal point

Y

точка

X

Z,z

Неподвижна Stationary я узлова я nodal point

Неподвижна Stationary я узлова я nodal point

точка

точка

Zci

Y

y

Zci

Fig. 3-11 The mode of vibrations of an infinite rotating rod in an inertial (fixed) coordinate system

Chapter 3. Bending vibrations of the drill string in a vertical well

85

They differ from the frequencies of stationary rods (3.27) only in terms of summands r Z . At the same time, the shape of oscillations is changed significantly. In this case, the waves are stationary and have fixed nodal points. However, each element of the rod moves in a circle at velocity ci in rotating coordinate system Oxyz (Fig. 3-10) and at velocity Z  ci in stationary system OXYZ (Fig. 3-11).

Dispersion curves for frequencies c1 , c3 and the dependence of phase velocities ci k on the wave length at ω = 4s–1 are shown in Fig. 3-12. We should note that in this case there are no non-transmission zones, and the marked periodic movements exist in the entire range of wave number k .

ci , s 1

1000

vi , m s

1000

800

800

600

600

c1

400

400

v1

c3

200

v3

200

k , m1 0

k , m1

0 0

0.4

0.8

1.2

1.6

0

0.4

0.8

a

1.2

1.6

b

Fig. 3-12 Dispersion curves for frequencies c1 , c3 (a) and phase velocity dependences v1 , v3 on wave number k (b) (T

0 , Mz

0, Z

4 s -1 , V

0)

It is also characteristic that since, together with solution (3.37), system (3.36) also has a solution in the form of

uz, t Asinkz ˜ sinct,

vz, t Asinkz ˜ cosct ,

(3.39)

and other similar ones, then using a superposition of these particular solutions, we can also build particular solutions in the form of progressive waves (3.33)

uz, t Acoskz  ct ,

vz, t Asinkz  ct ,

which we will not, however, analyse or comment on.

Modelling emergency situations in drilling deep boreholes

86

3.2.6. Analysis of the effect of internal fluid flow on the free vibration shape of a tubular rod If the flow of an ideal incompressible fluid moves in the cavity of a tubular rod at constant velocity V , the rod loses the possibility of free periodic movements in the modes of stationary waves. To confirm this position, let's write down the equations of the rod dynamics with the fluid flow arising from system (3.1)

w 2u w 2u w 2u w 4u 2  U 2 U U U 0,     V F V F F F f f f f r r f f wzwt wz 2 wt 2 wz 4 (3.40) w 4v w 2v w 2v w 2v 2  U r F r  U f F f 2 0. EI 4  V U f F f 2  2VU f F f wzwt wz wz wt As can be seen, these equations are no longer coupled. Each of these equations admits the simplest solution in the form of a planar progressive harmonic wave in plane xOz u  z , t Acoskz  ct (3.41) for the first equation, and wave in plane yOz v z , t Bcos kz  ct (3.42) for the second equation. For each of these equations, the same characteristic equations follow: EI

EIk 4  V 2 U f F f k 2  2VU f F f kc  Ur Fr  U f F f c 2 EIk  V U f F f k  2VU f F f kc  Ur Fr  U f F f c 4

2

2

2

0,

(3.43)

0,

with the same frequency values

c1, 2

c3, 4

VU f F f k

U r Fr  U f F f

r



k V 2 U p2 Fp2  U r Fr  U f F f EIk 2  V 2 U f F f



U r Fr  U f F f

(3.44)

for each of the waves. Therefore, in the future, we will only analyse waves u z , t (3.41) in plane xOz . It is typical that, if in all the above considered cases frequencies c1 and c2 (and c3 and c4 ) differ just in signs, then here they differ numerically. Thus, formerly, waves with frequency c1 propagated in the direction of axis Oz , waves with frequency c2 c1 propagated in the opposite direction. However, in this case, it is not like this. In fact, the wave velocities are calculated by formulas

v1, 2

c1, 2

VU f F f

k

U r Fr  U f F f

r



V 2 U 2f F f2  U r Fr  U f F f EIk 2  V 2 U f F f

U r Fr  U f F f

Analysis of the radical expression shows that the value of the wave number kcr

V U r Fr U f F f EI U r Fr  U f F f

.

(3.45)

Chapter 3. Bending vibrations of the drill string in a vertical well

87

is critical because the second summand in right part of (3.45) is converted to zero, and it will be possible to propagate only one wave at velocity v1 VU f F f / U r Fr  U f F f .

At k  kcr , the radical expression in (3.45) is negative, and the propagation in a tube with the liquid of waves with length O ! 2S / kcr is impossible. As k increases, starting from k cr , it becomes possible to propagate only in one direction of the two waves with velocities v1 and v2 in (3.45). Further, when in (3.45) the summands in the right part are equal, we have 2VU f F f U r Fr  U f F f ,

v1

v2

0

and further increase in k leads to conditions v1 ! 0 , v2  0 . In this case, one wave

propagates in the direction of flow, the second, against, and v2  v1 . ci , s 1

500

vi , m / s

500

c1

v1

250

250

k , m1

k , m1 0

0 0

0.4

0.8

1.2

1.6

0

c2

-250

0.4

0.8

1.6

v2

-250

-500

1.2

-500

ci , s 1

20

vi , m s

80

v1

70

15

60

c1

10

50

5

k, m 0

-10

k , m 1

40 30

0.04 -5

1

k cr

0.08

0.12

Wave nontransmission zone

c2

0.16

0.04

0.2 20 10 0

-15

k cr

0.08

0.12

Wave nontransmission zone

0.16

v2

-10

-20

-20

a

b

Fig. 3-13 Dispersion curves for frequencies c1 , c2 (a) and phase velocities v1 , v2 dependence on wave number k (b) ( T 0 , M z 0 , Z 0 , V 40 m/s)

0.2

Modelling emergency situations in drilling deep boreholes

88

The graphs of functions ci k , vi k for the above selected values of the mechanical parameters of the system and V 40 m/s are shown in Fig. 3-13. Here, you can see an important effect, which is that velocities v1 , v 2 of the bending wave of the same length depend on the direction of propagation. This effect is similar to the Doppler effect but has a different nature. 3.2.7. Running bending waves in prestressed rotating tubular rods with fluid flow (general case) The characteristic equation for the case of the dynamics of a tubular prestressed rod rotating around its axis was constructed and analysed above. However, to simplify the transformations and make it clearer, analysis of the effect of the inertia forces of the internal fluid flow in these studies is omitted, and simplified equations (3.2) are considered, and the effect of the flow is studied separately in subsection 3.2.6. Therefore, this subsection considers the problem of the propagation of bending waves in the general case when all the above factors are present simultaneously. Then, the substitution of the solution in form (3.5) in (3.1) leads to a homogeneous system of two algebraic equations

>EIk  T  V 4

2

U f F f k 2  2VU f F f ck  U r Fr  U f F f Z 2 

@ >

@

 U r Fr  U f F f c 2 A  M z k 3  2( U r Fr  U f F f )Zc B

>M k z

3

@ >





0,

 2U r Fr  U f F f Zc A  EIk  T  V U f F f k  4

2

@

2

 2VU f F f ck  U r Fr  U f F f Z 2  U r Fr  U f F f c 2 B

(3.46)

0,

from which the characteristic dispersion equation follows:

>EIk  T  V 4

2

U f Ff k 2  2VU f Ff ck  Ur Fr  U f Ff Z 2 

@ >

 Ur Fr  U f Ff c 2 2  M z k 3  2( Ur Fr  U f Ff )Zc

@

2

0,

(3.47)

It is equivalent to two second-degree equations in respect of frequencies ci





EIk 4  T  V 2 U f F f k 2  2VU f F f ck  U r Fr  U f F f Z 2 

 U r Fr  U f F f c 2  M z k 3  2U r Fr  U f F f Zc





0,

EIk  T  V U f F f k  2VU f F f ck  U r Fr  U f F f Z 2  4

2

2

 U r Fr  U f F f c 2  M z k 3  2U r Fr  U f F f Zc

(3.48)

0,

Therefore, each value of wave number k corresponds to four values of cyclic frequency ci

Chapter 3. Bending vibrations of the drill string in a vertical well

c1, 2 c3, 4

VU f F f k

U r Fr  U f F f

Z r

89

D1 , U r Fr  U f F f

(3.49)

VU f F f k

D2 Z r , U r Fr  U f F f U r Fr  U f F f

where D1

>

>VU

f

@

F f k  U r Fr  U f F f Z 2  U r Fr  U f F f u





@

u EIk  T  V U f F f k  U r Fr  U f F f Z 2  M z k 3 , D2

>

4

>VU

f

2

2

@

F f k  U r Fr  U f F f Z  U r Fr  U f F f u





2

(3.50)

@

u EIk 4  T  V 2 U f F f k 2  U r Fr  U f F f Z 2  M z k 3 ,

and four phase velocities

i



1,4 . (3.51) Equalities (3.49), (3.50) are of the most common kind and allow us to analyse forms of periodic motions of infinite rotating tubular rods prestressed by axial forces and torques and containing internal flows of the movable fluid. Depending on the values of the parameters that determine the above factors, periodic movements of such rods can exhibit the properties described above for each of the situations considered. Therefore, it can be argued that, in general, periodic bending movements of drill strings (which are described by such equations) can only be realised in the shapes of progressive circular cylindrical spirals (left or right), and each wavelength of this form of motion will correspond to four values of cyclic frequencies and four values of phase velocities. For some of them, there may be domains of nontransmission, and then the spiral wave of this orientation and this direction may not exist. It is also interesting to note that the velocity of the spiral wave propagation changes its value when the direction of propagation changes. For illustration, in Figs. 3-14−3-15, characteristic (dispersion) curves and vi

ci k ,

phase velocity graphs are shown for the values of T

Z

2˜106 N, M z

4.2 ˜ 107 N·m,

4 s-1 , V 40 m/s, and T 5˜106 N, M z 4.2 ˜ 107 N·m, Z 10 s-1 , V 40 m/s. By analysing these graphs, it can be concluded that the change in the sign of longitudinal force T led to a qualitative change in the velocity graphs only for small values of wave number k (that is, for large wavelengths O ). In the rest of the k domain, there has only been some qualitative change in functions ci k and vi k .

Modelling emergency situations in drilling deep boreholes

90

a

b

Fig. 3-14 Dispersion curves for frequencies c1 , c2 , c3 , c4 (a) and phase velocity v1 ,

v2 , v3 , v4 dependences on wave number k (b) 2 ˜106 N, M z

(T

4.2 ˜107 N·m, Z

4 s -1 , V

40 m/s)

а б Fig. 3-15 Dispersion curves for frequencies c1 , c2 , c3 , c4 (a) and phase velocity v1 ,

v2 , v3 , v4 dependences on wave number k (b) (T

5˜106 N, M z

4.2 ˜107 N·m, Z 10 s -1 , V

40 m/s)

Chapter 3. Bending vibrations of the drill string in a vertical well

91

3.3. Small free bending vibrations of rotating drill strings 3.3.1. Formulation of the problem of free bending vibrations of a drill string in a vertical well Current scientific literature mainly deals with the theoretical modelling of longitudinal and torsional vibrations of vertical strings, while the analysis of bending vibrations is performed only for the construction of the bottom of drill strings without taking into account its interaction with the top of the tubular structure. The problem of free vibrations of the whole body of a long length DS without taking into account its contact interaction with the well walls is set and solved below. Despite the fact that the constructed solutions are of limited use for oil and gas drill strings due to the collisions of their pipes with the well walls in places of large amplitude vibrations, they are, however, of interest for establishing the places of possible bending vibrations of drill strings and for estimating the value of the period of these vibrations [5, 6, 11, 13, 18, 24, 27]. In addition, the obtained solutions are of significant interest for the drill strings of coal mine wells with diameters reaching up to 5 m and lengths up to 2,000 m. The problem of DS free vibrations ˗ like the problem of their stability ˗ refers to the Sturm-Liouville boundary value problem, but it is more difficult because, as shown in paragraph 3.2, the modes of vibrations are not stationary, and, in its solution, it is necessary to analyse a system of four coupled ordinary differential equations of the fourth order that are also singularly perturbed. At this point, it is solved using the above developed methodology of the combined application of the initial parameters method, the splicing procedure, and Godunov orthogonalisation. Let's write down again equations (3.1) of vibrations of a rotating beam with an internal fluid flow, stressed by longitudinal force T and torque M z EI

wv wv · w 4u w § wu · w 2 §  ¨ T ¸  2 ¨ M z ¸  ( UF  U f F f )Z 2u  2( UF  U f F f )Z  4 wt w w w z z z wz ¹ © ¹ wz ©

 V 2 U f Ff

w 2u w 2u w 2u 2 U ( U U )    F F V F f f f f wzwt wt 2 wz 2

0,

(3.52) wu wu · w 4v w § wv · w 2 §  EI 4  ¨ T ¸  2 ¨ M z ¸  ( UF  U f F f )Z 2v  2( UF  U f F f )Z wt wz ¹ wz © wz ¹ wz © wz  V 2 U f Ff

w 2v w 2v w 2v 2 U ( U U )    F F V F f f f f wzwt wt 2 wz 2

0.

Remember that here u ( z , t ), v( z , t ) are the functions of transverse displacements of the DS pipe in the direction of axes Ox, Oy rotating with the string.

Modelling emergency situations in drilling deep boreholes

92

The presence of summands (3.52) with coefficients M z and UF  U f F f Z makes this system coupled, which excludes the possibility of DS vibrations along the planar modes with one common phase. To solve the problem of free vibrations of the DS, eight boundary conditions need to be added to equations (3.52). Let at ends z 0 , z L the string be hinge supported, then the boundary equalities u (0) v(0) 0, u czzc 0 vczzc 0 0, u ( L) v( L) 0, u czzc L vczzc L 0

(3.53)

are realised. A homogeneous system of relations (3.52), (3.53) determines the DS free vibrations. The values of its parameters, where it has non-trivial periodic solutions in time, are eigen values, and the solutions themselves are modes of free vibrations. They can only be presented in the form of

uz, t U s z sin ct  U c z cos ct , vz, t Vs z sin ct  Vc z cos ct ,

(3.54)

where c is the frequency of free vibrations; U s ( z), U c ( z), Vs ( z), Vc ( z) are the free vibrations mode functions to be determined. By substituting (3.54) in (3.52) and equating the sum of summands with sinct and cosct multipliers to zero separately, we obtain a system of four ordinary differential equations d 4U s d 2U s d 3V EI T  M z 3s  ( UF  U f F f )Z 2U s  2( UF  U f F f )Z cVc  4 2 dz dz dz 2 d Us dU  V 2 U f Ff  2VU f F f c c  ( UF  U f F f )c 2U s 0, dz dz 2 4 2 3 d Uc d Uc dV T  M z 3c  ( UF  U f F f )Z 2U c  2( UF  U f F f )Z cVs  EI dz dz 4 dz 2 2 d Uc dU s  V 2 U f Ff  2VU f F f c  ( UF  U f F f )c 2U c 0, dz dz 2 (3.55) d 4Vs d 2Vs d 3U s EI 4  T  Mz  ( UF  U f F f )Z 2Vs  2( UF  U f F f )Z cU c  dz dz 2 dz 3 d 2Vs dV  V 2 U f Ff  2VU f F f c c  ( UF  U f F f )c 2Vs 0, 2 dz dz d 4Vc d 2Vc d 3U c EI T  Mz  ( UF  U f F f )Z 2Vc  2( UF  U f F f )Z cU s  dz 4 dz 2 dz 3 d 2Vc dV  V 2 U f Ff  2VU f F f c s  ( UF  U f F f )c 2Vc 0. 2 dz dz

Chapter 3. Bending vibrations of the drill string in a vertical well

93

It corresponds to the system of boundary conditions arising from (3.53)

U s 0 U c 0 Vs 0 Vc 0 0 , U s L U c L Vs L Vc L 0 , U scc, zz 0 U ccc, zz 0 Vscc, zz 0 Vccc, zz 0 0 , U scc, zz L U ccc, zz L Vscc, zz L Vccc, zz L 0 .

(3.56)

Values ci , where system (3.55), (3.56) has trivial solutions together with non-trivial ones, are eigen values. They correspond to the natural vibration frequencies of the DS. To calculate frequencies ci at fixed values T , M z , Z, V , we use the item-byitem examination method. When using it, we write system (3.55), (3.56) in vector form: G dy G G G (3.57) F ( z ) y  c 2Gy  cHy , dz G G Ay (0) 0, By ( L) 0. (3.58) G Here, y (z ) is the sixteen-dimensional vector of indeterminate values U s ( z), U c ( z), Vs ( z), Vc ( z) that includes the required variables and their derivatives; F (z ) is the variable matrix, G and H are the constant matrices of dimension 16 u 16 ; A, B are the 8 u 16 constant matrices built on the basis of boundary conditions (3.56). The solution of the Sturm-Liouville problem for linear system (3.57), (3.58) is carried out using the method described above for studying drill string stability with the method of initial parameters and Godunov orthogonalisation. The only difference is that in this case the general order of ordinary differential equations (3.55) (or (3.57)) is brought to sixteen. As above, the solutions of system (3.57) are built in the form of Cauchy G G (3.59) y( z) Y ( z)C, where Y (z ) is the Cauchy matrix of dimension 16 u 16 of the system (5.10) particular solutions with initial conditions Y (0) E ; E is the unit matrix; G C (c1 , c2 ,...,c16 )T is the desired 16-dimentional vector. Its components are found from a system of linear algebraic equations, which is built by substituting right part (3.59) into the left parts of conditions (3.58). Values ci at which the determinant of the coefficient matrix in (3.58) vanishes are equal to the frequencies of free vibrations of the elastic system. The modes of free vibrations are built using equality (3.59) by G substituting the right part of vector C . When this approach is implemented, matrix Y (z ) is constructed by integrating system (3.57) using the Everhart method. We must

Modelling emergency situations in drilling deep boreholes

94

emphasise that this method has increased accuracy compared to the Runge-Kutta method. Its application in this case is justified as for long strings the problem is (as noted in Chapter 2) singularly perturbed [7]. Therefore, its solution does not converge well, and more accurate approaches are advisable to ensure the required accuracy. 3.3.2. Testing the developed method To verify the reliability of the calculations, a simpler case is considered in which system (3.57), (3.58) allows an analytical solution. Let's assume that M z 0 , T const , and there is no internal flow of drilling fluid. Then, system (3.55) is divided into two uncoupled subsystems d 4U s d 2U s EI  T  UFZ 2U s  2 UFZ cVc  UFc 2U s 0 , dz 4 dz 2 d 4Vc d 2Vc (3.60) EI T  UFZ 2Vc  2 UFZ cU s  UFc 2Vc 0 ; 4 dz dz 2 d 4U c d 2U c EI T  UFZ 2U c  2 UFZ cVs  UFc 2U c 0 , 4 2 dz dz d 2Vs d 4Vs EI 4  T 2  UFZ 2Vs  2 UFZ cU c  UFc 2Vs 0 . (3.61) dz dz Subsystem (3.60) has solution nSz nSz U s ( z ) U sn sin Vc ( z ) Vcn sin , , (3.62) L L where U sn , Vcn are the required coefficients. Substituting (3.62) into (3.60), we obtain a homogeneous system of algebraic equations n 4S 4 n 2S 2 [ EI 4  T 2  UF (Z 2  c 2 )]U sn  2 UFZ cVcn 0 , L L n 2S 2 n 4S 4 2 UFZ cU sn  [ EI 4  T 2  UF (Z 2  c 2 )]Vcn 0 . (3.63) L L It has non-trivial solutions under condition n 2S 2 n 4S 4 ( EI 4  T 2  UFZ 2  UFc 2 ) 2  4 U 2 F 2Z 2 c 2 0 , (3.64) L L where the expressions for free vibration frequencies cn of the rotating rod come from

· 1 §¨ n 2S 2 (3.65) EI 2  T ¸¸ . ¨ UF © L ¹ To test the method of the numerical study of free vibrations of drill strings, the values of free vibration frequencies of tubular rods, which rotated and were stretched by constant force T , were found numerically. The case of L = 7,000 m, d1 0.355 m, c1,2,3, 4

d2

0.327 m, ω = 4s–1, T

rZ r

nS L

8˜106 N was considered. With formula (3.65), the first

Chapter 3. Bending vibrations of the drill string in a vertical well

95

frequencies c1,1 4.11736365 s 1 for n 1 and c1, 2 3.76527228 s 1 for n 2 were found. These values coincide up to the eighth digit with values c1,1 and c1, 2 , which were found numerically according to the proposed method. 3.3.3. Analysis of the influence of the angular velocity of DS rotation on the frequencies and modes of its free vibrations Using the proposed method based on relations (3.52)–(3.59), the problems of determining the frequencies of free vibrations of the DS prestressed by torque M z and longitudinal force T  z , linearly varying along axis OZ , are solved. With its lower end, the DS rests on the bottom of the drill hole. So it is affected by the compressive force of reaction T ( L) R 1.6 ˜105 N and longitudinal tensile force T (0) G  R acts on the upper end of the DS, where G is the gravity of the entire DS calculated taking into account the action of the hydrostatic lifting force from the drilling fluid. At first, the effect of fluid flow on free vibrations was not taken into account. It is easy to notice that free vibration frequencies c i are significantly dependent on angular velocity Z of the DS rotation as well as on torque M z . Let's investigate the influence of angular velocity Z on the free vibration frequencies for strings with length L = 1,000 m and L = 7,000 m. We can find out how the frequencies of free vibrations change when the angular velocity of the string increases, taking the values Z 2 s 1 , Z 4 s 1 , Z 6 s 1 , Z 10 s 1 . In each of these cases, the calculations are made for torque values of M z

8 ˜ 10 4 N·m and

M z 15.8 ˜104 N·m. The dependencies of free vibration frequencies c i of the drill string on its angular velocity are shown in Table 3-1 (for L = 1,000 m) and Table 3-2 (for L = 7,000 m). Table 3-1 Values of frequencies of drill string free vibrations (L = 1,000 m)

Z (rad/s) 2 4 6 10

Mz (N·m) 8·104 15.8·104 8·104 15.8·104 8·104 15.8·104 8·104 15.8·104

c1 (rad/s) 0.23067 0.05914 0.04519 0.11196 0.15206 0.08326 0.02084 0.09766

c2 (rad/s) 0.45599 0.17295 0.12681 0.14811 0.42212 0.20544 0.17151 0.25179

c3 (rad/s) 0.67822 0.28684 0.21549 0.36801 0.70278 0.37659 0.32907 0.44217

Modelling emergency situations in drilling deep boreholes

96

From Table 3.1, it can be seen that for a string with a length of L = 1,000 m and torque M z 8˜104 N·m with increasing angular velocity Z from 2 s–1 up to 4 s–1 the first three free vibration frequencies are significantly reduced. With a further increase in Z (from 4 s–1 to 6 s–1), they increase and then, when the angular velocity reaches high values (ω = 10 s–1), decreases again. So, the lower vibration frequency c1 of DS when Z increases from 2 s–1 to 10 s–1 decreases by an order. If twice as much torque is applied to the DS with a length of L = 1,000 m, ( M z 15.8 ˜104 N·m), then the dependence of the frequencies of free vibrations on the angular velocity becomes different. So, lower frequency c1 with an increase in Z from 2 s–1 up to 4 s–1 increases about twofold. With a further increase in the angular velocity of Z to 10 s–1, it, slightly decreasing, remains at a constant level. Table 3.2 Values of frequencies of drill string free vibrations (L = 7,000 m)

Z (rad/s) 2 4 6 10

Mz (N·m) 8·104 15.8·104 8·104 15.8·104 8·104 15.8·104 8·104 15.8·104

c1 (rad/s) 0.010677 0.02661 0.02745 0.02153 0.02574 0.009339 0.01678 0.03625

c2 (rad/s) 0.03101 0.03630 0.03769 0.04360 0.04161 0.05799 0.05473 0.10799

c3 (rad/s) 0.05223 0.08942 0.09251 0.06491 0.06729 0.07674 0.08822 0.17982

As can be seen from Table 3.2, for drill strings with a length of L = 7,000 m at

8 ˜ 10 4 N·m and M z 15.8 ˜104 N·m, the first three frequencies of free vibrations take small values and with increasing angular velocity Z from 2 s–1 up to 4 s–1 vary slightly. Fig. 3-16 and 3-17 represent the modes of free vibrations of the drill string with a length of L = 1,000 m for different values of its angular velocity and torque. Fig. 3-16 shows the modes of free vibrations of DS with L = 1,000 m torque M z

corresponding to the first three frequencies for the case ω = 2 s–1, M z

8 ˜ 10 4 N·m.

Chapter 3. Bending vibrations of the drill string in a vertical well

c1 =0.23067 s-1

c 2 =0.45599 s-1

97

c 3 =0.67822 s-1

Fig. 3-16 Modes of free vibrations of a drill string (L=1 km, ω=2 s-1, Мz=8·104 N·m)

c1 =0.09766 s-1

c 2 =0.25179 s-1

c 3 =0.44217 s1

Fig. 3-17 Modes of free vibrations of a drill string (L=1 km, ω=10 s-1, Мz=15.8·104 N·m)

Modelling emergency situations in drilling deep boreholes

98

As can be seen from this figure, the upper half of the drill string is slightly deformed. Its lower half represents a spiral curve, in the upper part of which the steps of the spirals and the amplitude of the vibrations are much larger than in the lower one. Fig. 3-17 illustrates the free vibration modes of a DS with a length of L = 1,000 m corresponding to the first three frequencies for case ω = 10 s–1, M z 15.8 ˜104 N·m. Comparing Fig. 3-16 and 3-17, we must note that with increasing angular velocity Z from 2 s–1 to 10 s–1 the shapes of free vibrations vary significantly. With high angular velocity, the string sections, which have the shape of a spiral curve, periodically alternate with short sections, which deform slightly. And with an increase in Z , the lower part of the spiral curve (in the vicinity z L ) changes little. Fig. 3-16 and 3-17 below show the projections of the DS axial line in the deformed state on horizontal plane xOy of the rotating coordinate system for a string with a length of L = 1,000 m at different angular velocities. The same projections are shown in Fig. 3-18 a, b.

a

c1 =0.09766 s-1

c 2 =0.25179 s-1

c 3 =0.44217 s-1

b Fig. 3-18 Vibration modes of the drill string with a length of 1 km (projections on a horizontal plane)

Chapter 3. Bending vibrations of the drill string in a vertical well

99

By analysing Fig. 3-19–3-22, we can get an idea of the effect of the angular velocity of DS rotation with a length of L = 7,000 m on the shape of its free vibrations. These pictures show the shapes of free vibrations of a drill string with a length of L = 7,000 m, corresponding to the first three vibration frequencies, at ω = 2 s–1 (Fig. 3-19), ω = 4 s–1 (Fig. 3-22), and ω = 10 s–1 (Fig. 3-20). All these modes have the shapes of spiral curves. At low angular velocities (ω = 2 s–1, ω = 4 s–1) in the upper part of the DS (Fig. 3.19 and 3.22), the steps of the spirals are larger than in its lower part and the nodal points appear on the spirals. With an increase in angular velocity Z from 2 s–1 to 4 s–1, the amplitudes of the vibration in the lower part of the DS increase. With increasing angular velocity Z to 10 s–1, the shape of DS free vibrations becomes more complex (Fig. 3-20). Spiral curves have variable amplitude, the number of nodal points increases.

c1 =0.010677 s-1

c 2 =0.03101 s-1

c 3 =0.05223 s-1

Fig. 3-19 Shapes of free vibrations of a drill string with a length of L 7,000 m, ω = 2 s–1, M z 8˜104 N·m

Modelling emergency situations in drilling deep boreholes

c1 =0.03625 s-1

c 2 =0.10799 s-1

100

c 3 =0.17982 s-1

Fig. 3-20 Modes of free vibrations of a drill string with a length of L 7,000 m, ω = 10 s–1, M z 15.8 ˜104 N·m

c1 =0.06511 s 1

c 2 =0.32974 s 1 c 3 =0.62039 s 1

Fig. 3-21 Modes of free vibrations of a drill string with a length of L 500 m, ω = 4 s–1, M z 8 ˜ 104 N·m

Chapter 3. Bending vibrations of the drill string in a vertical well

c1 =0.02745 s 1

101

c 2 =0.03769 s 1 c 3 =0.09251 s 1

Fig. 3-22 Modes of free vibrations of a drill string with a length of L 7,000 m, ω = 4 s–1, M z 8˜104 N·m

Changes in the trajectories of motion during one period of the points of a drill string with a length L = 7,000 m in the planes of cross sections with increasing of angular velocity Z from 4 s–1 to 10 s–1 are shown in Fig. 3.23–3.26. The trajectory of DS different points was built on them at angular velocities ω = 4 s–1 (Fig. 3.23, 3.24) and ω = 10 s–1 in rotating (Fig. 3-25) and fixed (Fig. 3-26) coordinate systems. y

Y

x

O

a

X O

b

Fig. 3-23 Trajectories of the drill string element motion at point z = 3,885 m in rotating (a) and stationary (b) coordinate systems ( c 2 = 0.03769 s–1, ω = 4 s–1, M z 8˜104 N·m, L = 7,000 m)

Modelling emergency situations in drilling deep boreholes

1.50

102

1.50

1.00

1.00

0.50

0.50

0.00

0.00

-0.50

-0.50

-1.00

-1.00

-1.50

-1.50

-1.50

-1.00

-0.50

0.00

0.50

1.00

-1.50

1.50

-1.00

a

-0.50

0.00

0.50

1.00

1.50

b

Fig. 3-24 Trajectories of the drill string element motion at point z = 5,691 m in rotating (a) and stationary (b) coordinate systems ( c1 = 0.02745 s–1, ω = 4 s–1, M z 8 ˜ 104 N·m, L = 7,000 m) 2.00

1.00

0.00

-1.00

-2.00 -2.00

-1.00

0.00

1.00

2.00

Fig. 3-25 Trajectories of the drill string element motion at point z = 6,188 m in the rotating coordinate system (L=7,000 m, ω=10 s-1, Мz=15.8·104 N·m, с=0.10799 s-1)

Chapter 3. Bending vibrations of the drill string in a vertical well

103

2.00

0.20

1.00

0.00

0.00

-1.00 -0.20

-0.20

0.00

0.20

-2.00 -2.00

-1.00

a

0.00

1.00

2.00

b

Fig. 3-26 Trajectories of the drill string element motion at point z = 6,188 m in the stationary coordinate system (L=7,000 m, ω=10 s-1, Мz=15.8·104 N·m, с=0.10799 s-1)

t=0

t=T/4

t=T/2

t=3T/4

t=T

Fig. 3-27 The nature of changes in the shape of DS motion during one period (L = 7,000 m, ω=10 s–1, M z 15.8 ˜ 104 N·m, с=0.10799 s-1)

In conclusion, we should draw attention to the nature of changes in the shapes of DS movement during one period. Fig. 3-27 shows the shapes of DS free vibrations with a length of L = 7,000 m rotating at angular velocity ω=10 s–1, with time

Modelling emergency situations in drilling deep boreholes

104

moments t=0, t=T/4, t=T/2, t=3T/4, and t=T, where T 2S c is the period of oscillations, c = 0.10799 s-1 is the second oscillation frequency. It can be seen that these modes have the shapes of ribbons twisted around the axial line and rotating relative to it. References to Chapter 3 1. Aarrestad Thor Viggo, Kyllingstad Age. An experimental and theoretical study of a coupling mechanism between longitudinal and torsional drill string vibrations at the bit// SPE Drilling Engineering. – March 1988. – P. 12–18. 2. Ashley H., Haviland G. Bending vibrations of a pipeline containing flowing fluid // Journal of Applied Mechanics. – 1950. – V.17. – P. 229-232. 3. Babakov I.M. Theory of Oscillations // – M.: Nauka.1968. 559 p. (in Russian). 4. Barakat Elie R., Stefan Miska, Takach N. The effect of hydraulic vibrations on initiation of buckling and axial force transfer for helically buckled pipes at simulated horizontal wellbore conditions // SPE / IADC Drilling Conference, 20-22 February 2007, Amsterdam, Netherlands. 5. Beederman V.L. Mechanics of Thin-Walled Structures // – Moscow “Mashinostroenie”. – 1977. – 488 p. (in Russian). 6. Borshch E.I. Vashchilina E.V., Gulyayev V.I. Spiral progressive waves in elastic rods // Proceedings of the Russian Academy of Sciences. Solid Вody Mechanics 2009 No. 2, pp.143-149. (in Russian). 7. Chang K, Howes F. Nonlinear Singularly Perturbed Boundary Value Problems// M., Mir. – 1988. – 247 p. 8. Chen David C-K, Mark Smith, Scott LaPierre. Advanced drill-string dynamics system integrates real-time modelling and measurements// SPE Latin American and Caribbean Petroleum Engineering Conference, 27– 30 April 2003, Port-ofSpain, Trinidad and Tobago. – P. 97– 104. 9. Christoforou A.P., Yigit A.S. Dynamic modelling of rotating drill strings with borehole interactions // Journal of Sound and Vibration. – 1997. – V.206. – №2.– P. 243–260. 10. Dashevskiy D., Dahl T., Brooks A.G., Zurcher D., Lofts J.C., Dankers S. Dynamic depth correction to reduce depth uncertainty and improve MWD/LWD log quality// SPE Drilling & Completion, 2008, V. 23, № 1, – P. 13– 22. 11. Den-Hartog J. P. Mechanical Vibrations// – M.: GIFML, 1960. – 580 p. (in Russian). 12. Elishakoff I., Kaplunov J, Nodle E., Celebrating the centenary of Timoshenko's study of effects of shear deformation and rotary inertia// Applied Mechanics Reviews 2015, V. 67, p. 1 – 11. 13. Feodosyev V.I. Selected problems and questions on the resistance of materials // – M.: Nauka, 1967. – 237 p. (in Russian).

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14. Ford Brett J. The genesis of torsional drill string vibrations// SPE Drilling Engineering. – September 1992. – V.7. – P. 168–174. 15. Glazunov S.M. Torsion oscillations of deep drill strings in viscous liquid medium// Bulletin of the National Transport University 2015, №1(31). p. 96101. (in Ukrainian). 16. Gulyayev V.I., Borshch O.I., Free vibrations of drill strings in super deep vertical bore-wells. Journal of Petroleum Science and Engineering. – 2011 – V.78. – P.759 – 764. 17. Gulyayev V.I., Gaydaychuk V.V., Abdullayev F.Y. Self-excitation of unstable oscillations in tubular systems with moving masses // Prikladnaya Mekhanika. – 1997. – No 3. – pp. 84-90. (in Russian). 18. Gulyayev V.I., Vashchilina E.V., Borshch E.I. Cylindrical spiral waves in rotating twisted elastic rods // Prikladnaya Mekhanika. – 2008. Vol. 44, No. 3. pp. 125-134. (in Russian). 19. Gulyayev, V.I., Tolbatov E.Yu. Dynamics of spiral tubes containing internal moving masses of boiling liquid. Journal of Sound and Vibration 2004. – 274(2), P. 233–248. 20. Gulyayev, V.I., Tolbatov E.Yu. Forced and self-excited vibrations of pipes containing mobile boiling fluid clots. Journal of Sound and Vibration 2002. – 257(3), 425–437. 21. Iyoho A.W., Meize R.A., Millheim K.K., Crumrine M.J. Lessons from integrated analysis of GOM drilling performance // SPE Drilling & Completion. – March 2005. – P. 6–16. 22. Jansen J.D. Whirl and chaotic motion of stabilized drill collars// SPE Drilling Engineering. – June 1992.– V.7. – №2. – P. 107–114. 2011.-78. P. 759-764. 23. Jonggeun Choe, Jerome J.Schubert, Hans C. Juvkam-Wold. Well-control analyses on extended-reach and multilateral trajectories // SPE Drilling & Completion. – June 2005. – P. 101–108. 24. Kerimov E.G. Dynamic Calculations of the Drill String // – M.: Nedra, 1970. – 157 p. (in Russian). 25. Myslyuk M.A., Rybchych I.I., Yaremiychuk G.S. Drilling of wells. Vol. 3, Vertical and directional drilling // – Kiev: “Interpress LTD”. – 2004. – 294 p. (in Ukrainian). 26. Skrypnik S.G., Golovin A.A., Ryabikhina S.M. et al. Some questions of complex research of dynamic loads in the system of the lifting mechanism of the drilling rig – drill string// Machines and Oil Equipment. 1977. – №10. – pp. 21-24. (in Russian). 27. Tsigler G. Fundamentals of the theory of structural stability // – M.: Mir, 1971. – 192 p. (in Russian).

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EXCITATION OF TORSIONAL SELF-OSCILLATIONS OF STRINGS IN DEEP WELLS

One of the least investigated phenomena in the practice of deep-well drilling is self-excitation of the string torsion vibrations as a result of periodic interruption of the contact grip of bit cutters with the rock being destroyed. Such effects are widely found in nature and technology in the form of sound squeaks and disruptive selfoscillations in the process of cutting various parts and materials. The uncertainty in the understanding of this phenomenon in the drilling process is due to two factors. The first one is related to the complexity of the process that includes the essentially non-linear almost wave nature of the oscillations of the extended DS and bifurcation transitions from the stationary rotation of the bit to its oscillatory movements, which, moreover, are relaxation (almost discontinuous). The second factor is stipulated by the complexity of the mathematical model that describes these vibrations. It includes the equation of propagation of torsion deformation waves along the length of the string with essentially non-linear boundary equation at the lower end, which is formulated from the condition of dynamic friction interaction of the bit with the drill well wall. Using special transformations, it is reduced to a non-linear ordinary differential equation with a delay argument, which is also singularly perturbed due to the relations between the geometric and mass parameters of the system. These factors lead to rather complex forms of solutions of the constructed equations that have the shape of relaxation (almost discontinuous) oscillations and are transforming as a result of transitions through the states of birth and loss of limit cycles (Poincare–Andronov–Hopf bifurcations). As shown in this book, the solutions under construction undergo additional complications caused by overlays on periodic functions with large-scale discontinuities of additional small-scale discontinuities of velocities quantised with constant time intervals equal the time of passage by the torsional wave of doubled length of the DS. Other models of these processes, which have a simpler structure but are associated with greater difficulties in implementing their solutions due to the additional consideration of the viscous friction of the string in a liquid medium, are also considered. Since all features of the occurrence of torsional vibration in drill strings in respect of the considered task have not been studied in combination but analysed in different formulations based on the examples of different objects and phenomena, we shall look at a review of different aspects of this problem that are available in the scientific literature.

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4.1 Self-oscillating systems and methods of their analysis Self-oscillations are continuous oscillations of a non-linear dissipative dynamic system supported by external energy sources, whose type and properties are determined by the system itself and do not depend on initial conditions (at least in the finite boundaries). Despite the fact that the phenomenon of excitation of mechanical selfoscillations has been known for a long time [1, 29, 30], the beginning of its systematic study can be attributed only to the beginning of the 20th century. A wellknown confirmation of this is the fact that the terms ‘self-oscillation’ and ‘selfoscillating systems’ were first introduced by A. A. Andronov in 1928 in the work ‘Limit cycles of Poincare and the theory of self-oscillations,’ which he reported at the Fourth Congress of Russian physicists [1, 30]. In non-linear dynamical systems, relaxation self-oscillations (self-oscillations of the first kind) and self-oscillations close to harmonic ones (self-oscillations of the second kind, or Thomson ones [6, 28, 29, 38, 41]), are distinguished. In this case, if their internal difference is stipulated by the values of the coefficients of the resolving equations of the system motion, the external difference appears in the form of motion. So, Thomson self-oscillations have a shape close to sinusoidal, while the motion shape of relaxation self-oscillations is sharply non-sinusoidal or even almost discontinuous (for velocities function). Due to this peculiarity and taking into account the functional properties of the non-linear factor [28,38, 41], it becomes possible to study relaxation or harmonic self-oscillating processes in electrical and mechanical systems that are described qualitatively by different mathematical models (singular perturbed or regular non-linear differential equations). Electric relaxation self-oscillations are widely used in measuring equipment, remote control, automation, and other electronics. Various generators are used to create them, for example, a blocking generator, a multi-vibrator, RC-generators, etc. The study of these oscillations began in the 1920s in the works of B. van Der Pol [41]. They have been studied in detail in this area thanks to the possibility of using experimental methods of analogue modelling. One of the most important causes of self-oscillations in mechanical systems is the presence of non-linear friction forces. Frictional—that is, stipulated by friction— mechanical self-oscillations are quite a common class of phenomena in nature that play a fundamental role in technical applications. A clear and beautiful illustration of this phenomenon is the vibrations of the violin string under the pressure of a bow that is moving evenly. This is also true for the voice sounds of people and the sounds of birds and animals carried out through the self-oscillations of the muscular tissues of the throat in their frictional interaction with the air flow. The same effect is present in the self-oscillations of the water tap

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for certain water flows. This is accompanied by unpleasant sounds for the human ear. More serious (emergency) consequences may be caused by the self-oscillating flutter of different structural elements of aircraft and the self-oscillation of high-rise structures of unstable aerodynamic shape when air flows around them at a certain speed. Negative self-oscillating effects can also be attributed to the oscillation of the cutters of lathes and planers. These self-oscillating effects prompted by nature can play a positive role. They are used in the creation of wind musical instruments (organ, harmonica, flute, and many others). A less obvious but more ingenious application of self-oscillations takes place in the creation of time mechanisms, technical devices (mechanical regulators), and various generators of electrical and electromagnetic oscillations used in electrical engineering, radio engineering, and electronics. Self-oscillations in mechanical systems caused by friction are analysed in detail in the works of V. O. Kononenko [29]. The presentation of various aspects of the problems of theoretical and experimental research on self-oscillations of the noted types can be found in comprehensive and original works [1, 28, 29, 30] that provide a wide bibliography of this scientific subject. The review shows that researchers studying the self-excitation mechanism of frictional self-oscillations—at least to the mid-1960s—were limited to considering mainly model tasks for mechanical systems with one degree of freedom, choosing as a source, as a rule, either the Froude pendulum (Fig. 4-1,a) or the simplest single-mass Van der Pol model [41]. Such systems can also include a disc model with brake pads (Fig. 4-1,b) and a conveyor belt with a load (Fig. 4-1,c). After processing and systematisation of the obtained scientific data, it was found that the main role in the effect of the selfexcitation of mechanical self-oscillations belongs to the non-linear force of dry sliding friction of frictionally contacting bodies. However, when choosing the type of sliding friction curve and its analytical description, a wide variety was allowed [1, 6, 30]. There is no single point of view on this issue today. As for the possibilities of studying mechanical self-oscillating systems that possess two or more degrees of freedom, they found that they were only noted in a small number of works [29, 30]. At the same time, studies on self-oscillation in systems with aftereffect, which are described by equations with an argument that is delayed, look rather fragmented [36, 37]. Dynamic processes inherent to frictional mechanical self-oscillating systems are modelled by non-linear differential equations, for which a number of mathematical methods of non-linear analysis have been developed. Its priority areas include topological (qualitative) methods of graphic integration [1, 6, 30, 39] applicable to arbitrary systems with both large and small non-linearity as well as

Chapter 4. Excitation of torsional self-oscillations of strings in deep wells

Z

109

3

4

M

c

x

M

v

Z 2 1 a)

b)

c)

Fig. 4-1 The simplest mechanical models of self-oscillating systems: a = Froude pendulum b = disc with rotating brake pads c = conveyor belt with a load

analytical methods of successive approximations [1, 6, 29, 30, 38, 39, 41] that have limited applicability and are intended primarily for finding solutions to differential equations close enough to linear. Moreover, if qualitative integration methods (due to the ‘extreme complication of the topological pattern of the integral curves caused by an increase in the number of measurements of the phase space’) are associated with the limited capabilities of the study of self-oscillating systems with no more than two degrees of freedom, the general deficiency of analytical methods based on the hypothesis of the presence of the generating solution is a non-strictness of boundaries of the obtained results validity due to the relative uncertainty of the small input positive parameter [6]. The Thomson type of frictional mechanical self-oscillating systems, which are described by regular quasi-linear differential equations, belong to the class of oscillatory systems that allow their relatively complete analytical study by using different mathematical methods of perturbation theory. As is known, one of the most developed versions of these methods is the strictly justified asymptotic Krylov– Bogolyubov methods that are an effective mathematical device for a fairly wide range of problems of quasi-linear dynamics [39]. However, it must be recognised that these methods have not yet been properly applied in the general theory of frictional self-oscillations. We should mention that the self-oscillation of the drill string bit by its nature is closest to the self-oscillation of the cutters of a lathe or planer where the role of frictional forces plays the cutting force. Therefore, it is possible that the qualitative results obtained during analysis of the vibrations of cutters and other mechanical systems can be applied in the study of torsional self-oscillations of the DS bit. It is

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also important to take into account the small number of publications that exist on the subject. Despite the fact that torsional vibrations of the drill string bit are a dynamic phenomenon that occurs in the practice of drilling more often than others, it is only discussed in a few scientific publications [14–18, 19–23]. It would seem that this is because of its complexity. This issue is almost not touched upon in Ukrainian and Russian-language scientific literature. These questions began to be studied in a rather simple way in the 1960s (J. J. Bailey, I. Finnie [3, 1960], I. Finnie, J. J. Bailey [13, 1960], D. W. Dareing, B. J. Livesay [11, 1968]). It was later established (R. I. Leine, D. H. van Campen, L. van den Steen [31, 1998]) that the main factor determining the complexity of the phenomenon under consideration and its mathematical model is the discontinuous nature of the function of the static or kinetic mechanism of friction, which underlies this phenomenon. Most of the studies used a static friction model (G. W. Halsey, A. Kyllingstad, T. V. Aarrestad, D. Lysne [24, 1986], N. Challamel, E. Sellami, E. Chenevez, L. Gossuin [8, 2000]). It is noticeable that these oscillations were simulated on the model of a torsion pendulum with one degree of freedom (R. Dawson, Y. Q. Lin, P. D. Spanos [12, 1987], Y. Q. Lin, Y. H. Wang [32, 1991], J. F. Brett [5, 1991], J. D. Jansen [26, 1992]). Although such simplified models led to some understanding of the effect of torsional oscillations of the DS, they did not make it possible to establish their general patterns, because they ignored the distributed nature of the DS mechanical properties. Therefore, models with several degrees of freedom were used later. Aarrestad Thor Viggo et al. studied the torsional vibration perturbation by means of the motion of the drill bit over the grooves of multi-petal profile on well surface and analysed their effect on longitudinal oscillations. J. Ford Brett [5] showed that if the coefficient of static friction exceeds the coefficient of dynamic friction, then the torsional self-oscillations with drill string sticking (seizing) and slipping are perturbed. Besaisow Amjad A. [4] used simplified models of connected torsional, longitudinal, and flexural vibrations of the BHA to show that the main causes of vibration disturbance are imbalance, geometric imperfections, initial curvature of the tube, and DS movement along indents of boreholes with a multi-petal profile. R. W. Tucker et al. [40] studied the effect of friction on connected flexural–torsional– longitudinal vibrations using a six-degree of freedom model. The same authors proposed a mathematical model of active damping of torsional vibrations. A similar mathematical model is analysed by J. D. Jansen et al. in [27]. A. P. Christoforou et al. [10] proposed a simplified model of frictional interaction between the DS and the well wall. They used it to show that oscillations excited by such an interaction can be chaotic. N. Challamel [7], on the basis of the model problem for torsional–

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longitudinal oscillations of a system with two degrees of freedom, studied the effect of friction on the DS dynamic stability. Using the Lyapunov method and direct numerical integration, he established the possibility of generating unstable oscillations with sticking. In the works of A. W. Iyoho et al. [25], R. W. Tucker [40], attempts were made to study torsional self-oscillations of drill strings using the theory of running waves; however, even here, despite the fact that these waves do not disperse—that is, they do not change their profile during propagation—they are approximated by the superposition of harmonics. Therefore, completing this analysis, a conclusion can be made about the importance of the problem that arises in connection with the study of torsional self-oscillations of drill strings based on wave models without imposing restrictions on the modes of solutions that are associated with the nature of wave rearrangement during their movement. 4.2 Mathematical aspects of the problem of self-oscillation of deep-drilling strings 4.2.1

Hopf bifurcations in oscillatory and wave dynamic systems

As noted above, any non-conservative system is self-oscillating, in which, as a result of the development of instabilities, it is possible to establish undamped wave or oscillatory motions, the parameters of which are determined by the system itself and do not depend on the final change in the initial conditions. They arise due to the fact that the oscillatory energy in these non-conservative systems can dissipate due to loss and also be replenished due to the instability associated with the system’s imbalance. Therefore, the properties of the generated oscillations do not depend on when and from which initial state the system began. At the same time, they are stable both in relation to external perturbations and to changes in initial conditions. A. A. Andronov called systems that possess the ability to generate such oscillations self-oscillating, for the first time providing them with a clear mathematical content and linking selfoscillations with Poincare limit cycles [1]. The limit cycle—the closed phase trajectory to which all the neighbouring trajectories aspire—is the image of periodic self-oscillations that we will talk about in this section. Self-oscillations in a dynamic system can be not only periodic but also quasi-periodic and even stochastic. The cycle birth bifurcation or Hopf bifurcation corresponds to the occurrence of self-oscillations from stationary states in the mathematical description of the dynamic process. As noted above, historically, this problem returns to the works of A. Poincare conducted in 1892. Then, it was intensively discussed by A. A. Andronov and

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A. A. Vitt [1] from 1930. Hopf's main work dedicated to this problem appeared in 1942. Although the term ‘Hopf bifurcation’ is the most common, the term ‘Poincare– Andronov–Hopf bifurcation’ is often used to describe this phenomenon, since Hopf's principal contribution is related to the generalisation of the results of this phenomenon from the two-dimensional case to the higher dimensions [33]. To establish self-oscillations, the principle is the presence of non-linearity that controls the flow and loss of energy source. The frequency characteristics of the source do not play a fundamental role. Self-oscillations differ from their own (free) oscillations, whose frequency is determined by the system parameters, and the amplitude and phase, by the initial conditions, as well as forced oscillations, whose amplitude, phase, and frequency are determined by an external force and do not depend on the initial conditions, and the phase does not play a significant role. The first studies of self-oscillating processes were performed on systems with one degree of freedom that simulate electro- and radio engineering devices. Of special significance is the Van der Pol generator, which is described by the Van der Pol equation [41]

x  D 1  E x2 x  Z02 x 0 .

(4.1)

About a century after its appearance, it still serves as the main model for selfoscillation systems with one degree of freedom. The existence of limit cycles for it is shown in a relatively simple way. The value of parameter D in (4.1) shows how much the generator is perturbed (at D  0 perturbation conditions are not met). Value E characterises the amplitude of self-oscillations; the smaller E the greater the amplitude. By introducing dimensionless variables and parameters W obtain

Z0t , x E 1/ 2U , P DZ0 , we finally

x  P 1  x2 x  x 0 .

(4.2)

This raises the question: How does the shape of the limit cycle depend on P ? When P 0 , the system becomes linear conservative. It is natural to hope that at small P ( P  1) self-oscillations should differ from harmonic oscillations and nonlinear friction only ‘selects’ the amplitude of a stable limit cycle. With large P , the oscillation shape may differ significantly from sinusoidal. In respect of the Van der Pol generator, the mode of occurrence of selfoscillations does not need an initial impact, so it is called a ‘soft’ excitation mode. For generators with one degree of freedom, such a mode corresponds to the phase portrait shown in Fig. 4-2,a.

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113

x

x

2

1

x

x

1 a)

b)

Fig. 4-2 Phase portraits of self-oscillating systems: a = ‘soft’ excitation; b = ‘hard’ excitation 1 = stable limit cycle 2 = unstable limit cycle

There are also systems with ‘hard’ excitation of self-oscillations. These are systems where oscillations spontaneously increase from a certain initial perturbation. For transition of systems with hard excitation to a mode of stationary generation of oscillations, initial excitation with amplitude greater than a critical value is necessary. The phase portrait of such generator is shown in Fig. 4.2,b. We can see that for the trajectory to reach a stable limit cycle, the initial point on the phase plane must lie outside the region of attraction of the steady equilibrium state. Hence, the physical content of unstable limit cycles is also clear: they serve as a boundary between the areas of initial conditions, from which the system moves to different stable modes of motion (on the phase plane, such motions correspond to trajectories that attract—that is, attractors; for example, stable equilibrium states or limit cycles). The size of the limit cycle determines the amplitude of oscillator selfoscillations, the time of motion of the imaging point along the cycle determines their period, and the shape of the limit cycle determines the shape of oscillations. Therefore, the problem of studying periodic self-oscillations in the system is reduced to the problem of finding limit cycles in the phase space and determining their parameters. There is no uniform method for finding them (as, for example, for determining the coordinates and types of equilibrium states), even for second-order systems. In the more general case, when the motion of the mechanical system occurs in n

n -dimensional space R , bifurcations of the cycle birth are determined by special

solutions of the autonomous system of ordinary differential equations

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dx f  x;v , (4.3) dt where x  Rn , and v is the real parameter from interval I . The main assumptions are that system (4.3) has an isolated fixed point x x*  v , and that the Jacobi matrices § wf · Dx f  x*  v ;v ¨ i  x*  v ;v ; i, j 1, ,n n¸ ¨ wx ¸ © j ¹ have a pair of complex-conjugated eigen values O1 and O2 A v

O1  v O2  v D  v  iZ  v ,

(4.4)

(4.5)

such that at some value v vc  I

Z  vc Z0 ! 0, D  vc 0, D c  vc z 0 .

(4.6)

Number vc is called the critical value of parameter v . If all eigenvalues A  vc , except for riZ0 , have strictly negative real parts, then condition (4.6) means that with the transition of v through value vc a stability loss of stationary point x*  v

occurs. For this system, the bifurcation of the cycle birth is called the birth of periodic solutions from the equilibrium (stationary) state. Not long ago, specialists from different fields have focused their attention on the system of equations (4.7) x V  x  y , y  xz  rx  y, z xy  bz bz, which was given the name of Lorenz model. System (4.7) was derived from the Navier-Stokes equations in the problem of thermal convection, so parameters r, V ,b have a rather definite hydrodynamic content: r is the Rayleigh number, V is the Prandtl number, and b characterises the size of the system. In this system, using Lorentz computations on a computer when r 28 , V 10 , b 8 / 3 , a complex chaotic behaviour of trajectories was revealed, which pointed to the possibility of the existence of fundamentally new steady-state modes-stochastic oscillations different from self-oscillations and beats. This kind of motion is of interest in connection with the explanation of the turbulence phenomenon. The birth of periodic solutions from stationary ones also occurs in functional and integro-differential equations and in partial differential equations. The latter area can be attributed to the results of bifurcation studies of the Navier-Stokes equations solutions: from the Couette flow to Taylor vortices in the fluid flow between two coaxial cylinders. Using the notion of the birth cycle bifurcation, an attempt was made to explain the effect of the turbulence origin [33].

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This book, based on the partial differential equation (wave equation), investigates bifurcations of the birth and loss of cycles in the problem of selfexcitation of torsional vibrations in long drill strings. With the birth of the limit cycle from a focus type equilibrium state (or contraction to it), an important question arises associated with dynamic system behaviour with parameter values close to the boundary of the stability of the equilibrium state (or periodic motion) and the different nature of the boundaries of the range of oscillations (‘dangerous’ and ‘safe’ boundaries). Since with the growth of the characteristic parameter at the self-oscillation boundary the equilibrium state from stable becomes unstable, the representative point can break from the equilibrium state and be thrown quite far. Therefore, it is important to investigate the nature of the stability region boundaries, which can be twofold. ‘Safe’ boundaries are such boundaries, the violation of which leads to only small (as small as possible with sufficiently small violations) changes in the state of the system. It can be shown that in this case the system coordinates will also experience only rather small periodic changes that are superimposed on the equilibrium (now unstable) position of the system. ‘Dangerous’ borders are such borders, any small violation of which leads to the transition of the system to a new state, which cannot be approached by the initial choice of sufficiently small violations of the border. The described situations have a simple physical content and correspond, for example (in a particular case), to the soft and hard occurrence of self-oscillations. These situations for a system of two equations were first described by A. A. Andronov in 1931 in the report ‘Mathematical problems of the theory of selfoscillations’ at the First All-Union conference on oscillations [1]. We can conclude that the problem of determining the nature of the ‘danger’ and ‘safety’ of the boundaries of self-oscillation ranges plays an important role in the studies of torsional self-oscillations of drill strings. 4.2.2

Typical non-linear models of resistance, friction, and cutting forces in tasks for analysing self-oscillating processes

The characteristic properties of self-oscillating systems are stipulated by the non-linearity of differential equations that describe their behaviour. The right parts of these differential equations contain non-linear functions of phase variables x in system (4.3). The physical content and type of these functions are determined by the physical process, which is modelled [6]. Electrical and radio systems have the most complex nature of these dependencies. Fig. 4-3 shows some characteristics of the relay elements that are most frequently encountered [6].

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Functions y  x in Fig. 4-3,a,d have zones of ambiguity—that is, they have a hysteresis character. The middle part of the curves in Fig. 4-3,b,c indicates the existence of a dead zone. y

y

y

x

a)

x

y

x

x

b) c) d) Fig. 4-3 Non-linear characteristics of relay elements

The dependence of anode current ix of the electronic lamp on mains voltage u g is a curve shown in Fig. 4-4,a. This curve can be approximated as two half-lines (Fig. 4-4,b) or three segments of straight lines (Fig. 4-4,c) depending on which mode the lamp is operating. ix

ix

ug

ix

ug

ug

a)

b)

c)

Fig. 4-4 Functions of the dependence of the anode current on the grid voltage

F fr

I

I

U a)

v

U b)

c)

Fig. 4-5 Graphs of the dependence between current I and voltage U (a, b) and the graph of the non-linear force of viscous friction F fr (c)

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The tunnel diode used in modern radio systems has characteristics shown in Fig. 4-5,a,b, where I is the current, U is the voltage. These and other similar nonlinearities, which possess areas with a negative slope of the tangent for their implementation, need external energy sources. The same form has a graph of the dependence of viscous friction force F fr on velocity gradient v in a non-Newtonian fluid (Fig. 4-5,c). Fig. 4-6 presents the graphs of functions that reflect typical non-linearities encountered when considering many mechanical self-oscillating systems. F fr  M

fr



F fr  M fr

Fmax

F fr  M fr Fstat

v Z

v Z

v Z

Fmin a)

b)

c)

Fig. 4-6 Characteristics of the Coulomb friction

The characteristic of the force of dry (Coulomb) friction has the form shown in Fig. 4-6, a, where v is the relative speed of rubbing surfaces. In many cases, this dependence can be approximated by the so-called z -characteristic (Fig. 4-6,b). However, these friction curves are used in the simplified formulation. The issues regarding simulation of frictional interaction mechanics were actively discussed in the middle of the last century. It was concluded that a more general model is the curve shown in Fig. 4-6,c, in which the value of static ( Fstat ) and extreme ( Fmax , Fmin ) friction forces can vary widely. Such characteristics are used in models of cutting forces (friction) in lathes, they are also used to describe the moment of shear forces (friction) applied to the drill bit [40]. 4.2.3

Regular and singularly perturbed equations with delay argument. Thomson and relaxation self-oscillations

As noted in paragraph 4.2.2, the equations of mathematical physics with a small parameter at a higher derivative are singularly perturbed. This type of equation includes partial differential equations for torsional selfoscillations of super deep-drilling strings, which are considered in this book. They are caused by low bit inertia (the moment of inertia of the body that vibrates) in comparison with the moment of inertia of the entire drill string, which plays the role of an elastic element (torsion waveguide). The complexity of this equation’s solution

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significantly exceeds the complexity of the singularly perturbed equations known in the scientific literature. x x

t

Fig. 4-7 A typical shape of relaxation selfoscillations

x

Fig. 4-8 Typical phase portrait of relaxation self-oscillations

First, the complexity of the problems in the book is stipulated by the low mass of the body, which vibrates (bits), in comparison with the mass of the entire drill string, which in this case—by virtue of the adopted design scheme—plays the role of an elastic element. Therefore, the bit motion equation is singularly perturbed. The main feature of oscillation systems described by singularly perturbed equations is that their periodic solutions have the form of inclined polylines (Fig. 4-7) with a rectangular phase portrait (Fig. 4-8). This type of oscillations is called ‘relaxation’ [35]. Second, the equation of the dynamic system considered in the book is of a wave type; therefore, as our studies have shown, when the viscous friction forces are not taken into account, the bit oscillations are of a so-called quantised nature, in which the function of the bit torsional self-oscillation rate is stepwise with the step length equal to the time of the torsion wave propagation of the drill string doubled length. As a result of the demonstration of these two factors, the bit angular velocity function is almost discontinuous and includes both large-scale and small-scale jumps. Naturally, when taking into account the forces of viscous friction of the string in the liquid medium, the quantised steps disappear. In connection with the above, these features lead, on the one hand, to an extremely complex problem of constructing solutions for non-linear differential equations but, on the other hand, to very beautiful and unexpected solutions describing the mechanical phenomena under study. In addition to the above-mentioned features, the problem under consideration has another property that allows us to simplify the understanding of additional complex effects accompanying the processes. This property follows from the wave

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nature (not taking into account viscous friction) of the dynamic system “drill string−bit” string, and, as the torsion wave propagating in the string does not disperse and retains its profile, the wave equation with partial derivatives can be replaced by an ordinary differential equation with a delayed argument (as shown below) with some transformations. Since this equation is also singularly perturbed, its solution includes a number of effects, one of which (in our opinion) is associated with the presence of a delayed argument. Let us remember [36] that a differential equation with a delayed argument is an equation in which, in addition to argument t , the desired function and its derivatives are taken, generally speaking, at different values of t . Such an equation is delayed (otherwise, called an equation with an argument that is delayed) if the value of the highest derivative at any value t t is determined by the value of the lowest derivatives at t d t . Such equations and their systems describe processes whose speed is determined by their previous state. Depending on the circumstances, these processes are called the process with lateness, time-delayed processes with aftereffect, etc. Let's consider the simplest case in which an ordinary differential equation of the first order resolved in respect of the derivative can be written as (4.8) xc f  t,x , or in a more expanded record

xc  t

f  t,x  t .

(4.9)

The transition to a differential equation with a delayed argument means that instead of equation (4.9), the equations

xc  t

f  t,x t  W t

W t t 0

(4.10)

are considered. Here, f  t,x and W  t are the given functions, and x  t is the desired function. So, for equation (4.8), the speed of the process at each moment is determined by its state at the same time, and for (4.10), by its state at one of the previous moments. Under this interpretation, the role of delay inalienability requirement becomes clear: the speed of the process cannot be determined by its state in the following moments. To illustrate the simplest properties of differential equations with a delayed argument, we shall limit ourselves to linear equations but generalise the problem considering that derivative xc at argument value is determined by the function value not at one argument value t  W  t , which is delayed, but at several such values. Then, from equation (4.10), we move on to the equation with concentrated delays:

Modelling emergency situations in drilling deep borehole

xc  t

120

k0

¦ a t x t  W t  f t . k

k

(4.11)

k 1

Here, functions W k  t t 0 , ak  t  k 1, ,kk0 : k0 t 1 and f  t are given, and x  t is the desired function. Differential equations with a deviating argument are widely used in the theory of automatic control systems where the control signal from the control device reaches the controlled object on a finite path with a certain time delay W ; in the theory of selfoscillating systems; in the study of problems associated with rocket engine combustion; in the problems of long-term economic forecasting when crisis phenomena caused by force majeure only appear after time W ; in a number of biophysical problems and in many other fields of science and technology, the number of which is constantly expanding. Many applications stimulate the intensive development of the theory of differential equations with a deviating argument, and at this time this theory belongs to the rapidly developing sections of mathematical analysis. The presence of deviation-delay in the system under study often causes phenomena that significantly affect the course of the process. For example, in automatic control systems, delay is a time interval that in principle always exists, which is necessary for the system to respond to the input pulse. The presence of a delay in the automatic control system can cause the appearance of self-excited oscillations, an increase in over-regulation and even system instability. The reason for the instability of combustion in liquid rocket engines is—as is commonly believed— the presence of the delay time required to convert the fuel mixture into combustion products, etc. One of the most noticeable demonstrations of the delay effect in the theory of mechanical system oscillations is associated with the torsion dynamics of the drill string—drill bit system. This arises in connection with the wave nature of the propagation of torsional perturbations along the length of the string and is similar to the effect of delay of the wave signal in controlled systems. The delay of the wave pulses, which are applied to the bit of the extended drill string and then pass the path along the waveguide to the top of the drill string, are reflected and returned back through the period of time 'W , leads to the complication of the forms of its motion, the most noticeable of which is the self-tuning of the system to the quantised oscillations with time quantum 'W equal to the time of the torsion wave running through the double length of the drill string. This effect is consistent with the possibility of discontinuous solutions in the differential equation with a delayed argument indicated in [36]. Note that for a drill string, one value 'W of the delayed argument takes place in the case of its homogeneity. If the drill string is composed of several sections with

Chapter 4. Excitation of torsional self-oscillations of strings in deep wells

121

different values of torsional stiffness, the oscillation equation includes several concentrated delays 'W i . In this case, at each point of connection of dissimilar tubes of the string, there is an additional reflection-refraction of the wave pulse; it is split into two weaker pulses, one of which returns on a shorter path with a small delay 'W i , the second one is forced to overcome additional loops of the trajectory and return with a greater delay 'W j . As a result, wave effects from pulses with different delays are added (signals with different delays are superimposed), and the system behaviour becomes less ordered. The considered specificity of the problem of self-excitation torsional vibrations of the extended drill string bit is due to the design features of the system and the nature of the contact interaction of the bit with the processed rock. In the derivation of differential equations of the bit motion, these features lead to a violation of their regularity. First, this is stipulated by the low weight of the drill bit (flywheel) in comparison with the mass of the entire string (elastic element), which is caused by the relaxation nature of self-oscillations, in which the angular velocity of rotation is a discontinuous (‘almost discontinuous’) function. Second, since the role of the elastic element in this oscillator is played by the waveguide, wave pulses (wave signals) pass through the waveguide from the source (drill bit) and back with some delay. Therefore, the constructed differential equation of rotation of the bit contains summands with a delayed argument. As is known [36], solutions of such equations can also contain discontinuities of velocities. As our studies have shown, in the cases considered, both of these factors occur simultaneously. Moreover, the discontinuities caused by the singular form of the equations are large-scale and can be observed directly on the oscillation graphs. While the discontinuities (‘almost discontinuities’) stipulated by the presence of the delay effect are small-scale and occur at equal intervals 'W . As a rule, these discontinuities are visible only at a certain increase in image scale. Since both types of these discontinuities appear simultaneously and are superimposed on the general trajectory of the system, its rotation acquires a complex shape with fast and slow movements, although it retains certain patterns. It is clear that the task of studying these patterns is associated with significant difficulties. 4.3. Hopf bifurcations in problems of torsional self-oscillations of deep drilling strings 4.3.1. Structural diagram of drill string torsion vibrations The rotary drilling method is currently the most widely used method in oil and gas well drilling. For its implementation, special drilling rigs are used, which are a

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122

complex of drilling equipment and facilities. The composition of the drilling rig components and their design are determined by the purpose of the drilling wells as well as the conditions and method of drilling. As noted above, a drilling rig for exploration and development of oil and gas fields usually consists of drilling facilities (drilling rig); lifting and lowering equipment (winch); power equipment (drive winch); rotor and drilling pumps; equipment for rotating the drill string (rotary table); drilling fluid circulating in the cavity of the drill string, and a drill bit (Fig. 4-9). Lifting device

Rotary table

Z

Winch

L

L

Drill string

Flywheel

Bit

Fig. 4-9 Structural diagram of a drilling rig

M Fig. 4-10 Kinematic diagram of a drill string

One of the dynamic effects contributing to the occurrence of an abnormal situation in the drilling process is the self-excitation of torsional vibrations of a rotating drill string (DS). As the DS is a torsion pendulum (Fig. 4-10), where there is an outflow of energy from the drive mechanism to the environment in its lower part due to the friction interaction of the bit and the rock being destroyed, its transition from stationary rotation state to torsional self-oscillations can occur if the conditions of this outflow are violated.

Chapter 4. Excitation of torsional self-oscillations of strings in deep wells

123

The cause of self-excitation of torsional vibrations on drilling rigs is the bifurcation disruption of the balance of elastic forces moments in the string and nonlinear friction forces between the drill bit and the borehole wall. In DS dynamics problems, the parameter determining the stationary and selfoscillating modes of motion is angular velocity Z of the string rotation. With regard to the phenomena accompanying the DS rotation, the study of the possibility of generating self-oscillations allows us to answer three important questions: At what system parameter values and its functioning is it possible to generate the torsional self-oscillations; What type of self-excitation oscillation mode (soft or hard) takes place; What measures can eliminate these self-oscillations. For a DS in relatively shallow wells, the answers to these questions can be obtained using a simplified mathematical model of a rotating torsion pendulum, to whose flywheel (bit) non-linear friction forces of its frictional interaction with the rock being destroyed are applied. It can be assumed that the flywheel and all the elements of the string carry out rotary vibrations with the same phase; therefore, the entire elastic system can be replaced by one oscillator with one degree of freedom [5, 7, 27, 40]. Such models can be applied for drill strings used in the coal industry for drilling shallow boreholes with a diameter of up to 5 m, because the periods of rotary vibrations of their bits are large, and the time for the passage of the torsion wave length of the string is small; thus, the fluctuations of all elements of the system occur with one common phase. However, if the length of the DS is not small, the application of the torsion oscillatory pendulum model to analyse its dynamics is not justified as the oscillations of its elements can fail to be in-phase, and their simulation should be based on the wave theory. The need to apply this theory is indicated in [5, 40], although their attempt to solve this problem is based on the approximation of the in-going and reflected torsion waves by monochromatic harmonics. In real conditions, this simplification is not generally performed as the run time of the wave with the length of the DS is not a multiple of the period of oscillations of the lower flywheel, and its motion can acquire a complex shape. The complication of the shapes of flywheel motion can greatly contribute to the sticking effect of its vibrations inherent in systems with dry friction. It consists of short-term stops of flywheel motion in time intervals where the sum of all moments of active forces and moments of inertia forces appears less than a threshold moment of friction which it is necessary to overcome for the flywheel to start moving. In these periods of time, the drive unit at the upper end of the DS continues to rotate at angular speed Z , the DS spins and accumulates the potential energy of elastic deformations. After the elastic moment in the DS reaches a value equal the threshold value of the moment of friction, the lower flywheel (bit) begins to rotate,

Modelling emergency situations in drilling deep borehole

124

the DS is untwisted, and its potential energy begins to pass into the kinetic energy of the string and flywheel rotation. This rotation lasts until the sum of the elastic moment in the DS and the moment of inertia of the bit again becomes less than the threshold value of the friction forces, as a result of which the flywheel stops again, and so on. Since the described motion of the functions of angular velocities and accelerations becomes almost discontinuous, DS rotation attains the character of strikes that represent a significant danger to the dynamic strength and stability of the entire system. It is obvious that it is not rational to describe such irregular oscillations using trigonometric functions, so special approaches should be applied. A factor complicating this phenomenon and the formulation of the problem is also the effect of the torsion waves on the flywheel that reached the upper end of the DS, reflected from it with a delay equal to the time of their run, and returned to the lower end. The impact of this effect has not yet been studied. Three statements of the problem of torsional drill string self-oscillations are considered below. The first model assumes that all elements of the string and the drill bit rotate synchronously. Because of this, it can be considered to be a system with one degree of freedom. This model does not consider the wave effects of the wave propagation of torsion deformations and the viscous friction between the string and the drilling fluid. In the second model, the string is considered to be a system with distributed parameters in which the processes of transmission of elastic deformations have a wave character. The third model is the most common. The elastic torsional oscillations of the DS are analysed taking into account the forces of viscous friction between the string and the drilling fluid. This model applies to the cases of inclined wells. To analyse in more detail the mechanism for the generation of torsional selfoscillations of drill strings, let's first turn to the simplest case when the length of the drill bit is small, the bit has a large moment of inertia, and the time of passage of the torsion wave of DS length L is small in comparison with the period of the drill bit self-oscillations. Then, this system can be modelled as a torsion pendulum with one degree of freedom (Fig. 4-11). Consider the case of stationary rotation of the upper end of the DS with constant angular velocity Z . Introduce inertial coordinate system OXYZ with the origin in the bit centre of mass, axis OZ of which coincides with the axial line of the DS. Relative to this, system Ox1 y1 z1 rotates at speed Z . Let's connect coordinate system Oxyz to the bit that rotates with it and axis Oz that coincides with axis OZ . Then, the rotation angle of the bit relative to system OXYZ will be Zt  M , where Zt is the rotation angle of the upper end of the DS and coordinate system Ox1 y1 z1 , t is

Chapter 4. Excitation of torsional self-oscillations of strings in deep wells

125

the time, M  z,t is the angle of elastic twisting DS, and M  0,t is the angle of the elastic twisting of the bit. Z , z1 , z g z, t

Z

y O

y1

Y X Zt x1

f z, t

M fr

M x

Fig. 4-11 A design model of the drill string

Fig. 4-12 Scheme of torsion wave propagation

If we neglect the inertial properties of DS compared with the inertia of a bit, conventionally, to separate it from the DS and consider its dynamic equilibrium, then the equation of the elastic torsional oscillations of a pendulum can be represented in the form of the D'Alembert principle (4.12) M in  M fr  M el 0 . in in Here, M M M is the moment of inertia forces that act on the bit;

Z  M is the moment of friction between the bit and the rock to be destroyed; M el M el M is the moment of elasticity forces that act on the bit when M

fr

M

fr

whirling the DS. The point above M denotes differentiation with respect to t . Value M in is calculated by the formula M in  J ˜ M ,

(4.13)

where J is the drill bit moment of inertia. Moment M el is determined by equality

Modelling emergency situations in drilling deep borehole

M el

GI z

126

wM , wz

or (4.14) M el GI zM / L , where G is the modulus of elasticity of the material DS in shear, I z is the moment of inertia of the cross-sectional area of DS relative to axis Oz . The task of determining moment M fr is more complicated. Depending on the tribological properties of the materials of the contacting bodies and the conditions of their frictional interaction, different models of the connection between M fr and velocity Z  M of their relative motion are chosen. Their analytical demonstrations will be discussed below. After replacing (4.13), (4.14), we rewrite equation (4.12) in the form: GI JM  M fr Z  M  z M 0 . (4.15) L This ordinary differential equation has a second order and a simple structure. Its solution for given Z depends on the type of function M fr Z  M . It can be constructed numerically for specific initial conditions in respect of M  0 and M  0 . These conditions should be selected accounting for the fact that self-excitation of oscillations can be both soft and hard. Once again, we should emphasise that the stated formulation of the problem can only be correct for relatively short drill strings with large moments of drill bit inertia—that is, for systems in which wave effects and propagation delays of torsion waves can be neglected. 4.3.2. Modelling drill string torsional self-oscillations on the basis of wave theory The rotation dynamics of a bit suspended at the end of a long drill string have the specificity inherent in waveguide systems. As in such systems the disturbance applied to one end of the waveguide reaches the other end after a finite period of time, it is necessary to take into account the delay of such action. In fact, if, for example, the DS is made of steel, the propagation velocity of the longitudinal and transverse waves in it, which are expressed through elastic modules E , G and density U , are ߙ ൌ ඥ߃Ȁߩ ൎ ͷǡͳͲͲ m/s, ߚ ൌ ඥ‫ܩ‬Ȁߩ ൎ ͵ǡʹͲͲ m/s, respectively. Then, for L = 6,000 m, torsion perturbation applied to one of its ends reaches another one and returns back after only 4 s. Therefore, the dynamics of torsional vibrations of such a DS should be studied on the basis of the wave equation

Chapter 4. Excitation of torsional self-oscillations of strings in deep wells

127

U I z w 2M / wt 2  GI z w 2M / wz 2 0 , which describes the propagation of a torsion wave in an elastic rod. After performing substitution E G / U , we bring it to the standard form: w 2M w 2M (4.16)  E 2 2 0. 2 wt wz It has a solution in the form of functions depending on two phase variables z  E t , z  E t . Let's present it in the form of D'Alembert (4.17) M  z,t f  z  E t  g  z  E t ,

where

f  z  Et ,

g  z  E t are the arbitrary continuous, not necessarily

differentiable, functions. The first of them determines the wave propagating at velocity E in the positive direction of axis Oz , and the second one, in the opposite direction (Fig. 4-12). As these waves are not dispersive, they move without changing their profile, which greatly simplifies the solution of the problem. Indeed, in this case, functions f  z  E t , g  z  E t at t ! 0 are determined only by the initial conditions f  z  0

f0  z , g  z  0 g0  z

and boundary conditions F[ f  0  E t ,g  0  E t ]

0,

f  L  Et  g  L  Et 0 ,

(4.18) (4.19)

where F is the non-linear differential operator describing the motion of the drill bit. The first condition of system (4.19) is formed by equation (4.12) in which M in is found using formula (4.13), M fr is represented by one of the diagrams for the moment of shearing (friction), and M el is calculated using the equality (4.20) M el GI z wM / wz , where the angular deformation wM / wz is calculated as

wM w ª f  z  E t  g  z  E t º¼ z 0 . wz z 0 wz ¬ On the basis of the second equality of system (4.19), we have g  L  Et  f  L  Et . Then, M f  z  E t  f  2L  z  E t f u  f  w , . u z  E t, w 2 L  z  E t. At z 0 , we have ª § 2L ·º M  0,t f   E t  f  2 L  E t f   E t  f « E ¨ t  ¸ » . E ¹¼ ¬ ©

(4.21)

(4.22)

Modelling emergency situations in drilling deep borehole

128

Therefore, at point z 0 of the attachment of the bit to the DS, angle of DS twisting M  0,t is determined by the current value of function f  E t , and its previous value f  2L  E t , which took place at this point in time, shifted to the past by value T

2 L / E . This means that angle M  0,t is not only a function of current

argument value t but also of time-delayed argument  t  2L / E . The fact that the torsion waves propagate in the pipe without changing profile allows us to put in expression (4.20) for M el derivative wM / wz through wM / wt and write the equation of the bit motion in the form of an ordinary differential equation. Indeed, it follows from equality (4.21) wM wf  u wf  w wf  u wf  w   wz wz wz wu ww . w f u w f w w f u wM     E   E wf  w wt wt wt wu ww Comparing these relations, we can express wf  z  E t wf  2L  z  E t wM   . (4.23) wz Ewt Ewt Therefore, ª wf  z  E t wf  2 L  z  E t º M el GI z «  » Ewt Ewt ¬ ¼z 0 (4.24) GI z ª wf   E t wf  2 L  E t º   » wt E «¬ wt ¼z 0 Substituting (4.13), (4.14), and (4.24) into (4.12) and taking into account (4.22), we obtain a second-order non-linear ordinary differential equation with a delayed argument GI J ª¬ f   E t  f  2 L  E t º¼  M fr  z ª¬ f   E t  f  2 L  E t º¼ 0 . (4.25) E This is completely equivalent to the wave equation system with partial derivatives (4.16) and boundary conditions (4.19) arising from the conditions of torsion oscillations of the drill bit at z 0 and interaction of the in-going and reflected waves in a rigid (non-rotating) arrangement at z L . We can see that equation (4.25) does not contain function f  t and depends only on its first and second derivatives. This circumstance allows us to lower the order of the equation by one, but in practice the method of construction a complete solution to this equation is not simplified because of its significant non-linearity. It is more convenient to apply the Runge-Kutta method directly to equation (4.25).

Chapter 4. Excitation of torsional self-oscillations of strings in deep wells

129

The found solutions allow us to set the drilling modes, at which the mode of self-excitation of torsion vibrations of the bit and the entire string is implemented, to construct the modes of these oscillations, and to select the drilling conditions that exclude self-oscillation of the system. The problem is related to the case of stationary rotation when Z const . However, its formulation can be easily extended to the non-stationary states of DS rotation associated with accelerating or damping modes. Let's look at two important circumstances related to the structure of obtained equation (4.25). First, coefficient J before higher derivative M in this equation is much less than the coefficients before the other two summands ( M and M ) due to the fact that the moment of drill bit inertia is much less than the moment of inertia of the entire drill string, which plays the role of an elastic waveguide element. Therefore, this equation belongs to the class of singularly perturbed ones and has solutions in the form of functions containing large-scale discontinuities of first derivatives M (angular torsion velocities). The second feature of this equation is that it includes an argument  t  2L / E that lags with delay 'W 2L / E equal to the time of passage of the torsion wave of twice the length of the drill string. Therefore, as shown below, small-scale discontinuities of velocities occur in the solution functions at the same time intervals (quanta) 'W 2L / E . To identify both types of these discontinuities, it is necessary to perform numerical integration of the equation more accurately. 4.3.3. Modelling the drill bit cutting torque by non-linear viscous forces and Coulomb friction moments As noted above, the main reason for the self-excitation of torsional oscillations of the drill string is the bifurcation imbalance between the elastic torque of the drill string and the non-linear friction interaction of the drill bit with the wall of the drill well. At present, only the general characteristics of this phenomenon are known since the cutting torque depends on many parameters, and the effect of each of them on these processes represents a complex theoretical and experimental task. Therefore, in our book, the main goal is to establish the most common patterns of the occurrence of self-oscillations of the drill bit and to determine the factors that most affect the generation of these oscillations. As revealed by experimental analysis, the vibration self-excitation processes of cutting tools in metal-working machinery as well as the formation of cutting forces and their moments in the interaction of the drill bit with rock depend on the speed of the relative motion between the cutting tool (bit) and the processed material and can be modelled by diagrams shown in Fig. 4-6. In this case, the formation of the moment

Modelling emergency situations in drilling deep borehole

130

of cutting torque (moment of friction M fr ) in the drill string is associated with many circumstances that significantly impact the nature of the curves represented in these graphs. These include the design, geometric shape, and diameter of the bit; the materials of bit cutters (high-strength steel or diamonds) and the degree of wear; the force with which the bit is pressed against the bottom of the drill hole; mechanical and strength properties of the treated rock (parameters of elasticity and plasticity, brittleness, strength, etc.), as well as the composition of the drilling fluid, the pressure in it, and the conditions of its expiration through the outlet holes in the bit. These factors are not only extremely different, but also the numerical values of their characteristics constantly change throughout the drilling process. Therefore, it is clear that there are no universal functions of friction moments in drill strings, and they cannot be built with acceptable accuracy. In this regard, it can be assumed that the universal numerical models of the phenomenon under study are not even necessary, and it is enough to study only some of the most common and typical phenomena accompanying the drilling process and determine the conditions and boundaries of their implementation. Therefore, as shown below, in general, it can be assumed that the cutting forces can only be implemented in the form of viscous diagrams (Fig. 4-5,b) or Coulomb (Fig. 4-6,с) friction and analyse the effect of the positions of extreme points on these functions on the state of stability. Therefore, we will consider the moments of friction that are described by the law of non-linear viscous friction (Fig. 4-13) or a function having a vertical section (Fig. 4-14), which is implemented at the time of static friction with no slippage between the contacting bodies and which can only be found from the conditions of fr static equilibrium of the system. After reaching limit value M lim , the moment of friction M

fr

becomes viscous. The second type of friction is called Coulomb friction. fr M Cou

fr M visc

Tmin Tmax T fr

dM !0 dT

Tmin Tmax M

fr M max

fr M min

Fig. 4-13 Function of non-linear viscous friction torque

M

fr lim

fr stat

fr

fr M max

fr M min

fr M dyn

dM !0 dT

T

Fig. 4-14 The non-linear friction torque with a segment of static friction (Coulomb friction)

Chapter 4. Excitation of torsional self-oscillations of strings in deep wells

131

The type of the moment of friction diagram and the position of extreme points fr on it can be easily established with a help of simple full-scale or model M , M max experiments. All you need to do in the experiments is to gradually increase torque M z and track the angle and speed of drill bit rotation. This technique can be used up fr min

to the first extreme state M z



fr as the dependence of torque M fr T becomes M min

unstable, and torsion oscillations are self-excited. To continue further testing of friction interaction, it is necessary to overcome the range of generated selfoscillations and to resume measurements after restoration of the stationary rotation mode. Some features of friction M fr T moment function can be determined in the laboratory when testing a new type of developed drill bit designed for a particular rock. However, to achieve more complete results, it is necessary that the data from laboratory and field experiments should be combined with the results of computer simulations performed, for example, using our method. In this case, experimental and theoretical approaches could complement and clarify each other. The tests should be performed at different values of the bit pressure force on rock, which usually ranges from 10 to 103 kN, while friction moment M z and angular velocity Z reach values 500 kN·m and 15 rad/s, respectively. It is important to note that both viscous and Coulomb moments M fr have fr fr , M max and segment Tmini d T d Tmax between them within experimental points M min which derivative dM fr / dT ! 0 (Fig. 4-13, 4-14). The presence of this segment, as a rule, is the main reason for the origin of self-oscillations. In this case, extreme values fr fr , M max and range Tmini d T d Tmax are usually less than their limit values M z = M min 500 kN ˜ m and ωlim = 15 rad/s. Therefore, during experimental and theoretical modelling of dependence M fr T , we can restrict ourselves to the definition of just

fr fr four parameters M min , M max , T mini , T max that can be determined by relatively simple means. It is also important to establish whether function M fr T is determined by the

viscous dependence (Fig. 4-13) or Coulomb friction (Fig. 4-14). If the second case fr occurs, it is also necessary to measure the limit value of static friction M stat (Fig. 4-14). The procedure for this analysis is also simple. Taking into account the given arguments, the moment of viscous friction can be represented by the following approximating function a T  a3T 3  a5T 5  a7T 7  a9T 9 fr m ˜ 1 M visc , (4.26) 1  a2T 2

Modelling emergency situations in drilling deep borehole

132

in which coefficients ai  i 1,2, ,9 are chosen using the search through method fr fr based on the found values M min , M max , T mini , T max . If we consider that in our case T k ˜ ª¬Z  f  E t  f  2L  E t º¼ ,

(4.27)

then instead of (4.26) we can write M

fr visc

m ˜

^

` ^

`

3

a1 k ˜ ª¬Z  f   E t  f  2 L  E t º¼  a3 k ˜ ª¬Z  f   E t  f  2 L  E t º¼ 

^

`

1  a2 k ˜ ª¬Z  f   E t  f  2 L  E t º¼

 a7

^

`

k ˜ ª¬Z  f   E t  f  2 L  E t º¼

^

7

 a9

^

`

The model of Coulomb friction M

(4.28)

`

k ˜ ª¬Z  f   E t  f  2 L  E t º¼

1  a2 k ˜ ª¬Z  f   E t  f  2 L  E t º¼ fr Cou

2

9

2

(Fig. 4-14) can be constructed by adding

fr can be a vertical segment in curve (4.26) at T 0 . Then, at T ! 0 , moment M Cou represented, for example, by the following formula: fr M Cou

fr fr .  M lim  e M visc

(4.29)

fr lim

fr Cou

Here, constants M , e , and m determine the change of function M along the ordinate axis. They are selected from the conditions of satisfying the given values fr fr , M max , T mini , T max . As noted above, the shape and values of the parameters M min determining the moment of friction vary depending on the system operation regime and the degree of drill bit wear. It can be considered that dependence (4.28) is realised for more worn bits with blunt cutters, while formula (4.29) is suitable for unworn bits. Therefore, it is important to monitor the general patterns of self-excitation of whirling vibrations and their proceeding for both models of the friction process. 4.3.4. Wave model of drill string torsion self-oscillations The expressions for the moments of viscous forces (4.28) and Coulomb friction (4.29) built in paragraph 4.3.2 allow us to bring the equation of oscillations of a bit to the final form. For the case of viscous friction, equation (4.25) takes the form:

^ `  1  a ^ k ˜ ª¬Z  f   E t  f  2 L  E t º¼` a ^ k ˜ ª¬Z  f   E t  f  2 L  E t º¼` GI ª f   E t  f  2 L  E t º¼   E ¬ 1  a ^ k ˜ ª¬Z  f   E t  f  2 L  E t º¼`

J ª¬ f   E t  f  2 L  E t º¼  m ˜

a1 k ˜ ª¬Z  f   E t  f  2 L  E t º¼

2

2

9

9

z

2

2

0.

(4.30)

Chapter 4. Excitation of torsional self-oscillations of strings in deep wells

133

When numerically integrated, it is reduced to a system of two first-order equations. Let's set q1  t f   E t ; q2  t f   E t ; (4.31) p1  t f  2 L  E t ; p2  t f  2 L  E t . q1

By substituting (4.31) into (4.30), we obtain q2 ;

(4.32) 9 a1 ¬ª k ˜ Z  q2  p2 ¼º   a9 ª¬ k ˜ Z  q2  p2 º¼ GI z q2 m ˜   q2  p2  p2 , 2 JE J ª1  a2 ¬ª k ˜ Z  q2  p2 ¼º º ¬ ¼ where variables q1  t , q2  t are unknown; functions p1  t , p2  t , p2  t are known, they are equal to, respectively, functions q1  t  2L / E , q2  t  2L / E , q2  t  2L / E calculated earlier in time t  2 L / E .

If the Coulomb friction conditions are realised, then from relations (4.25) (4.29) it follows тр  J ª¬ f  E t  f  2L  E t º¼  M lim + e m ˜

^

`

a1 k ˜ ª¬Z  f   E t  f  2 L  E t º¼ 

^

^

`

1  a2 k ˜ ª¬Z  f   E t  f  2 L  E t º¼



`

 a9 k ˜ ª¬Z  f   E t  f  2 L  E t º¼ 2

GI z

ª f   E t  f  2 L  E t º¼ 0. E ¬

(4.33)

q1

The equivalent system of two equations of the first order has the form q2 ;

q2

тр  e m ˜ – M lim –e

a1 ª¬ k ˜ Z  q2  p2 º¼   a9 ªk ª¬ k ˜ Z  q2  p2 º¼  2 J ª1  a2 ª¬ k ˜ Z  q2  p2 º¼ º ¬ ¼ 9

(4.34)

GI z  q2  p2  p2 . JE As in equations (4.30), (4.33), the coefficients at the higher (second) derivatives are very small, they refer [9, 35] to the class of singularly perturbed ones. In addition, they contain a delayed argument [16, 18]. Therefore, their solutions can contain (and will contain) rapidly and slowly varying functions and functions close to broken ones. In all cases, systems (4.32) and (4.34) are integrated by the Runge-Kutta method with a time step selected from the conditions for the convergence of the computational process at constant angular velocity Z and given initial conditions 

9



Modelling emergency situations in drilling deep borehole

134

q1  0 q10 , q2  0 q20 . If the beginning of the drilling process is modelled (when

the rotating DS is tripping-in, and the drill bit comes into contact with the rock at the bottom of the borehole), it is possible to put q1  0 q10 , q2  0 q20 , p1  2L / E 0 , p2  2L / E 0 , p2  2L / E 0 . Then, after time 'W

2L / E ,

variables p1  t , p2  t , p2  t need to be set with their found values. The built solutions allow us to set the drilling modes at which the selfexcitation of the torsion vibrations of the bit and the entire string is implemented, to build the forms of these oscillations, and to select the drilling conditions that exclude the self-oscillation of the system. The calculations analysed the dependence of the self-oscillating process nature on DS length DS L , the bit moment of inertia J , and the values of characteristics тр , e , m , and k determining the position of extreme points on the friction M lim moment diagrams (viscous or Coulomb). Our studies consider both models. 4.3.5. Investigation of general regularities of the phenomena of limit cycle generation and loss in rotating strings Depending on the selected drilling mode during operation, the drill string can be both in a state of stationary rotation with constant angular velocity Z and torsional self-excited elastic oscillations. The types of these states are dictated by solutions of equations (4.30), (4.33) that are primarily determined by laws (4.28), (4.29) of changes in the moment of friction, the moment of inertia J of the bit, length L of the drill string, and the selected value of the angular velocity of rotation Z . Through the values of these parameters, all the other mechanical characteristics of the system can be determined, such as DS elastic stiffness during whirling, its inertial properties, the velocity of propagation of the whirling wave E G / U , and the time of delay 2L / E . Usually, value J is determined depending on the chosen drilling technology and the diameter of the drilling well. For example, oil and gas wells have relatively small diameters (up to 0.4 m ), while the diameters of coal mine wells reach 5 m. Therefore, value J can vary from 1 kg•m2 up to 5,000 kg·m2. However, the biggest change in the drilling process is drill string length L , which for a deep well can vary from 0 km up to 10 km. Therefore, in this subsection, the phenomenon of transition from DS stationary rotation to periodic self-oscillation mode for strings of different lengths is studied, and the dependence of the main characteristics of self-excited oscillations on string length L is analysed. The study was performed with the following characteristic parameters: G 80.77 GPa , J 3.1 kg ˜ m2 , U 7.8 ˜103 kg / m3 , I z 3.12 ˜105 m4 (at

Chapter 4. Excitation of torsional self-oscillations of strings in deep wells

r1

0.08415 m , r2

135

0.07415 m ). It is carried out for the case when Coulomb

fr had the form represented by equality (4.29) in Fig. 4-14 friction force moment M Cou

௙௥

for values ‫ܯ‬௟௜௠ = 41,250 N·m, a1 = 2,400 N·m·s, a2 = 225 s2, a3 = 15,000 N·m·s3, a5 = 1 N·m·s5, a7 = 4 N·m·s7, a9 = –130 N·m·s9, e = 143.61 m = –1,000, k = 0.1. Fig. 4-15 shows the diagram of this moment. 123750

fr M Cou ,N ˜m

82500 41250 0 -41250 -82500

Z  M , rad / s

-123750 -15

-10

-5

0

5

10

15

Fig. 4-15 Function of non-linear Coulomb friction torque fr ( M lim 41,250 N ˜ m , m 1,000 , k 0.1 )

The integration of system (4.34) was performed using the Runge–Kutta method with initial conditions M  0 0 , M  0 0 with the step for integration Δt = 6.5·10–6 s chosen from the conditions for the convergence of the calculations. 4.3.6. Dependence of the torsion self-oscillation nature of a drill string on its length Depending on the inertial and stiffness properties of the DS and the selected drilling mode, it may be in a state of stationary rotation or self-excited torsion oscillations. The type of these oscillations is determined by solutions of equation fr (4.33) that depend primarily on type (4.29) of function M Cou T M Coufr Z  M , angular velocity Z , and length L . As noted above, in the theory of non-linear differential equations, the periodic solution corresponding to self-oscillations is called a cycle, and the replacement of a stationary equilibrium solution with a periodic one when a certain characteristic parameter passes through a critical value is the birth of a

Modelling emergency situations in drilling deep borehole

136

cycle or Hopf bifurcation. In this case, the parameter determining the stationary and self-oscillating modes of the DS is the angular velocity of rotation Z . To determine the type of self-excited torsional oscillations in a drill string and to analyse the influence of the DS size on their amplitude D and periods T , the strings are selected with the length of L = 1,000 m, L = 2,000 m, L = 8,000 m. Hopf bifurcation (cycle birth bifurcation)

Hopf bifurcation (cycle loss bifurcation)

O

Zone of pre-critical stationary rotation of the drill string

Z

Zу

Zр Zone of stable selfoscillations

Zone of post-critical stationary rotation of the drill string

Fig. 4-16 Graphical representation of the range of stable selfoscillations and Hopf bifurcation states

In all cases, system (4.34) with initial conditions q1  0 0 , q2  0 0 was integrated using the Runge–Kutta method with time step Δt = 6.4742•10–6 s at selected fixed value Z . Calculations continued up to the moment when either the stationary state of the dynamic balance of the bit with certain angle M st of its turn was established, or it passed into the mode of steady-state periodic self-oscillations. By fr varying value Z , it is found that for given function M Cou Z  M there is range

Zb d Z d Zl beyond which the system from the initial position reaches a quasi-static stable equilibrium state q1 M st , q2  t 0 without excitation of torsional selfoscillations. When the parameter is passing inside range Zb d Z d Zl through value Z Zb (Fig. 4-16), Hopf bifurcation occurs, and a stable cycle (self-oscillating process) is born, and solution M M st —although it remains—becomes unstable. This property is preserved for all values Zb d Z d Zl , and when parameter Z goes beyond this range through value Z Zl , bifurcation of the cycle loss occurs, the selfoscillations fail to excite, and solution M M st becomes stable again. Let's comment on the features of the dynamic behaviour of the system in the vicinity of bifurcation state Z Zb based on the example of a DS with a length of L = 1,000 m with bit inertia J 3.1 kg ˜ m2 . In this case, to the left of bifurcation point Zb

0.71 rad / s (at Z Zb  0 ),

the drill bit, starting to move from position M  0 0, M  0 0 , quickly passes into

Chapter 4. Excitation of torsional self-oscillations of strings in deep wells

137

stationary state Mst 32.63 rad , Mst 0 and then continues to rotate at angular velocity Zb 0.71 rad / s without oscillation. However, at value Z Zb  0 , the drill bit moving at the initial stage along the same trajectory reaches value M 32.63 rad (Fig. 4-17,a) and then breaks into the oscillatory process. These fluctuations immediately take a steady character and occur with amplitude D 4.13 rad and with period T 14.25 s around the average position Mav 30.56 rad . Fig. 4-17 shows these vibrations in a larger scale. It is important to note that the shape of these oscillations in each period can be divided into separate stages corresponding to the slow and fast motions of the system. These peculiarities are particularly clearly visible on angular velocity time dependence graph M  t (Fig. 4-18), which may be considered as discontinuous. In the theory of self-oscillations, such movements are called relaxation movements in contrast to harmonic trajectories, which are called the Thomson trajectories. We should remember that such movements are quite rare in the problems of mechanical systems dynamics. In addition, a new effect inherent only to waveguide systems is revealed in the considered problem. It is found that the oscillations are almost quantised in time. So, if section 62 ≤ t ≤ 75 s of the curve in Fig. 4-18, a is in a larger scale (Fig. 4-18,b), it turns out that it is piecewise constant, and the length of each time step is 'W | 0.623 s , which is equal to the time of passage of the whirling wave of path length 2L from the bottom of the bit up to the DS suspension point and back—that is, 'W 2L / E | 0.623 s . In this case, we called self-oscillations almost quantised because they are superimposed by even smaller disturbances. Fig. 4-19 shows them in a larger scale. Below, in the following sections, more ordered quantised selfoscillations without additional perturbations are revealed. It should be noted that the almost discontinuous nature of the change in bit elastic rotation velocity M is demonstrated in this model despite the fact that the coefficients in system (4.34) and non-linearity in equality (4.29) are continuous, smooth, and differentiable. Apparently, this is because the system self-adjusts to the generation of weak shock waves in the torsion waveguide—that is, waves with discontinuities of the first derivatives, which in portions (quanta) propagate upwards from the bit, are reflected from the upper end of the DS, return, beat the bit, cause its quantisation motion, etc. As in the places of discontinuities, the velocity and acceleration functions (as well as torques) acquire pulse values, such modes of selfoscillation of the bit represent a significant danger to the dynamic strength of the system and cannot be considered acceptable.

Modelling emergency situations in drilling deep borehole

M , rad

0

138

M , rad

-28

-29

-10

-30

1 2

-20

T -31

Mav

-30

D

M st

3

t, s

-40 0

100

200

300

-32

t, s

-33 40

80

120

160

200

240

280

a) b) Fig. 4-17 Shape of drill bit self-oscillations at L = 1,000 m and angular velocity ω = 0.71 rad/s (cycle birth bifurcation): a = at the initial scale b = at an enlarged scale

M , rad/s рад / c

8

8

6

6

4

4

2

2

0

0

t, s

-2 0

100

200

300

M , rad/s рад / c

'W 'W 'W

t,sc

-2 62

64

66

68

70

72

a) b) Fig. 4-18 Graph of changes in drill bit angular velocity M t (L = 1,000 m, Z 0.71 rad / s ; cycle birth bifurcation): a = at the initial scale b = at an enlarged scale

74

76

Chapter 4. Excitation of torsional self-oscillations of strings in deep wells

139

We must also emphasise that the almost discontinuous nature of the self– oscillations of the bit drill string system is consistent with the properties of the singularly perturbed equations [9, 35] and differential equations with a delayed argument [36]. Therefore, the small inertia of the oscillatory system and the presence of a waveguide elastic element in it led to the fact that the self-oscillations generated in it acquired two types of discontinuities (quasi-breaks) of the angular velocities of the bit. The first of them owes its origin to a small inertia of the system and is large-scale (relaxation oscillations). The second factor is stipulated by the presence of a waveguide elastic element. Thanks to this, small-scale velocity discontinuities are generated in the form of quantised shock pulses. The relaxation character of selfoscillations is also clearly seen in their phase portrait (Fig. 4-20,a,b), which resembles a rectangular phase portrait (Fig. 4-8) exceptional for relaxation self-oscillations. It has zones where angle M t changes at constant velocity M t , and, vice-versa, velocity M t changes at a constant value of M t . This mode of motion is the main feature of relaxation oscillations. fr Fig. 4-21 presents function M Cou Z  M corresponding to given coefficients ai . A solid line is given to the area in which it changes its values in the process of fr periodically passes through two of its steady oscillations. It can be seen that M Cou extreme values. The found solutions also allow us to construct the forms of torsional vibrations of the drill string itself at different time moments. Fig. 4-22 shows the graphs of function M z in states 1, 2, 3 corresponding to the selected points 1, 2, 3 on the curve of Fig. 4-17, a of the drill bit self-oscillations. Although they represent superposition (4.17) of two wave movements M  z , t f  z  Et  g  z  Et , in Fig. 4-22, the wave nature of these movements is not noticeable, the whirling vibrations of the DS points are represented as occurring in one phase, and functions M z themselves are linear. This allows us to conclude that the wave nature of torsion oscillations of the drill string points can be ignored and the simplified (oscillatory) model described above can be used. However, it is not quite possible to agree with this argument because, if we consider the presented graphs of the torsion velocity of the DS points, for example, in Fig 4-23 for case 1, it can be seen that the function of the angular velocities of DS points is a superposition of smooth (almost constant and almost zero) and almost discontinuous (impulse) curves. This effect can only be if function M z has local peaks (singular perturbations) in the zones where function pulses occur. Derivatives of these singular perturbations also give bursts (pulses) of the angular velocity functions.

Modelling emergency situations in drilling deep borehole -0.28

M , rad / s

140

-0.3

M , rad / s

-0.32

-0.32

-0.34 -0.36

-0.36

-0.38

'W

-0.4

'W

-0.4 -0.42

-0.44

-0.44

t, s

-0.48 62.8

63.2

63.6

64

64.4

7

64.8

t, s

-0.46

65.2

64.01

64.02

64.03

64.04

64.05

64.06

M , rad / s

6 5 4

'W | 0.623 s

3 2 1 0

t, s

-1 71.4

71.6

71.8

72

72.2

72.4

Fig. 4-19 Specific details of the quasi discontinuous change of angular velocity M t of the drill bit (L = 1,000 m, Z 0.71 rad / s ; cycle birth bifurcation): a = almost quantised oscillations in the slow motion zone b = function M t change in a short period of time c = almost quantised oscillations in the fast motion zone

Chapter 4. Excitation of torsional self-oscillations of strings in deep wells 8

φ, rad / s

6

4

4

2

2

0

0

φ, rad

-2 -40

-30

φ, rad / s

8

6

-20

-10

141

φ, rad

-2

0

-33

-32

-31

-30

-29

-28

a) b) Fig. 4-20 Phase portrait of self-oscillation of a drill bit (L = 1,000 m, Z 0.71 rad / s ; cycle birth bifurcation): a = at the initial scale b = at an enlarged scale

123750

fr M Cou ,N ˜m

0

82500

fr M Cou ,N ˜m

-20625

41250

-41250

0

-61875

-41250 -82500

-82500

-123750 -15

-10

-5

0

5

10

15

Z  M , rad / s

0

2

4

6

8

10

12

Z  M , rad / s

a) b) fr Fig. 4-21 Dependence of torque M Cou on angular velocity Z  M (L = 1,000 m, Z 0.71 rad / s ): a = at the initial scale b = at an enlarged scale

Modelling emergency situations in drilling deep borehole

142

z, m 1000

z, m 1000

1

3 800

800

600

600

400

400

200

200

2

1

-35

-30

-25

M , rad

-20

-15

-10

-5

0

-1

Fig. 4-22 Graphs of function M z of the angle of elastic twisting at states 1, 2, 3

z, m 2

M , rad / s

0

1

-3

-2

3

Fig. 4-23 Graph of angular velocity function M z of the DS elements at state 1

z, m

1000

3

800

M , rad / s

2

800

600

600

400

400

200

200

M , rad / s -1

0

1

1000

-0.8

-0.6

-0.4

-0.2

0

a) b) Fig. 4-24 Graph of angular velocity function M z of the DS elements: a = for case 2 b = for case 3

0.2

0.4

Chapter 4. Excitation of torsional self-oscillations of strings in deep wells

143

Fig. 4-22 shows that the singular perturbations are so small that they are visually imperceptible. Wave pulses propagating along the length of the DS are the main cause of discontinuous and quantised movements of the drill bit. This can be seen in Fig. 4-24 illustrating the position of the pulses of the angular velocity waves of the DS points at states 2 and 3. We should emphasise that analysis of such a dynamic process can be carried out only with the help of the wave model of self-oscillations. The results of calculations of the drill bit-string system self-oscillations describe the behaviour of the system at state Z Zb of the limit cycle birth—that is, when moving from a stationary bit rotation with a constant angular velocity to periodic (self-oscillating) motion. The subsequent increase in angular velocity Z leads to the transition of the system into range Zb d Z d Zl (Fig. 4-16) of the angular velocity change in which oscillatory periodic phenomena occur. If Z keeps on increasing, then with Z Zl the loss of the limit cycle occurs and the drill string with the bit returns to steady rotation mode with a constant angular velocity. In this regard, it is interesting to see how the system behaves in bifurcation state Z Zl . Bifurcation value Zl 3.775 rad / s is determined by numerical calculations, and the behaviour of the system in this state is investigated. Figs. 4-25–4-31 illustrate the basic properties of self-oscillation of the bit at Z Zl 3.775 rad / s . Although in this case there was a significant restructuring of the motion forms in relation to the above state Z Zb , the general properties of the dynamic process are preserved. The oscillations also have a relaxation character (Fig. 4-25) and the function of changing angular velocity of the drill bit M t turned out to be almost discontinuous (quantised) with the same value of time quantum ο߬ ൎ ͲǤ͸ʹ͵‫( ݏ‬Fig. 4-26), although the phase portrait changed slightly (Fig. 4-27). fr The amplitude of the oscillations covering a larger zone of the diagram M Cou Z  M (Fig. 4-28) also increased. The character of the DS torsion wave motions along its length has not undergone any qualitative changes, since function M z remained almost linear (Fig. 4-29). At the same time, the waves of angular twisting velocity M z changed essentially: they became piecewise constant with additional oscillation perturbations at the transition points from one step to another (Fig. 4-30). In addition to the studies, analysis was performed of relaxation oscillations of drill strings with lengths L = 2,000 m and L = 8,000 m in the vicinity of their bifurcation states of cycle limit birth and loss.

Modelling emergency situations in drilling deep borehole

φ, rad

0

-29

144

φ, rad

-30

-10

-31

-20

1 2

-32

-30

-33

3

t, s

-40 0

10

20

30

40

50

60

t, s

-34

70

240

242

244

246

248

250

252

254

а) b) Fig. 4-25 Mode of drill bit self-oscillations with a length of L=1,000 m and angular velocity ω=3.775 rad/s (cycle loss bifurcation): a – at the initial scale; b – at an enlarged scale

M , rad / s

8

8

6

6

4

4

2

2

0

0

-2

-2

t, s

-4 0

10

20

30

40

50

60

70

'W 'W 'W

M , rad / s

t, s

-4 240

242

244

246

248

250

252

а) b) Fig. 4-26 Graph of changes in drill bit angular velocity M t of drill bit (L=1,000 m, ω=3.775 rad/s;cycle loss bifurcation): a – at the initial scale; b – at an enlarged scale

254

Chapter 4. Excitation of torsional self-oscillations of strings in deep wells

6

φ, rad / s

0

4

145

fr M Cou ,N ˜m

-20625

2

-41250

0

-61875

-2 -82500

-4 -34

-33

-32

-31

-30

0

-29

2

4

6

8

Fig. 4-27 Phase portrait of selfoscillations of the drill bit (L=1,000 m, ω=3.775 rad/s; cycle loss bifurcation)

10

12

Z  M , rad / s

φ, rad

fr Fig. 4-28 Dependence of torque M Cou on angular velocity Z  M (L=1,000 m, ω=3.775 rad/s)

z, m 1000

z, m 1000

1

3 800

800

600

600

400

400

200

200

2

1

-40

φ, rad

-30

-20

-10

0

Fig. 4-29 Graphs of elastic torsion angle function M z at states 1, 2, 3

-2

-1

φ, rad / s

0

1

2

Fig. 4-30 Graph of a function M z of angular velocity of DS elements at state 1

Modelling emergency situations in drilling deep borehole

φ, rad

0

146 8

φ, rad / s

6

-20

4

1

-40

2

2 -60

0

3

t, s

-80 0

50

100

150

200

-66

250

M , rad / s

8

6

6

4

4

2

2

0

0

t, s

-2 0

50

100

150

200

250

-64

-62

-60

-58

-56

Fig. 4-32 Phase portrait of selfoscillations of the drill bit (L=2,000 m, ω=0.71 rad/s; cycle birth bifurcation)

Fig. 4-31 Mode of drill bit selfoscillations for a length L=2,000 m and angular velocity ω=0.71 rad/s (cycle birth bifurcation)

8

φ, rad

-2

M , rad / s

'W

-2 160

'W 170

t, s

'W 180

a) b) Fig. 4-33 Graph of changes in drill bit angular velocity M t (L=2,000 m, ω=0.71 rad/s; cycle birth bifurcation): a – at the initial scale; b – at an enlarged scale

190

200

Chapter 4. Excitation of torsional self-oscillations of strings in deep wells 0

M , rad / s

0.1

-0.1

147

M , rad / s

0

-0.2 -0.1 -0.3 -0.2 -0.4 -0.3

-0.5

'W

t, s

-0.6 175

180

185

190

195

'W

-0.4

200

100

104

'W

108

t, s

112

116

а) b) Fig. 4-34 Specific details of the quasi discontinuous change of angular velocity M t of the drill bit (L=2,000 m, ω=0.71 rad/s; cycle birth bifurcation): а – fragment of angular velocity function M t , interval 175 d t d 200 s ; b – fragment of angular velocity function M t , interval 100 d t d 115 s

0

fr M Cou ,N ˜m

2000

3

-20625

1600

2

-41250

1200

800

-61875

400

1

-82500

0

2

4

6

8

10

12

Z  M , rad / s fr Fig. 4-35 Dependence of torque M Cou on angular velocity Z  M (L=2,000 m, ω=0.71 rad/s)

-80

-60

-40

-20

0

Fig. 4-36 Graphs of elastic torsion angle function M z at states 1, 2, 3

Modelling emergency situations in drilling deep borehole

φ, rad

0

148

M ,rad / s

8 6

-20 4

-40

1

2

2

0

-60

-2

3

t, s

-80 0

10

20

30

40

50

Fig. 4-37 Mode of drill bit self-oscillations for length L=2,000 m and angular velocity ω=3.775 rad/s (cycle loss bifurcation)

M , rad / s

-3.4

-4 0

10

20

30

40

50

t, s

Fig. 4-38 Graph of changes in the drill bit angular velocity M  t (L=2,000 m, ω=3.775 rad/s; cycle loss bifurcation)

0

fr M Cou ,N ˜m

-20625

-3.5

-41250

-3.6

-61875 -3.7

'W

'W

'W

-3.8 9

10

11

12

13

14

-82500

t, s 15

16

17

Fig. 4-39 Specific details of the quasi discontinuous change of angular velocity rad/s M  t of the drill bit (L=2,000 m, ω=3.775 rad/s; cycle loss bifurcation)

0

2

4

6

8

10

12

Z  M , rad / s

fr Fig. 4-40 Dependence of torque M Cou on angular velocity Z  M (L=2,000 m, ω=3.775 rad/s)

Chapter 4. Excitation of torsional self-oscillations of strings in deep wells

z, m 2000

149

φ, rad

0

3 1600

2

-100 1200

1 800

2

-200

1

400

3

t, s

-300 -80

M , rad

-60

-40

-20

0

0

Fig. 4-41 Graphs of elastic torsion angle function M z at states 1, 2, 3

-0.12

400

800

1200

Fig. 4-42 Mode of drill bit self-oscillations with a length of L=8,000 m and angular velocity ω=0.71 rad/s (cycle birth bifurcation)

M , rad / s

8

-0.16

6

-0.2

4

-0.24

2

-0.28

φ, rad / s

0

'W

'W

'W

-0.32 410

420

430

t, s 440

Fig. 4-44 Specific details of the quasi discontinuous change of angular velocity M t of the drill bit (L=8,000 m, ω=0.71 rad/s; cycle birth bifurcation)

φ, rad -2 -270

-260

-250

-240

-230

-220

Fig. 4-45 Phase portrait of selfoscillations of the drill bit (L=8,000 m, ω=0.71 rad/s; cycle birth bifurcation)

Modelling emergency situations in drilling deep borehole

0

150

fr M Cou ,N ˜m

z, m

8000

3 -20625

6000

2 -41250

4000

-61875 2000

1 -82500

0

2

4

6

8

10

12

Z  M , rad / s

fr Fig. 4-46 Dependence of torque M Cou on angular velocity ω  φ (L=8,000 m, ω=0.71 rad/s)

-300

φ, rad

-200

-100

0

Fig. 4-47 Graphs of elastic torsion angle function M z at states 1, 2, 3

Figs. 4-31–4-41 represent the forms of bit and DS elements movements for the length of L = 2,000 m in states Z Zb (Figs. 4-31–4-36) and Z Zl (Figs. 4-37– 4-41). The comparison of the string movement shapes with the shapes of string oscillations with a length of L = 1,000 m allows us to conclude that they remained practically unchanged and experienced only large-scale changes in the amplitudes and periods of oscillations as well as the lengths of quanta 'W . At the same time, the quantised nature of these oscillations became more pronounced, as with an increase in length L the values of bit inertia moment J in respect of the inertial properties of the entire string proportionally decreased, and the delay time doubled and became equal to ο߬ ൎ ͳǤʹͶ͵‫ݏ‬. The quantised nature of these oscillations is most clearly seen in Figs. 4-34,c and 4-39. During these studies, the greatest difficulties arose in modelling oscillations of a DS with a length of L = 8,000 m. It took 700 hours of computer time on a modern Quad-core personal computer over five months to solve the problem. Therefore, it was necessary to limit the study of this system to state Z Zb 0.71 rad / s . The results of the studies of this system are shown in graphical form in Figs. 4-42–4-47. It can be seen that they only differ from the shapes of motion of strings with a length of L = 1,000 m and L = 2,000 m in the scales of the amplitudes and periods of oscillations as well as the length of time quantum ο߬ ൌ ʹ‫ܮ‬Ȁߚ ൎ ͶǤͻ͹ʹ‫ݏ‬.

Chapter 4. Excitation of torsional self-oscillations of strings in deep wells

151

Table 4-1 Values of self-oscillating processes parameters of drill strings with a length of L = 1,000 m, L = 2,000 m, L = 8,000 m L  m

1,000

2,000

M

Z  rad / s

fr lim

Zl

Zb

8,000

41, 250  N ˜ m , k = 0.1, m = –1,000

Zl

Zb

Zb

Mst  rad

0.71

3.775

0.71

3.775

0.71

-32.63

-33.31

-65.25

-66.62

-261.03

Mav  rad

-30.56

-31.20

-61.14

-62.69

-244.62

D  rad

4.13

4.22

8.22

7.87

32.82

T s

14.25

2.51

28.50

4.99

115.63

'W  s

0.623

1.243

4.972

T1  s

1.246

2.486

9.944

The values of the generalised characteristics of the studied processes for strings L = 1,000 m, L = 2,000 m, L = 8,000 m are given in Table 4-1. It shows bifurcation values Zb , Zl for cases L = 1,000 m, L = 2,000 m and value Zb for string length L = 8,000 m as well as the limit values of the DS torsion angle at lowest point ߮௦௧ ; average value of the DS torsion angle ߮௔௩ , relative to which there are selfoscillations; amplitudes D of these oscillations and their periods T (see Fig. 4-17). Analysis of the obtained data allows us to conclude that the bifurcation values of angular velocity Zb , Zl do not depend on the length of the DS and are determined only by the type of moment of friction‫ܯ‬஼௢௨ Z  M diagram and the positions of ௙௥

extreme points on it. Limit value M st of the bit torsion angle in stationary rotation for bifurcation values Zb , Zl of each string varies little, but with increasing DS length L value M st increases proportionally to L . The same can be also noted about value φcr. Almost the same properties have the values of oscillation amplitude D , although its value for L = 2,000 m at Z Zl turned out to be less than the corresponding value at Z Zb . With an increase in DS length L , periods of oscillations T also increase proportionally, although their values for the string of one selected length at Z Zl are significantly less than the corresponding values of these magnitudes at Z Zb . It is interesting to compare the range of self-excitation frequencies of DS torsion vibrations with frequencies of its own oscillations. Taking into account the small mass of the bit relative to the mass of the entire string, when calculating its

Modelling emergency situations in drilling deep borehole

152

frequencies, we assume that at the upper end z L the DS is rigidly clamped, and M L 0 , at the lower end z 0 free edge condition wM / wz z 0 0 is realised. Then, for lowest frequency k1 , the solution of the wave equation 2 w 2M 2 w M  E wz 2 wt 2

0

(4.35)

can be represented in the form:

M z, t A sin

S

L  z ˜ sin k1t . 2L By substituting (4.36) in (4.35), we obtain the characteristic equation S2  k12  E 2 2 0 . 4L From there, it flows 2S 4 L SE T1 , . k1 k1 E 2L

(4.36)

(4.37)

(4.38)

For the considered DS, values k1 and T1 are equal to k1 = 5.052 s–1, T1 = 1.246 s (L = 1,000 m), k1 = 2.526 s–1, T1 = 2.486 s (L = 2,000 m), k1 = 0.632 s–1, T1 = 9.944 s(L = 8,000 m). It is interesting to note that found values T1 are exactly twice the value of 'W . Therefore, DS self-oscillations occur in the period ranges that significantly exceed the periods of their free oscillations (see Table 4-1). 4.3.7. Dependence of the drill string torsional self-oscillations on the nature of non-linear frictional interaction of a bit with a rock As mentioned above, in the general case, self-oscillations represent stationary periodic motion of a non-linear dissipative system supported by an external nonvibration energy source. In paragraph 4.3.6, it is established that the modes of these oscillations do not qualitatively depend on the length of the DS, and when it changes they only transform in accordance with certain conditions of similarity and scaling. In this regard, it can be assumed that the nature of self-oscillations is largely determined by the type of function M fr Z  M of the cutting forces moment (the moment of friction). However, it is known that these functions depend on a large number of factors, among which, in our case, the greatest role is played by the design shape and diameter of the bit; the force of the contact pressure of the bit on the bottom of the drill hole; the stiffness and strength properties of the treated rock; tribological properties of the drilling fluid; the material of the structure, the degree of wear of the bit cutting tool, and many other factors. As it is rather difficult to trace the

Chapter 4. Excitation of torsional self-oscillations of strings in deep wells

153

dependence of the type of function M fr Z  M on each of them, it seems appropriate to study the effect of some generalised characteristics of this function on the processes of the occurrence and flow of self-oscillations. Intuitively, it can be assumed that such generalised characteristics are parameters that determine the position of extreme points on diagram M fr Z  M . Therefore, in this subsection, we shall analyse the self-oscillating phenomena for different variants of function M fr Z  M with different abscissa and ordinates of their extreme points. Since, as shown by the calculations in paragraph 4.3.6, the effect of self-oscillations is only realised in the zones of the upper part of diagram M fr Z  M , for which the position of the static friction segment is insignificant, only the Coulomb friction law will be used in this subsection. Let's look at diagrams of the friction forces selected for the study of the laws of friction moments (Fig. 4-48). fr They show the three functions of the moment of friction M Cou Z  M : fr M Cou

u

M u

240,000 x  15,000 x3  0.01x5  0.0004 x 7  0.00013 x9 , 1  2.25 x 2 fr Cou

( 4.39c )

2) (Fig. 4-48,b): 82,500  144 u

960,000 x  60,000 x3  0.04 x5  0.0016 x 7  0.00052 x9 , 1  2.25 x 2

fr M Cou

u

1) (Fig. 4-48,a): 41,250  144 u

( 4.39cc )

3) (Fig. 4-48,c): 41,250  144 u

120,000 x  1,875 x3  0.0003125 x5  0.000003125 x 7  0.0000002539 x9 ( 4.39ccc ) . 1  0.563x 2

Curve ( 4.39c ) in our case is standard (see Fig. 4-48, a, which coincides with Fig. 4-15). Using this curve, studies were performed in paragraph 4.3.6. On curve ( 4.39cc ), the ordinates of its points (and the extreme points) are duplicated (see Fig. 4-48,b). On curve ( 4.39ccc ), ordinates of points remained unchanged compared to curve ( 4.39c ), but the values of their abscissa are doubled (see Fig. 4-48,c). When modelling the non-stationary processes of the system output to the modes of stationary rotation or steady self-oscillations, it was assumed that the drill string bit system was removed from contact with the bottom of the drilling well and in this state is accelerated to the selected value of angular velocity Z . Then, it fell to

Modelling emergency situations in drilling deep borehole

165000

154

fr M Cou ,N ˜m

165000

123750

123750

82500

82500

41250

41250

0

0

-41250

-41250

-82500

-82500

-123750

-123750

-165000

fr M Cou ,N ˜m

-165000 -24

-18

-12

-6

0

6

12

18

24

-24

Z  M , rad / s

-18

-12

a

-6

0

6

b

165000

fr M Cou ,N ˜m

123750 82500 41250 0 -41250 -82500 -123750 -165000 -24

-18

-12

-6

c

0

6

12

18

12

18

24

Z  M , rad / s

24

Z  M , rad / s

fr Fig. 4-48 Dependence of torque M Cou on angular velocity Z  M : a = standard, case I b = with doubled ordinate, case II c = with doubled abscissa, case III

Chapter 4. Excitation of torsional self-oscillations of strings in deep wells

155

the position in which the bit came into contact with the rock surface at the bottom of the drilling well and began its relative rotation also transmitted to the string. As a result of computer simulation of the subsequent movement, the condition (stationary rotation or stationary self-oscillations) which the system reaches was determined. In all cases, system (4.34) integration was carried out using the Runge-Kutta method with integration step Δt = 6.474•10–6 s. Dynamic behaviour of DS with lengths L = 1,000 m and L = 2,000 m was analysed. Results of computer simulation of DS dynamic behaviour with the selected fr lengths for standard diagram M Cou Z  M (case I, Fig. 4-48,a) are given in paragraph 4.3.6 in graphical form in Figs. 4.17–4.31 and in Table 4-1. Table 4-2 shows the calculations results for DS with a length of L = 1,000 m for all three cases presented in Fig. 4-48. Table 4-2 Values of self-oscillating process parameters of drill strings fr of length L = 1,000 m for different diagrams of moment M Cou Z  M of friction Graph type of the moment M

fr

of Coulomb friction

Mfrlim = 41,250 N·m Mfrlim =82,500 N·m k =0.1, m= –1,000 k = 0.1, m = –4,000 Case I

Mfrlim =41,250 N·m k = 0.05, m = –1,000

Case II

Case III

L(m)

1,000

Zb

Zl

Zb

Zl

Zb

ω

ω

ω

ω

Zl

ωst, (rad)

0.71 -32.63

3.775 -33.31

0.71 -78.76

3.775 -79.74

1.42 -32.65

1.43 -32.59

3 -32.85

6 -31.50

9 -31.51

9.35 -34.89

ωsr (rad)

-30.56

-31.20

-74.29

-74.49

-32.63

-29.65

-29.95

-29.81

-29.82

-31.82

D (rad) Т(s) Δ τ (s)

4.13 14.25

4.22 2.51

8.95 34.50

0.05 1.25

5.89 12.01

5.80 3.10

3.39 1.25

3.39 1.26

6.14 1.88

ω (rad/s)

Т1(s)

0.623

7.5; 10.51 6.43 0.623

1.243

1.243

0.623 1.243

For case II self-oscillation, the systems were investigated at the states of birth and loss of the limit cycle. Analysis of these data leads to the conclusion that with the fr doubling of the limiting values of friction moment ( M Cou Z  M ) bifurcation values

Zb and Zl of the angular velocity of rotation of the DS have not changed. At the same time, the numerical values of all other characteristic parameters of the system ( M st , Mav , D , T ) more than doubled, although the exact dependence of their growth in this case is not revealed.

Modelling emergency situations in drilling deep borehole

156

Figs. 4-49–4-54 illustrate the features of system self-oscillations at the state of the birth of the limit cycle (ω=0.71 rad/s). Their comparison with the modes of motion for case I presented in Figs. 4-17–4-24 indicates that these modes have not undergone any qualitative changes. Some differences in the modes of motion with increasing ordinates of function fr M Cou Z  M can be observed at the state of loss of the limit cycle (case II, Figs. 4-55–4-60). Their main feature is that self-oscillations are not fully established, they do not have a clearly defined period and occur with large amplitudes D 75 and 10.51rad , differently ‘alternating’ through two or three conventional periods T=6.43 s. The phase portrait of the described motion is presented in Fig. 4-57. Note that in applied mathematics and in the theory of dynamical systems such attractors are called strange. fr Case III, in which the scale of function M Cou Z  M was doubled along independent variable Z  M (equation ( 4.39ccc ), Fig. 4-48,c), was convenient for the study of the dynamic behaviour of the system not only at range boundaries Zb d Z d Zl , in this case a component part 1.43 d Z d 9.35 rad / s , but also within this range. The numerical results of these calculations are also shown in Table 4-2. It is important to note that bifurcation value Zb is also doubled in this case, while Zl 9.35 rad / s is more than twice the value of Zl 3.775 rad / s for cases I and II. It is worth noting that the values of parameters M st , Mav , D , and T at state Z Zb and Z Zl have not changed significantly. For case III, the bifurcation state of the cycle birth ( Z Zb 1.42 rad / s ) is considered more accurately. The results of the calculations are displayed in Figs. 4-61–4-63. It seems that it was possible here to establish the very beginning of the self-oscillations birth process, which occur with a fairly small amplitude D=0.05 rad and period T 1.25 s (Fig. 4-61). Therefore, in Fig. 4-61, a diagram of the output of these oscillations is represented by almost quasi-static mode of the drill bit with fairly small ‘tremors’ noticeable only by increasing the image scale (Fig. 4-61,b). These ‘tremors’ are visually represented as pronounced functions of the triangular sine, the derivative of which, however, has the form shown in Fig. 4-62. The graph of function M t in this figure shows that its primitive M t in Fig. 4-61, b is not completely broken but contains small-scale bridge segments at angular points, making this function differentiable. Fig. 4-63 shows the phase portrait of excited selfoscillations.

Chapter 4. Excitation of torsional self-oscillations of strings in deep wells

φ, rad

0

10

157

M , rad/s rad / s

8 -20

6 -40

4

1 2 3

-60

2 0

-80 0

100

200

300

120

t, s

Fig. 4-49 Mode of drill bit selfoscillations with a length of L 1,000 m and angular velocity ω=0.71 rad/s fr 82 ,500 N ˜ m , k 0.1 , ( M lim m 4,000 , case II)

10

t, s

-2 160

200

240

280

Fig. 4-50 Graph of changes in drill bit angular velocity M t (L=1,000 m, fr 82 ,500 N ˜ m , k 0.1 ω=0.71 rad/s, M lim m 4,000 , case II)

M , rad / s

165000

fr M Cou ,N ˜m

123750

8

82500

6

41250 0

4

-41250

2

-82500

0

-123750 -165000

-2 -80

-78

-76

-74

-72

-70

-68

M , rad

Fig. 4-51 Phase portrait of self-oscillation of the drill bit (L=1,000 m, ω=0.71 rad/s, fr M lim 82 ,500 N ˜ m , k 0.1 , m 4,000 , case II)

-12

-8

-4

0

4

8

12

Z  M , rad / s

fr Fig. 4-52 Dependence of torque M Cou on angular velocity Z  M (L=1,000 m, fr 82 ,500 N ˜ m , ω=0.71 rad/s, M lim k 0.1 , m 4,000 , case II)

Modelling emergency situations in drilling deep borehole

158

z, m 1000

1000

800

800

600

600

400

400

1

200

-80

-60

-40

-20

0

-1

0

M , rad / s

Fig. 4-53 Graphs of elastic torsion angle function M z at states 1, 2, 3 (L=1,000 m, ω=0.71 rad/s, k 0.1 , fr M lim 82 ,500 N ˜ m , m 4,000 , case II)

1

2

3

4

5

Fig. 4-54 Graph of angular velocity function

M z of the DS elements at state 1 (L-1,000 m, ω=0.71 rad/s, k 0.1 , fr M lim 82 ,500 N ˜ m , m 4,000 , case II)

φ, rad

0

200

M , rad / s

8

-20

4 -40

1 2 3

-60

0

-80

tt,s ,s

-4 0

20

40

60

80

100

t, s

Fig. 4-55 Mode of drill bit self-oscillations with a length of L=1,000 m and angular velocity ω=3.775 rad/s fr 82 ,500 N ˜ m , k 0.1 , m 4,000 ) ( M lim

20

40

60

80

100

Fig. 4-56 Graph of changes in drill bit angular velocity M t (L=1,000 m, fr 82 ,500 N ˜ m ω=3.775 rad/s, M lim k 0.1 , m 4,000 , case II)

Chapter 4. Excitation of torsional self-oscillations of strings in deep wells

φ, rad

8

165000

159

fr M Cou ,N ˜m

123750 82500

4

41250 0 -41250

0

-82500 -123750 -165000

-4 -80

-76

-72

-12

-68

φ, rad

Fig. 4-57 Phase portrait of drill bit self-oscillation (L=1,000 m, ω=3.775 rad/s, fr M lim 82 ,500 N ˜ m , k 0.1 , m 4, 000 , case II)

2

-60

-40

0

4

-20

8

12

Z  M , rad / s

fr Fig.4-58 Dependence of torque M Cou on angular velocity Z  M (L=1,000 m, fr 82 ,500 N ˜ m , ω=3.775 rad/s, M lim k 0.1 , m 4, 000 , case II)

z, м 3

800

1

φ, rad

-4

z, м 1000

3

-80

-8

1000

800

600

600

400

400

200

200

0

Fig. 4-59 Graphs of elastic torsion angle function M z at states 1, 2, 3 (L=1,000 m, ω=3.775 rad/s, fr M lim 82 ,500 N ˜ m , k 0.1 , m 4,000 , case II)

-4

0

φ, rad / s

4

8

12

Fig. 4-60 Graph of angular velocity function M z of the DS elements at state 1 (L=1,000 m, ω=3.775 rad/s, fr M lim 82 ,500 N ˜ m , k 0.1 , m 4,000 , case II)

Modelling emergency situations in drilling deep borehole

φ, rad

0

160

-32.6

φ, rad

-10 -32.62

-20 -32.64

-30

t, s

-40 0

10

20

30

40

50

60

t, s

-32.66 30

70

32

34

36

38

40

а) b) Fig. 4-61 Mode of drill bit self-oscillations for length L=1,000 m and angular velocity ω=1.42 rad/s at the initial stage of the process of the cycle birth ( M limfr 41, 250 N ˜ m , k 0.05 , m 1,000 , case III): а – in initial scale; b – in enlarged scale

0.2

M , rad / s

0.2

0.1

0.1

0

0

-0.1

-0.1

t, s

-0.2 30

32

34

36

38

Fig. 4-62 Graph of changes in drill bit angular velocity M t (L=1,000 m, ω=1.42 rad/s, M limfr 41, 250 N ˜ m , k 0.05 , m 1,000 , case III)

40

φ, rad / s

φ, rad

-0.2 -32.66

-32.64

-32.62

-32.6

Fig. 4-63 Phase portrait of drill bit self-oscillations (L=1,000 m, ω=1.42 rad/s, M limfr 41, 250 N ˜ m , k 0.05 , m 1,000 , case III)

Chapter 4. Excitation of torsional self-oscillations of strings in deep wells

161

It resembles a rectangular phase portrait for singularly perturbed systems presented in Fig. 4-8, where, however, the corner points are rounded and contain additional oscillations. This feature can be explained by the fact that the solutions presented in Figs. 4-7, 4-8 are typical of singularly perturbed systems only at H o 0 , in our case this parameter is not quite close to zero. As noted, the described case refers to the very initial stage of the process of the birth of torsion self-oscillations, and in the further simulation of this phenomenon it was enough to slightly change value of the angular velocity from Z Zb 1.42 rad / s to value

Z Zb 1.43 rad / s , and the parameters of the self-oscillations changed sharply. The range of oscillations increased significantly (see Table 4-2 and Fig. 4-64), and on curve M t (Fig. 4-64) the seizing area appeared with value Mst 32.59 rad / s . However, with a more detailed examination of this state, we can see that it is also the motion of the system, although at a very low speed. This fact can be established by analysing the graph of function M t derivative (Fig. 4-65). A further increase in parameter Z within range Zb d Z d Zl (Table 4-2) leads to the restructuring of the vibration modes, while remaining relaxation (Figs. 4-66, 4-67). We should emphasise that although the considered motions are presented as discontinuous, in fact the graphs of characteristic functions presented on a larger scale are differentiable curves. -26

1

M, rad

12

M, rad / s

8

-28

2 -30

4

-32

0

3

t, s

-34 30

40

50

Fig. 4-64 Mode of drill bit selfoscillations for length L=1,000 m and angular velocity ω=1.42 rad/s ( M limfr 41, 250 N ˜ m , k 0.05 , m 1,000 , case III)

60

t, s

-4 30

40

50

60

Fig. 4-65 Graph of changes in drill bit angular velocity M t (L=1,000, ω=1.42 rad/s, M limfr 41, 250 N ˜ m , k 0.05 , m 1,000 , case III)

Modelling emergency situations in drilling deep borehole -26

M, rad

162

12

-28

8

-30

4

-32

0

-34

t, s 10 12 14 16 18 20 22 24 26 28 30

Fig. 4-66 Mode of drill bit selfoscillations for length L=1,000 m and angular velocity ω=3 rad/s ( M limfr 41, 250 N ˜ m , k 0.05 , m 1,000 , case III)

M, rad / s

t, s

-4

10 12 14 16 18 20 22 24 26 28 30

Fig. 4-67 Graphs of bit angular velocity M t change (L=1,000 m, ω=3 rad/s, M limfr 41, 250 N ˜ m , k 0.05 , m 1,000 , case III)

The change in the characteristic parameters of these self-oscillations with change Z within range Zb d Z d Zl can be viewed in Table 4-2. As can be seen, values M st and Mav change insignificantly, while the amplitudes of self-oscillations and their periods tend to decrease. At Z Zl 9.35 rad / s , the values D and T again slightly increased. The behaviour of the system in the overcritical condition at Z 9.36 rad / s that exceeds Zl 9.35 rad / s deserves attention. It has the usual state of selfoscillation of the bit relative to the stationary rotation of the bit with almost constant angular velocity (Figs 4-68, 4-69). This state can be considered as a transition between the states of stable self-oscillations and stationary rotation. To confirm the conclusions made in this sub-paragraph on the example of a DS with a length fr L = 1,000 m of the influence of diagram type M Cou Z  M on the nature of selfoscillations, we shall carry out the same analysis for DS length of L = 2,000 m.

Chapter 4. Excitation of torsional self-oscillations of strings in deep wells

M, rad

0

163

M, rad d/s

15

10

-10

5 -20 0 -30

-5

t, s

-40 0

10

20

30

40

50

60

70

Fig. 4-68 Mode of drill bit selfoscillations at L=1,000 m and angular velocity ω=9.36 rad/s ( M limfr 41, 250 N ˜ m , k 0.05 , m 1,000 , case III)

t, s

-10 3

6

9

12

15

Fig. 4-69 Graph of changes in drill bit angular velocity M  t (L=1,000 m, ω=9.36 rad/s, M limfr 41, 250 N ˜ m , k m 1,000 , case III)

0.05 ,

Table 4-3 The values of the parameters of self-oscillation processes in drill strings with a length fr of L = 2,000 m for different moment M Cou Z  M of friction diagrams fr Type of moment of friction M Cou Z  M diagram

fr M lim 41,250 N ˜ m k 0.1 , m = –1,000

fr M lim 82,500 N ˜ m k 0.1 , m = –4,000

fr M lim 41,250 N ˜ m k 0.05 , m = –1,000

Case I

Case II 2,000

Case III

L  m

Zb

Zl

Zb

Zl

Zb

Zl

Mst  rad

0.71 -65.25

3.775 -66.62

0.71 -157.52

3.775 -159.49

1.43 -65.25

9.35 -69.75

Mav  rad

-61.14

-62.69

-148.61

-148.99

-59.33

-63.63

D  rad

8.22 28.50

7.87 4.99

17.82 65.33

20.99 13.01

Z  rad / s

T (s) Δτ (s) T1 (s)

1.243 2.486

1.243 2.486

11.85 12.25 23.95 3.78 1.243 2.486

Modelling emergency situations in drilling deep borehole

164

The results of the calculations are given in Table 4-3 for the sequence of fr parameter M Cou Z  M variation corresponding to its change in Table 4-2 (see Fig. 4-48,a–c). The forms and features of self-oscillations corresponding to the standard diagram (case I) can be traced in Figs. 4-31–4-41. Modes of DS self-oscillations with a length of L = 2,000 m for case II are shown in Figs. 4-70, 4-71 (cycle birth bifurcation) and Figs. 4-72–4-75 (bifurcation of cycle loss). These graphs are of interest due to the not quite periodic nature of motion at the state of the cycle loss bifurcation (see Figs. 4-72–4-73) which, moreover, occurs in a more clearly defined scenario of quantised self-oscillations (see Fig. 4-74). The features of relaxation self-oscillations of a DS with a length of L = 2,000 m for case III are illustrated in Figs. 4-76–4-78 (limit cycle birth bifurcation) and Figs. 4-79–4-80 (limit cycle loss bifurcation). The presented diagrams show that the fluctuations of the drill bit self-excited at these values of the characteristic variables differ from the above for case L = 1,000 m, as a rule, only by changes in the scale of their graphs along the axis of ordinates and abscissa with the preservation of the general similarity of the oscillatory processes. The values of the similarity and scale transformation coefficients can be established by comparing the values of the characteristic parameters for cases I, II, III in Tables 4-2, 4-3. d/s 0 10 M , rad M, rad 8 -40

6 -80

4 2

-120

0

t, s

-160 0

100

200

300

400

500

600

Fig. 4-70 Mode of drill bit selfoscillations for length L=2,000 m and angular velocity Z 0.71 rad / s fr 82,500 N ˜ m , k 0.1 , ( M lim m 4,000 , case II)

700

t, s

-2 300

350

400

450

500

550

600

Fig. 4-71 Graph of changes in drill bit angular velocity M  t (L=2,000 m,

Z 0.71 rad / s , M limfr 82,500 N ˜ m , k

0.1 , m

4,000 , case II)

Chapter 4. Excitation of torsional self-oscillations of strings in deep wells

M, rad

-135

165

M, rad / s

8

-140 4

-145

-150 0

-155

t, s

-160 40

80

120

(M

8

80

120

160

Fig. 4-73 Graph of changes in drill bit angular velocity M  t (L=2,000 m,

Z 3.775 rad / s , M limfr 82,500 N ˜ m ,

82,500 N ˜ m , k 0.1 , 4,000 , case II)

m

40

160

Fig. 4-72 Mode of drill bit selfoscillations for length L=2,000 m and angular velocity Z 3.775 rad / s fr lim

t, s

-4

4,000 , case II)

0.1 , m

k

z, m

M, rad d/s

2000

3 1600

2

4

1200

800

0

1 400

t, s

-4 50

54

58

62

66

70

Fig. 4-74 Specific details of the quasi discontinuous change of angular velocity M  t of the drill bit (L=2,000 m,

Z 3.775 rad / s ,

M

82,500 N ˜ m , k 0.1 , m 4,000 , case II)

fr lim

-160

M, rad

-120

-80

-40

0

Fig. 4-75 Graphs of elastic torsion angle function M  z at states 1, 2, 3 (L=2,000 m, Z 3.775 rad / s , fr M lim

82,500 N ˜ m , k 0.1 , m 4,000 , case II)

Modelling emergency situations in drilling deep borehole

M, rad

0

166

M, rad / s

10 8

-20

6 -40

4 2

-60

0

t, s

-80 0

40

80

120

50

Fig. 4-76 Mode of drill bit selfoscillations for length L=2,000 m and angular velocity Z 1.43 rad / s (M k

fr lim

0.05 , m

75

100

k

0.05 , m

-0.7

-0.8

-0.9

'W

-1

t, s

-1.1 42

44

46

48

50

Fig. 4-78 Specific details of the quasi discontinuous change of angular velocity M  t of the drill bit (L=2,000 m,

Z 1.43 rad / s , M limfr 41, 250 N ˜ m , k

150

Z 1.43 rad / s , M limfr 82,500 N ˜ m ,

M, rad d/s

40

125

Fig. 4-77 Graph of changes in drill bit angular velocity M  t (L=2,000 m,

82,500 N ˜ m , 1,000 , case III)

-0.6

t, s

-2

160

0.05 , m

1,000 , case III)

1,000 , case III)

Chapter 4. Excitation of torsional self-oscillations of strings in deep wells -56

M, rad

15

167

M, rad / s

10

-60

5 -64 0 -68

-5

t, s

-72 40

50

60

70

80

Fig. 4-79 Mode of drill bit selfoscillations for length L=2,000 m and angular velocity Z 9.35 rad / s k

fr 41, 250 N ˜ m , ( M lim 0.05 , m 1,000 , case III)

t, s

-10 40

90

50

60

70

80

90

Fig. 4-80 Graph of changes in drill bit angular velocity M  t (L=2,000 m, Z 9.35 rad / s , fr M lim

41, 250 N ˜ m , k 0.05 , m 1,000 , case III)

4.3.8. Dependence of string torsion self-oscillations on the magnitude of the moment of the bit inertia In paragraphs 4.1 and 4.2, it is noted that the main cause of relaxation (almost discontinuous) self-oscillations in dynamic systems is a small factor before the second derivative in the motion equation (see equation (4.33)). Such oscillations are inherent in low-inertia systems with a small mass and with a relatively large stiffness of the elastic element. They are most often found in electronic and radio systems, and, thanks to the possibility of using analogue modelling methods to analyse them, they are studied in sufficient detail for different, rather complex laws of non-linear processes of outflow and inflow of energy arising in them. Low-inertia oscillators are quite rare in mechanical systems. This is possibly why the issues of analysis of their relaxation oscillations are paid relatively little attention to. One example of an oscillator with low inertia in mechanical systems is the considered deep-drilling string where the mass of the oscillator (bit) is much less than the mass of the elastic element (the drill string itself). In this regard, a pronounced property of torsional self-oscillations generated in drill strings is their relaxation character. The main features of such oscillations are established in paragraphs 4.3.6 and 4.3.7; it is shown that they are accompanied by large-scale and small-scale discontinuities of bit rotation velocities. Small-scale discontinuities are demonstrated in the form of quantised impact pulses. Therefore, it is also interesting

Modelling emergency situations in drilling deep borehole

168

to study the self-oscillating processes in systems with not very low inertia, which are implemented in drilling technologies. These include drill strings for drilling vertical shafts in coal mines used to lift extracted coal, lower and lift coal miner crews and for ventilation. The diameters of such shafts reach up to five metres, and the moments of inertia of the drill bits of such drill strings reach large values. It is therefore interesting to analyse on model problems how self-oscillating processes evolve with increasing moment of inertia J in equation (4.25) and in systems (4.32) and (4.34). This point studies, by comparison, the features of the self-oscillation phenomena of systems with drill string length L = 1,000 m and L = 2,000 m with moments of inertia of drill bits J 3.1 kg ˜ m2 and J = 1,000 kg·m2 for the case of non-linear viscous friction described by the law M

fr visc

m ˜

^

` ^

`

3

a1 k ˜ ª¬Z  f   E t  f  2 L  E t º¼  a3 k ˜ ª¬Z  f   E t  f  2 L  E t º¼ 

^

`

1  a2 k ˜ ª¬Z  f   E t  f  2 L  E t º¼

^

`

 a7 k ˜ ª¬Z  f   E t  f  2 L  E t º¼

^

7

^

2

`.

 a9 k ˜ ª¬Z  f   E t  f  2 L  E t º¼

`

1  a2 k ˜ ª¬Z  f   E t  f  2 L  E t º¼

2

(4.40)

9

The values of the coefficients included here are a1 = 2,400 N·m·s, a2 = 225 s2, a3 = 15,000 N·m·s3, a5 = 1 N·m·s5, a7 = 4 N·m·s7, a9 = –130 N·m·s9, e = 143.61, m = – 1,000, k = 0.1. The features of dynamic behaviour of a DS with a length of L = 1,000 m for values J 3.1 kg ˜ m2 and J = 1,000 kg·m2 can be set according to the calculation result presented in Table 4-4. They show that bifurcation values Zb and Zl do not depend on the moment of bit inertia and were equal for both types of bits. However, if we analyse the forms of self-oscillations, then, as follows from the illustration below, with an increase in the value of bit inertia moment J , the oscillation equation becomes less singularly perturbed, and there is a smoothing of the peculiarities of the resolving functions in the places of their discontinuous change. For DS length L = 1,000 m this can be confirmed at the state of the birth of the cycle limit by comparing Figs. 4-81–4-84 ( J 3.1 kg ˜ m2 ) and Figs. 4-85–4-90 (J = 1,000 kg·m2). As can be seen, in case J 3.1 kg ˜ m2 , the self-oscillations are implemented according to the scenario typical of relaxation oscillations with pronounced discontinuities of angular velocities and the presence of a quantum effect (Fig. 4-83).

Chapter 4. Excitation of torsional self-oscillations of strings in deep wells

φ, rad

0

10

169

M, radd / s

8

1

-10

6

2

4

-20

2 -30

0

3

t, s

-40 0

50

100

150

200

250

300

40

350

Fig. 4-81 Shape of drill bit selfoscillations with a length of L=1,000 m, angular velocity Zb 0.71 rad / s and moment of drill bit inertia J 3.1 kg ˜ m2

80

120

160

200

240

Fig. 4-82 Graph of changes in drill bit

φ(t ) (L=1,000 m, angular velocity φ( Zb 0.71 rad / s , J 3.1 kg ˜ m2 )

M, radd / s

-0.62

t, s

-2

10

-0.63

φ, rad / s

8

-0.64

6

-0.65 4

-0.66 2

'W

-0.67

0

-0.68

t, s

-0.69 4

6

8

10

12

Fig. 4-83 Specific details of the quasi discontinuous change of angular velocity M  t of the drill bit (L=1,000 m,

Zb

0.71 rad / s , J

3.1 kg ˜ m2 )

-2 -36

-32

-28

-24

-20

φ, rad

Fig. 4-84 Phase portrait of selfoscillations of the drill bit (L=1,000 m, Zb 0.71 rad / s , J=3.1 kg·m2)

Modelling emergency situations in drilling deep borehole

φ, rad

0

170

M, radd / s

10

1 -10

8

2

6 4

-20

2 -30

3 -40 0

100

200

0

t, s

50

300

Fig. 4-85 Shape of drill bit selfoscillations with a length of L=1,000 m, angular velocity Zb 0.71 rad / s and moment of drill bit inertia

100

150

200

250

300

Fig. 4-86 Graph of changes in drill bit angular velocity M  t (L=1,000 m,

Zb

0.71 rad / s , J

1,000 kg ˜ m2 )

1,000 kg ˜ m2

J

10

t, s

-2

M, radd / s

-0.06

8

M, rad / s

-0.08

6 -0.1

4 -0.12

2 -0.14

0

t, s

-2 244

246

248

250

t, s

-0.16 235

236

237

238

239

240

a) b) Fig. 4-87 Specific details of the quasi discontinuous change of angular velocity M  t of the drill bit (L=1,000 m, Zb 0.71 rad / s , J 1,000 kg ˜ m2 ): a = fragment of angular velocity function M  t on time interval 244 ≤ t ≤ 250 s

b = fragment of angular velocity function M  t on time interval 235 ≤ t ≤ 240 s

Chapter 4. Excitation of torsional self-oscillations of strings in deep wells 10

171

z, m

φ, rad / s

8

1000

800

6

600

1

4

400 2

200

0

φ, rad / s

φ, rad

-2 -36

-32

-28

-24

-20

-1.6

-16

Fig. 4-88 Phase portrait of selfoscillations of the drill bit (L=1,000 m, Zb 0.71 rad / s , J 1,000 kg ˜ m2 )

-1.2

-0.8

-0.4

0

Fig. 4-89 Graph of angular velocity function M  z of the DS elements (L=1,000 m, Zb

J

0.71 rad / s , 1,000 kg ˜ m2 )

At the same time, an increase in moment of inertia J , although it did not lead to a significant quantitative change in parameters such as Zb , M st , Mav , D , T , was, however, associated with a significant smoothing of motion shapes (see Figs. 4-85− 4-90). Table 4-4 The values of the self-oscillation process parameters in a drill string with a length of L = 1,000 m for different moments of inertia values of bits J L  m

J  kg ˜ m

1,000 2



3.1

1,000

Zl

Zb

Mst  rad

0.71 -32.51

2 -32.91

3.775 -33.12

0.71 -32.55

2 -33.09

3.775 -33.49

Mav  rad

-26.27

-26.43

-26.81

-25.33

-25.92

-26.76

D  rad

12.49 48.02

12.96 10.25

12.62 8.025 0.623 1.246

14.45 54.67

14.34 12.13

13.47 8.94

Z  rad / s

T (s) Δτ (s) T1 (s)

Zb

Zl

Modelling emergency situations in drilling deep borehole

172

So, a burst of angular velocity momentum in Fig. 4-87, a took the form of a differentiable function compared to the quantised pulse of this function in Fig. 4-82, and the oscillating change of this variable in Fig. 4-87, b replaced the step (broken) curve shown in Fig. 4-83,b. The phase portrait of self-oscillations shown in Fig. 4-88 also underwent significant changes. In contrast to the broken outline in Fig. 4-84, it became smoothed retaining only slightly noticeable bulges (Fig. 4-88) in places of vertices coming out of the corners in Fig. 4-84. fr The performed calculations M visc , N ˜m 123750 showed that the functions of the angles of DS elastic whirling retained their 82500 almost rectilinear form, while their derivatives (Fig. 4-89) significantly 41250 smoothed out and lost their discontinuous character. In Fig. 4-90, 0 the bold line shows the range of change Z M -41250 in variable where the oscillations of the system are implemented at the moment of the cycle birth bifurcation. This has no -123750 qualitative differences from the case of -12 -8 -4 0 4 8 12 Z  M , rad / s small value J 3.1 kg ˜ m2 . The fr marked regularities remain in force in Fig. 4-90 Dependence of torque M visc on the oscillatory state associated with the angular velocity Z  M ( L = 1,000 m, loss of the limit cycle (see Table 4-4). 2 Zb 0.71 rad / s , J = 1,000 kg·m ) Figs. 4-91–4-94 show the most typical graphs of oscillation of the drill bit with a small moment of inertia established at Zl 3.775 rad / s . They also occur under the scenario of relaxation oscillations and are accompanied by both large-scale and small-scale discontinuities of angular velocity functions M t . Modes of self-oscillations of the bit with a large moment of inertia at the same state are given in Figs. 4-95–4-99. Comparison of Figs. 4-91, 4-92 with Figs. 4-95, 4-96 leads to the conclusion that with an increase in the moment of inertia J the general behaviour of the system changes slightly. However, the representation of angular velocity function M t at an increased scale shown in Fig. 497 made it possible to establish that the singular features presented in Fig. 4-93 were lost, and the process of self-oscillations became more regular. The phase portrait of bit oscillation has undergone some changes (Fig. 4-98). -82500

Chapter 4. Excitation of torsional self-oscillations of strings in deep wells

φ, rad

0

8

173

M, radd / s

6 -10

4 -20

2 0

-30

-2

t, s

-40 0

40

80

-4

120

35

Fig. 4-91 Mode of drill bit self-oscillations with a length of L=1,000 m, angular velocity Zl 3.775 rad / s and moment of drill bit inertia J

Zl

6

-3.6

4

-3.65

2

-3.7

0

'W t, s 1

2

3

4

5

Fig. 4-93 Specific details of the quasi discontinuous change of angular velocity M  t of the drill bit (L=1,000 m, Zl

J

55

60

65

70

75

t, s

3.775 rad / s , J

3.1 kg ˜ m2 )

-2

-3.8 0

50

φ, rad / s

8

-3.55

-3.75

45

Fig. 4-92 Graph of changes in drill bit angular velocity M  t (L=1,000 m,

3.1 kg ˜ m2

M, radd / s

-3.5

40

3.775 rad / s ,

3.1 kg ˜ m ) 2

-4 -35

-30

-25

Fig. 4-94 Phase portrait of selfoscillations of the drill bit (L=1,000 m, Zl 3.775 rad / s ,

J

3.1 kg ˜ m2 )

-20

φ, rad

Modelling emergency situations in drilling deep borehole

174

M , rad

0

M, radd / s

8

1 -10

6

2

4 2

-20

0

-30 -2

3

t, s

-40 0

20

40

60

80

-4 10

100

Fig. 4-95 Shape of drill bit selfoscillations with a length of L=1,000 m, angular velocity Zl 3.775 rad / s and moment of drill bit inertia

J

8

30

40

50

60

70

80

90 100

t, s

Fig. 4-96 Graph of changes in drill bit angular velocity M  t (L=1,000 m,

Zl

3.775 rad / s , J

1,000 kg ˜ m2 )

1,000 kg ˜ m2

M, rad / s

8

6

6

4

4

2

2

0

0

-2

-2

φ, rad / s

-4

-4 34

36

38

40

42

44

46

t, s

Fig. 4-97 Specific details of the quasi discontinuous change of angular velocity M  t of the drill bit (L=1,000 m,

Zl

20

3.775 rad / s , J

1,000 kg ˜ m2 )

-34

-32

-30

-28

-26

-24

-22

-20

M , rad

Fig. 4-98 Phase portrait of selfoscillations of the drill bit (L=1,000 m, Zl 3.775 rad / s , J 1,000 kg ˜ m2 )

Chapter 4. Excitation of torsional self-oscillations of strings in deep wells

z, m

175

1000

If for a bit with a small moment of inertia the self-oscillations are set within 800 almost one period, and the portrait immediately takes the form of a limit cycle (Fig. 4-94), then for a bit with a 600 3 large moment of inertia at the considered time interval the oscillations are 400 1 intermittent, and the limit cycle is not 2 formed. In addition, the portrait becomes 200 more smoothed and, perhaps, represents a strange attractor, although close to regular. To confirm this assumption, it is -6 -4 -2 0 2 4 M , rad / s necessary to consider fluctuations over a longer period of time. Fig. 4-99 Graph of angular velocity The profiles of angular velocity function M  z of the DS elements at states waves that propagate along the axial line 1, 2, 3 (L = 1,000 m, Zl 3.775 rad / s , of the DS are also noticeably smoothed 2 J = 1,000 kg•m ) (Fig. 4-99). Table 4-5 Values of self-oscillating process parameters of drill strings with a length of L = 2,000 m for different values of drill bit moment of inertia J L  m

J  kg ˜ m

2,000 2



3.1

Mst  rad

Zb 0.71 -65.48

Mav  rad D  rad

Z  rad / s

T (s) Δτ (s) T1 (s)

1,000

2 -65.85

Zl 3.775 -65.95

Zb 0.72 -65.21

2 -66.11

Zl 3.775 -66.81

-52.80

-52.89

-53.49

-51.58

-52.53

-53.66

25.36 94.85

25.91 20.75

24.93 27.26 16.33 107.67 1.243 2.486

27.17 21.38

26.31 17.20

Formulated conclusions on the influence of the value of the bit moment of inertia on the nature of the system’s self-oscillations obtained from analysis of the calculation results of DS with a length of L = 1,000 m are also given in the table and illustrations below on the calculation of a string with a length of L = 2,000 m (Table 4-5 and Figs. 4-100–4-108).

Modelling emergency situations in drilling deep borehole

φ, rad

0

176

M, radd / s

10 8

-20

6 -40

4 2

-60

0

t, s

-80 0

150

300

450

120

Fig. 4-100 Mode of drill bit selfoscillations with a length of L=2,000 m, angular velocity Zb 0.71 rad / s and moment of drill bit inertia J 3.1 kg ˜ m2

10

M, rad / s

t, s

-2 180

240

300

360

420

Fig. 4-101 Graph of changes in drill bit angular velocity M  t (L=2,000 m,

Zb 0.71 rad / s , J

3.1 kg ˜ m2 )

φ, rad

0

8 -20

6 -40

4 2

-60

0

t, s

-2 150

155

160

165

170

Fig. 4-102 Specific details of the quasi discontinuous change of angular velocity M  t of the drill bit (L=2,000 m, Zb

J

0.71 rad / s , 3.1 kg ˜ m2 )

t, s

-80 0

100

200

300

400

500

Fig. 4-103 Mode of drill bit selfoscillations with a length of L=2,000 m, angular velocity Zb 0.72 rad / s and moment of drill bit inertia J 1,000 kg ˜ m2

Chapter 4. Excitation of torsional self-oscillations of strings in deep wells

M, rad / s

10

8

6

6

4

4

2

2

0

0

-2

-2

0

100

200

300

400

500

Fig. 4-104 Graph of changes in drill bit angular velocity M  t (L=2,000 m,

Zb 0.72 rad / s , J 1,000 kg ˜ m2 )

φ, rad

0

M , rad / s

10

8

177

t, s 148

152

156

160

164

168

Fig. 4-105 Specific details of the quasi discontinuous change of angular velocity M  t of the drill bit (L=2,000 m,

Zb 0.72 rad / s , J 1,000 kg ˜ m2 )

M, rad / s

8

-20

4 -40

0 -60

t, s

-80 0

40

80

120

Fig. 4-106 Mode of drill bit self-oscillations with a length of L=2,000 m, angular velocity Zl 3.775 rad / s and moment of drill bit inertia J

1,000 kg ˜ m2

-4 10 20 30 40 50 60 70 80 90 100 110

t, s Fig. 4-107 Graph of changes in drill bit angular velocity M  t (L=2,000 m,

Zl

3.775 rad / s , J

1,000 kg ˜ m2 )

Modelling emergency situations in drilling deep borehole 8

M, radd / s

-3.2

178

M, radd / s

-3.3 4

-3.4 0

'W

-3.5

-4

t, s

-3.6 70

72

74

76

78

80

82

t, s

65

66

67

68

69

70

71

a) b) Fig. 4-108 Specific details of the quasi discontinuous change of angular velocity M  t of the drill bit (L=2,000 m, Zl 3.775 rad / s , J 1,000 kg ˜ m2 ): a = fragment of angular velocity function M  t on a time interval 70 ≤ t ≤ 82 s

b = fragment of angular velocity function M  t on a time interval 65 ≤ t ≤ 71 s

Analysis of these results shows that—as in the case of a string with a length of L = 1,000 m — the increase in the moment of inertia J did not significantly affect the values of the generalised characteristics of self-oscillations because values Z p , M st ,

Mav , D , T changed little. At the same time, the forms of periodic motion experienced qualitative changes. As for the case considered above, with an increase of J in the fast motion area, the function of angular velocity M slightly smooths (see Figs. 4-102 and 4-105 for the cycle birth bifurcation state), although it has undergone significant changes in the modes of slow motion, the transformation of stepped (broken) outlines in oscillating curves with a conditional period equal to the quantum of time 't . For the state of the cycle limit loss bifurcation, the marked feature is more noticeable (see Fig. 4-108). Here, small-scale oscillations have acquired an even more complex form with the same conditional period 'W equal to 1.243 s. It is also worth noting the outlines of the wave profiles of angular velocities propagating along the axial line of the DS. They also became more regular. If we summarise the obtained results, it can be concluded that in paragraph 4.3 based on the wave model of torsion pendulum developed without taking into account dissipative effects the methodology of a theoretical study of the phenomena of the torsional oscillations origin of the vertical deep-well strings and the transition of

Chapter 4. Excitation of torsional self-oscillations of strings in deep wells

179

modes of motion through the bifurcation states (the Hopf bifurcation of birth and loss of limit cycles) is proposed. The basic evolution regularities of the dynamic processes with change of characteristic parameters of system are established. A wave model of torsional self-oscillations of a homogeneous drill string in the form of a torsion pendulum is developed. On the basis of the D'Alembert solution arising from the condition of the absence of torsion wave dispersion, the wave differential equation with partial derivatives is reduced to a non-linear ordinary differential equation with a delayed argument. It is shown that the formulated differential equation is singularly perturbed. Therefore, a conclusion is made about the possibility of the existence of periodic relaxation solutions characterised by discontinuities (in our case almost breaks) in the angular velocity of drill bit rotation. By solving problems for the two most typical kinds of function of the drill bit cutting forces moment in the form of non-linear viscous Coulomb friction moments, it is shown that ranges of values of the drill string rotation angular velocity exist in which the system is in torsion oscillation mode. The boundaries of these ranges (states of birth bifurcation and loss of limit cycles—Hopf bifurcation) are found, and the forms of self-oscillations at these states are investigated. It has been established that the studied oscillations are relaxation and are accompanied by the presence of fast and slow movements with large-scale discontinuities of the angular velocities of the drill bit rotation. The function of the elastic torsion angle of the string elements varies almost linearly along its length with the presence of small short-wave disturbances, which, however, leads to almost discontinuous functions of changing their angular velocities. It is shown that the shapes of drill bit torsion self-oscillations do not qualitatively depend on the length of the drill string, and when it changes they only transform in accordance with certain conditions of similarity and scaling. The bifurcation values of the angular velocity of the homogeneous drill string are determined mainly by the position of the falling section and extreme points on the diagram of the cutting torque dependence on the angular velocity of rotation. 4.4. Torsional oscillations of composite drill strings In the previous sections, studies were made of the self-excitation processes of torsional oscillations of homogeneous drill strings, whose stiffness and inertia properties do not change along their axial coordinates (Fig. 4-109,a). For drill strings in deep bore-holes, the homogeneity property of the DS is not usually fulfilled as the DS is prestressed by gravity forces and is therefore impacted by the action of an axial force variable along the DS length.

Modelling emergency situations in drilling deep borehole

180

Thus, to provide the string with equal strength, they are made, as a rule, heterogeneous, with increased cross-section sizes in the upper parts (Fig. 4-109,b). Quite often, the property of heterogeneity due to local thickening is also given to the drill string in its lower part to locally increase its weight and give more local rigidity, which makes it possible to ensure the project straightness of the drilling well at the drilling site.

l2

Z, z1 z

L

Z, z1 z

L

l1

y O

X Zt x1

y

y1

O

Y X Zt

M x

x1

y1

Y

M x

a) b) Fig. 4-109 Diagrams of homogeneous (a) and composite (b) drill strings

Naturally, the waveguide properties of composite drill strings are fundamentally different from the wave characteristics of homogeneous strings due to the occurrence of multiple acts of wave reflection–refraction at the points of abrupt change in stiffness at the junctions. This also leads to the restructuring of the wave profiles of oscillatory and self-oscillating processes of drill bit torsion movements. Since the torsional stiffness function of a string at the point of connection of two different sections is discontinuous, the function of the torsion deformation wave also

Chapter 4. Excitation of torsional self-oscillations of strings in deep wells

181

experiences additional discontinuities, which−although diffracted on the inhomogeneities−still carry out an impact effect on its components. These effects lead to a noticeable increase in shear stresses, increased drill bit wear, disruption of drilling technology, and the risk of unwinding threaded connections of different pipe segments. Therefore, these effects should be studied specifically. 4.4.1. Formulation of the composite drill string torsion self-oscillation problem To formulate the problem of diffraction of the elastic torsional wave at the discontinuities of torsion stiffness uniformity GI z , we shall first consider a homogeneous drill string in the form of a torsion pendulum, to the lower end of which a drill bit is attached (Fig. 4-109,a). The upper end of the DS rotates at given constant speed Z . Let's introduce inertial coordinate system OXYZ with the origin in the bit centre of the mass, axis OZ of which coincides with the axial line of the DS. Let's connect coordinate system Oxyz to the bit rotating with it and Oz coinciding with axis OZ . Then, the rotation angle of the bit relative to system OXYZ will be Zt  M , where Zt is the rotation angle of the upper end of the DS, t is the time, M is the angle of elastic torsion of the DS. As underlined above, when ignoring the effects of energy dispersion, DS elastic torsional motions are described by the wave equation w 2M w 2M (4.41) UI z 2  GI z 2 0 , wt wz where U is the density of the DS material, G is its shear modulus of elasticity, I z is the polar moment of inertia of the cross-sectional area of the DS pipe. Denoting

E G / U , where E is the velocity of the elastic shear wave (torsional wave), equation (4.41) is reduced to the standard form: w 2M w 2M  E 2 2 0. (4.42) 2 wt wz It has a solution in the D'Alembert form (Fig. 4-110) M  z , t f  z  Et  g  z  Et , (3.3)   g z  E t f z  E t where , are arbitrary continuous, not necessarily differentiable functions. The first of them determines the wave propagating at velocity E in the positive direction of axis Oz , and the second one, in the opposite direction. These waves do not disperse, which is why they move without changing their profile. The specified functions have boundary conditions F[ f 0  Et , g 0  Et ] 0, (4.44)

Modelling emergency situations in drilling deep borehole

182

f L  Et  g L  Et 0 , (4.45) where F is the non-linear differential operator describing the motion of the drill bit.

f 2r  z, t

f z, t

g z, t

g 2i  z , t

f 2t  z , t g1t  z , t

f1i  z , t g1r  z , t

z y

x Fig. 4-110 The scheme of torsion waves propagation in a homogeneous drill string

Fig. 4-111 Scheme of torsion waves propagation in a non-uniform composite drill string

Condition (4.44) is formed by the equation of balance of moments of inertia forces M in , friction forces M fr , and elastic forces M el (4.46) M in  M fr  M el 0 , which is derived from the D'Alembert principle, written for a drill bit, conventionally separated from the DS pipe. Here, M in is the moment of inertia forces of the drill bit calculated using the formula M in  J ˜ M . (4.47)

M el is the moment of elasticity forces determined by the equality

Chapter 4. Excitation of torsional self-oscillations of strings in deep wells

wM . wz is the moment of friction forces calculated using the formula M el

M M

тр вяз

fr

m ˜

183

^

GI z

` ^

(4.48)

`

3

a1 k ˜ ª¬Z  f   E t  f  2 L  E t º¼  a3 k ˜ ª¬Z  f   E t  f  2L  E t º¼ 

^

`

1  a2 k ˜ ª¬Z  f   E t  f  2 L  E t º¼

^

`

 a7 k ˜ ª¬Z  f   E t  f  2 L  E t º¼

^

7

^

`

 a9 k ˜ ª¬Z  f   E t  f  2 L  E t º¼

`

1  a2 k ˜ ª¬Z  f   E t  f  2 L  E t º¼

(4.49)

2

2

9

,

or fr M Cou

fr  M lim  e M elfr .

(4.50)

As waves f z  E t and g z  E t are not dispersive, and their profiles remain unchanged, it is possible to express functions f z, t and g z, t through phase variables z  E t and z  E t . This property makes it possible to move from partial derivative wM / wz to derivative wM / wt (4.23) in equality (4.48), to express functions g z  E t through functions f 2 L  z  E t with a delay argument, and to bring equation (4.46) to ordinary differential equation (4.25) J ª¬ f   E t  f  2 L  E t º¼  M fr 

GI z

ª f   E t  f  2 L  E t º¼ 0 E ¬

(4.51)

with independent variable t . This algorithm is effective for a homogeneous drill string as wave f z  E t is diffracted once at the upper end z L , and the magnitude of its delay when returning to the bit is 2L / E . However, the situation is much more complicated if the DS is heterogeneous, and torsion waves experience additional refraction-reflection acts at the junction of the DS segments of different stiffness. Then, there are acts of secondary, tertiary, etc. reflections–refractions of already reflected and refracted waves both at the discontinuity point of the torsion stiffness of the DS and at its upper end (at interface points). In this case, the refracted–reflected waves can ‘wander’ between interface points experiencing multiple reflection–refraction acts and reach the bit with different time delays. Therefore, it is hardly possible to avoid the wave formulation of the problem with one ordinary differential equation of type (4.25), and it seems necessary to directly monitor the movement of waves f z, t and g z, t . This is easy to do, provided that the waves are not dispersed.

Modelling emergency situations in drilling deep borehole

184

To make this transition, we turn to wave propagation scheme f z, t and

g z, t in a two-section string (Fig. 4-111). To simplify, we assume that both sections are made of the same material, and therefore propagation velocity E of torsion waves in them is the same. Let's assume that the lower part is the first, the upper part is the second. Then, wave f z, t rising up will be born at z 0 as a result of the impact of the bit on the lower end of the DS. This wave will propagate without changing its profile, up to the interface point and, therefore, when it reaches it, will play the role

of an incident wave. Let's define this wave as f1i  z , t . If this wave is diffracted at the point of separation z l1 of the first and second sections, it will be transformed to a reflected wave and a refracted (transmitted) wave. Let's mark them as g1r  z , t and f 2t  z , t (Fig. 4-111). Reflected wave g1r  z , t will descend at velocity E ; refracted

wave f 2t  z , t will propagate up at the same speed. At the top point z

L , wave g 2r  z, t is formed that propagates down at

velocity E along the second section and plays the role of an incident wave for the interface point.

When this point is reached, the g 2i  z , t wave disintegrates into refracted wave

g1t  z , t propagating down in the first section, and reflected wave f 2r  z, t , moving up

in the second section. In this case, wave g1t  z , t is superimposed on wave g1r  z , t , and they act together on the drill bit, and wave f 2r  z, t is superimposed on wave f 2t  z , t , and they, rising up, then take part in the reflection from the top point turning

into the g 2i  z , t wave. It should be noted again that none of these waves are dispersive; therefore, they all propagate without changing their profile. The described scheme of the propagation of refracted and reflected f z, t and g z, t waves is used in the formulation of the problem of the self-excitation of drill string and the drill bit torsion oscillations. It is used with just the difference that in equation of motion (4.46) function g z  E t is not replaced by function f z, t ,2 L / E with the delay argument, and in expression (4.43) for M  z, t z

g z, E t z

0

g1r z, t

z 0

 g1t z, t

z 0

0

function

is used directly. Corresponding changes are

implemented for derivatives wM / wz , wM / wt , w 2M / wz 2 , w 2M / wt 2 . This algorithm of the problem solution leads to the need to build mobile wave profiles of functions

f1i  z , t on section I, f 2t  z, t  f 2r  z , t on section II, g 2i  z , t on section II, and

Chapter 4. Excitation of torsional self-oscillations of strings in deep wells

185

g1r  z , t  g1t  z , t on section I. We should again emphasise that the procedure for their

construction does not cause great difficulties since these waves are not dispersive and propagate without changing their shapes. Therefore, the summands of equality (4.46) take the corresponding forms. The moment of inertia forces of the bit is recorded in its initial form: w2 (4.52) M in  J ˜ M  J 2 ª¬ f   E t  g  E t º¼ . wt The moment of viscous friction forces (4.26) after substitution (4.53) T k Z  M z 0 k Z  f  E t  g E t

>

is replaced with the equation M

fr el

m ˜

^

@

` ^

`

3

a1 k ˜ ª¬Z  f   E t  g  E t º¼  a3 k ˜ ª¬Z  f   E t  g  E t º¼ 

^

`

1  a2 k ˜ ª¬Z  f   E t  g  E t º¼

^

`

 a7 k ˜ ª¬Z  f   E t  g  E t º¼

^

7

^

2

`.

 a9 k ˜ ª¬Z  f   E t  g  E t º¼

`

2

9

(4.54)

1  a2 k ˜ ª¬Z  f   E t  g  E t º¼ The moment of Coulomb friction forces, which is described by the expression fr M Cou

fr  M lim  e M elfr ,

(4.55)

is also subject to corresponding changes. To derive function M el  t , as above, we use the dependence

wM . (4.56) wz It differs from equality (4.52) in that it contains a partial derivative of function M with respect to z . To move from this derivative to the derivative with respect to t , we shall write the following dependencies wM  z, t wf  z  Et wg  z  Et wf wu wg ww wf wg  ˜  ˜  , wz wz wz wu wz ww wz wu ww (4.57) wM  z, t wf  z  Et wg  z  Et wf wu wg ww wf wg  ˜ E E .  ˜ wu ww wt wt wt wu wt ww wt From there, it leads to wM 1 wf 1 wg   . (4.58) wz E wt E wt Taking into account (4.58), we write down § 1 wf 1 wg · M el GI z ¨  (4.59) ¸. © E wt E wt ¹ M el

GI z

Modelling emergency situations in drilling deep borehole

186

Variables f  Et and g Et used in relations (4.52), (4.54), (4.59) are defined as

f  Et

f1i 0, t ,

(4.60) g Et g1r 0, t  g1t 0, t . We substitute resulting expressions (4.52), (4.54), and (4.59) into equality (4.46). Taking into account (4.60), we obtain in the final form a non-linear ordinary differential equation of the bit torsion motion GI 0, t  g1r  0 0, t  g1t  0 0, t º¼  M fr  z ª¬ f1i  0 0, t  g1r  0 0, t  g1t  0, 0 t º¼ . (4.61) J ª¬ f1i  0 E It was used to study self-excitation of the oscillations of composite drill strings.

In it, functions g1r t , g1t t of the incoming waves are considered known, and the function of initial wave f1i t is unknown. It is found by numerical integration of this equation.

M 2 i

E2 l2

l2

E 2 't

M 2

E 2 't

E2

t

E2

L

E 2 't

M 2 r

L

M1r E1' t l1

E1

E1

z

E1' t l1

M1i

E1' t

y x Fig. 4-112 Diagram of diffraction of incident torsion wave M1i  t at the junction point of two segments of the DS

E1 M1t

z y x Fig. 4-113 Diagram of diffraction of incident torsion wave M 2i  t at the junction point of two segments of the DS

Chapter 4. Excitation of torsional self-oscillations of strings in deep wells

187

To set functions g1r t , g1t t , it is necessary to monitor the motion of the whole system of waves in sections I and II and simulate their transformation at interface point z l1 (Fig. 4-111). The mechanical characteristics of the string in these areas shall be E1 , U1 , I1

and E 2 , U 2 , I 2 , respectively. Then, the elements of wave f z  Et spreading from point z 0 and reaching point z l1 will be exposed to refraction–reflection

impacts. To calculate the intensities of the corresponding refracted and reflected waves, we consider the process of diffraction of the wave element with length E1't over time 't (Fig. 4-112). We distinguish the elements of the DS in the incident, reflected, and penetrated waves participating in this interaction with angular i r t velocities M1 , M1 , M 2 and the lengths of elements E1't , E1't , E 2 't , respectively. Here, indices i, r , t , as above, are marked, respectively, as incident, reflected, and refracted (transformed) waves. r t i Assuming that function M1 is known, we calculate M1 , M 2 . To do this, we use the condition of the moment of movement conservation of the selected elements relative to axis Oz before and after the impact. It has the form: i

r

'K1

t

'K1  'K 2 ,

(4.62)

where i

M1i ˜ U1I1E1't ,

r

M1r ˜ U1I1E1't ,

'K1 'K1

t

(4.63)

t

'K 2 M 2 ˜ U 2 I 2 E 2 't. Enhancing equation (4.62) with the condition of continuity of angular velocities M1i  M1r M 2 t , (4.64) we obtain a system of two equations (4.65) E1 U1 I1M1r  E 2 U 2 I 2M 2 t  E1 U1 I1M1i , r t to calculate M1 and M 2 . M1r  M 2 t M1i . It has a solution E 2 U 2 I 2  E1 U1 I1 i M1r M , E1 U1 I1  E 2 U 2 I 2 1

2 E1 U1 I1 M i . (4.66) E1 U1 I1  E 2 U 2 I 2 1 Torsion angles in the reflected and penetrated waves are calculated from the conditions of continuity of the torque and the torsion angle. They are defined by similar formulas

M 2 t

Modelling emergency situations in drilling deep borehole

188

E 2 U 2 I 2  E1 U1 I1 i 2 E1 U1 I1 M1 , M 2t M i. (4.67) E1 U1 I1  E 2 U 2 I 2 E1 U1 I1  E 2 U 2 I 2 1 When considering the diffraction of wave g z  Et in a cross-section z l1 , M1r

wave g 2  z  E 2t in the second section approaching this section is incident and is i

considered as known, and reflected and refracted waves g 2  z  E 2 t ( z t l1 ) and r

g1  z  E1t ( z d l1 ) are to be determined. They are modelled in accordance with the t

diagram presented in Fig. 4-113 for wave M z, t incoming from above to interface section z l1 . A system of equations

E1 U1 I1M 2 r  E 2 U 2 I 2M1t

i

 E 2 U 2 I 2M 2 ,

(4.68)

M 2 r  M1t M 2 i .

is used to determine them. These kinematic characteristics are calculated by the above method using equations 2E 2 U 2 I 2 E1 U1 I1  E 2 U 2 I 2 i M 2 r M , M1t M i , (4.69) E1 U1 I1  E 2 U 2 I 2 2 E1 U1 I1  E 2 U 2 I 2 2

M2r

E1 U1 I1  E 2 U 2 I 2 i M , E1 U1 I1  E 2 U 2 I 2 2

M1t

2E 2 U 2 I 2 M2i . E1 U1 I1  E 2 U 2 I 2

(4.70)

As a result of diffraction of waves M1  z  E1t and M 2  z  E 2 t , superposition i

i

of penetrated (transmitted) M 2  z  E 2 t and reflected M 2  z  E 2 t waves makes up r

t

wave f 2 z  E 2 t in the second section, and sum M1  z  E1t  M1  z  E1t , wave g1 z  E1t in the first section. With this in mind, we obtain the initial conditions at point z l1 for wave f 2 z  E 2 t in segment l1 d z d L r

t

2 E1 U1 I1 E U I  E2 U2 I 2 r t g l  E1t , f l  E1t  1 1 1 E1 U1 I1  E 2 U 2 I 2 2 1 E1 U1 I1  E 2 U 2 I 2 1 1 (4.71) 2 E1 U1 I1 f l  E t f t l  E t  E1 U1 I1  E 2 U 2 I 2 g r l  E t , 2 1 2 1 1 E1 U1 I1  E 2 U 2 I 2 2 1 E1 U1 I1  E 2 U 2 I 2 1 1 and the initial conditions at this point for wave g1 z  E1t in segment 0 d z d l1 2E 2 U 2 I 2 E1 U1 I1  E 2 U 2 I 2 r t g l  E 2t , f l  E1t  g1 l1  E1t E1 U1 I1  E 2 U 2 I 2 2 1 E1 U1 I1  E 2 U 2 I 2 1 1 (4.72) 2E 2 U 2 I 2 E1 U1 I1  E 2 U 2 I 2  r t g l  E 2t . f l  E1t  g1 l1  E1t E1 U1 I1  E 2 U 2 I 2 2 1 E1 U1 I1  E 2 U 2 I 2 1 1 Equations (4.42), (4.43) together with boundary conditions (4.45), (4.46) and interface conditions (4.71), (4.72) describe a three-point boundary value problem in f 2 l1  E 2t

Chapter 4. Excitation of torsional self-oscillations of strings in deep wells

189

respect of variable z with conditions at points z 0 , z l1 , and z L . It is solved using the Runge-Kutta method. To present the solution method, we shall write down the defining equations and describe the algorithm for constructing their solutions. An important factor simplifying the method of solution is that torsion waves M z, t propagating along the axial line of the DS adhere to wave equation (4.41). Solution M z, t of this equation represents a superposition M z, t f z  Et  g z  Et of two non-dispersing waves, in which wave f z  Et propagates up, and wave g z  Et , down. These waves do not change their profile while moving within each segment of the DS. Therefore, there is no need to reconstruct the profile of each of these waves when they move on segments 0  z  l1 and l1  z  L , but it is sufficient—by using the conditions of their transformation at points z 0 , z l1 and

z L —to use the values of the reflected and refracted waves as initial conditions for modelling their subsequent movement on each of selected segments 0  z  l1 , l1  z  L . We can give a summary of the defining relations. At

g z, t

g1r

point

z

0,

the

condition

of

diffraction

z, t  z, t and its transformation into wave g1t

f1i

of

z, t

incident

wave

are determined by

the equation of elastic torsion vibrations of the drill bit: GI 0, t  g1r  0 0, t  g1t  0 0, t º¼  M fr  z ª¬ f1i  0 0, t  g1r  0 0, t  g1t  0, 0 t º¼ . (4.73) J ª¬ f1i  0 E In it, functions g1r 0, t , g1t 0, t are considered to be given, and function f1i 0, t is unknown. It is found by integrating equation (4.73) using the numerical method and its subsequent transfer up without changing the shape (Fig. 4-111). After

reaching point z l1 , wave f1i  z , t diffracts at the interface and splits into one

reflected wave g1r  z , t and one penetrated wave f 2t  z , t . Wave g1r  z , t , falling down, again falls on the drill bit and again takes part in the formation of wave f1i  z , t , while penetrated wave f 2t  z , t continues to propagate up in section

l1  z  L . The intensity of waves g1r l1 , t and f 2t l1 , t are calculated using the formulas E U I  E1 U1 I1 i f l  E1 t , g1r l1  E1 t  2 2 2 E1 U1 I1  E 2 U 2 I 2 1 1 (4.74) 2 E1 U1 I1 f1i l1  E1 t , f 2t l1  E 2 t E1 U1 I1  E 2 U 2 I 2

derived from formulas (4.66), (4.67). Upon reaching point z

L , wave f 2t L, t

diffracts at this cross-section and is reflected as wave g 2i L, t . The intensity of the

Modelling emergency situations in drilling deep borehole

190

reflected wave is determined from the condition of interaction of the incident wave with a rigid restraint f 2t L,t  g 2i L, t 0

or

g 2i L, t  f 2t L,t .

(4.75)

Wave g 2i L, t reflected from the rigid clamping becomes an incident wave for section l1  z  L (Fig. 4-111). It propagates downwards, without changing its profile, to interface point z l1 , diffracts at it, and splits into reflected wave f 2r l1 , t superimposed on wave f 2t l1 , t and into penetrated wave g1t l1 , t , which, falling down together with wave g1r  z , t , becomes an incident wave for the drill bit (Fig. 4-113). The intensities of the waves formed at interface point z l1 are determined by the equations E U I  E2 U2 I 2 i g l  E 2 t , f 2r l1  E 2 t  1 1 1 E1 U1 I1  E 2 U 2 I 2 2 1 (4.76) 2E 2 U 2 I 2 t i g l  E 2 t , g1 l1  E1 t E1 U1 I1  E 2 U 2 I 2 2 1 arising from relations (4.69), (4.70). Therefore, the process of solving the problem is reduced to just the integration of one non-linear ordinary differential equation (4.73) and calculation by formulas (4.74)–(4.76) at each step of calculating the reflected and penetrated waves when passing through the interface point. Equation (4.73) is integrated using the RungeKutta method, integration step 't is selected from the condition of convergence of the computational process. 4.4.2. Testing of the constitutive relations For a more complete justification of the method for modelling the phenomenon of whirling wave propagation in a composite string and testing the conditions of their diffraction and rearrangement at the places of discontinuous changes in the mechanical properties of the DS obtained in paragraph 4.4.1, we shall analyse relations (4.74), (4.76) determining the boundary conditions. To this end, we shall write again equations (4.74) for point z l1 E U I  E1U1I1 i g1r  l1  E1t  2 2 2 f l  E t , E1U1I1  E 2 U 2 I 2 1 1 1 (4.77) 2 E1U1I1 f 2i  l1  E 2t f1i  l1  E1t . E1U1I1  E 2 U 2 I 2 Let's consider the limiting case when the mechanical properties of the DS links in segments I and II become the same—that is,

Chapter 4. Excitation of torsional self-oscillations of strings in deep wells

U1

U2 ,

E1 E 2 ,

I1

191

I2 .

(4.78)

z, t passing It is clear that then the string will be homogeneous, wave through this point will spread further down without changing its intensity, and the phenomenon of reflection of the incident wave at this point will not take place. That is, from relation (4.77), it should follow g1r l1  E1 t 0, (4.79) f 2t l1  E 2 t f1i l1  E1 t . In fact, using equation (4.78), we obtain E U I  E1 U1 I1 i f l  E1 t 0, g1r l1  E1 t  1 1 1 E1 U1 I1  E1 U1 I1 1 1 f1i

2 E1 U1 I1 f i l  E1 t f1i l1  E1 t . E1 U1 I1  E1 U1 I1 1 1 So, the incident wave will pass through point z l1 without feeling any of its influence. The second limit case will be modelled assuming that the parameters of torsion stiffness of the DS in the second segment tend to infinity, which corresponds to the condition of rigid clamping of the first section at point z l1 . Let's rewrite equation (4.77) for this case as f  E1 U1 I1 i f l  E1 t  f1i l1  E1 t , g1r l1  E1 t  E1 U1 I1  f 1 1 2 E1 U1 I1 f i l  E1 t 0. f 2t l1  E 2 t E1 U1 I1  f 1 1 f 2t l1  E 2 t

As expected, when interacting with rigid clamping, reflected wave g1r l1  E1t retains its intensity while changing only its sign to the opposite. Naturally, no wave

penetrates in a perfectly rigid restraint; therefore, we obtain f 2t l1  E 2t 0 . In the third test case, we assume that the stiffness numerical characteristics of the DS in the second segment tend to zero. This corresponds to the free end of the drill string at point z l1 of segment I. As is known, when reflected from the free end, the torsion waves do not change their intensity or orientation. Indeed, putting E 2 U 2 I 2 o 0 in (4.77), we get 0  E1 U1 I1 i g1r l1  E1 t  f l  E1 t f1i l1  E1 t , E1 U1 I1  0 1 1 which is consistent with the above considerations. It is interesting to note that the intensity of the penetrated wave

Modelling emergency situations in drilling deep borehole

f 2t l1  E 2 t

192

2 E1 U1 I1 i f l  E1 t 2 f1i l1  E1 t E1 U1 I1  0 1 1

is duplicated. However, this result has no physical content since at z ! l1 the medium is actually absent, and the question of the propagation of the wave in it makes no sense. In this case, although wave f 2t  z, t 0 and penetrates into another medium with a doubled amplitude, its energy is zero because U 2 0 and I 2 0 . Such conversions and checks can be performed with conditions (4.76). They also confirm the validity of the results obtained in paragraph 4.4.1. 4.4.3. Analysis of the self-oscillations of composite drill strings using the viscous friction force moment model As accentuated above in the analysis of self-oscillations of homogeneous drill strings, the solutions of equation (4.30) have several characteristic properties arising from the structure of function M fr Z  M . The most important of them is that this

equation has stationary solutions f t const or M t const for any value Z . In addition, there is a range of values Zb d Z d Zl where, in addition to these stationary solutions, stable non-stationary periodic solutions are also added, while the stationary ones become unstable. Beyond this range, stationary solutions in the form of equilibrium rotation Z const , M const are stable. States Z Zb and Z Zl where stationary rotations pass into self-oscillation modes and vice versa are called bifurcations of birth and loss of limit cycle or Hopf bifurcations. In addition, the main characteristic that affects the process of self-excitation of torsion oscillations and its properties is the law of dependence of friction moment M fr on total angular velocity T Z  M of rotation of the drill bit. This function is determined by many factors, among which we distinguish the forms of dependence M fr on T and highlight only the most common patterns of the occurring self-oscillating process.



In this section, we shall consider the function of moment M fr T in the form



of viscous friction. Formula (4.40) M fr T for this case is presented in subsection 4.3.8 where values of quantities ai and constants e , m , k determining the shape of its graph are given. In the simulation of self-excitation of oscillations, it was considered that the drill bit was removed from contact with the bottom of the well, and the string put in rotational motion at constant velocity Z . Then, the DS is lowered until the bit enters into contact interaction with the well bottom, and the process of unsteady movement of the bit and the string occurs, which is modelled using a system of equations (4.73),

Chapter 4. Excitation of torsional self-oscillations of strings in deep wells

193

(4.74), (4.76). Computer simulation of such movement is carried out until the stationary rotation of the drill bit with a constant angular velocity (which corresponds to the drilling mode), or the drill bit begins to perform stable torsional selfoscillations. Such modelling is implemented at different values Z and makes it possible to find parameters Zb , Zl separating the ranges of stable drilling with constant angular velocity of drill bit Z and the established periodic relaxation selfoscillations with constant period T and amplitude D . The stability of these processes is checked directly by computer simulation. Equation (4.73) is integrated using the Runge-Kutta method under initial conditions M 0 0 , M 0 0 . The integration step was chosen equal to Δt = 6.4742•10–6 s. Three main combinations of design parameter values of a DS with a length of L = 1,000 m and composed of two sections were considered: case I: l1 L / 3 , l2 2 L / 3 case II: l1 l2 L / 2 case III: l1 2 L / 3 , l2 L / 3 The mechanical and geometric parameters of the pipe of the first (lower) section are: G 80.77 GPa , U 7.8 ˜103 kg / m3 , J 3.1 kg / m2 , r1 0.05049 m , 0.41 ˜ 105 m4 . For the second (upper) section, the mechanical properties of the pipe material and the moment of inertia of the drill bit remained unchanged, and the tube radii acquired values r2 0.08415 m , r2 0.07415 m , while I z 3.12 ˜105 m4 . We should note that in the previous sections, homogeneous strings with these characteristic parameter values were studied. The results of the calculations for the three cases are given in Table 4-6. For comparison, the numerical characteristics of self-oscillations of a homogeneous pipe with the previously considered parameters at Zb 0.71 rad / s are given again from r2

0.04449 m , while I z

Table 4-4. It should be noted that, as shown by numerical analysis, in all considered cases bifurcation of birth ( Zb ) and loss ( Zl ) of the limit cycle occur at the same values Zb | 0.71 rad / s , Zl | 3.775 rad / s . The graphic materials for these states in case I are shown in Figs. 4-114–4-119 ( Zb | 0.71 rad / s ) and Figs. 4-120–4-125 ( Zl | 3.775 rad / s ). We can see that for this construction the dynamic process proceeds according to the scenario of relaxation self-oscillations with large-scale and small-scale velocity discontinuities, although significant differences have acquired the effect of their quantisation in time. If for a homogeneous string the quantum length is equal to the ratio of the doubled length of the DS to the wave velocity, then in this case, due to the additional formation of numerous reflection–refraction acts in the interface point of the string

Modelling emergency situations in drilling deep borehole

φ, rad

0

194

M, rad / s

12

8

1

-40

T 2

4

Mср

-80

D

Mст

-120

3

0

200

t, s 400

M, rad / s

0

'W 3

200

400

600

Fig. 4-115 Graph of changes in angular

velocity  of the drill bit of a composite DS (L=1,000 m, Zb 0.71 rad / s ,

M t

l1

12

-0.704

t, s

-4

600

Fig. 4-114 Shape of self-oscillations of a composite DS drill bit with a length of L=1,000 m at angular velocity Zb 0.71 rad / s ( l1 L / 3 , l2 2 L / 3 )

-0.702

0

L / 3 , l2

2L / 3 )

φ, rad / s

8

-0.706

4

'W -0.708

0

t, s

-0.710 0.5

1.0

1.5

2.0

2.5

3.0

Fig. 4-116 Specific details of the quasi discontinuous change of angular velocity M  t of a composite DS drill bit (L=1,000

m, Zb

0.71 rad / s , l1 )

L / 3 , l2

2L / 3

φ, rad

-4 -110

-100

-90

-80

-70

-60

Fig. 4-117 Phase portrait of self-oscillations of composite DS drill bits (L=1,000 m, Zb 0.71 rad / s , l1 L / 3 , l2 2 L / 3 )

Chapter 4. Excitation of torsional self-oscillations of strings in deep wells

195

z, m 1000

z, m

1000

800

800

600

600

400

400

3 2 1 200

200

1 -120

M , rad

-80

-40

-5

0

Fig. 4-118 Graphs of elastic torsion angle function M  z at states 1, 2, 3 of a composite DS (L=1,000 m, Zb 0.71 rad / s , l1 L / 3 , l2 2 L / 3 )

-2

-1

0

1

M , rad / s

8

1

-40

-3

Fig. 4-119 Graph of angular velocity function M  z of composite DS elements at state 1 (L=1,000 m, Zb 0.71 rad / s , l1 L / 3 , l2 2 L / 3 )

φ, rad

0

-4

M , rad / s

4

2 0

-80

3

t, s

-120 0

40

80

120

160

Fig. 4-120 Shape of self-oscillations of a composite DS drill bit with a length of L=1,000 m at angular velocity Zl 3.775 rad / s ( l1 L / 3 , l2 2 L / 3 )

t, s

-4 0

40

80

120

160

Fig. 4-121 Graph of changes in angular velocity M  t of the drill bit of a composite DS (L=1,000 m, Zl 3.775 rad / s , l1 L / 3 , l2 2 L / 3 )

Modelling emergency situations in drilling deep borehole

M , rad / s

-3.73

196

8

-3.74

M t

φ, rad / s

4

-3.75

'W

-3.76

0

t, s

-3.77 0.5

1.0

1.5

2.0

2.5

3.0

Fi. 4-122 Specific details of the quasi discontinuous change of angular velocity M  t of a composite DS drill bit (L=1,000 m, Zl 3.775 rad / s , l1 L / 3 , l2 2 L / 3 )

z, m

3

M , rad

-4 -110

-100

-90

-80

-70

-60

Fig. 4-123 Phase portrait of self-oscillations of a composite DS drill bit (L=1,000 m, Zl 3.775 rad / s , l1 L / 3 , l2 2 L / 3 )

z, m

1000

1000

800

800

600

600

2

1

400

400

200

200

1 -120

M , rad

-80

-40

-4

0

-3

M , rad / s

-2

-1

0

1

Fig. 4-124 Graphs of elastic torsion angle function M  z at states 1, 2, 3 of a composite

Fig. 4-125 Graph of angular velocity function M  z of composite DS elements at state 1

DS (L=1,000 m, Zl 3.775 rad / s , l1 , l2 2 L / 3 )

(L=1,000 m, Zl

L/3

3.775 rad / s , l1 l2 2 L / 3 )

L/3 ,

Chapter 4. Excitation of torsional self-oscillations of strings in deep wells

197

distant from the bit by the length of one-third of the DS, the quantum 'W 2L / E set above turned out to be divided into three smaller quanta 'W 3 'W / 3 (Figs. 4-116, 4-122). Therefore, smaller steps have also appeared on phase portraits (Figs. 4-117, 4-123). Significant changes have also acquired a function of the dependence of DS torsion angle on length L . It became discontinuous (Figs. 4-118, 4-124) with discontinuity point z L / 3 dividing it into two almost straight sections. The derivatives of this function retain their low-ordered character (Figs. 4-119, 4-125). Similar qualitative changes in the self-oscillations of the system also occurred when both sections of an inhomogeneous DS had equal lengths l1 l2 L / 2 (Table 4-6). Table 4-6 The values of the parameters of the oscillatory processes of a composite drill string with a length of L = 1,000 m under the action of viscous friction forces on the drill bit

The only noticeable qualitative change in this case could be established for the form of quantum fragmentation 'W associated with the location of interface point z L / 2 . This led to time quantum 'W divided into two equal quanta 'W 2 'W / 2

for case Zb 0.71 rad / s . As for case I, here, the graph of function M z turned out to be broken with break point z L / 2 . The case where l1 2 L / 3 , l2 L / 3 is reflected in column 3 (case III) of Table 4-6. For it, quantum 'W

2L / E is also divided into three equal quanta 'W 3

'W / 3 .

Modelling emergency situations in drilling deep borehole

198

Therefore, a noticeable difference between the self-oscillations of a composite drill string and a homogeneous one is that torsion angle function M z has become a polyline with a break point located at the interface point, and that the small-scale discontinuities of function M t caused by the wave nature of the propagation of the dynamic perturbations of the torsion waves, in contrast to the case of a homogeneous string, have experienced additional fragmentation with time quanta 'W 2 , 'W 3 equal to half or one-third of main quantum 'W . At the same time, the modes of selfoscillations in increased scales have not received any special changes and are implemented according to the relaxation auto-oscillations scenarios inherent to mechanical systems described by singularly perturbed equations. It is also necessary to note a significant qualitative change in the numerical values of the simulation results caused by a decrease in torsion stiffness of one of the DS sections. As analysis of Table 4-6 shows, the transition from a homogeneous string to a composite string is associated with a noticeable increase in average torsion angle Mav , in respect of which the self-oscillations of the drill bit occur, as well as amplitudes D of these self-oscillations. The period of self-oscillations at angular velocity Z Zb 0.71 rad / s also increased noticeably. The influence of heterogeneity of DS mechanical properties on the effects of time-based self-oscillation quantisation occurs according to the scenarios described above. The study showed that the self-oscillation mode does not depend on the initial conditions; therefore, self-excitation is soft. 4.4.4. Analysis of self-oscillations of a composite drill string using the model of a moment of the Coulomb friction force The examples of self-excitation of torsion self-oscillations considered in the previous subsection are of applied importance because strings of different lengths are generally used in deep-well drilling. Such design solutions are caused by the inhomogeneity of stress fields in the DS along the length due to the action of gravity on each of its elements. Since the longitudinal stresses formed by them have a maximum value at the DS suspension point and decrease linearly to zero in the direction of its lower end, when designing their structures, usually, the upper sections are given larger diameters and cross-section areas in comparison with their lower sections. This design feature was taken into account when conducting research in paragraph 4.4.3. However, it is also usual to increase mechanical stiffness of the DS structures in the lower sections. This design solution makes it possible to reduce the flexural

Chapter 4. Excitation of torsional self-oscillations of strings in deep wells

199

buckling of the lower part of the DS, which is exposed to compressive axial forces in the drilling process and to provide the necessary accuracy (straightness) of the centreline of the drilling wells. Therefore, it is interesting to study the features of whirling self-oscillations of strings with increased stiffness in the lower sections. This subsection presents the results of numerical studies of DS self-oscillations with six combinations of the location of the drilling string sections with different stiffnesses and ratios of the upper and lower part lengths. Table 4-7 The values of the parameters of the oscillatory processes of a composite drill string with a length of L = 1,000 m under the action of the Coulomb friction forces on a drill bit

The studies were performed for the case of the interaction of the drill bit with the well wall according to the law of dry friction (Coulomb friction). The results of numerical studies for three combinations of ratios between the lengths of the string segments are presented in Table 4-7. They are gathered into three groups of two cases. In each group, the ratios of lengths l1 and l 2 are fixed, and the cases differ only in the torsion stiffness of the upper and lower segments. For odd cases, the pipe of the lower segment has smaller cross-sections ( I z ,1 0.41 ˜ 105 m4 ) compared to the size of the pipe cross-sections on the upper segment ( I z ,2 3.12 ˜105 m4 ). For cases with even numbers, the inverse combination ( I z ,1

3.12 ˜105 m4 , I z ,2 0.41 ˜ 105 m4 ) takes place.

Modelling emergency situations in drilling deep borehole

200

The table also gives the values of the main characteristics of the dynamic process for a homogeneous pipe with a polar moment of inertia I z 3.12 ˜105 m4 . As for the case of a composite DS considered in paragraph 4.4.3, here, due to the inclusion in the pipe of a segment with reduced torsion stiffness, the values of all parameters that determine the amplitudes and periods of oscillations significantly increased. At the same time, the influence of the order of the pipes with the same length and moment of inertia on the quantitative characteristics of the oscillation parameters turned out to be insignificant (compare the calculation data for cases 1 and 6; 2 and 5; 3 and 4). The results of the calculations again show that the self-excitation of the oscillations is accompanied by their quantisation with the formation of the main quantum duration Δτ = 0.623 s and multiple quanta 'W 3 'W / 3 . In conclusion, it can be noted that in this subsection a new problem is set on the propagation of torsion waves and self-excitation of oscillations in composite drill strings. Equations of diffraction of torsion waves in the connection points of the segments in the strings with different stiffness are formulated. Studies have been carried out for cases where the drill string is composed of two segments of equal length or with lengths at a ratio of 1:2. As a result of the performed analysis, the ranges of angular velocities of rotation are found where the effect of self-oscillations occurs. It is established that large-scale forms of self-oscillations have not received any special changes in comparison with the oscillations of homogeneous drill strings and are implemented according to the scenarios of relaxation self-oscillations inherent to mechanical systems described by singularly perturbed equations. However, in contrast to the nature of self-oscillations of a homogeneous string accompanied by an almost linear change in the angle of elastic torsion along its length, the graph of the torsion angle of a composite string represents a broken line with two almost straight links within each of its two segments. Small perturbations superimposed on this graph are less ordered and lead to additional discontinuities (quasi-breaks) of the angular velocities of rotation. It is also typical that in contrast to the case of a homogeneous string quantised self-oscillations experience additional fragmentation with multiple time quanta 'W i equal to half or one-third of the value of main quantum 'W for the considered cases. 4.5. Self-excitation of torsional vibrations of deep drill strings in a viscous liquid medium Continuous and discrete models of oscillatory systems are generally used in the analysis of self-excitation forms of oscillations in mechanics. Yet, despite the large

Chapter 4. Excitation of torsional self-oscillations of strings in deep wells

201

amount of work on the theory of self-oscillating processes, researchers usually do not go beyond one- and two-mass systems. Real objects usually include continuous links with distributed parameters of inertia, elasticity and friction, which can lead to significant inaccuracies in the description of self-oscillating phenomena as well as to the occurrence of new effects that cannot be revealed by means of computational schemes consisting of discrete links. Three models of the drill string self-oscillations that are built in the form of an elastic torsion pendulum in liquid medium are analysed below. It is found that the influence of hydrodynamic forces of viscous friction of the drilling fluid on the selfoscillating process is small at the real values of the determining parameters (angular velocity of DS rotation, cutting moment, drilling fluid viscosity, elastic pliancy of the DS pipe during torsion, length of the DS). The effects of bifurcation transitions (Hopf bifurcations) from the state of stationary rotation of the system to elastic torsional self-oscillations and back are studied. It is shown that the forms of these self-oscillations differ significantly from harmonic (Thomson) ones and have a relaxation character consisting in the alternation of fast and slow movements. Besides, quantised character of these oscillations is lost. 4.5.1. Self-oscillation models of a torsion pendulum with distributed parameters The process of excitation of torsion vibrations of the DS drill bits represents a demonstration of one of the most common effects of mechanics associated with the generation of self-oscillations during realization of the Hopf bifurcation. These oscillations have a rather complex structure and at the initial stages of theoretical studies were analysed using very simplified approaches [5, 10, 31]. In works [16, 21, 23], it was noted that the equations of torsion oscillations of the DS describe running waves, and, with this in mind, a wave model of an elastic torsion pendulum was proposed. It was used to obtain a number of results related to bifurcation transitions from stationary rotation to torsion oscillations and back [9, 39]. Related issues in this area were analysed in works [7, 11]. However, since the wave model did not take into account the frictional effects caused by the environment of the DS viscous liquid, it was pointed out that it was necessary to develop a dissipative model. This was proposed in [14,15] with the application of the basic provisions of the theory of viscous fluid [2, 34]. The purpose of this subsection is to develop a mathematical model of torsion self-oscillations of a drill string in a dissipative medium. In general, the construction of such a model must be based on the equation of the torsion dynamics of the rod placed in the liquid

Modelling emergency situations in drilling deep borehole

GI

w 2M wM · w 2M §  k ¨Z  ¸  UI 2 2 wz wt ¹ wt ©

202

0

0 d z d L .

(4.80)

Here, M is the angle of elastic twisting of the DS pipe element; G is the elastic modulus of the pipe material at shear; Z is the angular velocity of the pipe; k is the coefficient taking into account the viscous friction in the liquid; I is the polar moment of inertia of the cross-sectional area of the pipe; z is the axial coordinate; L is the length of the DS; t is the time. Boundary conditions are written for equation (4.80) (4.81) M  0 0 , M in  L  M el  L  M fr  L 0 , where M in  L is the moment of inertia forces acting on the drill bit; M el  L is the elastic torque transmitted to the drill bit from the DS end; M fr  L is the moment of friction represented by the cutting diagram for this pair of the drill bit and rock. These moments are calculated using the formulas (4.82) M el  L GI ˜wM / wz z L , M in  L  J M  L , where J is the moment of inertia of the drill bit relative to axis Oz . The first equation of system (4.81) means that point z 0 is selected for the beginning of the reference of angle change M  z ; the second equation is a condition of the dynamic equilibrium of all moments applied to the drill bit in respect of axis Oz . In publications [16, 18, 20, 21, 23], it was assumed that the contribution of the distributed moments of viscous friction, which are described by summand wM · § k ¨ Z  ¸ , is small in the total balance of all moments in comparison with the wt ¹ © moment of friction acting on the drill bit. Therefore, this summand can be ignored. After discarding it, equality (4.80) becomes a pure wave equation and is used to reduce the second equation of system (4.81) to an ordinary differential equation with delayed argument 2L  E t [16, 18, 21, 23]

J ª¬ f  E t  f  2L  E t º¼  M fr  GI / E ˜ ª¬ f  E t  f  2L  E t º¼ 0 , (4.83)

where f  z  E t is the function of the elastic torsional wave departing from the drill bit; E

G / U is the velocity of the transverse (torsional) wave in the DS pipe. This equation is a model of a torsion wave pendulum. Its properties are determined by the form of dependence of friction moment M fr on total angular velocity D Z  M and the values of coefficients J and GI / E . The type of function M fr Z  M depends on the strength of the treated rock as well as on the design and

degree of wear of the drill bit cutters. In general, it may have a different character, but

Chapter 4. Excitation of torsional self-oscillations of strings in deep wells

203

as shown by tribological studies, friction function curve F fr  v usually has extreme point F extr , which is the beginning of the descending curve (Fig. 4-126,a). The mandatory presence of such parts for the functions of the friction forces acting on the drill bit is indicated in works [10, 32]. Ffr

extr

Pcut

P

Fextr v

vb

Mfr

Ffr Fextr

D1 D 2

v

D Z  M

Mextr v Fig. 4-126 Diagram of friction and cutting forces

However, it should be noted that in the drilling process there is a combination of two phenomena—the friction interaction of the drill bit with the rock and the effect of cutting the rock with diamond cutters. Scientific literature does not contain separate descriptions of the mechanisms of these phenomena, but in the theory of cutting the diagram of the dependence of the cutting force on the relative velocity of motion shown in Fig. 4-126,b is generally accepted. As can be seen, at small values Q , this force is not determined, but with increasing Q it reaches maximum P extr at v v1 followed by a descending section. In extreme cases, for some types of processed materials in the extremum zone, the friction function becomes a polyline, and the descending section is converted into a vertical segment (Fig. 4-126,c). Nevertheless, it is considered [10] that even if for each drill bit cutter the cutting function has the form presented in Fig. 4-126,c, the total function of the moment of friction M fr Z  M for the entire drill bit in the results of averaging over all the cutters available on its surface acquires a more smoothed form shown in Fig. 4-126,d.

Modelling emergency situations in drilling deep borehole

204

The effect of the values of the coefficients of equation (4.81) on the nature of self-oscillations is noticeable due to the smallness of J compared to GI / E , as in this case the role of the first summand with higher derivative f becomes significant only at high accelerations—that is, at transitions from low to high speeds and back. Such forms of oscillations are called relaxation, and equation (4.83) describing them is singularly perturbed [9]. To establish the peculiarities of the phenomenon of DS torsion oscillation selfexcitation without taking into account the forces of its viscous friction with the surrounding flow of the drilling fluid, a numerical simulation of the rotation of the steel string with the parameters L = 8,000 m, I 3.12 ˜105 m4 , J 3.1 kg ˜ m2 ,

G 8.077 ˜ GPa , U

7.8 ˜103 kg / m3 in range of angular velocity 0 d Z d 4 rad / s

changes is performed. The accepted form of dependence M fr Z  M is shown in Fig. 4-126,d at Mextr = –82,500 N•m, D1 0.72 rad / s , D 2 3.8 rad / s . In all cases, with selected fixed value Z , equation (4.83) was integrated using the Runge-Kutta method under initial conditions f  0 0 , f  0 0 with time integration step 't 6.5 ˜106 s . Calculations have shown that—depending on values Z —the drill bit with the string can be either at a state of stationary rotation turning to certain angle Mst of elastic twist or go into a mode of self-excited torsion oscillations. In the theory of non-linear differential equations, the periodic solution corresponding to selfoscillations is called the limit cycle (or attractor), and the effect of changing the stationary equilibrium solution to a periodic one when the characteristic parameter passes through critical value ( Zb ) is the birth of a cycle or Hopf bifurcation. The bifurcation transition from range of self-oscillations Zb d Z d Zl to the area of stationary rotations through point Zl is called the loss of the cycle. Values Zb , Zl are determined by the type of diagram M fr Z  M . At the same time, as shown by the calculations, value Zb corresponds to the first extreme point in this diagram, value Zl is located on this curve after passing the second extremum. In the case under consideration Zb 0.72 rad / s , Zl 3.8 rad / s , these values and self-oscillations forms are independent on the initial conditions. This self-excitation is called soft. Fig. 4-127 shows a diagram of the change in angle M  0,t of DS elastic torsion at point z 0 of attachment of the drill bit at Z Zb

0.72 rad / s . Its

movement is initiated from the rest state of the whole system at M  0 0 , M  0 0 . After the string is given a rotational motion at upper point z = 8,000 m with fixed velocity Z 0.72 rad / s , the DS begins to whirl along its entire length and, when

Chapter 4. Excitation of torsional self-oscillations of strings in deep wells

205

maximum angle M  t 265 rad is reached at lower point z 0 , started to oscillate with period T 341.2 s around average value Mav 223 rad . At the same time, the amplitude of oscillations was D 84 rad . It can be seen, that drill bit self-oscillations are relaxation (in contrast to Thomson's auto-oscillations described by harmonic functions) and have the shape of broken curves. Therefore, on the graph of angular velocity function M  t (Fig. 4-128), areas of fast and slow motions have appeared, with slow motions occurring at most of the period and occurring at negative velocity M  t  0 with modulus approximately equal to angular velocity Zb . This means that in these parts of the motion total angular velocity Z  M of the drill bit is approximately zero, and the oscillations occur almost with stops. The effect established by calculation is observed in the practice of drilling, and the noted effects of when drill bit rotation stops when the upper part of the DS rotates at constant velocity Z are called ‘sticking’ [10, 31].

M , рад

0

M ,rad/s рад / с

8

T

4

-150 0

Mсрav

D -4

-300

t, t,s c 0

400

800

1200

Fig. 4-127 Diagram of angle change

1600

t, st, c

-8 0

400

800

1200

1600

Fig. 4-128 Graph of angular velocity function

It is also interesting to note that despite the complex nature of periodic movements of the drill bit and the used wave model of DS torsion oscillations, wave functions M  z,t built with the help of the proposed wave model of DS torsion oscillations are quickly smoothed and averaged over variable z . As a result, they take the form of linear functions, in which only the angle of their inclination to axis Oz changes over time (Fig. 4-129).

Modelling emergency situations in drilling deep borehole

z, m

8000

206

,rad

0

6000

-100 4000

-200 2000

,rad -300

t, s

-300 -200

-100

0

Fig. 4-129 Diagram of twisting angle function

0

200

400

600

Fig. 4-130 Diagram of function ( )

The modes of self-oscillations of the drill bit in other states within range Zb d Z d Zl also have similar peculiarities. Figs. 4-130, 4-131 show the graphs of oscillations of the drill bit at Z 3.55 rad / s . Their difference from the diagrams of the previous case is that here the sticking effect takes place twice over the period, at M | 0 and at M | Z rad / s . However, function M  z,t remained almost linear along variable z . 8

,rad

4

0

t, s

-4 0

200

400

600

Fig. 4-131 Diagram of the function ( )

At the same time, the neglect of the energy dissipation effect due to the forces of the viscous frictional interaction of the DS pipe with the liquid washing it left open the question of the influence of these forces on the self-oscillating process. Therefore, in paper [15], a model of string self-oscillations in a dissipative medium was proposed based on the study of complete equation (4.80) with boundary conditions (4.81) at the ends. Since this model is associated with the need to solve linear partial differential rad/s equation (4.80) and a non-linear boundary condition at the lower end, it describes a non-linear dynamic boundary value problem

Chapter 4. Excitation of torsional self-oscillations of strings in deep wells

207

for a system with distributed parameters. Therefore, the constructed equations can only be solved by numerical methods. As a rule, for the integration of such systems, the most effective method is the time implicit finite–difference scheme. Its advantage is that it is always stable, and computational accuracy is provided by the choice of a sufficiently small value of integration step 't . To verify the reliability of the constructed models, the studies conducted above using the wave model were repeated using the second model at k 0 in equation (4.80). They coincided with high accuracy with the results of the analysis presented in Figs. 4-127–4-131. It is also important to accentuate that according to both models and despite the almost discontinuous forms of self-oscillations and periodic changes of fast motion with slow ones, functions M  z of changing the angles of rotation of the DS elements along its length are almost linear. If we take this fact into account, it is possible to construct another model of torsional self-oscillations (with one degree of freedom), which, unlike the two previous ones, is described by one non-linear ordinary differential equation of the second order and does not contain a delay argument. 4.5.2. Model with one degree of freedom of elastic torsion pendulum in a dissipative medium The noted property of linear change of the function of the DS elastic twisting angle along its length allows us to simplify the mathematical model of its dynamics. To create the model, we use the following initial hypotheses: At the suspension point, the DS rotates at constant angular velocity Z . The drill bit is impacted by the moment of shear forces (friction) M b fr . The DS is in a liquid (dissipative) medium, each of its elements is impacted by the moment of viscous friction forces m fr . The angle of elastic twisting M  z of the DS elements as well as angular velocity M  z and acceleration M  z are linearly dependent on z . The last condition is formulated in the form: M t M t M t (4.84) M  z,t д  L  z , M  z,t д  L  z , M  z,t д  L  z . L L L Here, Mb , Mb , Mb are the angle of elastic twisting, angular velocity, and angular acceleration of the drill bit; L is DS length; z is the coordinate directed along the DS axis.

Modelling emergency situations in drilling deep borehole

208

To model self-oscillations of the drill bit (and the entire drill string), which are self-excited as a result of non-linear frictional interaction with the destroyed rock, we shall conditionally separate the DS from the holding connections at its suspension point z L and apply elastic moment compensating for their action M c el  L (Fig. 4-132). Let's consider the dynamics of torsion motion of the entire drill string and the drill bit under the action of the moment of elastic forces M el ; distributed moments m fr of viscous friction forces acting on the elements of the DS; the moment of rock cutting forces M b fr ; distributed moments min of inertia forces acting on the elements of the DS; moment M b in of inertia forces acting on the drill bit. Fig. 4-132 Diagram of a drill Then, the equation of dynamic equilibrium string in a dissipative medium of the system can be represented as M cin  M bin  M c fr  M b fr  M c el 0 . (4.85) Here, the moment of all the forces of inertia applied to the drill string is equal to L

M c in

³m 0

L

in

dz

 ³ U IM  z dz 0

L

³ U I 0

Mb L

zdz

1  U ILMb . 2

(4.86)

The moment of inertia forces of the drill bit is (4.87) M bin  JMb , the moment of all friction forces applied to the drill string is calculated as follows: L

L

z ·º 1 ª § fr ³0 m ds ³0 k «¬Z  Mb ¨©1  L ¸¹»¼ dz kZ L  2 kLMb , the moment of friction applied to the drill bit is determined by equality M b fr M b fr Z  Mb , M c fr

(4.88)

(4.89)

the moment of elastic forces applied to the drill string at its suspension point is calculated by the formula wM M M c el GI GI b . (4.90) L wz

Chapter 4. Excitation of torsional self-oscillations of strings in deep wells

209

Taking into account (4.86)–(4.90), from equation (4.85), we obtain an ordinary second order differential equation 1 1 GI § · (4.91)  ¨ J  U IL ¸Mb  M b fr Z  Mb  kZ L  kLMb  Mb 0 . 2 L 2 © ¹ As can be seen, this equation has a simple structure since the coefficients before desired function Mb contained in the equation and its derivatives Mb , Mb are constants. Its possible complexity is determined by function M b fr Z  Mb , which— as noted above—depends on many factors. These include the force of pressing the drill bit to the bottom of the well, the design of the drill bit, the degree of wear and dulling of its cutters, the strength of the rock, the composition of the drilling fluid, etc. These factors change as the well progresses, so it is hardly possible to establish a universal type of function M b fr Z  Mb . However, it is possible to choose the most typical forms of this function to analyse the general regularities of the drill bit selfoscillation generation process and its flow modes. The studies presented below are based on the values of the parameters used in the previous paragraph. They differ only in terms of the moment of viscous friction forces presented in (4.91) by summands kZ L , kLMb / 2 . With given function M b fr Z  Mb and known initial conditions, the Cauchy problem is posed for this equation. As noted above, it is solved using the RungeKutta numerical method. To study the effect of the viscosity of the drilling fluid on the nature of DS twisting self-oscillations, it is generally necessary to consider that it moves up in the cavity between the two cylinders, the inner cylinder rotates with an angular velocity varying in time. The Z problem of analysing such movement of the fluid and its power of influence on the torsional oscillations of the inner cylinder represents a separate complicated m fr problem. For this reason, this book only raises the question about the qualitative assessment of this impact Fig. 4-133 Scheme of the and the verification of the need to account for it or the DS rotation in the fluid possibility of ignoring it. Therefore, we shall further medium ignore the influence of the axial component of the flow motion on the rotational motion of the DS (Fig. 4-133). When modelling the circumferential motion of the fluid in the cavity between cylindrical surfaces, we shall assume that it is steady since even the smallest period of self-oscillation of the drill bit T | 80 s is relatively large. Then, in each ring crosssection of the cavity occupied by the fluid, the motion can be considered planar.

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Under the theory of incompressible viscous fluid, the velocity of its particles in the circumferential direction is determined by the formula

Z1 § r12 r22

·  r12 r ¸ (4.92) ¨ r r © r ¹ Here, r is the radial coordinate of the liquid particles, r1 is the radius of the outer surface of the DS pipe, r2 is the radius of the surface of the well wall; v  r is vr

2 2

2 1

the speed of the drilling fluid particles in the circumferential direction. By using equality (4.92), it is possible to find magnitude dv  r1 / dr on the surface of the pipe and calculate the distributed force of viscous friction acting on it dv  r1 . (4.93) p fr K dr Here, K is the coefficient of plastic viscosity of the drilling fluid. Its value generally depends on the composition of the fluid, its temperature, the suspended mud particles present in it, and other factors. Value K is usually in range 0.006 d K d 0.01 Pa ˜ s . Force p fr creates a moment of friction

dv  r1 wM · § 2 k ¨ Z  (4.94) Z  M . ¸ 2S r1 ˜K wt ¹ dr © Comparing the second and third terms in this double equality, after taking into account dependence (4.93), we obtain § r2  r2 · k 2S r12K ¨ 12 22 ¸ . (4.95) © r2  r1 ¹ m fr

To estimate value k for real systems, we shall calculate its values at r1 0.1 m , r2

0.15 m , 0.006 d K d 0.01 Pa ˜ s . Then k lies within 1 ˜103 d k d 1.63 ˜103 N ˜ s .

The study of DS self-oscillations using the second and third models (system (4.80)−(4.82) and equation (4.91), respectively) with the initial data taken in the previous section and calculated coefficient k showed that in this case the application of three different models led to almost the same result reflected in Figs. 4-127–4-131. This means that the moment of friction (shear) M b fr applied to the drill bit has a decisive influence on the self-oscillating process, and the action of the viscous friction of the drilling fluid on the DS pipe can be ignored. This conclusion can be confirmed by the solution to the problem of the DS elastic twisting by moment m fr in conditions M b fr 0 while removing the drill bit from contact with the rock at the well bottom. Then, there are no self-oscillations,

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M 0, M 0 , and equation (4.80) has an analytical solution. In fact, in this case, it takes the form GI

d 2M dz 2

kZ

(4.96)

Its solution kZ 2 (4.97)  L  z2 2GI indicates that DS twisting angle M  z caused by viscous friction varies quadratically

M  z 

along variable z . At first glance, this appears to be contrary to the results presented in the previous section (see Fig. 4-129) and the basic prerequisite used in the construction of the third model. However, as it turned out, with real initial data, values M  z in (4.97) are so small that this function can be ignored in comparison with the amplitude values of the twisting angles stipulated by the self-oscillations of the system. So according to formula (4.97), the angle of rotation of the end z 0 of the DS was M  0 | 0.091 rad , although at self-oscillations its greatest value equals

M  0

max

| 262 rad (Fig. 4-127).

The obtained results allow us to conclude that for the law used in scientific literature of the dependence of the torque applied to the drill bit during drilling on the drill bit rotation angular velocity the influence of the drilling fluid viscous friction forces moment on the self-excitement process of the string torsion oscillation is insignificant. Therefore, the considered mathematical models with distributed parameters and with one degree of freedom—based on taking into the account and not taking into account the viscous friction forces—lead to almost the same results. All of them confirm the most destructive effect accompanying these oscillations. This is associated with the drill bit stick-slip mode of motion that significantly reduces its durability. During the stick period, when the bit rotation speed is almost zero, the drill string continues to rotate in its upper part, and it accumulates the energy of elastic deformations sufficient to continue cutting the rock. Then, there is a rapid local destruction of the rock in the area of its contact with the cutters, the bit ‘comes loose’ and begins to slip with an angular velocity several times higher than the speed of its rotation in the normal mode. These sudden impacts may cause the drill bit cutters to fracture or at least to accelerated dulling and wear in the abrasive environment of the rock. In more severe cases, they may also be accompanied by the destruction of other elements of the DS or a decrease in overall drilling efficiency. As practice shows, the negative effects occurring in the drilling design are further strengthened by the wear of the drill bits.

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Summing up the results, we can conclude that in the simulation of twisting self-oscillations of drill strings for practical use, a model with one degree of freedom, which has the greatest simplicity, can be recommended. Nevertheless, it should be noted that the conclusions are related to the case of using the smooth (differentiable) function of the dependence of the cutting torque on the angular speed of drill bit rotation, which is generally accepted in the scientific literature on drilling. At the same time, the general theory of cutting and tribology considers cases when these functions are broken or even discontinuous. It is clear that the question of the suitability of the findings in these cases should be studied separately. References to Chapter 4 1. Andronov A.A., Vitt A.A., Khaikin S.E. Theory of Oscillations. M.: Fizmatgiz, 1959. – 915 p. (in Russian). 2. Astarita J., Marrucci J. Fundamentals of Hydro Mechanics of Non-Newtonian Fluids. – M.: Mir, 1978. (in Russian). 3. Bailey J.J., Finnie I. An analytical study of drill string vibration // Journal of Engineering for Industry, ASME Transaction. ― 1960. ― V. 82, No. 2. ― pp. 122 – 128. 4. Besaisow A.A., Payne M.L. A study of excitation mechanisms and resonances inducing bottomhole-assembly vibrations // SPE Drilling Engineering. ― March 1988. ― pp. 93 – 101. 5. Brett F.J. The genesis of bit induced torsional drill string vibrations // SPE 21943, Proceedings of the SPE/ IADC Drilling Conference, Amsterdam. ― March 11 – 14. ― 1991. 6. Butenin N.V., Neymark Yu.I., Fufayev N.A. Introduction to the Theory of Non-linear Oscillations. – M.: Nauka, 1987. ― 384 p. (in Russian). 7. Challamel N. Rock destruction effect on the stability of a drilling structure // Journal of Sound and Vibration. ― 2000. ― V. 233, No. 2. ― pp. 235 – 254. 8. Challamel N., Sellami E., Chenevez E., Gossuin L. A stick-slip analysis based on rock/bit interaction: theoretical and experimental contribution // SPE 59230. Presented at the IADC/SPE Drilling Conference, Orleans, LA. ― 2000. 9. Chang K., Howes F. Non-linear Singularly Perturbed Boundary Value Problems // M., Mir. – 1988. – 247 p. (in Russian). 10. Christoforou A.P., Yigit A.S. Dynamic modelling of rotating drill-strings with borehole interactions // Journal of Sound and Vibration. ― 1997. ― V. 206, No. 2. ― pp. 243 – 260. 11. Dareing D.W., Livesay B.J. Longitudinal and angular drill string vibrations with damping // Journal of Engineering for Industry, ASME Transaction. ― 1968. ― V. 90, No. 6. ― pp. 671 – 679.

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12. Dawson R., LinY.Q., Spanos P.D. Drill string stick-slip oscillations // Proceedings of the Spring Conference of the Society for Experimental Mechanics, Houston, TX. ― 1987. 13. Finnie I., Bailey J.J. An experimental study of drill string vibration // Journal of Engineering for Industry, ASME Transaction. ― 1960. ― V. 82, No. 2. ― pp. 129 – 135. 14. Glazunov S.M. Conservative and dissipative models of torsional oscillation of a drill string. // Bulletin of the National Transport University. – 2013. - No 28. ― pp. 88 – 94. (in Ukrainian). 15. Glazunov S.M. Torsion oscillations of deep drill strings in viscous liquid medium. // Bulletin of the National Transport University. – 2015. – No 1 (31). – pp. 96-101. (in Ukrainian). 16. Gulyaev V.I., Glushakova O.V., Khudoliy S.M. Quantised attractors in wave models of torsion vibrations of a drill string // Bulletin of the Russian Academy of Sciences. Solid Body Mechanics. ─ 2010. ─ No. 2. ― pp. 134 – 147. (in Russian). 17. Gulyaev V.I., Lugovoi P.Z., Glushakova O.V., Glazunov S.N. The torsional vibrations of a deep drill string in a viscous liquid medium // International Applied Mechanics. – 2016. – V 52 (2). ― pp. 64 - 77. 18. Gulyaev V.I., Gaidaichuk V.V., Glushakova O.V. Andronov-Hopf bifurcations in wave models of torsional vibrations of drill strings // International Applied Mechanics. – 2010. – V 46 (11). ― pp. 73 - 83. 19. Gulyayev V., Hudoliy S., Glushakova O. Quantized attractors in the wave torsion models of super deep drill columns // International Symposium RA08 on Rare Attractors and Rare Phenomena in Non-linear Dynamics. Riga, Latvia. ─ 2008. ─ p. 33. 20. Gulyayev V.I., Glushakova O.V., and Glazunov S.N. Stationary and nonstationary self-induced vibrations in waveguiding systems // Journal of Mechanics Engineering and Automation. − 2014. − V. 4, No..3. − pp. 213-224. 21. Gulyayev V.I., Glushakova O.V. Large-scale and small-scale self-excited torsional vibrations of homogeneous and sectional drill strings // Interaction and Multiscale Mechanics. ― 2011. ― V. 4, No. 4. ― pp. 291 – 311. 22. Gulyayev V.I., Glushakova O.V. The Poincare – Andronov – Hopf bifurcations in the torsion wave models of super deep drill columns // Proceedings. The Third International Conference on Non-linear Dynamics. Kharkov. ― September, 21-24, 2010. ― pp. 296-301. 23. Gulyayev V.I., Hudoliy S.N., Glushakova O.V. Simulation of torsion relaxation auto-oscillations of drill string bit with viscous and Coulombic

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friction moment models // Journal of Multi-body Dynamics. ― V. 225. ― pp. 139 – 152. 24. Halsey G.W., Kyllingstad A., Aarrestad T.V., Lysne D. Drill string torsional vibrations: comparison between theory and experiment on a full-scale research drilling rig. // SPE 15564, Proceedings of the SPE Annual Technical Conference and Exhibition, New Orleans. ― October 5 – 8. ― 1986. 25. Iyoho A.W., Meize R.A., Millheim K.K., Crumrine M.J. Lessons from integrated analysis of GOM drilling performance // SPE Drilling and Completion. ― 2005. ― V. 20, No. 1. ― pp. 6 – 16. 26. Jansen J.D. Whirl and chaotic motion of stabilized drill collars // SPE Drilling Engineering. ― 1992. ― V. 7, No. 2. ― pp. 107 – 114. 27. Jansen J.D., van den Steen L. Active damping of self-excited torsional vibration in oil well drill strings // Journal of Sound and Vibration. ― 1995. ― V. 179. ― pp. 647 – 668. 28. Kharkevich A.A. Self-oscillations. – M.: GITTL, 1954. – 170 p. (in Russian). 29. Kononenko V.O. Non-linear Oscillations of Mechanical Systems. – K.: Naukova dumka, 1980. ― 382 p. (in Russian). 30. Landa P.S., Self-oscillations in Distributed Systems. – M.: Nauka, 1983. ― 320 p. (in Russian). 31. Leine R.I., Van Campen D.H., van den Steen L. Stick-slip vibrations induced by alternate friction models // Non-linear Dynamics. ― 1998. ― V. 16. ― pp. 41 – 54. 32. Lin Y.Q., Wang Y.H. Stick-slip vibration of the drill strings // Journal of Engineering for Industry, ASME Transaction. ― 1991. ― V. 113. ― pp. 38 – 43. 33. Marsden G., McCracken M. The Hopf Bifurcation and its Applications. – M.: Nauka, 1980. ― 368 p. (in Russian). 34. Mirzadzhanzade A.H., Mirzoyan A.A., Gewinnyan G.M., Sairdza M.K. Hydraulics of Clay and Cement Mortars. – M.: Nedra, 1966. (in Russian). 35. Mishchenko E.F., N.H. Rozov. Differential Equations with Small Parameter and Relaxation Oscillations. – M.: Nauka, 1975. ― 248 p. (in Russian). 36. Myshkis A.D. Linear Differential Equations with Delay Argument. – M.: Nauka, 1972. ― 352 p. (in Russian). 37. Rubanik V.P. Oscillations of Quasilinear Systems with Delay. – M.: Nauka, 1969. ― 288 p. (in Russian). 38. Teodorchik K.F. Self-oscillating Systems. – M. – L.: GITTL, 1952. – 271 p. (in Russian). 39. Tihiro Hayashi. Non-linear Oscillations in Physical Systems. – M.: Mir, 1968. ― 432 p. (in Russian).

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40. Tucker R.W., Wang C. An integrated model for drill-string dynamics // Journal of Sound and Vibration. ― 1999. ― V. 224, No. 1. ― pp. 123 – 165. 41. Van Der Pol. Non-linear Theory of Electrical Oscillations. – M.: Svyazizdat, 1935. ― 42 p. (in Russian). 42. Vandiver K.J., Nicholson J.W., Shyu R.-J. Case studies of the bending vibration and whirling motion of drill collars // SPE Drilling Engineering. ― December 1990. ― pp. 282 – 290.

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CHAPTER 5. SELF-EXCITATION OF DRILL STRING BIT WHIRLING The study of oscillatory and wave phenomena in elastic extended rod structures under the influence of force, inertia, and kinematic disturbances is one of the classical problems of mechanics. It is also widely used in the development of equipment and technologies for drilling oil and gas wells. Due to the great depths of modern wells and complex impacts on the strings during drilling of the combination of longitudinal gravity forces, torques, gyroscopic inertia forces of the rotational motion of the string, the forces of inertia from the internal flows of the drilling liquid as well as due to the non-conservative nature of the interaction of the bottom hole assembly with the processed rock, the regimes of well drivage may be accompanied by emergency situations, including the self-excitation of the bit whirling vibrations. These vibrations are caused by the complex friction and contact forces of the surface pressure of the drill bit against breaking rock. They have complex irregular modes associated with the rolling of the bit on the surface of the well bottom and can lead to critical states of the system and cause its destruction. The results of field observations established [21] that up to 40% of the total length of all channels of oil and gas wells are drilled under conditions of whirling vibration proceeding. There are still no methods for the physical and computer modelling of these effects and the detection of critical dynamic states of these systems. This situation is associated with the high complexity of the studied phenomena caused by the long length of the drill string, the complex mechanical scheme of the system under consideration, and the conditions of the contact or kinematic (nonholonomic) interaction of its bit with the surface of the well bottom. An essential part of the problem of predicting critical states of a drill string during its whirling vibrations is the construction of a mathematical model describing its dynamic behaviour during operation. Taking into consideration the high accident rate of drilling production, large economic losses, and high levels of environmental pollution in recent time, one can conclude that the issues of a theoretical simulation of possible critical dynamic states accompanying the process of deep well drilling become very important. 5.1. Existing models of drill string whirling vibrations An important class of transverse drill string vibrations is directly related to its rotation, since even small imbalances in its mass or small distortions of its axis can lead to whirling, similar to the beatings of unbalanced centrifuges. The theory that explains this type of self-excited vibration is known in mechanics as rotor dynamics. This theory has a long history.

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217

However, the phenomena that underlie the process of elastic bending vibrations of the drill string bottom have significant differences from the effects revealed in the theory of elastic shafts with rotors. These differences are caused by the presence of constant separated and unseparated contact interactions of the bit with the well bottom and its wall. Fig. 1-2 shows various modes of these motions. As this takes place, one of the most complex types of vibrations is caused by drill string bending deformations accompanied by the continuous contact of the bit and the rock. As a result of this interaction, the bit is rolling over the surface of the bottom and wall (Fig. 1-2, d), entering into the so-called whirl vibration (whirling) mode, moving in the direction of rotation of the string (forward whirling) or in the opposite direction (backward whirling). In this chapter, mathematical models of whirling vibrations known in scientific literature are analysed, new models taking into account frictional and nonholonomic effects are proposed, and computer simulation of whirling vibrations at different values of characteristic parameters is carried out. The detailed study of whirling vibrations probably began with the work of Jansen [11]. Based on the ‘mass-spring’ mathematical model where the bit is replaced by a hard disk, and the elastic drill string is replaced by a spring, an attempt was made in different works to explain the vibrations of forward and backward whirling. However, forward whirling is explained as a result of the usual circular motion in the fixed coordinate system of the bit having an imbalance of mass. In this statement, the contact of the disk with the well bottom and wall is not taken into account (Fig. 5-1). Backward whirling was modelled as the effect of rolling the disk rim along the well wall. An assertion was made about the possibility of using this model to explain the mechanism of self-excitation of forward and backward whirling. These conclusions are unlikely to be accepted as the proposed model is too simplistic and cannot reveal the main effects inherent in the system. In fact: 1. The main force acting on the bit during drilling is axial force T that presses the bit to the bottom of the well. It is directed vertically (Fig. 1-2, d). When the drill string is bent and tilted, force T is also tilted, and its horizontal component Thoriz appears. With this force, the bit can be pressed to the well wall. Usually, this force varies within 104 d T d 106 N, and even in the most adverse cases Thoriz does not exceed 10% of T . Therefore, in our opinion, when analysing the balance of forces acting on the bit, it is necessary to consider the action of vertical force T . 2. When creating a mathematical model, it is first necessary to take into account the interaction of the bit with the well bottom, and only with the development of the oscillatory process, when the transverse displacements become large and

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exceed the inter-tube gap, one can start taking into account the contact of the bit with the well wall. Driving engine

Spinning drill bit-disk

Well wall

Drill string Eccentric

Disk

а)

Drill string

Eccentric

b)

Fig. 5-1. Side view a) and top b) models of whirling vibrations

3. In these cases, the nature of the bit motion significantly depends on the geometry of the bit and the well bottom. It is hardly suitable in this case to model a drill bit as a round flat disk. Nevertheless, the ‘disk-spring’ model is widely used by many authors. Leine et al. [13] used this model to describe the sliding (friction model) and rolling without sliding (roller model) of the bit on the well wall (Fig. 5-2). It was used to study the stability of this process. The next more complex whirling disk vibration model was obtained due to the elastic flexibility of the drill string tube at the bottom. This model takes into account the friction and impact interactions of the bit with the well wall. On its basis, the equations of the system vibration are constructed, which are then integrated by numerical methods. Among publications in this field, a special place is occupied by the article by Kovalyshen [12] where it is noted for the first time that the shape of the whirling vibrations to a great extent depends on the bit geometry. He considered the imbalance of the bit mass and its shape and developed a simplified mathematical model with a finite number of degrees of freedom. With their help, forward and backward whirling vibrations were simulated. Completing the review of scientific publications in this area, it should be noted that they are based on a simplified disk model and do not take into account the action of the longitudinal compressive force, the geometry of the bit, and its frictional

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219

interaction with the well bottom. Meanwhile, as noted in Stroud et al. [22], whirling vibrations are the most typical dynamic process that accompany—as shown by practical statistics—40% of all drilled wells. At the same time, the frequency of whirling vibrations can be 5 to 30 times higher than the angular speed of rotation of the drill string itself and lead to fatigue and destructive effects on the structure of the lower part of the drill string. ωt

Drill string

Fcont

Fel

ɸ

Fin

Wall

x

y

Well bottom

FbT

ɸ

Tb

Reaction

а) b) Fig. 5-2. Diagram of the whirling motion (disk model): а—schematic model; b—forces

Our book proposes a new model that considers the initial process of whirling vibrations, and the drill string is subject to small elastic bending. In addition, a bit of different shapes deviates from its operating state by a small displacement and slides along the curved surface of the well bottom, without coming into contact with its wall but carrying out friction interaction between the contacting surfaces. 5.2.

The main aspects of the whirling model based on the effects of friction and nonholonomic rolling of the bit

As already noted, the bending vibrations of the bottom of the drill string have one of the most complex mechanisms of motion caused by the action on the bit of the time-varying normal and tangential forces of the contact and friction interaction of the bit with the well bottom and wall (Fig. 1-2, d). In this case, the geometric centre of the bit begins to move around the centreline of the well, overtaking or lagging

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behind the rotational motion of the string itself. Similar motions are carried out by the gyroscope or the centrifuge rotor of an old design washing machine under the influence of gyroscopic forces of inertia. In mechanics, they are called precession oscillations. Papers [6, 11, 13] note that the motion of the bit centre described above has a different nature and the term ‘whirling’ is used for its definition. As stated in paragraph 5.1, it was studied on the basis of the very simplified physical and mathematical models with one or two degrees of freedom under different laws of friction interaction of the bit with the well wall and bottom. These models are very far from a real system and poorly reflect real dynamic processes. At the same time, experiments and observations show that under some vibration modes the bit begins to roll along the curved surface of the well bottom (Fig. 5-3), while its centre moves along a quite complicated trajectory—reminiscent of a multi-petal flower—with the formation of a system of troughs on the surface of the well wall unacceptable in terms of the drilling technical conditions. It is possible to implement two types of bit motions. In one of them, the rolling of the bit occurs with slippage, and between its surface S 2 and well bottom S1 at contact point G a friction force appears directed along a tangent to the motion path of the contact point. In this case, simulation of the bit motion should be carried out using a friction model of rolling with sliding. The study of such self-oscillations was performed in works [8–10]. Pipe of the string

D

Surface S 2 of the drill bit

R C x

x2

G

D

z2

Surface S1 of the well bottom

Z

Fig. 5-3. Geometry of contact of the bit surface S2 with surface S1 of the well bottom

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In the second type of motion, the rotating bit is rolled without slipping along the surface of the well bottom, obeying the conditions of kinematic (nonholonomic) correlations. The study of self-oscillations of such a system can be carried out only by nonholonomic mechanics [1, 16, 19]. Our work is devoted to the problem of computer modelling and prediction of the phenomenon of bit and drill string oscillations using both friction and nonholonomic models. The study of dynamics of the body that rolls without sliding on a solid surface largely determined the development of analytical dynamics of nonholonomic systems (i.e., systems with differential non-integrable constraints) in the late 19th and early 20th centuries [16]. The simplest example of a nonholonomic system is a ball that moves without sliding on a plane. The beginning of nonholonomic mechanics is associated with the works of J. Lagrange and M. V. Ostrogradskiy. However, the qualitative difference between holonomic and nonholonomic systems was clearly established only at the turn of the 19th and 20th centuries when it was found that the motion of a nonholonomic system, unlike a holonomic one, could not be described by standard Lagrange equations of the second kind. The development of many issues regarding nonholonomic system mechanics is closely related to the application of methods of the theory of differential equations and differential geometry. The general geometric interpretation of the motion problems of such systems contributed to the creation of a new section of differential geometry—nonholonomic geometry—the basis of which is the problem of rolling without sliding one surface over another [19]. It is formulated as follows. There is fixed surface S1 and movable surface S 2 that is in contact with S1 at touch point G (Fig. 5-4). The vector-function ω(t ) of surface S 2 instantaneous angular velocity ω dependence on time t is given. It is necessary to build motion paths l1 and l2 of point G on each of surfaces. Thus, if you identify surfaces S1 and S 2 in Fig. 5-3 with surfaces S1 and S 2 in Fig. 5-4, it can be concluded that the problem of nonholonomic geometry is to a great extent similar to the problem of rolling a bit along the bottom of a well. When solving this task, it should be taken into account that at each moment the velocity field of the points of moving surface S 2 is the same as if it rotates with angular velocity ω around an axis passing through the contact point. Depending on the direction of the instantaneous axis of rotation, pure or actual rolling and the socalled spinning are distinguished. Pure rolling occurs when the instantaneous axis of rotation of the moving surface lies in a plane tangent to both surfaces, and spinning occurs when the instantaneous axis of rotation is normal to these planes. In general, surface S 2 rolling on surface S1 can be arranged into pure rolling and pure spinning in accordance with the decomposition of vector ω into component

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ωW that lies in the tangent plane, and component ω n , normal to the surface (Fig. 54). Z Zп

S2

ZW G

l2

S1

l1

Fig.5-4. Rolling with twisting of surface S 2 on surface S1

When modelling this process, to write differential equations of rolling relations of one surface over another, it is necessary to find a relation connecting increment dT of the rotation angle of the moving surface in the vicinity of its contact point with the fixed surface with linear elements dl1 and dl2 that are described by the contact point on curves l1 and l2 . To do this, it is convenient to decompose rotation angle dT of the movable surface into angles of pure rolling dT r and pure spinning dT s . As shown in work [19], angle dT r for given value dl is determined by normal curvature radii R1n and R2n of lines l1 and l2 , while angle dT s depends on radii

R1g and R2g of their geodesic curvatures. In a particular case, if surfaces S1 and S 2 are spheres, radii R1n and R2n are constants, and one only has to look for radii R1g and R2g for given ω n and ωW . The problem of rolling with spinning is much more complicated if vectors ωW (t ) and ω n (t ) are not given but should be determined by some additional conditions. In nonholonomic mechanics, it is considered that surfaces S1 and S 2 limit solid rough bodies that have masses, and their mutual motion without sliding is carried out as a result of the application to them of forces that depend (or do not depend) on time t . Then, the dynamic equations of the motion of bodies are formed, for which the kinematic conditions of their contact interaction play the role of

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nonholonomic constraints. In this arrangement, the problems of rolling without sliding bodies of simple shapes with simple surfaces are solved [19]. In particular, it is shown that, depending on the initial conditions, the rolling of a rough ball along a rough spherical surface can take unexpected modes and be accompanied by the movement of the contact point along some smooth sinusoidal paths, curves with cuspidal points, and loop-like curves (Fig. 5-5, a, b, c, respectively). At the same time, it is shown that the problems of nonholonomic dynamics become more complicated with the variable geometry of contacting bodies, and the most unexpected dynamic effects can be realised with the motion of the system that is determined by their solutions. Perhaps, the most striking of the known examples of nonholonomic systems is a two-wheeled and even single-wheeled (monocycle) bikes that maintain the stability of their vertical positions due to the presence of nonholonomic controlling constrains.

l1

l1 S2

S1

S2

S1

x

x

b)

a)

l1 S1

S2 x

c)

Fig. 5-5. Loci of positions of points of ball S 2 contact with segment S1 of an immovable sphere

Rolling with spinning conditions can also appear in the drill bit/well bottom system. They are provided by the presence of diamond inclusions on the surface of the bit that play the role of solid indenters when rolling as they are pressed into the rock on the surface of the well bottom and prevent the bit from sliding along it. As the bit surface and the well bottom can have different geometric shapes, during drilling, the possibilities exist of the bit motion transition from pure rotation (regular drilling process) to additional rolling, of departure of the bit contact point with the well bottom from the vertical and bending of the drill string axis. To study these

Modelling emergency situations in drilling deep boreholes

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phenomena, it is necessary to set the problem of elastic transverse vibrations of the drill string in which nonholonomic relations are boundary conditions for the equations of the DS motion. To conclude this subsection, it should be noted once again that there are two dominant models in mechanics illustrating the motion of one solid body on the surface of another. A more general formulation is to study the relative motion of solids taking into account the contact interaction and the friction force that can be, for example, dry (Coulomb) or viscous (Newtonian—proportional to velocity). However, such a statement, due to its complexity, often does not allow a detailed analysis. In fact, only a few simple solutions for the motion of a spherical body can be noted here. The nonholonomic model of body motion without slippage is simpler and clearer. But it can be noted that this model is less common in practice since the relative slippage of one body on the surface of another can be characteristic of almost any rough bodies. However, the motion of the bit on the surface of the well bottom in this case can be an exception, as the relative slip can be significantly prevented by diamond inclusions that exclude the possibility of one body sliding on the surface of another. In this connection, the absolute roughness of the contacting bodies and the conditions of the existence of complete adhesion between them are noted in the scientific literature. It should be noted that in our work the studies of the bit motion along the well bottom were carried out on the basis of both (frictional and nonholonomic) models. 5.3.

The problem of Celtic stone mechanics and its analogy with bit rolling dynamics

The examples of nonholonomic contact interaction of solids were dwelled upon above, and the frictional and kinematic models of the dynamics of this motion were discussed. There is another example of such an interaction that is directly related to our problem since it is associated with the forward and backward rolling of a body with a convex surface, as is the case with a bit of an elastic drill string. This is called the Rattleback (Celtic stones) problem. Here is its story. In the late 19th century, one of the most surprising dynamic effects of nonholonomic mechanics was discovered associated with the so-called Rattleback problem and consisted in the conceptual violation of the law of physics on the conservation of angular momentum. It is implemented for ellipsoidal solid bodies with a violation of geometric or inertial symmetry properties (rattlebacks) (Fig. 5-6). If such a body comes into contact with a rough horizontal surface and is spun relative to the vertical axis, it ceases to rotate after a while and—after short irregular oscillations in respect of a horizontal axis (dance) without external influence—begins to rotate again around the vertical axis but in the opposite direction.

Chapter 5. Self-excitation of drill string bit whirling

225

As this takes place, the point of contact of the body with the plane traces out rather complex trajectories. During rotational motion, they have the shapes of expanding or narrowing spirals, and during transition through the vibrational mode they have the mode of complex curves with loops and cuspidal points. For some shapes of solids, such changes in the rotation direction occur repeatedly.

Fig. 5-6. The kinematic model of a rattleback

The first description and physical explanation of this effect, discovered more than a hundred years ago by archaeologists excavating ancient Celtic dwellings, was given by the American physicist G. T. Walker in 1895 [23]. Later, two main dynamic models were used to analyse the unusual behaviour of the rattleback. A more general and complex formulation of the problem is to study the motion of a solid body on a horizontal plane taking into account the slip effect and the presence of friction forces at the contact point of the bodies. Because of its complexity, this was less attractive and productive. The nonholonomic model of rattleback motion is simple and clear, with which it was possible to identify the main properties and qualitative peculiarities of its behaviour. Without dwelling on the details of the historical and descriptive aspects regarding the motion of rattlebacks, it should be noted only that they are discussed in detail in the literature given in monograph [16]. Externally, similar motions can be carried out by the drill bit of deep drill strings, although the task of its motion is somewhat more complicated since the bit is connected with an elastic drill string. In this case, bit rotation is carried out by rotating the entire drill string as a result of the driving torque action on its upper end. In working conditions, when the string abuts the lower end in the bottom of the well, it is impacted by the compressive force of the vertical reaction, torque, and centrifugal forces of inertia of rotational motion that contribute to the reduction of the bending stiffness or can even lead to bifurcational buckling. As noted in works [9, 11, 13], as a result of such deformations, the alignment of the string and the bit is

Modelling emergency situations in drilling deep boreholes

226

violated, the bit axis rotates at an angle relative to the well axis, and the bit begins to touch its bottom at a certain distance from the axis of the system. It should be noted once again that in this case—as in the problem with the rattlebacks—there are two regimes of motion that differ in the nature of the interaction of the bit with the well bottom and are therefore described by various mathematical models. So, if the force of pressing the bit to the bottom is small, and the contact between them is broken, there is a usual frictional interaction that is modelled by the Amonton-Coulomb law of friction. The study of bit dynamics using this model is performed in work [8]. The situation changes significantly with an increase in the axial force pressing the bit to the bottom of the well and bringing it to Eulerian critical value for the drill string. Then, the bending stiffness of the string decreases, it is buckled, and diamond inclusions available on the surface of the bit go deep into the rock. In this case, the bit loses the ability to slide along the bottom of the well and begins to roll over its surface lagging or advancing the rotation of the string. As a result, the contact point of the bit with the reference plane can trace out rather complex trajectories containing loops, breaks, and cuspidal points and change the direction of movement, as is the case with rattlebacks. Based on this statement of the problem, article [9] studied the dynamics of rotation and rolling without sliding of a spherical bit on the spherical surface of the well bottom. It is noted that the vibrations of bit whirling can be accompanied by three types of stable and unstable motion associated with the forward and backward rolling of the bit and its pure spinning. Moreover, the modes of these motions largely depend on the bending flexibility of the drill string, which is determined not only by its mechanical stiffness but also by the proximity of its state to Eulerian instability. However, as shown in work [19], with the transition from spherical to ellipsoidal bodies, the problem of rolling without sliding the body on the plane is much more complicated. Nevertheless, since the bits in the shape of elongated and flattened ellipsoids are widely found in the design of deep drilling strings, the questions of studying the impact of their geometry on the forms of behaviour of whirling vibrations are of significant interest. These issues are discussed in this book. It should be noted that as rattlebacks are separate rotating solid bodies that are not associated with any elastic elements, as a rule, the problems of their motion are solved by analytical methods. However, the bit of the drill string is a more complicated system because its motion is due to the flexural oscillations of an elastic drill string. In this regard, numerical methods with a focus on the computer simulation of dynamic processes should be used to describe these vibrations.

Chapter 5. Self-excitation of drill string bit whirling

5.4.

227

Friction model of elastic oscillations of the system during rolling of a spherical bit on the spherical surface of the well bottom

5.4.1. Equations of elastic transverse oscillations of the drill string The leading position in the oil and gas well drilling technology is the rotary method where rock cutting is carried out by a bit fixed at the lower end of the rotating drill string. The problem of bending vibrations of prestressed rotating rods is directly applicable to the dynamics of deep drilling strings. These strings can reach up to 10 km long. Under operating conditions, they are exposed to longitudinal gravity forces, torque, inertia forces of rotational motion, and inertia forces from the internal flow of the drilling liquid (Fig. 5-7).

T

L

Z

D Centring devices

l

B

e

A

C Bit

Mf R Fig. 5-7. Computational model of a drill string in a deep well

Modelling emergency situations in drilling deep boreholes

228

To simulate the whirling vibrations of the system drill string/drill bit, it can be represented as a long tubular elastic rod stressed by longitudinal force T and torque M z and rotating at constant angular velocity Z around its longitudinal axis. In the channel of the drill string pipe, fluid moves at speed V with density U t . We investigate the vibrations of the rod in rotating coordinate system Oxyz in respect of axis Oz directed along the longitudinal axis of the undeformed rod. The equations of elastic bending vibrations of a drill string in a vertical well are given in Chapters 2 and 3. Let's write them down again EI

wv · w 4u w § wu · w 2 § 2  ¨T ¸  ¨ M z ¸  Ut F t  U f F f Z u  wz ¹ wz 4 wz © wz ¹ wz 2 ©

 2Ut F t  U f F f Z

wv w 2u w 2u w 2u  V 2 U f F f 2  2VU f F f  Ut F t  U f F f 2 wt wzwt wz wt

0,

(5.1) 4

EI

2

w v w § wv · w § wu · 2  ¨T ¸  ¨Mz ¸  Ut F t  U f F f Z v  wz ¹ wz 4 wz © wz ¹ wz 2 ©

 2Ut F t  U f F f Z

w 2v w 2v wu w 2v  Ut F t  U f F f 2  V 2 U f F f 2  2VU f F f wzwt wt wt wz

0.

Here, u ( z, t ) , v( z, t ) are the transverse elastic displacements of the string, z is the longitudinal coordinate, t is the time. The rest of the notation is deciphered in Chapter 2. As these equations are related, it can be concluded that the drill string cannot perform plane oscillations, and its mode of motion can only be spatial (see Chapter 3). Relations (5.1) must be supplemented by the corresponding conditions at the edges of the DS segment separated for the calculation and the conditions at the intermediate supports. As noted above, the boundary conditions at the lower end of the DS are formed based on the equations of the contact interaction of the bit with the rock. This can be frictional or kinematic (nonholonomic). To derive the boundary conditions, it is necessary to consider the rolling of the bit on the well bottom surface. 5.4.2. Boundary conditions at the ends of the drill string (friction model) Whirling oscillations of the bit rotating with angular velocity Z are accompanied by the involvement in the vibration process of the lower segments of the string located between the centring devices playing the role of additional supports (Fig. 5-7). As a rule, the number of such supports does not exceed five, and the distance between them is 9 m to 18 m. As the most intense bending vibrations of the drill string are observed in the span directly adjacent to the bit, in the analysis of the mechanism of the whirling vibrations excitation, we will ignore the influence of the

Chapter 5. Self-excitation of drill string bit whirling

229

upper part of the drill string and select its fragment of length l between two lower centring supports A and B , conditionally separating it from the top of the drill string, and the adjacent console section of length e with the bit at the end (Fig. 5-8). O j

Zt

i

k

X

x

Y

l

y

A

e

B C

Z, z Fig. 5-8. The bottom hole assembly scheme

The drill bit is conventionally shown in the shape of a solid body. The selected segment of the drill string is stressed by the torque applied to the bit and the longitudinal compressive force equal to the reaction of the bit support to the bottom of the well. The dynamics of this segment will be modelled based on the theory of rotating compressed-twisted rods [2, 6, 8, 9]. To do this, we introduce fixed coordinate system OXYZ and coordinate system Oxyz rotating together with the drill string at speed Z that have common origin O at support A . For the quantitative analysis of kinematically excited whirling vibrations, it is necessary to construct a dynamic equation of the total two-span beam ABC separated for the consideration that rotates and is prestressed by torque M z M fr and longitudinal compressive force T  R . Inside, the drill string pipe contains mud that is taken into account as the attached mass. We will study the beam oscillations in rotating system Oxyz . In respect of this system, the motion of each element of the drill string is compound, so

Modelling emergency situations in drilling deep boreholes

230

in vibrations the inertia forces of the relative and bulk motions as well as the Coriolis inertia forces should be properly allowed for consideration. The equation of dynamic elastic bending of the drill string in the presence of the marked factors is specified above. They have form (5.1). It should be noted that although system (5.1) is linear, the methods of its solution are hard, as the required functions have multipliers T and M z . Thanks to them, the modes of stability loss and vibration of drill strings have irregular shapes with a predominant buckling in their lower parts. To derive boundary conditions at support A , let us assume that the oscillations of two adjacent sections AB and AD (Fig. 5-7) occur in antiphase mode with inverse symmetry relative to point A of the centring device. In this case, you can enter boundary equations at the edge of A uz

0

vz

0

0,

Mx

z 0

My

z 0

0.

(5.2)

At support B ( z

l ) , the transverse deflections equal zero, therefore (5.3) u B vB 0 . Besides, the condition of rotation angles continuity provides the equalities дu дu дv дv . (5.4) , дz l 0 дz l 0 дz l 0 дz l 0 To formulate the boundary conditions at edge z l  e , as indicated above, we assume that the process of excitation of the whirling vibrations is just beginning, and the bit can move in the gap between it and the well wall without reaching the last one. At the same time, the character of the bit rolling and the boundary equations at point C are determined by the geometry of the surfaces of both bit ( S 2 ) and well bottom ( S1 ) (Fig. 5-4). In general, they can have the shapes of surfaces of rotation (Fig. 5-9) that can be approximated by spheres or ellipsoids. Let us first consider a simple case where both contacting surfaces S1 and S 2 are spheres with radii a and R , respectively (Fig. 5-10). To describe the elastic rotation of a bit, we also introduce coordinate system Cx1 y1 z1 rigidly connected with it, whose axes Cx1 , Cy1 are in the initial position parallel to axes Ox , Oy , respectively, and slew at angles vc C and u c C with the elastic drill string bending. The rolling of surface S 2 on surface S1 will be set in right movable coordinate system Gx2 y2 z2 , origin G of which coincides with the contact point of surfaces S1 and S 2 , axis Gz 2 is the continuation of segment CG , and axis Gy2 is perpendicular to the plane containing axis OZ and segment Gy2 and is oriented in the direction of rotation.

Chapter 5. Self-excitation of drill string bit whirling

231

Fig. 5-9. Geometrical shapes of the drill bits

Z S2

y1

z1

y2

Z

C

x1

S1

x

G

x2

z2

X

u

v

xG

y1

Zt x

x1

E

y2 z2

x2

y Zt Y Fig. 5-10. The scheme of the bit rolling on the surface of the well bottom

Modelling emergency situations in drilling deep boreholes

232

The rolling condition of the bit with slippage makes it possible to formulate two groups of boundary equations at point C that determine the dynamic equilibrium of all forces and torques relative to point G . We shall assume that displacements u , v and angles uc дu дz , vc дv дz are small. To determine the speed of bit centre C , we express the absolute angular velocities of the introduced coordinate systems through angular velocity ω of system rotation Oxyz , angles u c , vc of the elastic rotations of the bit, and angular velocities uc , vc of these rotations. The absolute angular velocity of system Oxyz , by definition, equals Ω ((00))

Zk .

(5.5)

The absolute angular velocity of system Cx1 y1 z1 in the projections on the axes of the same system is (5.6) Ω((11)) :((10))  u cj1  vci1 vci1  u cj1  Zk1 . The absolute angular velocity of system Cx1 y1 z1 in the projections on the axes of system Oxyz is Ω ((10))

 vc  Zuc i  u c  Zvc j  ωk .

(5.7)

The orientation of system Gx2 y2 z2 relative to system Oxyz is given by angle D between axes Oz and Gz 2 (Fig. 5-11) and angle E of plane x2Gz 2 rotation relative to plane xOz (Fig. 5-10)

D

R u 2  v2

a

C

x2

G

D

z2

Z Fig. 5-11. Diagram of axis Gz 2 orientation

Chapter 5. Self-excitation of drill string bit whirling

sin D

sin E

u 2  v2 , R  a

233

cosD

v

1

(u 2  v 2 ) , R  a 2

(5.8)

u

cos E

. (5.9) u v u  v2 Then, the absolute angular velocity of system Gx2 y2 z2 in projections on the axes of the same system is calculated as follows: (5.10) Ω(( 22)) (Z  E ) sin D ˜ i 2  D ˜ j2  (Z  E )k 2 . 2

2

,

2

To formulate the kinematic boundary conditions, we calculate the absolute velocity of centre C of rolling body S 2 in projections on the axes of system Oxyz [5] o

v C(0) v Gabс  Ω((10)) u GC ,

(5.11) o

where v Gabс is the absolute speed of bit point G ; GC is the vector determined by the formula o

GC

a sin D cos E i  a sin D sin E j  a cosD k .

(5.12)

T

Qx Qy

Q F in

C F con

F fr

Fig. 5-12. Diagram of forces acting on the drill bit

The kinematic relations built above allow us to start the calculation of all forces and moments acting on the bit (including frictional) and to consider their equilibrium. To do this, we conditionally separate the bit from the drill string and consider the balance of forces acting on it (Fig. 5-12). In general, these include friction force F fr , inertia force F in , axial force T , contact force F con normal to contacting surfaces, and shearing force Q acting on the bit from the side of the

Modelling emergency situations in drilling deep boreholes

234

separated drill string. Based on the d'Alembert principle, the vector sum of all these forces is zero. To simplify the problem, we assume that the bit inertia moment is small compared to the inertial characteristics of the whole system, so the forces of inertia relative to other forces can be ignored. Then, the equilibrium equation of the bit separated from the drill string can be written as Q  T  F con  F fr

We assume that F

con

0.

(5.13)

T , then equation (5.13) is simplified

(5.14) Q  F fr 0 . It is convenient to consider this equation in coordinate system Oxyz that rotates together with the drill string. In it, (5.15) Q Qx i  Qy j , where Qx

д 3u д 3v , (5.16) Q y EI 3 . 3 дz дz of the Coulomb friction between the bit and the well EI

Friction force vector F fr surface is calculated as follows:

F fr

P v Gаbс

P T ˜ v Gаbс / v Gаbс .

(5.17)

To calculate, we use formula (5.11). It follows from it v Gаbс

o

v C(0)  Ω((10)) u GC .

(5.18)

Vector v C( 0) used here is calculated taking into account the fact that point C participates in the elastic rod vibrations in the rotating coordinate system and in the rotational motion of the entire system at speed ω . Then, v C( 0) v Cel  ω u (ui  vj) 0 0 Z ui  vj  u v i

where

v Cel

0

(u  Zv)i  (v  Zu ) j ,

(5.19)

j k

ui  vj is the velocity of this point in rotating coordinate system

Oxyz . After calculating the product

Chapter 5. Self-excitation of drill string bit whirling

o

Ω ((10)) u GC

i  vc  Zu c  a sin D cos E

235

j u c  Zvc  a sin D sin E

k

Z

> a cosD uc  Zvc 

 a cosD

 aZ sin D sin E @i  > a cosD  vc  Zu c  aZ sin D cos E @j   > a sin D sin E  vc  Zu c  a sin D cos E u c  Zvc @k ª º u2  v2 v (u 2  v 2 ) c c   u v a   ˜ Z Z « a 1  »i  2 2 2 Ra R  a u  v »¼ «¬ º ª u 2  v2 u (u 2  v 2 ) c c   v u a  « a 1     ˜ Z Z »j  2 2 2 Ra R  a u  v »¼ «¬ ª º u 2  v2 v u 2  v2 u    « a ˜  vc  Zu c  a ˜ u c  Zvc »k Ra Ra u 2  v2 u 2  v2 «¬ »¼ ª v º ª (u 2  v 2 ) (u 2  v 2 ) c c    vc  Zuc      u v a a 1 Z Z i « a 1  » « R  a »¼ «¬ R  a 2 R  a 2 «¬ u º ª v  vc  Zuc  a u u c  Zvc º»k  aZ j  « a » Ra R  a¼ ¬ R  a ¼ we obtain vGabs ,x

u  Zv  a 1 

(5.20)

(u 2  v 2 )  u c  Zvc  aZv , Ra R  a 2

(5.21) vGabs ,y

v  Zu  a 1 

2

2

(u  v )

R  a

2

 vc  Zuc  aZu . Ra

Substituting equations (5.15)−(5.17) and (5.21) into equation (5.14), we obtain w 3u w 3v EI 3 i  EI 3 j  wz wz 2 2 § · PT ¨ u  Zv  a 1  (u  v )  u c  Zvc  aZv ¸i  (5.22)  2 2 2 ¨ R  a ¸¹ R  a  vGabs vGabs © ,x ,y

 



PT

v  v abs 2 G, x

abs 2 G, y

2 2 § · ¨ v  Zu  a 1  (u  v )  vc  Zu c  aZu ¸ j 0. 2 ¨ R  a ¸¹ R  a ©

Projecting vector equality (5.22) on axes OX , OY , we obtain the first group of boundary conditions for equations (5.1) of elastic oscillations of the drill string pipe

Modelling emergency situations in drilling deep boreholes

EI

PT

w 3u  wz 3

v  v abs 2 G, x

abs 2 G, y

236

2 2 § · ¨ u  Zv  a 1  (u  v )  u c  Zvc  aZv ¸ 0 , 2 ¨ R  a ¸¹ R  a ©

(5.23)

PT

3

EI

wv  wz 3

  2 vGabs ,x

§ · ¨ v  Zu  a 1  (u  v )  vc  Zu c  aZu ¸ 0 . 2 ¨ R  a ¸¹ R  a © 2

2 vGabs ,y

2

The dynamic boundary equations at point C follow from the condition of the dynamic equilibrium of elastic moments, moments of inertia forces, and reactions of constraints applied to the bit. In their construction, the choice of the pole and the axis system, in respect of which the moments acting on the bit are calculated, has a significant influence on the structure of these equations. It is usually most convenient to choose the contact point for the pole and a coordinate system for the reference frame where the axial moments of inertia of a moving body remain unchanged [19]. The first condition leads to the exclusion from consideration of the nonholonomic constraining reactions, the second leads to the avoidance of the need to differentiate the moments of inertia of the body in time. In this regard, we choose point G for the pole and coordinate system Gx2 y2 z2 for the reference system. To build the equations of motion, we use the theorem of change of the moment of the bit angular momentum relative to point G [19] ~ d K G( 2)  Ω (( 22))u K G( 2) M G( 2) , dt

(5.24)

where, K G( 2) is the angular momentum of the bit relative to point G presented in system Gx2 y2 z2 ; M G( 2) is the moment of elastic forces acting on the bit also recorded in the same system. Vector K G( 2) is calculated taking into account equalities (5.7)−(5.9) K G( 2)

J  ma 2 ­° ® (vc  Z uc)u  (uc  Z vc) v u 2  v 2 °¯

>



J  ma 2 2

u v

2

@ 1  (uR av) 2

> (vc  Z uc)v  (uc  Z vc) u @ j

2

2 2

Z

u 2  v 2 ½° ¾i 2  R  a °¿



(5.25)

­° 1 u 2  v 2 ½°  J ® (vc  Z uc)u  (uc  Z vc) v ˜  Z 1 ¾k 2 . Ra ( R  a) 2 °¿ °¯

>

@

Moment M G( 2) is expressed through internal bending moments EIu cc , EIvcc and shearing forces EIuccc , EIvccc at edge C of the drill string

Chapter 5. Self-excitation of drill string bit whirling

237

^ >ucc  auccc(cosD  uc sin D )@sin E  >vcc  avccc(cosD vcc   vc sin D )@cos E `cosD i  EI ^ > u cc  au ccc(cosD  u c sin D )@cos E   >vcc  avccc(cosD vcc vc sin D )@sin E `j  EI ^ >u cc  au ccccosD   u c sin D @sin E  >vcc  avccc(cosD vc sin D )@cos E `sin D k .

M G( 2)  EI

2

(5.26)

2

2

By substituting the right parts of equations (5.25), (5.26) in equation (5.24), we obtain a vector equation of equilibrium of moments at edge C . The projections of this equation on axes Gx2 , Gy 2 are used as dynamic boundary conditions for equations (5.1). The projection of equation (5.26) on axis Gz 2 approximately determines the additional influence of dynamic torque on the rotational motion of the drill string. This additional dynamic effect is not considered here, and it is assumed that the drill string rotates at constant angular velocity Z under the action of constant torque M z . In practical calculations, however, it is assumed that the moments of inertia of the bit relative to each of its central axes are relatively small. Therefore, the values of the vector K G( 2) components in formula (5.24) can be ignored. As a result, the sum of moments of elastic forces relative to each of the horizontal axes equals zero. This assumption leads to two boundary conditions in respect of moments >ucc  auccc(cosD  uc sin D )@sin E  > vcc  avccc(cosD  vc sin D ) @cos E 0, (5.27) > ucc  auccc(cosD  uc sin D )@cos E  > vcc  avccc(cosD  vc sin D ) @sin E 0 . Therefore, correlations (5.23) and (5.27) determine the boundary conditions at the lower end of the drill string. They describe the friction interaction of the bit with the bottom of the well. In the computer implementation of the computational process, they are replaced by their finite-difference analogues. 5.5.

Analysis of frictional whirling vibrations of the system

The above relations determine the three-point boundary value problem of the lower span dynamics of the drill string with a bit at the lower end. Its computational solution was performed by the method of finite differences in variable z using an implicit time integration scheme with step 't 10 4 s. As a result of the study, it was found that the self-excitation regime of these self-oscillations and their modes greatly depend on the bending stiffness of the drill string, values T , M z , and the geometry of the contacting surfaces of the bit and the well. It follows from this conclusion that by choosing different values of these parameters it is possible to both stabilise and destabilise these oscillations. The type

Modelling emergency situations in drilling deep boreholes

238

of model used for calculations can also play a significant role in the study. The calculations are performed with the following values of the system characteristic parameters: Ft

E

Ut

2.1˜1011 Pa,

S (r12  r22 ) 5.34 ˜103 m2,



Ff



S r22

Uf

7.8 ˜103 kg/m3,

2.01˜102 m2, 5

1.5 ˜103 kg/m3, 9 m,

l

e 1 m,

1.94 ˜10 m . 0.09 m, r2 0.08 m, I S Fig. 5-13 shows the modes of motion of bit centre C in rotating coordinate system Oxyz (left) and in fixed coordinate system OXYZ (right). The case of r14

r1

1˜ 105 N, M z Coefficient of friction

T

1˜ 10 4 N∙m, Z

u (t ), m

Motion along the limit circle (steady motion)

0.015 0.01

0.005

0.005

0

0

-0.005

-0.005

-0.01

0.015

0

0.005

0.01

0.015

v(t ), m

Motion along the limit circle (backward whirling, steady motion)

x

u (t ), m

-0.015 -0.015 -0.01 -0.005

0.015

0.01

0.01

0.005

0.005

0

0

-0.005

-0.005

0

0.005

0.01

0.015

v(t ), m

-0.015 -0.01 -0.005

0

0.005

0.01

0.015

0

0.005

0.01

0.015

0

0.005

0.01

0.015

x

u (t ), m

-0.015 -0.01 -0.005

0.015 0.01

0.005

0.005

0

0

-0.005

-0.005

X (t ), m x

-0.01

-0.01 -0.015 -0.015 -0.01 -0.005

Y (t ), m

-0.015

0.01

Y (t ), m

X (t ), m

-0.01

-0.01 -0.015

0.015

X (t ), m

-0.01

-0.015 -0.01 -0.005

μ=5

Trajectories of bit point C in the fixed coordinate system

0.01

-0.015

μ = 30

4

5 rad/s, a 0.12 m, R 0.75 m is

Trajectories of bit point C in the rotating coordinate system 0.015

μ = 0.2

 r24

0

0.005

0.01

0.015

v(t ), m

-0.015 -0.015 -0.01 -0.005

Y (t ), m

Fig. 5-13. Trajectories of bit centre C whirling in the moving (left) and fixed (right) coordinate systems (case T =-1·105 N, Mz =-1·104 N·m, Z = 5 rad/s, a = 0.12 m, R = 0.75 m, t =20 s)

considered. The simulation is performed in the time interval 0 d t d 20 s. It was considered that the motion of the system begins after a small deviation from the initial vertical position. Analysing these results, it can be noted that at different

Chapter 5. Self-excitation of drill string bit whirling

239

Coeffici ent of friction

coefficients of friction P the bit sliding along the well bottom is small, so the trajectory of centre motion C is practically the same in all three cases. Graphs of friction force FXfr change FXfr , N

FYfr , N

20000

μ = 0.2

Graphs of friction force FYfr change 20000

10000

10000

0

0

-10000

-10000

t, s

-20000 0

FXfr , N

4

8

12

16

fr

FY , N

500000

μ=5

4

8

12

16

20

0

4

8

12

16

20

0

4

250000

0

0

-250000

-250000

t, s

-500000 0

μ = 30

0

500000

250000

FXfr , N

t, s

-20000

20

4

8

12

16

fr

FY , N

3000000

1500000

t, s

-500000

20

3000000

1500000

0

0

-1500000

-1500000

-3000000 0

4

8

12

16

20

t, s

-3000000 8

12

16

20

t, s

Graphs of displacement function u(z)

Graphs of displacement function v(z)

z, m

0

-0.015

8 9

8

μ = 30

-0.015

8 -0.015

u ( z ), m 0.015

u ( z ), m 0.015

z, m

0

u ( z ), m

9 0

8 0

z, m

8

0.015

z, m

9 -0.015

u ( z ), m

9 0

0

9

0

-0.015

u ( z ), m

0.015

0

z, m

0

z, m

0

μ=5

μ = 0.2

Coeffici ent of friction

Fig. 5-14. Graphs of friction forces FXfr (t ) and FYfr (t ) change in the fixed coordinate system (case T =-1·105 N, Mz =-1·104 N·m, Z = 5 rad/s, a = 0.12 m, R = 0.75 m, t =20 s)

0.015

-0.015

9

8 0

0.015

v( z ), m

Fig. 5-15. Graphs of displacement functions u (z), v (z) in the rotating coordinate system (case T =-1·105 N, Mz =-1·104 N·m, Z = 5 rad/s, a = 0.12 m, R = 0.75 m, t =20 s)

Modelling emergency situations in drilling deep boreholes

240

It is important to emphasise that in the fixed coordinate system the bit rolls in the direction opposite to Z direction. These modes are called backward whirling of the bit. The motion is carried out on a closed circular path, so it is stable. According to experts, it is a danger to the system because after the disturbance the bit does not return to its original state. Fig. 5-14 shows the graphs of the friction forces Fxfr (t ) and Fyfr (t ) changes

Coefficient of friction

for the regimes of motion shown in Fig. 5-13 in the fixed coordinate system. It is seen that in this coordinate system the friction forces, in addition to the basic harmonic oscillations, also experience high-frequency periodic changes with small amplitudes. In this case, the drill string pipe is bent in simple modes (Fig. 5-15). It is also interesting to note that the oscillations for this case occur mainly in the direction of axis Ox . Trajectories of bit point C in the rotating coordinate system

μ = 0.2

0.015

u (t ), m

Unwinding spiral (unstable motion)

0.01

0.005

0.005

0

0

-0.005

-0.005

-0.01

-0.01

-0.015 -0.01 -0.005

0.015

μ=5

0.015

0.01

-0.015 0

0.005

0.01

0.015

v(t ), m

u (t ), m

Unwinding spiral (backward whirling, unstable motion)

x

-0.015 -0.01 -0.005

0.015 0.01

0.005

0.005

0

0

-0.005

-0.005

0

0.005

0.01

0.015

0.005

0.01

0.015

0.005

0.01

0.015

Y (t ), m

X (t ), m x

-0.01

-0.01

-0.015 -0.01 -0.005

0.015

X (t ), m

-0.015

0.01

-0.015

μ = 30

Trajectories of bit point C in the fixed coordinate system

0

0.005

0.01

0.015

v(t ), m

u (t ), m

-0.015 -0.015 -0.01 -0.005

0.015

0.01

0.01

0.005

0.005

0

0

-0.005

-0.005

Y (t ), m

x

-0.01

-0.01 -0.015 -0.015 -0.01 -0.005

0

X (t ), m

0

0.005

0.01

0.015

v(t ), m

-0.015 -0.015 -0.01 -0.005

0

Y (t ), m

Fig. 5-16. Trajectories of whirling motion of bit centre C in the rotating (left) and fixed (right) coordinate systems (case T =-1·105 N, Mz =-1·104 N·m, Z = 10 rad/s, a = 0.12 m, R = 0.75 m, t =20 s)

If for the considered system rotation speed Z is increased from Z 5 rad/s to Z 10 rad/s, the character of the motion of the bit whirling undergoes significant

Chapter 5. Self-excitation of drill string bit whirling

241

changes (Fig. 5-16), and with an increased coefficient of friction the trajectory of motion takes the mode of expanding spirals, which indicates their instability. As in the previous case, backward whirling occurs here representing the biggest danger for the system in question. In this case, the nature of the change of functions Fxfr (t ) and Fyfr (t ) , u (z ) and v(z ) roughly coincides with the graphs of the corresponding

Coefficie nt of friction

functions for the previous case. Trajectories of bit point C in the rotating coordinate system u (t ), m

Motion along the limit circle (steady motion)

0.02

μ = 0.2

0.01

X (t ), m

0.01

0

0

-0.01

-0.02

-0.01

0

0.01

0.02

v(t ), m

-0.02 -0.02

X (t ), m

0.02

μ=1

0

0

-0.01

-0.01

v(t ), m

-0.02

μ = 10

u (t ), m

-0.01

0

0.01

0

0.01

0.02

-0.01

0

0.01

0.02

-0.01

0

0.01

0.02

Y (t ), m

Y (t ), m

-0.02 -0.02

0.02

X (t ), m

0.02

0.02

0.01

0.01

0

0

-0.01

-0.01

-0.02 -0.02

-0.01

0.02

0.01

0.01

-0.02

Motion along the limit circle (forward whirling, stable motion)

0.02

-0.01

-0.02

u (t ), m

Trajectories of bit point C in the fixed coordinate system

-0.01

0

0.01

0.02

v(t ), m

-0.02 -0.02

Y (t ), m

Fig. 5-17. Trajectories of bit centre C whirling in the moving (left) and fixed (right) coordinate systems (case T = -1·105 N, Mz = -1·104 N·m, Z = 5 rad/s, a = 0.12 m, R = 0.25 m, t =20 s)

If the spherical bit is in contact with the bottom of the well of a smaller radius R 0.25 m, then at small values of the friction coefficient ( P 0.2 and P 0.5 ) the centre of the bit moves in a circle of a constant radius representing a stable trajectory of motion (Fig. 5-17). However, as in the previous case, with an increase in magnitude P ( P t 1), the whirling motion again becomes unstable, and it occurs along an expanding spiral. Here, there is the forward whirling. With a further increase in the angular velocity to Z 10 rad/s and Z 20 rad/s (Fig. 5-18), trajectories acquire the modes of unwinding spirals, so the motion is unstable. Graphs of functions FXfr (t ) and FYfr (t ) changes, which refer to the previous case, retain their appearance at these speeds. If the radius of the well is quite small ( R 0.15 m)—that is, slightly larger than the radius of the bit, then at angular

Coefficie nt of friction

Modelling emergency situations in drilling deep boreholes

Trajectories of bit point C in the rotating coordinate system

μ = 0.2

u (t ), m

Unwinding spiral (unstable motion)

0.02

242

Trajectories of bit point C in the fixed coordinate system X (t ), m

0

0

-0.01

-0.01

v(t ), m

-0.02 -0.02

-0.01

0

0.01

0.02

-0.02

μ=5

0

-0.01

-0.01

-0.01

0

0.01

v(t ), m

0.02

-0.02

0.02

Y (t ), m

x

Y (t ), m -0.01

0

0.01

0.02

0

0.01

0.02

0.02

0.01

0.01

μ = 30

0.01

-0.02

X (t ), m

0.02

0

0

-0.01

-0.01

-0.02 -0.02

0

0.01

0

-0.02

u (t ), m

-0.01

0.02

0.01

-0.02

x

-0.02

X (t ), m

0.02

Unwinding spiral (backward whirling, unstable motion)

0.01

0.01

u (t ), m

0.02

-0.01

0

0.01

0.02

v(t ), m

x

-0.02 -0.02

-0.01

Y (t ), m

Coefficie nt of friction

Fig. 5-18. Trajectories of bit centre C whirling in the moving (left) and fixed (right) coordinate systems (case T = -1·105 N, Mz = -1·104 N·m, Z = 10 rad/s, a = 0.12 m, R = 0.25 m, t =20 s) Trajectories of bit point C in the rotating coordinate system u (t ), m

0.015

Motion along the limit circle (stable motion)

μ=1

0.01

X (t ), m

0.015

0.0008

0.0007

0.01

0.0006

0.005

0.005

0

0

-0.005

-0.005

-0.01

-0.01

0.0005

0.0004

0.0003 -0.0159 -0.0158 -0.0157 -0.0156 -0.0155 -0.0154

-0.015 -0.015 -0.01 -0.005

μ = 10

u (t ), m

0

0.005

0.01

0.015

-0.015 -0.015 -0.01 -0.005

X (t ), m

0.01

0.01 0.005

0

0

-0.005

-0.005

-0.015 -0.01 -0.005

0

0.005

0.01

0.015

0.005

0.01

0.015

0

0.005

0.01

0.015

-0.01

v(t ), m

Y (t ), m

-0.015 -0.015 -0.01 -0.005

X (t ), m

0.015

0

Y (t ), m

0.015

0.005

-0.01

u (t ), m

v(t ), m

0.015

-0.015

μ = 30

Trajectories of bit point C in the fixed coordinate system

0.0008

0.015

0.0007

0.01

0.01

0.005

0.005

0

0

-0.005

-0.005

-0.01

-0.01

0.0006

0.0005

0.0004

0.0003 -0.014 -0.0139 -0.0138 -0.0137 -0.0136 -0.0135

-0.015 -0.015 -0.01 -0.005

0

0.005

0.01

0.015

v(t ), m

Y (t ), m

-0.015 -0.015 -0.01 -0.005

0

0.005

0.01

0.015

Fig. 5-19. Trajectories of bit centre C whirling in the rotating (left) and fixed (right) coordinate systems (case T = -1·105 N, Mz = -1·104 N·m, Z = 5 rad/s, a = 0.12 m, R = 0.592 m, t =20 s)

Chapter 5. Self-excitation of drill string bit whirling

243

velocity Z 5 rad/s of the trajectory the whirling motion of the bit centre acquires the mode of dense unwinding spirals for all values of the friction coefficient. The subsequent increase in the angular velocity to Z 20 rad/s leads to the fact that even for a fairly small period of time the trajectory of the bit is rapidly approaching the expanding spiral. In this regime of oscillations, the motion of the bit centre and the rotation of the drill string in the fixed coordinate system occur in one direction—that is, forward whirling takes place. An interesting case can be seen in Fig. 5-19 for well radius R 0.592 m and angular velocity Z 5 rad/s. In the rotating coordinate system, motion of the bit centre occurs along a trajectory that goes to the limit circle. However, in the fixed system, the bit almost stopped in place. In this situation, drilling path bending can happen, which is slightly shifted to the side. At a larger scale, the mode of motion resembles a ‘multi-petal flower.’ This motion is stable, but it seems to be nonstandard, because backward whirling is observed with it, and the trajectory of drilling can deviate from the given direction. 5.6.

Kinematic (nonholonomic) model of the spherical bit whirling on the spherical surface of the well bottom

If the force of friction interaction between the drill bit and the borehole bottom is large, and the rolling of the bit on the surface of the rock occurs without slipping, the absolute velocity of bit point G equals zero, then equality (5.11) will take the form: v C(0)

o

:((10)) u CG , o

or v C(0)  : ((10)) u CG

0,

(5.28) o

where v C( 0) is v C(0) (u  Zv)i  (v  Zu) j , while vectors : ((10)) and CG are determined by formulas (5.7) and (5.12) Ω ((10))  vc  Zuc i  uc  Zvc j  ωk , o

GC a sin D cos E i  a sin D sin E j  a cosD k The mechanical meaning of equation (5.28) is that the speed of the bit point contacting with fixed point G of the well bottom is also zero at the given time. We shall represent this property in the form:

Modelling emergency situations in drilling deep boreholes

o

Ω ((10)) u GC

i  vc  Zu c  a sin D cos E

j u c  Zvc  a sin D sin E

244

k

Z  a cosD

> a cosD uc  Zvc  aZ sin D sin E @i   > a cosD  vc  Zu c  aZ sin D cos E @j   > a sin D sin E  vc  Zu c  a sin D cos E u c  Zvc @k.

(5.29)

Here, cosD , sin D , cos E , sin E are calculated by formulas (5.8) and (5.9). Substituting the values of trigonometric values into equality (5.29) and considering equality (5.19), we obtain nonholonomic (kinematic) boundary conditions for the lower end of the drill string u  Z v  a 1 

u 2  v2  uc  Z vc   Z v Ra ( R  a) 2

u 2  v2  vc  Z uc  Z u v  Z u  a 1  2 Ra ( R  a)

0,

(5.30) 0.

The dynamic boundary equations for the drill string at point C are similar to dynamic boundary conditions (5.27) formulated when considering the friction model. Let's write them down again >ucc  auccc(cosD  uc sin D )@sin E  > vcc  avccc(cosD  vc sin D ) @cos E 0, (5.31) > ucc  auccc(cosD  uc sin D )@cos E  > vcc  avccc(cosD  vc sin D ) @sin E 0 . In this case, relations (5.30) and (5.31) determine the boundary conditions for equations (5.1). They are used in a nonholonomic model to analyse the whirling vibrations. 5.7.

Analysis of elastic vibrations of the system using the nonholonomic model of the bit rolling

Relations (5.30) and (5.31) together with boundary conditions (5.2)−(5.4) determine the three-point boundary value problem of the dynamics of the two bottom spans of the drill string with a bit. They are also supplemented by initial conditions that define the initial perturbation of the system. As in the case considered for the friction model, the computational solution of the problem is carried out by the finite difference method using the implicit time integration scheme. This scheme is stable for any time step value 't but has an acceptable accuracy only at its sufficiently small values. To ensure acceptable accuracy, the calculations were performed using relatively small steps of integration of 't 1 ˜ 104 s and 't 1 ˜ 10 5 s. Since the

Chapter 5. Self-excitation of drill string bit whirling

245

numerical results for the two statements coincided with an accuracy of 4, 5th significant figure, it can be considered that the achieved accuracy of the calculations is sufficient. Computational analysis of the problem was preceded by calculation of the stability of the drill string inside segment AB . For the simplest case, when T z 0 , Z z 0 , V z 0 , M z 0 , the relation for their critical values is as follows [8, 9]:

S4

EI  S 2Tcr  L2 ( Ut Ft  U f F f )Zcr2  S 2 U f F f Vcr2 0 . (5.32) L2 From this, it is possible to obtain a critical value for the longitudinal force S2 L2 Tcr  2 EI  2 ( Ut Ft  U f F f )Z 2  U f F f V 2 , (5.33) L S angular velocity

Zcr

r

S 2 EI  L2T  L2 U f F f V 2 , Ut Ft  U f F f

S 2

L

(5.34)

and the speed of the internal flow of the drilling liquid r

Vcr

S 4 EI  S 2 L2T  L4 ( Ut Ft  U f F f )Z 2 . S 2 L2 U f F f

(5.35)

The analytical solution for the critical state of the drill string under the action of torque exists when only axial force T acts on the drill string. Then, the critical value of force T at given value M z can be found from the equality 

S 2 EI



M z2 cr

. (5.36) 4 EI L2 Table 5-1 shows the force T critical values with different combinations of Tcr

values

Z,

V,

and

Mz

Ft

S r12  r22 5.34 ˜103 m2,

Uf

1.5 ˜103 kg/m3,

l

r14

 r24

a 0.2 m, I

S



8 m,

for

the

S r22

Ff

e 1 m,

1.94 ˜10

5

given r1

parameters:

2.01 ˜102 m2, 0.09 m,

r2

Ut

E

2.1˜1011 Pa,

7.8 ˜103 kg/m3,

0.08 m,

m 30 kg,

4

m.

As can be seen from this data, the critical values of the compressive force Tcr are markedly dependent on values V and M z . This suggests that the stiffness of the drill string is mainly determined by values T and Z . Therefore, it can be concluded that by choosing different values of these parameters, it is possible to stabilise or destabilise the dynamics of the bit whirling. Figs. 5-20–5-24 show the calculation results for T 1˜ 105 N, Mz

1˜ 10 4 N∙m. They represent the motion paths of bit centre C in rotating

Modelling emergency situations in drilling deep boreholes

246

coordinate system Oxyz (left) and fixed coordinate system OXYZ (right). Offsets X (t ) and Y (t ) in plane OXY are calculated by the formulas X (t ) u cosZt  v sin Zt , Y (t ) u sin Zt  v cosZt . Table 5-1 Critical values of static and kinematic parameters of drill string loads

1

Tcr (N) 3.834∙105

ω cr ( rad/s) 5

V cr (m/s) 0

Mz cr (N∙m) 0

2

2.378∙105

15

0

0

3

5

20

0

0

5

3.987 ˜10 3.896∙105

0

10

0

0

20

0

3.745∙10

5

0

30

0

4.017∙10

5

0

0

1∙103

8

4.016∙10

5

0

0

1∙104

9

4.010∙105

0

0

1∙105

No.

1.104∙10

4 5 6 7

The simulation shows that angular velocity ZC of point C in rotating coordinate system Oxyz is greater than zero, and it always moves clockwise in this system. However, in fixed coordinate system OXYZ , it behaves differently. In fact, if the surface of the well bottom is shallow, perturbations in the form of the initial displacement of the bit centre approach to the limit cycle with different directions in different coordinate systems (Fig. 5-20 for case R 1 m, Z 5 rad/s). u (t ), m X (t ), m 0.02 0.02 0.01

0.01

0

0

-0.01

-0.01

v(t ), m

-0.02 -0.02

-0.01

0

a)

0.01

0.02

Y (t ), m

-0.02 -0.02

-0.01

0

0.01

0.02

b)

Fig. 5-20. Trajectories of motion of the bit centre in rotating coordinate system Oxyz a) and in the fixed coordinate system b) (R = 1 m, ω = 5 rad/s)

Chapter 5. Self-excitation of drill string bit whirling

0.02

u (t ), m

0.02

X (t ), m

0.01

0.01

t

247

t

00 x -0.01

0

0

x

t otf

-0.01

t otf -0.02 -0.02

v(t ), m

-0.01

0

0.01

0.02

Y (t ), m

-0.02 -0.02

-0.01

a)

0

0.01

0.02

b)

Fig. 5-21. Trajectories of the bit centre motion in rotating coordinate system Oxyz a) and in the fixed coordinate system b) (R = 0.5 m, ω = 20 rad/s)

0.015

u (t ), m

0.015

0.01

0.01

0.005

0.005

0

0

-0.005

-0.005

-0.01

-0.01

v(t ), m

-0.015 -0.015 -0.01 -0.005

0

0.005

0.01

X (t ), m

Y (t ), m

-0.015 -0.015 -0.01 -0.005

0.015

a)

0

0.005

0.01

0.015

b) 0.0003

X (t ), m

0.0002

0.0001

0

-0.0001

Y (t ), m

-0.0002 -0.0105 -0.0104 -0.0103 -0.0102 -0.0101

-0.01

c)

Fig. 5-22. Motion paths of the bit centre (R = 0.669 m, ω = 5 rad/s): in rotating coordinate system Oxyz a); in the fixed coordinate system b); one complete revolution in the fixed coordinate system c)

Modelling emergency situations in drilling deep boreholes

248

It turned out that the drill string rotates counter clockwise, the centre of the bit moves clockwise in the coordinate system associated with the drill string (Fig. 5-20, a), and it moves in a circle counter clockwise relative to coordinate system OXYZ (Fig. 5-20, b). In the case of large angular velocity Z , the trajectory of the bit has the shape of a narrowing spiral approaching the state u 0 , v 0 (Fig. 5-21 for R 0.5 m, Z 20 rad/s). This mode is the most stable and favourable. Between these two phenomena, there is one intermediate state when R 0.669 m and Z 5 rad/s (Fig. 5-22). In this case, the angular velocities of rotation of the drill string and the motion of the centre of mass of the bit in system Oxyz are almost equal in magnitude but opposite in sign. Therefore, the absolute speed of the centre of the bit is almost zero, and it tends to some stationary point X X s , Y 0 (Fig. 5-22, b). During the intermediate state, the trajectory of point C (Fig. 5-22, c) is similar to the curve in Fig. 5-5, c. After reaching the limit state, the bit moves to the state of pure rotation and continues to drill, but in this case in the other direction. Therefore, this regime is also not acceptable. If to reduce value R to 0.5 m, the trajectory again takes the mode of a circle, but the displacement will occur clockwise in both coordinate systems (Fig. 5-23 for R 0.5 m, Z 10 rad/s). 0.02

X (t ), m

u (t ), m

0.02

0.01

0.01

0

0

-0.01

-0.01

v(t ), m

-0.02 -0.02

-0.01

0

a)

0.01

0.02

Y (t ), m

-0.02 -0.02

-0.01

0

0.01

0.02

b)

Fig. 5-23. Trajectories of the bit centre motion in rotating coordinate system Oxyz a) and in the fixed coordinate system b) (R = 0.5 m, ω = 5 rad/s)

The opposite is true when radius R is only slightly larger than value a . The motion of the bit centre in this case ceases to be stable, and it begins to move in the mode of an expanding spiral (Fig. 5-24 for R = 0.25 m, Z 20 rad/s). These modes are considered to be the most dangerous.

Chapter 5. Self-excitation of drill string bit whirling

0.06

249

X (t ), m

u (t ), m

0.06

0.04

0.04

0.02

0.02

0

0

-0.02

-0.02

-0.04

-0.04

v(t ), m

-0.06 -0.06 -0.04 -0.02

0

0.02

0.04

0.06

Y (t ), m

-0.06 -0.06 -0.04 -0.02

a)

0

0.02

0.04

0.06

b)

Fig. 5-24. Trajectories of the bit centre motion in rotating coordinate system Oxyz a) and in the fixed coordinate system b) (R = 0.25 m, ω = 20 rad/s)

Note again that the task is multi-parametric since the state of the system depends on EI , l , e , m , a , R , T , M z , and Z . In addition, even with a slight change in these parameters, as is typical for most nonholonomic systems, there is a significant restructuring of the modes of the bit motion, which is accompanied by a complication of the trajectory geometry and the generation of motion in the opposite direction in the fixed coordinate system. Therefore, attempts to establish general regularities of motion of the bit, even in a simple spherical shape, at values T , M z , Z , far from the critical, have not been successful. However, it is found that parameters T , M z , and Z have the greatest influence on the system dynamics and its stability when they approach the critical values, since in these cases the bending stiffness of the drill string is significantly reduced, the bit loses the constraints imposed on it, its motion becomes less stable, and it takes a complex mode.

0.02

u (t ), m

0.02

0.01

0.01

0

0

-0.01

-0.01

v(t ), m

-0.02 -0.02

-0.01

0

a)

0.01

0.02

X (t ), m

Y (t ), m

-0.02 -0.02

-0.01

0

0.01

b)

Fig.5-25. Trajectories of bit centre C in the vicinity of T, Mz, ω (R = 1 m, ω = 20 rad/s) close to critical

0.02

Modelling emergency situations in drilling deep boreholes

250

Fig. 5-25 shows the shapes of trajectories of the bit centre for values T 1˜ 105 N, M z 1˜ 10 4 N∙m, Z 20 rad/s, a 0.2 m, R 1 m, close to critical. In this case, point C moves along the line with the loops drifting away from the axis of the drill string. Similar unstable modes appear in close proximity to the critical values for other values T , M z , Z , and also a and R .

5.8.

Friction and nonholonomic rolling of the spherical bit on the ellipsoidal surface of the well bottom

The cases of the dynamic behaviour of a spherical bit were considered above when it is rolling along the spherical surface of the well bottom. However, in practice, there may be situations where the surface of the well bottom has the shape of an ellipsoid of revolution. For the quantitative analysis of kinematically excited whirling vibrations, in this case, it is necessary to compose the dynamics equation of the total separated for the consideration rotating two-span beam ABC , which is prestressed by torque and longitudinal compressive force T  R (Fig. 5-26). O A

X Zt

y

x

Y

Drill string

Spherical drill bit

B Ellipsoidal surface of the well bottom

C

S Z, z

ω

Fig. 5-26. The scheme of the bottom of the drill string with a spherical bit rolling on the ellipsoidal surface of the well bottom

As in the other considered cases, the equations of dynamic elastic bending of the drill string in the presence of the factors presented earlier have the form:

Chapter 5. Self-excitation of drill string bit whirling

EI

251

w 4u w § wu · w 2 § wv · 2  ¨T ¸  ¨ M z ¸  Ut F t  U f F f Z u  wz ¹ wz 4 wz © wz ¹ wz 2 ©

 2Ut F t  U f F f Z

wv w 2u w 2u w 2u  V 2 U f F f 2  2VU f F f  Ut F t  U f F f 2 wt wzwt wz wt

0, (5.37)

4

EI

2

wu · w v w § wv · w § 2  ¨T ¸  ¨ M z ¸  Ut F t  U f F f Z v  wz ¹ wz 4 wz © wz ¹ wz 2 ©

wu w 2v w 2v w 2v  V 2 U f F f 2  2VU f F f  Ut F t  U f F f 2 0. wt wzwt wz wt where, as above, uz, t , vz, t are the elastic displacements of the drill string pipe element in the directions of axes Ox , Oy , respectively; EI is the stiffness of the drill string pipe under bending; U t , U f are the densities of the rod material and the drilling  2Ut F t  U f F f Z

liquid, respectively; Ft , F f are the cross-sectional areas of the rod wall and its internal channel, respectively; t is the time. In the formation of boundary conditions at support A( z 0) , we assume that the vibrations of the drill string in the spans adjacent to this support occur in the antiphase, so the bending moment at point A is zero. Then, at edge z 0 , we have

uA

vA

0,

w 2u / w z 2

A

w 2v / w z 2

A

0.

(5.38)

At support B ( z l ) , the deflections of the beam equal zero, and the angles of rotation are continuous functions. These conditions can be written as дu дu дv дv u B vB 0 , . (5.39) , дz l 0 дz l 0 дz l 0 дz l 0 To derive the boundary conditions at the lower edge, we assume that the process of excitation of whirling vibrations is just beginning, and the bit can move in the gap between it and the well wall without reaching it (Fig. 5-26). At the same time, the rolling nature of the bit and the boundary conditions at the point are determined by the geometry of both the bit and the well bottom. Consider the case when the surface of the bit is spherical, and surface S of the well has the shape of an ellipsoid of revolution (elongated at b  c or flattened at b ! c ) (Fig. 5-27). As the shape of the well bottom is ellipsoid, then in the cross-section we have an ellipse, the equation of which is written in the form: X 2 Z2 (5.40)  2 1. b2 c The equation of the ellipse, which in Fig. 5-27 is represented by a dashed line, has the following form:

Modelling emergency situations in drilling deep boreholes

252

x2 z2 (5.41)  1. 2 (b  a) (c  a ) 2 Let's write this ellipse equation in parametric form: ­ X b cosT , ­ x (b  a) cosT , (5.42) ® ® ¯ Z c sin T , ¯ z (c  a) sin T . Find the distance between these ellipses (5.43) ( X  x) 2  (Z  z ) 2 (b  b  a) 2 cos2 T  (c  c  a) 2 sin 2 T a 2 . From equality (5.43), we see that the distance between the corresponding points of both ellipses with the same arguments is the same. b

b

D x c

u 2  v2

C G

x Z

D

x2

z2

Fig. 5-27. Geometric scheme of a spherical bit on the ellipsoidal surface of the well bottom

Using parametric formulas, we find derivatives dZ c dz (c  a )  ctgT ,  ctgT . (5.44) dX b dz (b  a) dZ c (c  a )  ctgT , and tgD a  ctgT , we express Considering that tgD dX b (b  a) tgD through tgD a (b  a) c tgD ˜ tgD a . (5.45) (c  a ) b But

tgD a 

dz dx



x (c  a ) ˜ (b  a) (b  a) 2  x 2

(c  a ) u 2  v 2 (b  a) (b  a) 2  u 2  v 2

,



u 2  v2 (c  a ) ˜ (b  a) (b  a) 2  (u 2  v 2 )

Chapter 5. Self-excitation of drill string bit whirling

253

whilst

c u 2  v2

tgD

b (b  a) 2  u 2  v 2

.

(5.46)

Knowing the equation of surface π, we can find the radius of its curvature [b 4  x 2 (c 2  b 2 )]3 2 b 4c . Given that x o 0 , we have Rcr b 2 c . The bit rolling on the well surface will be described in right moving coordinate system Gx2 y2 z2 , the origin of which coincides with contact point G , and axis Gz2 is the continuation of CG segment (Fig. 5-27). The condition of the bit rolling without sliding allows us to formulate two groups of boundary equations at the C point. These are two kinematic equations that define the point C velocity and two dynamic equations that determine the dynamic equilibrium of all moments relative to point G . The orientation of system Cx 2 y2 z 2 relative to system Oxyz is set by angle D between axes Oz and Gz 2 Rcr

c u 2  v2

sin D

b 2 (b  a) 2  (u 2  v 2 )(c 2  b 2 )

, (5.47)

2

2

b (b  a)  u  v

cosD

2

2

b (b  a) 2  (u 2  v 2 )(c 2  b 2 )

,

where a is the radius of the bit; b , c are the large and small semi-axes of the well. To build the equilibrium equations of the friction model, we use equality (5.13). Let's write it down again

Q  F fr

0,

(5.48)

where

Q Friction force vector F fr through the formula

д3u д 3v  EI i j. (5.49) дz 3 дz 3 between the bit and the well surface is calculated EI

F fr

P vGabs

P T ˜ vGabs / vGabs .

(5.50)

To find v Gabs we will use equality (5.11). Then, we have

vGabs o

o

v C(0)  Ω((10)) u GC ,

(5.51)

where vectors v C( 0) , Ω ((10)) , GC that are used here are calculated using correlations

Modelling emergency situations in drilling deep boreholes

Ω ((10)) o

v C(0) (u  Zv)i  (v  Zu) j ,

(5.52)

 vc  Z uc i  uc  Z vc j  ωk ,

(5.53)

acu



GC

2

b (b  a)  (u 2  v 2 )(c 2  b 2 ) 

254

2

i

acv b 2 (b  a) 2  (u 2  v 2 )(c 2  b 2 )

(5.54) j  a cosDk. o

In this case, substituting vector product Ω ((10)) u GC and equality (5.52) into equation (5.51), we obtain vGabs ,x

§ · cZ v ¸, u  Zv  a cosD ¨  uc  Z vc  2 2 2 2 2 2 ¨ ¸ b b a u v c b     ( ) ( )( ) © ¹

(5.55) § · cZ u ¨ ¸ v  Zu  a cosD  vc  Z uc  . ¨ b 2 (b  a) 2  (u 2  v 2 )(c 2  b 2 ) ¸¹ © Using equalities (5.49), (5.50), and (5.55) in equation (5.48), we have w 3u w 3v EI 3 i  EI 3 j  wz wz · § PT cZ v ¸i  ¨  uc  Z vc   (5.56) 2 2 2 2 2 2 2 ¸ abs 2 ¨ ( ) ( )( )     b b a u v c b  vGabs v ¹ © G, y ,x vGabs ,y

 



PT

v  v abs 2 G, x

abs 2 G, y

§ · cZ u ¨  vc  Z uc  ¸ j 0. ¨ b 2 (b  a) 2  (u 2  v 2 )(c 2  b 2 ) ¸¹ ©

Projecting vector equality (5.56) on axes OX , OY , we obtain equilibrium conditions for frictional rolling of the spherical bit on the ellipsoidal surface of the well bottom § · PT w 3u cZ v ¨  uc  Z vc  ¸ 0, EI 3  2 2 2 2 2 2 ¸ abs 2 abs 2 ¨ wz b b a u v c b     ( ) ( )( ) vG , x  vG , y © ¹

 

(5.57) 3

EI

wv  wz 3

PT

  2 vGabs ,x

2 vGabs ,y

· § cZu ¸ 0. ¨  vc  Zuc  2 2 2 2 2 2 ¸ ¨ b ( b a ) ( u v )( c b )     ¹ ©

When switching to the regime of nonholonomic rolling of a spherical bit on an ellipsoidal surface, equilibrium equations (5.57) are transformed into kinematic boundary conditions

Chapter 5. Self-excitation of drill string bit whirling

255

§ · cZ v ¸ 0, u  Zv  a cosD ¨  u c  Z vc  2 2 2 2 2 2 ¨ b (b  a)  (u  v )(c  b ) ¸¹ ©

(5.58) § · Z c u ¸ 0. v  Zu  a cosD ¨  vc  Z uc  ¨ b 2 (b  a) 2  (u 2  v 2 )(c 2  b 2 ) ¸¹ © The dynamic boundary equations at the C point are similar to the dynamic boundary conditions (5.27) presented above. Let's write them down again >ucc  auccc(cosD  uc sin D )@sin E  > vcc  avccc(cosD  vc sin D ) @cos E 0, (5.59) > ucc  auccc(cosD  uc sin D )@cos E  > vcc  avccc(cosD  vc sin D ) @sin E 0 . Here, cos E , sin E , cosD , sin D are determined by formulas (5.9) and (5.47). Correlations (5.37), (5.57)−(5.59) determine the three-point boundary value problem of the lower span dynamics of the drill string with a bit. They are also supplemented by initial conditions that define the initial perturbation of the system. The computational solution of the problem is carried out by the finite difference method using implicit time integration scheme.

X (t )

P 0.2

Y (t )

a

P 1

X (t )

P 30 x

P 0.5 Y (t )

b

P 10 Y (t )

c

e

X (t )

X (t ) Y (t )

d

X (t )

X (t )

Nonholonomic model

Y (t )

f

Y (t )

Fig. 5-28. The trajectories of the spherical bit motion on the ellipsoidal surface of the well bottom (T = - 1·105 N, Mz = - 1·104 N·m, ω = 5 rad/s): a) μ = 0.2; b) μ = 0.5; c) μ = 1; d) μ = 10; e) μ = 30; f) nonholonomic model

Modelling emergency situations in drilling deep boreholes

256

Using the derived relations of the spherical bit rolling on the elliptical surface of the well bottom, the bit trajectories were built in fixed coordinate system OXYZ for the case of T 1˜ 105 N, M z 1˜ 10 4 N∙m, Z 5 rad/s, a 0.12 m, b 0.22 m, c 0.18 m, t 20 s. Fig. 5-28 shows the solutions results. For the friction model, parameter P assumed the values 0.2 (a), 0.5 (b), 1 (c), 10 (d), 30 (e). As we can see, for all values of the friction coefficients starting from some initial excitation, the bit eventually moves along a spiral curve, striving for a state in which the drill string becomes rectilinear, and the bit occupies a position at the central point on the rotation axis. Therefore, such a regime is sustainable and most favourable for the drilling process. Some exceptions are the motion modes at small values of the friction coefficient ( P 0.2 and P 0.5 , positions a and b in Fig. 5-28). In this case, the friction contact between the bit and the bottom of the well is weak, and the friction forces are not enough to quickly reduce the elastic forces in the drill string. However, after a while, the transitional regime of the bit oscillation calms down and it also begins to approach the axis of rotation. It was believed that at the limit values of friction coefficient P , there is no slippage between the contacting bodies, and a nonholonomic contact model is attained between them. The trajectory of motion under this assumption is shown in Fig. 5-28, e. It is almost identical to the trajectories built for friction models with P 10 and P 30 (Fig. 5-28, d, e). It is interesting to note that in all the cases considered, with the selected geometric parameters of the contacting bodies, the bit is forward whirling. We should emphasise again that such regimes of motion can take place in real systems, but the model of the bit rolling on the lateral surface of the well that is used in foreign scientific literature is not suitable for describing this effect. As we can see, our model is more universal. 5.9.

The frictional rolling of ellipsoid bit on the bottom surface of a well

In paragraphs 5.4 and 5.2, it was noted that the phenomenon of the bit whirling on the surface of the well bottom could be considered as the effect of the relative motion of two absolutely solid bodies contacting at one point. Therefore, it should be investigated on the basis of methods of theoretical mechanics. The study of this phenomenon is associated with the possibility of using two basic mathematical models that differ in the initial assumptions about the smoothness and roughness of the surfaces of contacting bodies. In this regard, two models related to the assumption of friction and nonholonomic rolling of one surface on the other were considered when studying the problem of vibrations of the spherical bit whirling on the spherical surface of the well bottom.

Chapter 5. Self-excitation of drill string bit whirling

257

In theoretical mechanics, there is an opinion that the friction model is a more general and realistic model of the motion of a solid body on a rough surface, and the nonholonomic model is only suitable for a qualitative description of the rolling of a body in extreme cases, when their surfaces are absolutely rough and do not allow slippage [1, 16]. However, this is not the case for drill bit whirling dynamics because the possibility of slippage of the bit along the bottom surface of the wells is significantly reduced by the availability of diamond studs. This effect is enhanced by increasing the force of pressing the bit to the bottom of the well and reducing the bending stiffness of the drill string. Then, we can assume that the surfaces of the bodies are absolutely rough, there is pure rolling carried out with the spinning of the bit, and the constraints superimposed on the system are nonholonomic. With the wear of the bit and its clogging with rock particles, the adhesion between its surface and the rock is broken, and the bit begins to roll with slippage. To analyse such motion, it is necessary to use the frictional dynamical model. In this section, the frictional and nonholonomic models are used to simulate the whirling vibrations of the bits in the shape of ellipsoids (elongated and flattened). The influence of drill string bending compliance in its lower part on the trajectory shapes of the bit central point motion and on their stability is analysed. The main difference between the problems of the bit whirling on the well surface and of the solid rolling on an undeformed surface is that the bit is fastened to the elastic bending drill string and is not free. Therefore, the motion of the bit depends significantly on the elastic flexibility of the string, which is determined not only by the value of bending stiffness EI (here, E is the elastic modulus of the string rod material, I is the moment of inertia of its cross-sectional area) but also by the proximity of its stress state to the critical. In general, the occurrence of this state depends on the change in axial force T , torque M z , angular velocity Z of the drill string, and speed V of the internal flow of the drilling liquid [6]. The T force is determined by the distributed gravity forces that act on the elements of the drill string and vertical force R of the contact interaction of the bit with the well bottom. Therefore, in its upper part, the drill string is stretched, and in the bottom part it is compressed. This is why the drill string can lose its stability only in the lower edge. Usually, the lower sections of the string are the most deformed, so for the dynamic analysis of the whirling vibrations of the bit, we conditionally select a system that consists of the bit and two lower sections of the drill string (Fig. 5-29). Elastic vibrations of the drill string are studied using fixed ( OXYZ ) and rotating ( Oxyz ) coordinate systems (Fig. 5-29). Axes OZ and Oz of these systems coincide.

Modelling emergency situations in drilling deep boreholes

258

As shown in paragraphs 5.4 and 5.5, for typical values of the parameters that determine the dynamics of the internal flow of the drilling liquid its effect on the bending of the drill string is quite small. Therefore, this hydrodynamic effect can be ignored, and the equation of transverse vibrations of the drill string pipe can be used in form (5.1) recorded for the case of a spherical bit.

O A

X

Zt

y

x

l

Y

L

Centring devices

e

B

C

S ω

Z, z a)

b)

Fig. 5-29. Structural a) and computational b) models of whirling vibrations

For this system, the problem with initial conditions (the Cauchy problem) in respect of the independent time t variable and a three-point boundary value problem in respect of independent variable z in the region 0 d z d L (Fig. 5-29, b) are formulated. As in the formulation of the Cauchy problem for spherical bits, it is accepted that some initial deviation u (z,0) , v(z,0) is introduced into the axial line of the drill

Chapter 5. Self-excitation of drill string bit whirling

259

string, and some initial velocities u (z,0) , v(z,0) are applied to its elements. Then, the initial conditions for the case in question are formulated as follows: u( z,0) u0 ( z ) , v( z,0) v0 ( z ) , (5.60) 0 d z d L , u ( z,0) u0 ( z ) , v( z,0) v0 ( z ) where functions u0 ( z ) , v0 ( z ) , u0 ( z ) , v0 ( z ) are specified. Boundary conditions at points A and B are given as (5.38), (5.39). The question of the formulation of boundary conditions at the lower end of the drill string is associated with the need to take into account the conditions of the contact interaction of the bit with the rock at the well bottom. It is assumed that a completely solid drill bit has the shape of an ellipsoid of revolution with semi-axes a and b (Fig. 5-30). At point G , it is in contact with the flat horizontal bottom of the well and slides along plane S at speed v G . The drill string rotates at constant angular velocity Z resulting in the bit compound motion. Additional elastic longitudinal and torsional vibrations of the drill string are not taken into account, and it is believed that axial force T and torque M acting on the bit remain unchanged under oscillations.

Mlon

Flon

x

F con

vG

r

x

x

z2 Fig. 5-30. Diagram of forces and moments acting on the bit in the inclination plane

For analysis of the bit whirling vibrations, we will rigidly connect coordinate system Cx1 y1 z1 with it (Fig. 5-31). Axes Cx1 and Cy1 of this system in the undeformed state are parallel to axes Ox , Oy , axes Cz and Oz are collinear. We also introduce system Cx 2 y2 z2 whose axis Cx 2 lies in the plane of the bit inclination, and axis Cz 2 coincides with the axis of its symmetry. All the systems are right and have unit vectors i , j , k ; i1 , j1 , k 1 ; i 2 , j2 , k 2 , respectively.

Modelling emergency situations in drilling deep boreholes

X

X

u

O

260

Zt

C

Ux

Uy

D

y x

y1

G

x x1

V Y

Fig. 5-31. Top view on the location of points C and G and on the plane V slope trace

Conditionally, we separate the bit from the drill string and consider the dynamic balance of forces and torques acting on it (Fig. 5-30). The condition of zero of the resultant of all forces applied to the bit, in general, has the form: (5.61) Fel  T  Fcon  Fin 0 , el where F is the resultant of elastic shearing forces at point С of attachment of the bit to the drill string; F in is the resultant of forces of inertia; F con is the vector of contact forces applied to the drill bit at point G . This vector can be represented as (5.62) F con F fr  F norm , fr norm is the normal component of the contact force. where F is the friction force; F To formulate the boundary conditions for equations (5.1) at edge z L (at point C ), we project equality (5.61) on axes Ox , Oy of rotating coordinate system Oxyz . Vector F el can be represented as follows: w 3u w3v Fel Qelx i  Qely j EI 3 i  EI 3 j . (5.63) wz wz At the lower end, the drill string is rotated through angles дu / дz and дv / дz , so in system Oxyz force T acting on the bit is calculated as follows:

T

T uc ˜ i  T vc ˜ j  T 1  (uc) 2  (vc) 2 ˜ k .

Here, values (uc) , (vc) can be ignored in comparison with one. 2

2

(5.64)

Chapter 5. Self-excitation of drill string bit whirling

261

Friction force vector F fr is calculated based on the Amonton-Coulomb law, which is formulated for a moving body in fixed coordinate system OXYZ in the form of a relation F fr

P T ˜ vGabs vGabs ,

(5.65)

where P is the coefficient of dry friction; v Gabs is the absolute speed of bit point G in contact with the well bottom. It is calculated by the formula

vGabs vCabs  Ω u r ,

(5.66)

vCabs

is the absolute speed of bit point С ; Ω is the vector of the angular where velocity of reference system Cx1 y1 z1 in respect of the fixed coordinate system; r is the vector connecting points С and G (Fig. 5-31). To calculate speed v Gabs , it is assumed that rotation angles u c , vc of system Cx1 y1 z1 relative to system Oxyz are small. Then, the rotation angle vector can be entered (5.67) θ vci  ucj  0k T x i  T y j .

In this case, axis Cz1 turns in the vertical plane of inclination V by angle T

(uc) 2  (vc) 2 (Fig. 5-31). Angle D between planes xOz and V is

determined by the formula (Fig. 5-31)

D

arctg (T x T y ) .

(5.68)

v Cabs ,

With the help of equations (5.67), (5.68), vectors, Ω , and r determining the absolute speed of bit point G sliding along the well bottom are calculated. In the projections on the axes of rotating coordinate system Oxyz , they have the form:

v Cabs ui  vj  ω u ui  vj u  Zv i  v  Zu j , Ω vci  ucj  Zk , Ω u r u crz  Zry i  vcrz  Zrx j   vcry  u crx k .

(5.69)

Components rx , ry , rz of vector r used in these formulas are determined by the equality

r

(b 2  a 2 ) sin T y cosT y a 2 sin 2 T y  b 2 cos2 T y

i

(b 2  a 2 ) sin T x cosT x a 2 sin 2 T x  b 2 cos2 T x

j (5.70)

 a sin T  b cos T k. Correlations (5.65), (5.69), (5.70) provide the possibility to calculate the components of the vectors in rotating coordinate system Oxyz 2

2

2

2

Modelling emergency situations in drilling deep boreholes

vGabs ,x

u  Z v  u crz  Z ry ,

and find the projections of vector F Fxfr

vGabs ,y

262

v  Z u  vcrz  Z rx

(5.71)

fr

on axes Ox , Oy u  Z v  ucrz  Z ry , P T 2 2 vxabs  v abs y

 

(5.72)

Fyfr

P T

v  Z u  vcrz  Z rx

v  v abs 2 x

abs 2 y

.

As a result of equation (5.61) projecting on vertical axis Oz , we obtain F norm T . Considering the equilibrium of all forces applied to the bit in the horizontal plane and using equations (5.63), (5.65), (5.72), we obtain the first two boundary conditions for system (5.1) at point C u  Z v  ucrz  Z ry w 3u EI 3  P T 0, 2 2 wz vxabs  v abs y

 

(5.73) 3

EI

wv  PT wz 3

v  Z u  vcrz  Z rx

v  v abs 2 x

abs 2 y

0.

The second group of boundary equations at this edge is formulated from the condition of zero value of the resultant moment of all forces applied to the bit. The form of these equations depends on the choice of the reference centre of the considered forces. It is most convenient to consider the equilibrium of all moments relative to point G of bit contact with the rock, as in this case the moments of the normal and frictional components of the contact force are excluded from consideration. It should also be noted that the moment of inertia of the bit body is relatively small. Then, the moment of bit inertia forces will also be small, and in the equation of equilibrium of moments in respect of point G only the moment of the vector of elastic shearing forces will be preserved. In vector form, this equation is formulated as follows: MG (F el )  MGel

0.

(5.74)

Here

MG (F el ) (r ) u F el

(Try  Qy rz )i 

 (Trx  Qx rz ) j  (Qy rx  Qx ry )k ,

MGel

EI

w 2v w 2u i  EI 2 j  M zk . 2 wz wz

(5.75)

Chapter 5. Self-excitation of drill string bit whirling

263

Substituting (5.75) into (5.74) and projecting the resulting expression on axes Ox , Oy of the rotating coordinate system, we obtain

EI

д 3v д 2v  EI 3 rz  Try 2 дz дz

0, (5.76)

2

3

дu дu  EI 3 rz  Trx 0 . 2 дz дz Therefore, relations (5.1), (5.38), (5.39), (5.73), (5.76) represent the complete system of constitutive and boundary equations for the problem. Its solution is performed by replacing the derivatives with respect to z by their finite-difference analogues. An implicit finite-difference integration scheme is used to discretise them in respect of independent variable t . The choice of integration steps 'z , 't values is carried out stemming from the conditions of convergence of the computational process.  EI

5.10.

The analogy between the rolling dynamics of the ellipsoid bit and rattleback rotation

As established by experimental and theoretical studies, the free rotation of elongated ellipsoidal bodies with small geometric or mass defects (rattlebacks) tends to change the direction of their rotation. Similar effects are typical for drill bits. They can also make forward or backward whirling vibrations and change their direction, tracing the trajectory in the shape of multi-petal flowers that cut multichannel troughs in wells [22]. However, unlike rattlebacks, bits can have different geometry (including elongated and flattened ellipsoids), they are not free and are attached to an elastic string, they rotate at predetermined speed Z , and change the direction of their axes in accordance with the elastic bending of the axis of the drill string and its inclinations during vibration. Since the aim of this work is to understand the interaction between these geometric, kinematic, and structural factors, it is believed that a three-dimensional model that considers the relationship between the bending of the drill string and the vibrations of the elliptic bit is adequate. To identify the main reasons that affect the regime of the bit motion, we consider the simplest schemes of nonholonomic rolling on the plane of an ellipsoidal body of revolution attached to an elastic rod that rotates at angular velocity ω 0 . For clarity, we select the state in which plane CDG of the bit inclination coincides with plane XOZ (Fig. 5-32). Then, if the ellipsoid is extended, and angles u c(C ) , vc(C ) of inclination of its axis to vertical OZ are positive, then at the considered time velocities of displacement of contact point G( v G ) of the bit with plane S and vertex

Modelling emergency situations in drilling deep boreholes

264

D( v D ) of the ellipsoid are parallel to axis OY , and the bit moves around the string in the direction of its rotation (Fig. 5-32, a). Therefore, this mode corresponds to the effect of forward whirling. However, the situation changes if displacements u (C ) , v(C ) are positive, but angles u c(C ) , vc(C ) are negative (Fig. 5-32, b). In this case,

velocities v D , v G change their directions to the opposite, and the bit rolls around the string in the direction opposite to its rotation, performing the backward whirling mode. More difficult is the kinematics of motion of the flattened bit if its top D and contact point G are located in different sides of the string axis. In this case, depending on the sign of angles u c(C ) , vc(C ) , top D can move in the direction of rotation, while contact point G can move in the opposite direction (Fig. 5-32, c) or vice versa, point D can move in the direction of rotation, and point G , in the opposite direction (Fig. 5-32, d for backward whirling). B

B

x

S

C

xx

G

x

D ω

Z

ω

Dx C x G ω

S

GD X vG vD

ω

a)

x

Dx

C

G

DxC

S ω

x

v D vG X O D G

ω

Y b)

Z

G O D

vG c)

x

G

ω

Z

Y

Y

x

S

Z

O

B

B

X

ω

vD

vG

X

D O G

vD

Y d)

Fig. 5-32. Kinematics of forward and backward nonholonomic rolling of bits of ellipsoidal shape

Since in real conditions the bit with the shape of an ellipsoid is impacted by the forces and moments from the elastic oscillating string, it can constantly move from

Chapter 5. Self-excitation of drill string bit whirling

265

one kinematic scheme shown in Fig. 5-32 to another, thus changing the direction of circular motion as happens with rattlebacks. In complex cases, when these changes of direction occur many times, the trajectory of the bit points can trace complex shapes, including shapes that resemble multi-petal flowers (Fig. 5-5 and Fig. 5-22, c). The proposed model makes it possible to explain one characteristic feature of the whirling process—that is, it can take the most destructive modes in which the angular velocity of backward whirling is significantly higher than rotation speed Z of the drill string and can reach up to 5–30-multiplicity of its value [22]. To confirm this possibility, see Fig. 5-32, b. At the considered time, the bit is rolled with angular velocity ωW relative to the horizontal axis passing through the instantaneous centre of velocities G . Let C be the curvature centre of the surface section of the bit with the plane that contains point G and is normal to vector ωW , and r is the radius-vector of the normal built at point G . Then, its velocity v c is perpendicular to plane XOY and equals the value v c ωW u r ZW ˜ r ˜ j . As follows from work [9], speed v G of the instantaneous centre of the bit velocities is zero, but its trace on plane S moves at the speed vSG

vс .

Then, vGS

and the angular velocity of whirling Z

wh

Z wh

ZW ˜ r

of the bit is calculated by the formula

vcS d

ZW ˜ r d

,

(5.77)

where d is the distance between point G and axis OZ . It is clear that Z wh increases with a decrease in d , but not indefinitely as ωW also decreases. It is also important to note that the whirling speed in equation (5.77) depends on radius r , which is small for elongated bits and large for flattened ones. In this regard, it can be expected that flattened bits are more prone to rapid whirling compared to longitudinal ones. The kinematics of the complex motion of bit centre C becomes more obvious if studied in rotating coordinate system Oxyz (Fig. 5-33). In this case, relative velocity vector v r has Cartesian components xi ui , yj vj , so it can be expressed r ) and radial ( v rrad ) components in the corresponding in terms of circumferential ( v cir

polar coordinate system. Then, absolute speed ( v Cabs ) of point C can be represented by the expression (5.78) v Cabs v e  v r ,

Modelling emergency situations in drilling deep boreholes

266

where v e is the vector of the bulk velocity calculated by the formula ve e

r v cir

Therefore, if vectors v and

ω u ( xi  yj) .

are oriented in the same direction, the bit is

r have different rolled, overtaking the rotation of the drill string. If vectors v e and v cir directions, the bit rolling speed lags behind the rotation speed of the drill string at

r . It performs a pure rotation without rolling when v e v e ! v cir

r and rolls in v cir

r the direction opposite to the rotation of the drill string when v e  v cir .

y j Z

O xi

v rrad  radial C

x

Z

vr

O

x C

y j

vr

r v cir

v rrad

xi

r v cir  circular

y

y

b)

a)

Fig. 5-33. Top view of the kinematic schemes of the orientation of relative velocity vector v

r

Since in practice the ellipsoidal bit is exposed to elastic forces and moments from the vibrations of the drill string, it can constantly move from one kinematic scheme presented in Fig. 5-32 and Fig. 5-33 to another, meanwhile changing the direction of its rotational movement, as is the case with rattlebacks. In complex cases, when these changes of directions are repeated many times, due to the elastic vibrations of the drill string, the trajectory of the bit centre can acquire more complicated shapes. 5.11.

Dynamics of elastic bending of a drill string with the bit in the shape of an elongated ellipsoid

It is well known that the main cause of the effect of whirling vibrations is geometric imperfections of the bit imbalance, which lead to its leaving the axis of the drill string. In a balanced bit, the forces acting on it can be decomposed into an axial component, a cutting torque, and a small radial force called the bit imbalance force. It is usually expressed as a percentage of the vertical force acting on the bit. In normal

Chapter 5. Self-excitation of drill string bit whirling

267

Coefficient of friction

conditions, the bit is balanced with high accuracy with an imbalance of 2%, although 10% imbalances are also typical. To install a bit and drill string coaxially, centring devices (or centralisers) are installed in the lower part of the drilling rig. The bending stiffness of the drill string, which is achieved by using centralisers, is usually sufficient to return the drill string from the deflected state to its centred working position. However, if the stabilisers have even small gaps with the wall of the well, they can deviate by their value from the fixed position and stop supporting the bit in the desired position. As a result, the drill bit begins whirling, thereby increasing the gap and adding additional energy to the dynamic process. This trend allows us to simulate the self-excitation of whirling vibrations process, which can also occur at significant sizes of the well diameter in its lower part. Trajectories of bit point C in the rotating coordinate system

μ = 0.2

0.04

u (t ), m

0.02

0.02

0

0

-0.02

-0.02

-0.04

μ=1

0.04

-0.02

0

0.02

0.04

v(t ), m

u (t ), m

-0.04 -0.04

0.04

0.02

0.02

0

0

-0.02

-0.02

-0.04 -0.04

0.04

-0.02

0

0.02

0.04

v(t ), m

u (t ), m

-0.04

0.02

0.02

0

0

-0.02

-0.02

-0.04

-0.02

0

0.02

0.04

v(t ), m

-0.02

0

0.02

0

0.02

0

0.02

X (t ), m

-0.04

0.04

-0.04

Zb

X (t ), m

0.04

-0.04

μ = 30

Trajectories of bit point C in the fixed coordinate system

-0.02

X (t ), m

0.04

Zb

-0.02

Y (t ), м 0.94 rad / s

Y (t ), m Zb 1.41rad / s 0.04

-0.04 -0.04

3.74 rad / s

0.04

Y (t ), m

Fig. 5-34. The whirling trajectories of bit centre C in the moving (left) and fixed (right) coordinate systems (case T = -1·104 N, Mz = -1·104 N·m, ω = 5 rad/s, a = 0.1 m, b = 0.3 m, t = 20 s)

Modelling emergency situations in drilling deep boreholes

268

When modelling, we have assumed that there are some initial bending deviations in the geometry of the drill string, and further modelling of the system was carried out numerically. The study was conducted in the range of angular velocity 0 d Z d 20 rad/s. The following values of mechanical parameters of the system were selected: EI

4.07 ˜ 106 Pa∙m4,

Ut

7.8 ˜103 kg/m3, U f

S (r12  r22 ) 5.34 ˜103 m2,

Ft

2.01˜102 m2,

S ˜ r22

Ff

1.5 ˜103 kg/m3, L 9 m, e 1 m, r1

0.09 m, r2

0.08 m.

Coefficient of friction

As indicated in work [12], the geometry of the bit has a significant influence on the whirling vibrations. Therefore, in this paragraph, we consider the cases where the bit has the shapes of elongated ellipsoids of revolution, the semi-axes of which are a 0.1 m, b 0.3 m and a 0.1 m, b 0.5 m. In the study of the bit whirling vibrations using a friction model, the selection of friction coefficient P plays an important role. It is known that its value depends on the tribological properties of the rubbing bodies and the quality of the material processing. The graph of the friction force Fx fr(t) change

μ = 0.2

Fхfr , N

The graph of the friction force Fy fr(t) change Fyfr , N

2000

1000

1000

0

0

-1000

-1000

t, s

-2000 0

fr

Fx , N

4

8

12

16

Fyfr , N

μ=1

0

4

8

12

16

20

0

4

8

12

16

20

0

4

8

12

16

20

10000

5000

0

0

-5000

-5000

t, s

-10000 0

μ = 30

t, s

-2000

20

10000

5000

Fxfr , N

2000

4

8

12

16

Fyfr , N

150000

t, s

-10000

20

300000

300000

150000

0

0

-150000

-150000

t, s

-300000 0

4

8

12

16

20

t, s

-300000

Fig. 5-35. Graphs of the friction force Fxfr (t) and Fyfr (t) functions in the rotating coordinate system (case T = -1·104 N, M z = -1·104 N·m, ω = 5 rad/s, a = 0.1 m, b = 0.3 m, t = 20 s)

It must be emphasised that the coefficient of friction between the surfaces of the bit and the rock can have large values since the diamond cutters existing on the surface of the bit significantly increase the adhesion between the contacting bodies.

Chapter 5. Self-excitation of drill string bit whirling

269

Therefore, the simulation of whirling vibrations using the friction model was conducted at the P values in the range 0.2, 1 .0, 10 , 20, and 30 . In doing so, equations (5.1), (5.73), and (5.76) were used for that. Their integration was carried out by the method described above. Fig. 5-34 demonstrates trajectories of bit point C motion on the well bottom in the rotating coordinate system (left) and in the fixed coordinate system (right). The

Coefficient of friction

case of T 1˜ 104 N, M z 1˜ 10 4 N∙m, Z 5 rad/s, a 0.1 m, b 0.3 m, t 20 s is considered. It is important to point that in both coordinate systems the motion proceeds in the same direction, opposite to the Z direction. With values of friction coefficient P 0.2 , 0.5, 1, respectively, the oscillations become chaotic in nature, but their amplitudes decrease. Such drilling regimes can be considered as stable. At the same time, with an increase in P value ( P t 10 ), the trajectory of the Trajectories of bit point C in the rotating coordinate system

μ = 0.2

0.2

u (t ), m

0.2

0.1

0

-0.1

-0.1

v(t ), m -0.2

μ=1

0.2

-0.1

0

0.1

-0.2

u (t ), m

0.2

0.1

0

0

-0.1

-0.1

-0.2 -0.2

-0.1

0

0.1

0.2

v(t ), m

u (t ), m

0.1

0

0

-0.1

-0.1

-0.1

0

0.1

0.2

v(t ), m

0

0.1

X (t ), m

-0.2

0.1

-0.2

-0.1

0.2

Zb

-0.2

0.2

-0.2

Y (t ), m

-0.2

0.2

0.1

0.2

Zb 1.26 rad / s

X (t ), m

0.1

0

-0.2

μ = 30

Trajectories of bit point C in the fixed coordinate system

-0.1

0

0.1

X (t ), m

0.2

-0.2

-0.1

0

0.1

Y (t ), м

Zb

-0.2 0.2

3.4 rad / s

3.4 rad / s

Y (t ), m

Fig. 5-36. The whirling trajectories of bit centre C in the rotating (left) and fixed (right) coordinate systems (case T = -1·105 N, Mz = -1·104 N·m, ω = 5 rad/s, a = 0.1 m, b = 0.3 m, t = 20 s)

Modelling emergency situations in drilling deep boreholes

270

bit in the rotating system after certain oscillations takes the shape of an elongated ellipse, and in the fixed one the shape becomes similar to a multi-petal flower. In the fixed coordinate system, the bit moves counter clockwise, so in this case there is backward whirling. Average angular velocity Zb of the bit in system OXYZ is less than the rotation speed of the drill string. Fig. 5-35 shows the graphs of the friction forces Fxfr (t ) and Fyfr (t ) changes

Coefficient of friction

(formulas (5.72)) for the regimes of motion represented in Fig. 5-34. They have the mode of decaying oscillations. Trajectories of bit point C in the rotating coordinate system

μ = 0.2

u (t ), m

Trajectories of bit point C in the fixed coordinate system X (t ), m

0.2

0.1

0.1

0

0

-0.1

-0.1

-0.2 -0.2

u (t ), m

-0.1

0

0.1

0.2

v(t ), m

-0.2 -0.2

X (t ), m

0.2

μ=1

0

0

-0.1

-0.1

-0.2 -0.2

μ = 30

-0.1

0

0.1

0.2

Y (t ), m

0.2

0.1

0.1

u (t ), m

0.2

-0.1

0

0.1

0.2

v(t ), m

-0.2

X (t ), m

0.2

0.1

-0.2

-0.1

0

0.1

0.2

-0.2

-0.1

0

0.1

0.2

Y (t ), m

0.2

0.1

0

0

-0.1

-0.1

-0.2 -0.2

-0.1

0

0.1

0.2

v(t ), m

-0.2

Y (t ), m

Fig. 5-37. The whirling trajectories of bit centre C in the moving (left) and fixed (right) coordinate systems (case T = -1·105 N, Ms = -1·104 N·m, ω = 10 rad/s, a = 0.1 m, b = 0.5 m, t = 20 s)

In the fixed coordinate system, the character of the change of friction forces

FXfr (t )

and FYfr (t ) has a sinusoidal shape. In this case, the drill string pipe is bent in

simple shapes. If the longitudinal force is increased to T 1˜ 105 N (Fig. 5-36), the character of the motion of point C changes significantly. In the fixed coordinate system, the trajectory of motion after some unwinding goes into a circle of constant radius, while in the moving coordinate system the bit, having made few circles, moves to the side. The angular velocity of the bit changed from Zb 1.26 rad/s to

Zb

3.4 rad/s. The graphs of the frictional forces functions Fxfr (t ) and Fyfr (t )

changes have the modes of harmonic oscillations with high frequencies and amplitudes.

Chapter 5. Self-excitation of drill string bit whirling

271

If the drill bit is made even more oblong ( b

0.5 m), its motion becomes less

stable. Fig. 5-37 represents the case where T 1˜ 105 N, M z 1˜ 10 4 NM, Z 10 rad/s, a = 0.1 m, b = 0.5 m, t 20 s. In the rotating coordinate system, the trajectory of point C is similar to that shown in Fig. 5-36. In the moving system, the trajectories of bit motion are realised in the shape of an expanding spiral, which is why this motion is unstable. It represents backward whirling, since the bit and the drill string rotate in opposite directions. The results of studies at angular velocities of Z 5 rad/s and Z 20 rad/s are similar to the data presented in Fig. 5-37. 5.12.

Dynamics of elastic bending of a drill string with the bit in the shape of a flattened ellipsoid

Coefficient of friction

When drilling oil and gas wells, drill bits similar in shape to the flattened ellipsoid (rattlebacks) are widely used. The calculations of such a system were Trajectories of bit point C in the rotating coordinate system

μ = 0.2

u (t ), m

Trajectories of bit point C in the fixed coordinate system

v(t ), m

0

X (t ), m

u (t ), m

μ=1

7.54 rad / s

0

0

0

0

Y (t ), m

Zb

5.97 rad / s

0

0

v(t ), m

0

u (t ), m

μ = 30

Zb

X (t ), m

X (t ), m

0

Y (t ), m

Zb 1.88 rad / s

0

0

v(t ), m

0

Y (t ), m

Fig. 5-38. The whirling trajectory of bit centre C in the moving (left) and fixed (right) coordinate systems (case T = -1·104 N, Mz = -1·104 N·m, ω = 5 rad/s, a = 0.3 m, b = 0.1 m, t = 20 s)

Modelling emergency situations in drilling deep boreholes

performed with the time interval 6

4

272

0 d t d 20 s with the following values: 0.08 m, l 8 m, e 1 m, Ut

EI

4.07 ˜10 Pa∙m , r1

0.09 m, r2

Uf

1.5 ˜103 kg/m3, Ft

S (r12  r22 ) 5.34 ˜103 m2, Ff

S ˜ r22

7.8 ˜103 kg/m3,

2,01˜102 m2. In

Coefficient of friction

the simulation of whirling vibrations with the use of a friction model, we considered bits with semi-axes a 0.3 m, b 0.1 m and a 0.5 m, b 0.1 m. The trajectories of the flattened bits are very sensitive to changes in the friction coefficients. Therefore, if the longitudinal force is small T 1˜ 10 4 N (Fig. 5-38)

Trajectories of bit point C in the rotating coordinate system

Trajectories of bit point C in the fixed coordinate system

μ = 0.2

u (t ), m

0

v(t ), m

0

X (t ), m

u (t ), m

μ=1

5.34 rad / s

0

0

0

Y (t ), m

Zb

5.02 rad / s

0

0

v(t ), m

X (t ), m

0

Y (t ), m

0

u (t ), m

μ = 30

Zb

X (t ), m

Zb

0.94 rad / s

0

0

v(t ), m

0

Y (t ), m

Fig. 5-39. The whirling trajectories of bit centre C in the moving (left) and fixed (right) coordinate systems (case T = -1·105 N, Mz = -1·104 N·m, ω = 5 rad/s, a = 0.3 m, b = 0.1 m, t = 20 s)

and coefficient P d 1, then in the moving system the trajectory of point C after some unregulated oscillations passes into a straight line. In the fixed system, the drill bit traces the motion of an expanding spiral. With the further increase in the coefficient

Chapter 5. Self-excitation of drill string bit whirling

273

Coefficient of friction

of friction to P 10 , the trajectory of motion in a fixed system passes into a circle with a constant radius, and at P 30 , to a shape similar to a multi-petal flower. However, in the moving system, the trajectory represents a flattened ellipsoid. And it does not mean that this mode is favourable, because when the bit moves with small loops, it experiences large accelerations generated by intense dynamic loads. Trajectories of bit point C in the rotating coordinate system

Trajectories of bit point C in the fixed coordinate system

μ = 0.2

u (t ), m

0

v(t ), m

0

X (t ), m

u (t ), m

μ=1

7.85 rad / s

0

0

0

Y (t ), m

Zb 10.05 rad / s

0

0

v(t ), m

0

u (t ), m

μ = 30

Zb

X (t ), m

X (t ), m

Y (t ), m

Zb 10.05 rad / s

0

0

0

v(t ), m

0

Y (t ), m

Fig. 5-40. The whirling trajectories of bit centre C in the moving (left) and fixed (right) coordinate systems (case T = -1·105 N, Mz = -1·104 N·m, ω = 10 rad/s, a = 0.5 m, b = 0.1 m, t = 20 s)

Fig. 5-39 represents the case of T 1˜ 105 N, M z 1˜ 10 4 N∙m, Z 5 rad/s, a 0.3 m, b 0.1 m, t 20 s. If friction coefficient P d 1, the trajectory of the bit at the final stage in both systems is a circle of constant radius, while in the moving system the circle has a displaced centre of much smaller radius than in the fixed system. When the friction coefficient is increased to P 10 and P 30 , the shapes of motion in the rotating coordinate system change to a flattened ellipse, and in the fixed system, into a multi-petal flower. In the fixed coordinate system, the bit moves

Modelling emergency situations in drilling deep boreholes

274

counter clockwise, and the drill string moves clockwise, so in this case backward whirling takes place. It is advisable to analyse here average angular velocity ωb of the bit, which, when friction coefficient P increases, decreases from Zb 5.34 rad/s to

Zb

0.94 rad/s. If semi-axis b is increased to 0.5 m, the bit will become more flattened. This case is shown in Fig. 5-40. The trajectories of bit point C in the upper positions are similar to the case considered in Fig. 5-39. With P 30 , the trajectory of the bit in the moving system becomes a flattened ellipse, and in a fixed system the bit moves with large loops. Backward whirling occurs here, as the bit and drill string rotate in opposite directions. As coefficient P increases, average angular velocity Zb of the drill bit increases from Zb 7.85 rad/s to Zb 10.05 rad/s. 5.13.

Elastic oscillations of the drill string for nonholonomic rolling of an ellipsoid bit on the surface of the well bottom

5.13.1. Problem statement To study the phenomenon of rolling without sliding the bit on the surface of the well bottom, it is necessary to state the problem of elastic bending vibrations of the drill string, taking into account the limitations imposed by the boundary conditions of the dynamic equations of the drill string.

Fig. 5-41. The scheme of sections selected for analysis

Chapter 5. Self-excitation of drill string bit whirling

275

As above, we assume that it rotates at a constant speed. In its upper part, the drill string is stretched by gravity, in the lower segment it is compressed by the force of contact between the string and the well. To increase the stiffness of the drill string, centring devices are usually installed at the bottom of the string and serve as additional supports. As the most intense bending vibrations of the string prevail in the sections adjacent to the drill bit, the impact of the upper sections on the dynamics is not taken into account, and its two bottom sections are considered (segments AB and BC in Fig. 5-41), which are separated for modelling. When analysing the bending dynamics of a drill string, we introduce inertial coordinate system OXYZ with its origin at the end A and axis OZ coinciding with the string axis. We will also use system Oxyz that rotates together with the drill string pipe. Let i , j , k be the unit vectors of this system. We will take into account that the bending-stress state of the drill string is determined by displacements u (z ) , v(z ) in planes xOz , yOz of system Oxyz . In the construction of the differential equations of the drill string motion, we take into account that it is prestressed by internal axial force T and torque M z . Let the string rotate at angular velocity Z and the mud moves inside it with the speed of V . Its transverse oscillations are determined by equations (5.1). A characteristic feature of the problem in question is the method of recording its boundary conditions. They are formulated easily at points A and B . At point A located between the conditionally removed upper part and one of the centralisers, the equations have the form (5.38). At point B , the equations are written as (5.39). They follow from the condition of continuity of elastic displacements u and v . In this case, the boundary equations at point C are much more complicated, so they must be particularly considered. The proposed model considers the mechanism of pure rolling with spinning, and the motion of the instantaneous centre of bit speeds around the axis of the drill string is studied. The considered model makes it possible to simulate transitions from forward to backward whirling and to explain the significant excess in the whirling speed in comparison with the rotation speed of the drill string. After taking these initial assumptions, it becomes possible to formulate the equations of motion of the body under the action of the elastic force and the moments caused by the contact interaction between the bit body and the well bottom at the end of the drill string. Let a bit in the shape of an ellipsoid of revolution be attached to the lower end of the drill string (Fig. 5-41). We consider that bit centre C coincides with the lower end of the drill string. To describe the bit rolling on plane S , we rigidly connect it

Modelling emergency situations in drilling deep boreholes

276

with coordinate system Cx1 y1 z1 , the axes of which in the undeformed state of the system are parallel to the corresponding axes of system Oxyz (Fig. 5-42). We consider that with elastic bending of the drill string, rotation angle T of system Cx1 y1 z1 in respect of system Oxyz is small (Fig. 5-43), so we can introduce a rotation angle vector (5.79) θ vci  ucj T x i  T y j . In the deformed state, Cz1 axis slews by angle T in plane V passing through top D of bit, its centre C, and point G of the bit contact with plane S . To determine the coordinates of point G in Oxyz system, we introduce right coordinate system Cx2 y2 z2 , axis Cz 2 of which coincides with axis Cz1 , and axis Cx 2 lies in plane V (Fig. 5-43).

O

X C

a

x

x1

y

x2

Zt

y1

C

b

S

ρ xx

D

G

Y Fig. 5-42. Top view of the coordinate system

T

z1 , z2

Fig. 5-43. Position of contact point G in plane S

Then, according to formula

T r (uc) 2  (vc) 2 we can find angle T and then calculate angle α between planes V and xOz D arctg T x / T y . With the help of the introduced angles, we determine vector ρ connecting points C and G (Fig. 5-43) (b 2  a 2 ) sin T y cosT y (b 2  a 2 ) sin T x cosT x ρ i j a 2 sin 2 T y  b 2 cos2 T y a 2 sin 2 T x  b 2 cos2 T x (5.80)  a 2 sin 2 T  b 2 cos2 T k.

Chapter 5. Self-excitation of drill string bit whirling

277

The absolute velocity of bit point G is zero at the moment of contact with plane S , so (5.81) v Gabs v Cabs  Ω u ρ 0 , where v Cabs is the absolute velocity of drill bit point C ; Ω is the angular velocity vector of the system of axes Cx1 y1 z1 fixed in the body of the bit. Considering that the bit moves as a result of drill string rotation and elastic displacements and turns of its lower end, we can write v Cabs ui  vj  ω u ui  vj u  Zv i  v  Zu j , Ω vc i  ucj  Z k , (5.82) Ωuρ

i  vc

j u c

Ux

Uy

k

Z Uz

ucU z  ZU y i  vcU z  ZU x j   vcU y  ucU x k .

Substituting these equalities in equation (5.81) and projecting it on axes Ox , Oy , we obtain the kinematic (nonholonomic) boundary conditions for bending the drill string at point C u  Z v  ucU z  ZU y 0 , (5.83)

v  Z u  vcU z  ZU x 0 . After substituting expressions for U x , U y , U z , we obtain u  Z v  u c a 2 sin 2 T  b 2 cos2 T  Z

(b 2  a 2 ) sin T x cosT x a 2 sin 2 T x  b 2 cos2 T x

0,

(5.84)

v  Z u  vc a 2 sin 2 T  b 2 cos2 T  Z

(b 2  a 2 ) sin T y cosT y a 2 sin 2 T y  b 2 cos2 T y

0.

The built equations are nonholonomic because they are expressed through derivatives of unknown variables with respect to time. In general, the dynamic boundary equations can be obtained from the theorem of the change of the bit motion momentum. These equations are formulated as follows: ~ dKG  Ω u K G M Gel , (5.85) dt where K G is the moment vector of momentum relative to point G ; M Gel is the moment of elastic forces relative to the same point.

Modelling emergency situations in drilling deep boreholes

278

However, since the mass and moment of inertia of the drill bit are small compared to the inertial characteristics of the drill string, they can be ignored. Then equation (5.85) is simplified (5.86) M Gel 0 . Moment M Gel consists of moment M GM of elastic moments and moment M GF of elastic forces. Let's present them in the forms: (5.87) M GM M x i  M y j  M z k , M GF

 ρ u F

i

j

k

 Ux

Uy

Uz

Qx

Qy

T

 TU y  Qy U z i 

  TU x  Qx U z j  Qy U x  Qx U y k.

Elastic bending moments M x , M y and transverse forces Qx , Q y are calculated through the formulas д 2v д 2u M y EI 2 , , 2 дz дz (5.88) 3 3 дu дv Qx EI 3 , Q y EI 3 . дz дz After the corresponding transformations, vector equation (5.86) is rewritten in the form of two scalar equations д 2v д 3v EI 2  EI 3 U z  TU y 0, дz дz (5.89) д 2u д3u  EI 2  EI 3 U z  TU x 0. дz дz Taking into account equation (5.80), we convert equation (5.89) to the form: 2 2 T (b  a ) sin T y cosT y д 2u д3u 2 2 2 2  a sin T  b cos T  0, EI a 2 sin 2 T y  b 2 cos2 T y дz 2 дz 3 Mx

EI

(5.90)

дv дv 2 2 T (b  a ) sin T x cosT y  3 a sin T  b 2 cos2 T  2 EI a 2 sin 2 T x  b 2 cos2 T x дz дz 2

3

2

2

0.

We should emphasise that the unknown force of adhesion between the bit and the well bottom is not included in these relations. This effect became possible by

Chapter 5. Self-excitation of drill string bit whirling

279

using the elements of nonholonomic mechanics in this problem and the selection of the instantaneous centre of velocities (point G) as a polar point. Boundary conditions (5.83), (5.90) are kinematic and quasi-static constraints imposed on the bit movement. 5.13.2. Dynamics of the drill string in nonholonomic rolling of the bit in the shape of a flattened ellipsoid In accordance with the developed method, numerical investigation of the bit whirling vibrations was performed for the following parameters used in (5.1), (5.80), and (5.90): e 1 m,

Ft

E

S

2.1˜1011 Pa,

r12

 r22



Ut

5.34 ˜10

3

Uf

7.8 ˜103 kg/m3, 2

m,

Ff

S

˜ r22

1.5 ˜103 kg/m3, l

2.01˜10

2

2

m,

r1

8 m,

0.09 m,

r2 0.08 m. Here, r1 , r2 are the outer and inner radii of the drill string pipe. The values of parameters T , M z , and Z varied. As accented above, the modes of whirling vibrations significantly depend on the bending stiffness of the drill string. With its reduction, the drill string becomes flexible, and the axis of the bit acquires an additional opportunity to deviate from the vertical position and begin a circular movement. If we consider that the bending stiffness of the drill string tends to zero when approaching its critical (according to Euler) state, the expediency of estimating the proximity of the stress strain state to the critical one becomes clear. However, the problem of stability analysis of the drill string represents a complex interrelated multi-parametric problem, since the drill string is prestressed by torque and longitudinal axial force. In addition, it rotates and simultaneously carries out longitudinal, transverse, and torsional vibrations, and flows of mud move inside it. Since this problem does not have a general solution, it is interesting to analyse some simple variants of these loads that make it possible to estimate the rigidity of the drill string. There are two combinations of rod prestressing, under the action of which the critical states of the drill string can be determined analytically. For example, we consider the case where a pivotally supported rotating tubular rod is compressed by axial force T and the mud moves inside it at speed V . Then, the critical values of parameters T , V , and Z are connected by the analytical dependence [6]

S4

2

EI  S 2Tcr  l 2J tZcr2  S 2J f Vcr 0. (5.91) l2 An analytical solution to the problem of Euler stability can also be obtained for the case where a pivotally supported rod is prestressed by torque M z and constant axial force T

Modelling emergency situations in drilling deep boreholes

280

M z2,cr . (5.92) 4 EI l2 Therefore, equation (5.91) can be used for evaluation of the proximity of the lower segments of the drill string to their critical states when the torque value is small, and equation (5.92) can be applied if the rotation of the drill string and the fluid flow can be ignored. Tcr

u (t )



S 2 EI

X (t )

2vs

2u s

0



Zb 1.6 rad / s

0

v(t )

Y (t )

0

0

a)

b)

Fig. 5-44. Trajectories of bit centre motion in rotating coordinate system Oxyz a) and in the fixed coordinate system b) (case T = -1·104 N, Mz = -1·104 N·m, a = 0.3 m, b = 0.1 m, ω = 5 rad/s)

u (t )

Zb

X (t )

2vs

2u s

0

5.02 rad / s

0

Y (t )

v(t ) 0

0

a)

b)

Fig. 5-45. Trajectories of bit centre motion in rotating coordinate system Oxyz a) and in the fixed coordinate system b) (case T = -1·104 N, Mz = -1·104 N·m, a = 0.3 m, b = 0.1 m, ω = 20 rad/s)

Fig. 5-44 presents the case of T 1˜ 10 4 N, M z 1˜ 10 4 N∙m, a 0.3 m, b 0.1 m, Z 5 rad/s. We can see that the motion of the bit in the moving and fixed coordinate systems is performed in opposite directions in this case. In the rotating coordinate system, the trajectory of the motion is an ellipse with semi-axes a 2u s and b 2vs (Fig. 5-44, a), in the fixed coordinate system it becomes similar to a four-

Chapter 5. Self-excitation of drill string bit whirling

281

petal flower (Fig. 5-44, b). Average angular velocity ωb of the bit in fixed coordinate system OXYZ is 1.6 rad/s. 12

FYfr , MN

FXfr , MN

12 6

6 0

0

-6

-6

-12

t, s 0

4

8

12

16

-12

20

t, s 0

4

8

12

16

20

Fig. 5-46. Graphs of the friction force FXfr (t ) and FYfr (t ) functions in the fixed coordinate system (case T = -1·104 N, Mz = -1·104 N·m, a = 0.3 m, b = 0.1 m, ω = 5 rad/s)

However, if, with same values T , M z , the angular velocity is increased to Z 20 rad/s, no change in trajectory occurs in the rotating coordinate system, while in the fixed coordinate system the trajectory takes the shape of a multi-petal flower in one revolution. With further motion, these figures are rotationally superimposed on each other, and the overall trajectory becomes complicated. In this case, the motion occurs in the ring at average speed Zb 5.02 rad/s (Fig. 5-45). u (t )

X (t )

0

Zb

0.79 rad / s

0

v(t )

Y (t )

0

0

a)

b)

Fig. 5-47. Trajectories of bit centre motion in rotating coordinate system Oxyz a) and in the fixed coordinate system b) (case T = -1·105 N, Mz = -1·104 N·m, a = 0.3 m, b = 0.1 m, ω = 5 rad/s)

Fig. 5-46 shows the graphs of the friction forces functions Fxfr (t ) and Fyfr (t ) in the rotating coordinate system. We can see that after some initial oscillations they pass into a sinusoidal mode with a much smaller amplitude.

Modelling emergency situations in drilling deep boreholes

282

The results of the research presented in Fig. 5-47 are obtained for the values of longitudinal force T 1˜ 105 N, torque M z 1˜ 10 4 N∙m, and angular velocity Z 5 rad/s. In this case, the trajectory of the bit centre in the rotating coordinate system is an ellipse, and in the fixed coordinate system it is a multi-petal flower, and with an increase in the angular velocity the number of turns became larger. Increasing speed Z leads to an increase in speed Zb . For more clarity, the figure on the right shows the shape of the trajectory per revolution. With a further increase in the angular velocity to Z 20 rad/s (Fig. 5-48), the motion of the system becomes very complex. Therefore, it is incorrect to talk about angular velocity Zb , and it was not calculated. X (t )

u (t )

0

0

v(t )

Y (t )

0

0

a)

b)

Fig. 5-48. Trajectories of bit centre motion in rotating coordinate system Oxyz a) and in the fixed coordinate system b) (case T = -1·105 N, Mz = -1·104 N·m, a = 0.3 m, b = 0.1 m, ω = 20 rad/s)

Cases were considered above where torque remained equal to M z 1˜104 N∙m, and only the values of longitudinal force T and angular velocity Z varied. Let's analyse the shape of the motion of the bit centre where the torque is increased to M z 1˜105 N∙m. At longitudinal force T 1˜104 N and angular velocity Z 10 rad/s (Fig. 5-49), the trajectory of the bit is close to the ellipse in the rotating coordinate system, and in the fixed coordinate system it resembles a multipetal flower with elongated loops. For angular velocity Z 20 rad/s (Fig. 5-50), the trajectory of motion in coordinate system Oxyz (left position) is an expanding flattened ellipse, and in coordinate system OXYZ (position right) it is almost a starshaped curve of complicated configuration. Since the tops of the trajectory represent the instants of the bit hitting against the well wall, this mode is unfavourable for the bit and leads to a less smooth surface of the wall. The drill bit and the string in the fixed coordinate system move in a clockwise direction, so this motion is forward whirling.

Chapter 5. Self-excitation of drill string bit whirling

u (t )

283

X (t )

0

0

Y (t )

v(t ) 0

0

a)

b)

Fig. 5-49. Trajectories of bit centre motion in rotating coordinate system Oxyz a) and in the fixed coordinate system b) (case T = −1·104 N, Mz = −1·105 N·m, а = 0.3 m, b = 0.1 m, ω = 10 rad/s)

X (t )

u (t )

0

0

Y (t )

v(t ) 0

0

a)

b)

Fig. 5-50. Trajectories of bit centre motion in rotating coordinate system Oxyz a) and in the fixed coordinate system b) (case T = -1·105 N, Mz = -1·104 N·m, a = 0.3 m, b = 0.1 m, ω = 20 rad/s)

10

Fxfr , MN

10

5

5

0

0 -5

-5 -10

Fyfr , MN

t, c 0

4

8

12

16

20

-10

t, c 0

4

8

12

16

20

Fig. 5-51. Graphs of the friction force Fxfr (t) and Fyfr (t) functions in the rotating coordinate system (case T = -1·104 N, Mz = -1·105 N·m, a = 0.3 m, b = 0.1 m, ω = 5 rad/s)

Modelling emergency situations in drilling deep boreholes

u (t )

Zb 1.89 rad / s

X (t )

0

284

0

Y (t )

v(t ) 0

0

b) a) Fig. 5-52. Trajectories of bit centre motion in rotating coordinate system Oxyz a) and in the fixed coordinate system b) (case T = -1·105 N, Mz = -1·104 N·m, a = 0.5 m, b = 0.1 m, ω = 5 rad/s)

X (t )

u (t )

0

Zb

21.76 rad / s

0

Y (t )

v(t ) 0

0

b) a) Fig. 5-53. Trajectories of bit centre motion in rotating coordinate system Oxyz a) and in the fixed coordinate system b) (case T = -1·105 N, Mz = -1·104 N·m, a = 0.5 m, b = 0.1 m, ω = 20 rad/s)

Fig. 5-51 shows the graphs of the friction forces Fxfr (t ) and Fyfr (t ) changes in the rotating coordinate system for this case. It is seen that they have the modes of oscillations with increasing amplitude. Similar modes are observed in the fixed coordinate system. During the study, a bit with semi-axes a 0.5 m, b 0.1 m was also considered. Fig. 5-52 and Fig. 5-53 show the case of T 1˜ 105 N,

0.5 m, b 0.1 m. If the angular velocity is Z 5 rad/s, then in coordinate system Oxyz the bit moves along a closed ‘dumbbell-like’ trajectory (Fig. 5-52, position (a)), and at Z 20 rad/s it takes the shape of an oval (Fig. 5-53, a). In these cases, in fixed coordinate system OXYZ , point C moves in the annular zone (Fig. 5-52, Fig. 5-53, positions (b)). These movements are backward whirling, as the drill bit and the string rotate in opposite directions. In practice, this mode is Mz

1˜ 10 4 N∙m, a

Chapter 5. Self-excitation of drill string bit whirling

285

considered to be the most unfavourable as it is associated with significant dynamic loads on the bit and string. 5.13.3. Dynamics of the drill string in nonholonomic rolling of the bit in the shape of an elongated ellipsoid Via the use of the developed techniques, the study of bit whirling vibrations in the shape of an elongated ellipsoid is performed at geometric values of the system parameters: e 1 m,

Ft

E

2.1˜1011 Pa,

Ut

7.8 ˜103 kg/m3,

S r12  r22 5.34 ˜103 m2,

Ff

S ˜ r22

Uf

1.5 ˜103 kg/m3,

2.01˜102 m2,

r1

l

8 m,

0.09 m,

0.08 m. The angular velocity varied in the range 0 d Z d 20 rad/s. Simulation of the whirling vibrations using the nonholonomic (kinematic) model was carried out on the basis of equations (5.1), (5.38), (5.39), (5.86), and (5.90). As in other cases, their integration was carried out numerically using the finite-difference method in respect of spatial coordinate z and implicit finitedifference scheme in time t . Length l was divided into n finite-difference segments 'z l / n . At each step of discretised time t j and at each node point z i , the r2

derivatives in respect of z and t were replaced by their finite difference analogues. The modes of the ellipsoidal bit displacements depend to a great extent on longitudinal force T , torque M z , and the bit shape itself. Cases are considered where the bit had semi-axes a 0.1 m, b 0.3 m and a 0.1 m, b 0.5 m. Fig. 5-54 shows the calculation results for the case of T 1˜ 10 4 N, Mz

1˜ 10 4 N∙m, a 0.1 m, b 0.3 m, 0 d Z d 20 rad/s. In rotating coordinate

system Oxyz , the trajectory of the bit after several oscillations passes into an oblong ellipse. In fixed coordinate system OXYZ , the trajectory of motion is a multi-petal flower, which, with continued motion, turns by a certain angle, thus creating a complex path in the annular area. Moreover, with increasing angular velocity Z , the filling of the ring becomes denser. Here, it is advisable to analyse average angular velocity Zb of the bit that increased from Zb 1.26 rad/s (for the case Z 5 rad/s) to

Zb

5.34 rad/s (for the case Z 20 rad/s). For the considered mode, shown in Fig. 5-54, graphs of the friction forces

Fxfr (t ) and Fyfr (t ) functions in the rotating coordinate system are of interest. As seen in Fig. 5-55, they represent oscillating curves with small decreasing amplitudes. If the longitudinal force is increased to T 1˜105 N and the angular velocity changes within the limits 5 d Z d 20 rad/s, the trajectory of the bit in the moving coordinate system is a circle of a smaller radius compared to the circle in the fixed

Modelling emergency situations in drilling deep boreholes

u (t )

286

X (t )

0

Zb 1.26 rad / s

0

v(t )

Y (t )

0

0

a)

b)

Fig. 5-54. Trajectories of bit centre motion in rotating coordinate system Oxyz a) and in the fixed coordinate system b) (case T = -1·104 N, Ms = -1·104 N·m, a = 0.1 m, b = 0.3 m)

Fxfr , MN

2

2

1

1

0

0

-1 -2

Fyfr , MN

-1 0

4

8

12

16

20

t, s

-2

0

4

8

12

16

20

t, s

Fig. 5-55. Graphs of the friction force Fxfr (t) and Fyfr (t) functions in the rotating coordinate system (case T = -1·104 N, Ms = -1·104 N·m, a = 0.1 m, b = 0.3 m, Z = 5 rad/s) 0.2

u (t )

0.2

0.1

0.1

0

0

-0.1

-0.1

v(t )

-0.2 -0.2

-0.1

0

a)

0.1

0.2

X (t )

Zb

5.44 rad / s

Y (t )

-0.2 -0.2

-0.1

0

0.1

0.2

b)

Fig. 5-56. Trajectories of bit centre motion in rotating coordinate system Oxyz a) and in the fixed coordinate system b) (case T = -1·105 N, Ms = -1·104 N·m, a = 0.1 m, b = 0.3 m)

Chapter 5. Self-excitation of drill string bit whirling

u (t )

287

X (t )

0

0

v(t )

Y (t ) 0

0

b) a) Fig. 5-57. Trajectories of bit centre motion in rotating coordinate system Oxyz a) and in the fixed coordinate system b) (case T = -1·104 N, Ms = -1·105 N·m, a = 0.1 m, b = 0.3 m, ω = 5 rad/s)

X (t )

u (t )

0

0

Y (t )

v(t ) 0

0

a)

b)

Fig. 5-58. Trajectories of bit centre motion in rotating coordinate system Oxyz a) and in the fixed coordinate system b) (case T = -1·104 N, Ms = -1·105 N·m, a = 0.1 m, b = 0.3 m, ω = 20 rad/s)

X (t )

u (t )

0

0

Y (t )

v(t ) 0

0

a)

b)

Fig. 5-59. Trajectories of bit centre motion in rotating coordinate system Oxyz a) and in the fixed coordinate system b) (case T = -5·105 N, Ms = -1·105 N·m, a = 0.1 m, b = 0.3 m, ω = 5 rad/s)

Modelling emergency situations in drilling deep boreholes

288

X (t )

u (t )

0

0

v(t )

Y (t )

0

0

a)

b)

Fig. 5-60. The motion paths of bit centre in rotating coordinate system Oxyz a) and in the fixed coordinate system b) (case T = -5·105 N, Ms = -1·105 N·m, a = 0.1 m, b = 0.3 m, ω = 20 rad/s)

u (t )

Zb

X (t )

0

2.20 rad / s

0

v(t )

Y (t ) 0

0

b) a) Fig. 5-61. Trajectories of bit centre motion in rotating coordinate system Oxyz a) and in the fixed coordinate system b) (case T = -1·104 N, Ms = -1·104 N·m, a = 0.1 m, b = 0.5 m, ω = 5 rad/s)

coordinate system (Fig. 5-56). Angular velocity Zb of the bit varies from

Zb 5.44 rad/s to Zb 10.05 rad/s. Figs. 5-57 and 5-58 represent the cases for the values of T

1˜ 10 4 N,

M z 1˜ 10 N∙m, a 0.1 m, b 0.3 m, 0 d Z d 20 rad/s. Here, in both systems, the trajectories of the bit are complex loop-like curves that expand rapidly with increasing angular velocity. Velocities ωb were not calculated for them. 5

At T

1˜ 105 N, M z

1˜ 105 N∙m, a 0.1 m, b 0.3 m, Z

5 rad/s, the

trajectories of bit centre C in coordinate systems Oxyz and OXYZ are expanding spirals. Therefore, such motion can be considered unstable. For an angular velocity equal to Z 20 rad/s, the bit trajectory in both systems takes the shape of a loop curves in both systems. Here, it is interesting to note a significant increase in angular velocity ωb of the backward whirling of the bit relative to fixed coordinate system

Chapter 5. Self-excitation of drill string bit whirling

289

OXYZ , which grew to Zb 62.8 rad/s and significantly exceeded rotation speed Z of the drill string. This effect is also observed in practice. A further increase in the longitudinal force to T 5 ˜ 105 N leads to similar results (Fig. 5-55, Fig. 5-60). As we can see, the drill bit moves counter clockwise. Fig. 5-61 shows the results of the study of a more oblong bit ( a 0.1 m, b 0.5 m). The trajectory is an ellipse in the rotating coordinate system. In the fixed coordinate system, in one revolution, the trajectory of the bit takes the shape of a three-petal flower, which is then rotating forming a curve of a complex configuration. As in the fixed coordinate system, the drill string rotates clockwise, and the drill bit moves in the opposite direction, so there is backward whirling. This mode is dangerous for drilling.

References to Chapter 5 1. Borisov A.V., Mamaev I. S., Kilin A. A. Selected Problems of Nonholonomic Mechanics. – Moscow-Izhevsk: Institute of Computer Research, 2005.–290 p. (in Russian) 2. Borshch E. I., Vashchilina E. V., Gulyaev V. I. Helical traveling waves in elastic rods // Mechanics Solids.–2009.–№2.–P. 288–294. 3. Brett J.F., Warren T.M., Behr S.M. Bit whirl- a new theory of PDC bit failure // SPE Drilling Engineering.–1990.–V.5, №6–P. 275–281. 4. Chen S.L., Blackwood K., Lamine E. Field investigation of the effects of stickslip, lateral, and whirl vibrations on roller-cone bit performance // SPE Drilling & Completion.−2002.−V.17.−P.15–20. 5. Glazunov S. M. Torsion oscillations of deep drill strings in viscous liquid medium // Bulletin of the National Transport University.–2015.–Issue. 31.−P. 96−102. (in Ukrainian) 6. Gulyayev V.I., Borshch O.I. Free vibrations of drill strings in hyper deep vertical bore-wells // Journal of Petroleum Science and Engineering.–2011.– V.78.–P.759-764. 7. Gulyaev V.I., Glushakova O.V., Khudolii S.N. Quantized attractors in wave models of torsion vibrations of deep-hole drill strings // Mechanics Solids.– 2010.–V.45, №2.–P.264-274. 8. Gulyayev V.I., Shevchuk L.V. Drill string bit whirl simulation with the use of frictional and nonholonomic models // Journal Vibration Acoustics.‒2015.‒V.138, No.1.‒P.011021-011021-9. 9. Gulyayev V.I., Shevchuk L.V. Nonholonomic dynamics of drill string bit whirling in a deep bore-hole // Journal of Multi-body Dynamics.–2013.– V.227.–P.234-244. 10. Gulyayev V.I., Vashchilina O.V., Glazunov S.M. Incipient regimes of drill bit whirlings on uneven bottoms of deep bore-holes // Proceedings of 5th

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International Conference on Nonlinear Dynamics 2016 ND-KhPI, September 27-30, 2016, Kharkov, Ukraine, P. 312-317. 11. Jansen J.D. Whirl and chaotic motion of stabilized drill collars // SPE Drilling Engineering.–1992.–V.7, №2.–P. 107–114. 12.Kovalyshen Y. A simple model of bit whirl for deep drilling applications // Journal of Sound and Vibration.−2013.−V.332, №24.−P.6321-6334. 13. Leine R.I., Van Campen D.H., Keultjes W.J.G. Stick-slip whirl interaction in drillstring dynamics // Journal of Vibration and Acoustics.–2002.–V.124, April.–P.209-220. 14.Lindberg R.E., Longman R.W. On the dynamic behaviour of the wobblestone // Acta mechanica.–1983.–V.49, №1-2.–P. 81-94. 15. Majeed F. Abdul, Karkoub M., Karki H., Abdel Magid Y.L. Drill bit whirl mitigation analysis: an under actuated system perspective // International Journal of Sustainable Energy Development.−2012.−V.1, №1/2/3/4.−P. 36−40. 16. Markeev A. P. Dynamics of a Body Contacting with a Solid Surface. M.: Science, 1992.–336 p. (in Russian) 17. Mihajlović N., Van de Wouw N., Rosielle P.C.J.N., Nijmeijer H. Interaction between torsional and lateral vibrations in flexible rotor systems with discontinuous friction // Nonlinear Dynamics.−2007.−V.50, №3.−P.679−699. 18. Musa Nabil W., Gulyayev V.I., Shevchuk L.V. Whirl interaction of a drill bit with the bore-hole bottom // Modern Mechanical Engineering.−2015.−V.5.−P.41-60. 19. Neymark Yu. I., Fufayev N. A. Dynamics of Nonholonomic Systems. M.: Science, 1967.–519 p. (in Russian) 20. Pascal M. Asymptotic solution of the equations of motion for a Celtic stone // Journal of Applied Mathematics and Mechanics.–1983.–V.47, №2.–P.269-276. 21. Samuel R. Friction factors: what are they for torque, drag, vibration, bottom hole assembly and transient surge/swab analyses? // Journal of Petroleum Science and Engineering.−2010.−V.73.−P.258−266. 22. Stroud D., Pagett J., Minett-Smith D. Real-time whirl detector improves RSS reliability, drilling efficiency // Hart Exploration & Production Magazine.−2011.−V.84, №8.−P.42–43. 23. Walker G.T. On a curious dynamical property of celts // Proceedings of the Cambridge Philosophical Society.–1895.–V.8, №5–P.305-306. 24. Walker J. The mysterious “rattleback”: A stone that spins in one direction and then reverses // Scientific American.–1979.–V.241, №4.–P.144-149. 25. Yigit A.S., Christoforou A.P. Stick-slip and bit-bounce interaction in oil-well drillstrings // Journal of Energy Resources Technology.–2006.–V.128, №4.– P.268-274.

Chapter 6. Modelling resistance forces and drill string sticking effects…

291

CHAPTER 6. MODELLING RESISTANCE FORCES AND DRILL STRING STICKING EFFECTS IN CURVILINEAR BOREHOLE CHANNELS 6.1

Features of modern technologies of oil and gas fuel extraction and problems of mechanics of elastic bending of drill strings in curvilinear boreholes

The problems of analysing elastic bending of drill strings in deep curvilinear oil and gas boreholes are closely related to the development of effective technologies for the production of hydrocarbon fuels. In the world's leading countries, the oil and gas industry, as an important component of the fuel and energy complex, plays a significant role in the development of their economies and in maintaining energy security. The main factors that determine production levels of hydrocarbons are associated with natural factors—namely, volumes and reserves of hydrocarbons and distances from industrial centres. Other factors on which production volumes depend are the level of efficiency of their extraction, the price situation in the consumer markets, and the state tax policy. The problems associated with oil and gas complex development include improving the technology of curvilinear drilling and creating computer modelling software, which have become extremely relevant owing to the beginning of shale gas production in the 21st century. Shale gas is a natural gas produced from oil shale, consisting mainly of methane. Large-scale industrial production of shale gas was started by Devon Energy in the United States in the early 2000s at the Barnett Shale field. The company drilled a horizontal well in this field for the first time in 2002. Due to the sharp increase in shale gas production, what the media called the "gas revolution", in 2009, the United States became the world's top gas producer (745.3 billion cubic metres), with more than 40% from non-traditional sources (methane from coal seams and shale gas). In the first half of 2010, the world's largest fuel companies spent $21 billion on assets associated with shale gas production. At that time, some commentators expressed the opinion that the hype around shale gas—called the shale revolution—was the result of an advertising campaign inspired by several energy companies that had invested heavily in shale gas production projects and therefore needed additional funds. Whatever it was, after the appearance of shale gas on the world market, gas prices began to fall. The cost of production of shale gas is higher than for traditional gas and is $150 per thousand cubic metres. While in Russia, the cost of natural gas from old gas fields, including transportation costs, is about $50 per thousand cubic metres.

Modelling emergency situations in drilling deep boreholes

292

Horizon level Conventional gas field

The boundary of the geological formation Sand

Oil

Sands saturated with gas Shale gas

Fig. 6-1 Diagram of laying a curvilinear borehole in a shale rock formation

Horizontal drilling, hydraulic fracturing, and seismic modelling are used for shale gas extraction. The same technology is used to obtain coalbed methane. Although shale gas is contained in small amounts (0.2–3.2 billion m3 km 2 ), it can be obtained in significant quantities owing to the discovery of large areas. Factors that have a positive impact on the prospects of shale gas production include the proximity of deposits to sales markets, significant reserves, and the interest shown by several countries in reducing dependence on imports of fuel and energy resources. Shale gas also has a number of disadvantages that negatively affect the prospects of its global production. Such shortcomings include relatively high cost, unsuitability for long-distance transportation, rapid depletion of deposits, low level of proven reserves in the overall structure of reserves, and significant environmental production risks. In 2009, shale gas production in the United States made up 14% of total gas supply, and this percentage has increased, which in 2009 led to significant changes to the distribution of the world combustible gas market between countries and the formation of excess supply in the market by the beginning of 2010. As a result of the growth of shale gas production, liquefied gas import terminals in the United States remained idle. They are currently being converted for gas exports. Large deposits of shale gas were found in a number of European countries, in particular, in Austria, England, Hungary, Germany, Poland, Sweden, and Ukraine. However, the practical implementation of the technology for drilling wells of complex spatial orientation requires appropriate mathematical modelling for the design of their rational trajectories and the use of state-of-the-art equipment and technology for

Chapter 6. Modelling resistance forces and drill string sticking effects…

293

their drivage. As this takes place, the most interesting issues are those for determining the external and internal forces as well as the torques acting on the drill string (DS) in the curvilinear borehole in the processes of its tripping in/out and drilling [9, 10, 12, 15– 17, 22, 24, 30, 34, 36, 38–40]. Modelling resistance forces and dynamic phenomena accompanying well drilling makes it possible to solve such fundamental problems as obtaining the required trajectory of the well and reducing the longitudinal and transverse vibrations of the string as well as reducing the forces of contact and friction interaction between the string and the well wall, thereby achieving a reduction in wear of the DS and the lock joints, eliminating unplanned bending of the centreline of the well, and, as a consequence, eliminating complex abnormal situations in the drilling process [2, 4, 5]. As noted above, under conditions of geometric similarity, modern drill strings in deep curved wells are similar to a human hair. Therefore, in well planning, the designers usually simulate drill strings as completely flexible cables. Such problem statements are proposed in the works of scientists Aadnoy B. S. in joint works with Kenneth Larsen, Per C. Berg [6], Andersen K. [7]; Mohiuddin M. A., Khan K., Abdulraheem A., Al-Majed A., Awall M. R. [3], Choe Jonggeun, Schubert J. J., Juvkam-Wold H. C. [10]; Descant F. J., Fisk J. H., Pourcian R. D., Waltman R. B. [11]; Sawaryn S. J. with co-authors Sandstrom B., McColpin G. [28], and Thorogood J. L. [29]; Sheppard M. C. in joint works with Wick C., Burgess T. [31], and Riley R. H., Warren [32]. Nevertheless, such modelling is only valid for wells with perfectly smooth geometry. As wells with ideal geometry do not exist in practice, and they all have geometric imperfections in the shape of localised spirals, harmonics, and breaks because of local bending of the drill string in these areas, the forces of contact and friction between the string and the well wall significantly increase. As a rule, they are the main causes of the so-called drill string sticking equivalent to an emergency situation. The modelling of these phenomena can only be carried out on the basis of the nonlinear theory of deforming of flexible curvilinear rods. Two problems of elastic bending of DS in a deep curvilinear borehole can be set and solved on the basis of the model of the theory of flexible curvilinear rods. The first problem is related to analysis of elastic deforming of the DS during tripping in/out operations. As a result of its formulation and solution, strains and internal stresses of the DS as well as external forces of resistance (friction) that prevent its displacement can be determined. The second problem relates to the study of DS stability. Its solutions allow us to eliminate critical and post critical conditions of the DS.

Modelling emergency situations in drilling deep boreholes

6.2

294

Mathematical aspects of the problem of forced elastic bending of the drill string in the channel of a curvilinear borehole

One of the widely known methods of designing wells of complex geometry (including three-dimensional) is the minimum curvature method. An analysis of the advantages and features of its practical application is given in the compendium of S. J. Sawaryn et al. [29]. The essence of this method is that the trajectory of a curvilinear borehole is represented as a series of smoothly coupled circular arcs and segments of straight lines. The connection points and the planes of orientation are set from the condition of achieving the set geological goal, on the way to which geological faults and neighbouring boreholes can meet. This approach is used to create mathematical software, which is implemented in practice to plan the borehole trajectory. M. C. Sheppard et al. [31], S. F. Shepard et al. [32] propose combining the straight segments with chain lines. Previously, when analysing the mechanics of curvilinear drill strings, researchers limited themselves to rather simplified models where it was believed that a DS is a weighty absolutely flexible cable for which the forces of contact interaction with the wall of a curvilinear borehole were calculated either based on the prescribed conditions of the cable tension (Bernt S. Aadnoy et al. [7], A. S. Demarchos et al. [13]) or (for inclined drill strings) by projecting gravity forces on normal to the centreline of the string (Bert S. Aadnoy et al. [7]). After that, the tangent to the centreline of the string and the Coulomb friction force, which was taken as equal to the product of friction coefficient P and the normal pressure force, were determined. It is clear that even for an absolutely flexible cable neither of these approaches can be considered accurate since for a curvilinear cable the forces of normal interaction must be calculated based on solutions to the general equations of the theory of flexible cables, not to mention the fact that the string must be modelled by an elastic rod. Nevertheless, special computer products for designing curvilinear boreholes have been developed based on the theory of flexible cables. They are used to calculate the frictional resistance forces acting on the strings during tripping in/out operations and torques during drilling. So, based on the model of a heavy weight pipe, Bernt S. Aadnoy et al. [7] calculated the resistance forces for a DS with an axial line composed of circular arcs, chain lines, and straight segments. The need to account for the small-scale crookedness of the borehole was pointed out by F. Akgun in [1]. He noted that the distortions usually have short-wave sinusoidal shapes due to the structure of the bottom hole assembly (BHA). To eliminate the distortions, the BHA is calculated in the paper using the finite element method. A similar finite element programme for statics and dynamics of BHA in a threedimensional curvilinear borehole was created by Birades Michel [8].

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295

It should be noted that in addition to the algorithmic disadvantage associated with the fact that the connection problems do not have a single solution these methods also have a significant practical defect. The matter is that there are curvature discontinuities at the connection points of arcs and straight segments in the borehole trajectories; therefore, the bending moments in the DS become discontinuous at these ones. And this—as shown below—is only possible when two reactions of the contact interaction of the string and the borehole are exposed in these places in the form of a pair of forces with a moment equal to the magnitude of the rupture of the bending moment. These reactions contribute to a significant increase in the friction forces and can be reason for the string sticking or the destruction of the borehole wall. The features of the theoretical problem for modelling the mechanics of curvilinear drill string behaviour consist in the need to calculate both the internal longitudinal and transverse forces and moments and the external forces of contact and friction interaction of the string with the borehole wall [14, 23, 25, 27, 33–35, 37, 41]. To determine the internal forces, direct problems of structural mechanics are usually set, while external forces are calculated by setting inverse problems. These problems should be set simultaneously based on the theory of flexible curvilinear rods. Currently, these questions have not been formulated for curvilinear rods, and the authors of this work seem to be the first ones to pose them. They considered the problems of calculating the forces of resistance to axial and rotational motion of the drill string in a borehole with a relatively smooth trajectory, including geometric imperfections of the trajectory of a relatively regular structure. It should be noted, however, that, as a rule, when laying a curvilinear borehole, the shape of its trajectory is somewhat different from that planned, and under the influence of various technological factors localised short-wave geometric imperfections of sinusoidal or spiral character and smoothed breaks appear on it. As a result of the forced distortion of the string itself in the form of these imperfections, bending moments become obvious in it and significantly exceed the moments in the string of the planned geometry. Under the influence of these moments, the forces of contact and friction interaction increase essentially. These forces can only be calculated according to the theory of flexible curvature rods described, for example, in the monograph of V. I. Gulyayev et al. [18]. The main relations of this theory are formulated with the help of stationary and moving coordinate systems, methods of differential geometry, moving trihedrons, and the axes system associated with the considered cross section. Based on this approach, techniques for calculating the friction forces and torques of friction forces [17, 20, 21] have been developed. This makes it possible to determine these values with high accuracy in almost any configuration considering the geometric imperfections of drill strings and local deviations of the centreline of the borehole from that specified.

Modelling emergency situations in drilling deep boreholes

6.3 6.3.1

296

Stiff string drag and torque model of DS bending in curvilinear boreholes The features of the problem of elastic bending of DS in the channel of a curvilinear borehole during tripping in/out operations and drilling

As mentioned above, the increasing application of oil and gas production technologies using curvilinear boreholes is based on two main factors. First, the volume of unused hydrocarbon fuels remaining in underground reservoirs is decreasing, resulting in increased production efficiency. Second, the design and drilling of curvilinear boreholes are the main component of the technological processes of shale gas production. These boreholes are usually long. Therefore, the forces of contact interaction arising in them and the friction forces and moments of friction caused by them reach large values. Not only do these forces and moments increase the energy consumption during drilling and tripping in/out operations, but they are also one of the main causes of accidents during drilling. z x Localised imperfections

H Gas-bearing bed

L а

b

Fig. 6-2 Geometric diagram of a curvilinear borehole with centreline imperfections

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297

As these forces and moments are largely dependent on the geometric nonsmoothness of the borehole trajectory, the calculation must take into account the geometric imperfections of its centreline. This book dwells upon the problems of computer simulation of friction forces and moments of friction forces in curvilinear boreholes and in boreholes with simple regular geometric imperfections, in particular, localised spiral (Fig. 6-2) and harmonic imperfections, and imperfections in the form of local breaks. 6.3.2

Theoretical modelling of the directional borehole trajectory with localised geometric imperfections

When modelling the elastic deforming of a tubular drill string in a curvilinear borehole with known geometry during drilling and tripping in/out operations carried out under the action of longitudinal force Fz and torque Mz, at the point of its suspension as well as distributed gravity forces fgr, forces fc of contact between the DS and the borehole wall, forces ffr, and moments mfr of their friction interaction, let's assume that the axial line of the drill string coincides with the axial line of the borehole. Therefore, it can be assumed that the elastic line x(s), y(s), z(s) of the drill string is also known, and all its geometric characteristics can be found. Here, s is a natural parameter that is measured by the length of the string from an initial point to the current one. Assume that for technological reasons localised geometric distortions are introduced into the trajectory of the borehole, while the distortions of the well design outline cannot be accompanied by the formation of its centreline breaks, and it is differentiable by parameter s. However, if these imperfections are small-scale, curvature k R s and torsion kT s functions can gain larger values. We shall present the basic relations that determine the geometry of the centreline of a curvilinear borehole. Two parameterisation methods can be used to represent them. In the first method, the position of the point on the curve is given by coordinate s (natural parameterisation), in the second an arbitrary (dimensionless) parameter - is used, which can be more convenient for describing the general properties of the curve. Let the equations of the well centreline in the Cartesian coordinate system Oxyz have the form x xs , y ys , z z s (6.1) or (6.2) ρ ρs Here, ρ x i  y j  z k , where i, j, k are the unit vectors of the coordinate system Oxyz .

Modelling emergency situations in drilling deep boreholes

298

Let's write down the basic geometric relations used for the problem statement of drill string deforming in a curvilinear borehole and the construction of differential equations of its bending [19, 26]. To set the orientation of the external and internal forces applied to the string, let's introduce the movable Frenet trihedron τ, n, b , where τ is the unit tangent vector oriented in the direction of increasing s ; n is the unit vector directed along the main normal to the curve, and the unit vector b directed along the binormal so that τ , n , and b compose the right system of vectors. Unit vector τ is calculated by the formula (6.3) τ dρ ds . Using the second derivative of ρ with respect to s , we build the equality

d 2ρ ds 2 k R ˜ n , where k R is the curvature of the centreline determined by the relation

xcc 2   ycc 2 zcc 2 .

kR

(6.4) (6.5)

Here, the prime mark denotes differentiation with respect to s . The vector b is determined from the condition of orthogonality of system τ , n , b b τ un. (6.6) Its components are found from the following relations 1 § dy d 2 z dz d 2 y · ¸, ¨ ˜ bx  ˜ k R ¨© ds ds 2 ds ds 2 ¸¹ 1 § dz d 2 x dx d 2 z · ¸, ¨ ˜ by  ˜ (6.7) k R ¨© ds ds 2 ds ds 2 ¸¹ 1 § dx d 2 y dy d 2 x · ¸. ¨ ˜ bz  ˜ k R ¨© ds ds 2 ds ds 2 ¸¹ Together with curvature k R , an important geometric characteristic of the centreline of the borehole is its torsion kT . To calculate it, the formula is used dn · § τ ˜ ¨n u ¸. ds ¹ © The scalar form of equality (6.8) is used in the calculation xc y c z c 2 kT k R xcc y cc z cc . xccc y ccc z ccc kT

(6.8)

(6.9)

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299

In the process of the rod deformation and during the motion of the natural trihedron along its centreline along coordinate s , it is necessary to take into account the rotation of the trihedron. It is characterised by the Darboux vector (6.10) Ω k R b  kT τ , presenting a vector of the total curvature of the centreline and defined as a vector of the angular velocity of the natural trihedron rotation relative to its origin when the latter moves along the elastic line in the direction of increasing s with a unit linear velocity. It should also be noted that when using the Frenet basis it is necessary to take into account the important role of the first integrals (6.11) τun b τ ˜n 0, τ 1, n 1, arising from the condition of orthonormality of unit vectors τ , n , b . Introduce a moving system of principal axes of inertia ( u, v, w ) of the rod cross section, where axis w is directed along unit vector τ . Just like unit vectors n , b , they lie in a normal plane and are turned by angle F measured from unit vector n to axis u . Angular velocity vector ω F of the main trihedron ( u, v, w ) is determined by the

dF τ. ds The projections of vector ω F on the axes u , v , w represent, respectively, the

relation ω F

Ω

curvatures of the projections of element ds on the planes v, w and u, w and the twisting of the centreline. Let's denote them with the symbols p , q , r and calculate using the formulas dF r kT  . (6.12) p k R sin F , q k R cos F , ds If curve (6.1) is defined through parameter x x- , y y- , z z- , (6.13) then in the differentiation operator Dd- is used instead of ds , where the metric coefficient D is calculated using the formula

D

x 2  y 2  z 2 .

(6.14)

Here, the point symbol denotes differentiation with respect to - . It should be noted that to solve the problem of drill string bending in a borehole with complex shape imperfections, it is convenient to use parameters s and - at the same time applying them in parallel to perform various geometric transformations. Then, the basic relations that determine the external geometry of the centreline will be rewritten as dρ dτ 1 τ b τun . , n , (6.15) ˜ Ddk R Dd-

Modelling emergency situations in drilling deep boreholes

300

In this case, equations (6.16) and (6.17) are used instead of formulas (6.5) and (6.9), respectively, kR

x

2





2  y 2  z 2 x2  y2  z2  x x  y y  z z



x 2  y 2  z 2

k R2 ˜



3

x y z x y z x y z

,

(6.16)

. (6.17) 3  y 2  z 2 Equalities (6.1)–(6.17) fully determine the geometric characteristics of the borehole centreline and the elastic line of the drill string. kT

x

2



6.3.3. Constitutive equations of the elastic bending of a drill string in the channel of a curvilinear borehole during the drilling process and tripping in/out operations Consider the phenomenon of a drill string bending during its motion in the channel of the borehole during the drilling process or tripping in/out operations. Assume that curvature k R and torsion kT of curve (6.2) are so small that the deformation of the drill string occurs in the elastic stage. The stress state of each conditionally selected element of the drill string is determined by the resultant vectors of internal forces F(s) and moments M(s) in the cross sections of the drill string and by the distributed vectors of external forces and moments with intensities f (s) and m(s) . Force f includes gravity f gr (s) as well as the forces of contact interaction f c (s) and friction f fr (s) between the outer surface of the DS pipe and the borehole wall. The external distributed moment m(s) consists only of the moment of friction forces m fr (s) . In this regard, the vector equations of equilibrium [18] can be written as dF dM f gr  f c  f fr ,  τ u F  m fr , (6.18) ds ds which are invariant in respect of any coordinate system. In the general case, it is most convenient to record them in the movable axis system ( u, v, w ). Since it rotates as it moves along coordinate s , it is necessary to present total derivatives dF ds , dM ds in the moving system. Then ~ ~ dF d F dM d M (6.19)  ωF u F ,  ωF u M . ds ds ds ds Here, the sign ~ indicates the operation of local differentiation.

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301

By substituting the right parts of equations (6.19) in equations (6.18), we obtain ~ ~ dF dM ω F u F  f gr  f c  f fr , ω F u M  τ u F  m fr . (6.20) ds ds Let's write down three scalar equations of forces equilibrium corresponding to the first equation of this system dFu r ˜ Fv  k R cos F ˜ Fw  f ugr  f uc , ds dFv k R sin F ˜ Fw  r ˜ Fu  f vgr  f vc , (6.21) ds dFw k R cos F ˜ Fu  k R sin F ˜ Fv  f wgr  f wfr . ds From the last equality of system (6.12), we obtain dF r  kT . (6.22) ds Taking into account that the axes ( u, v, w ) are the principal axes of the bending and torsion of the rod element, we present moment M components M u , M v , M w as (6.23) M u A p A k R sin F , M v A q A k R cos F , M w C r , where A E I ; C G I 0 ; E, G are the drill string material elasticity modules under tension and shear; I , I 0 are the axial and polar moments of inertia of the cross section of the drill string pipe. Using equations (6.23) and (6.22), we reduce second vector equation (6.20) to a system of three scalar equations of moments equilibrium F dk R AC sin F  k R cos F ˜ r  kT ˜ k R cos F ˜ r  v , ds A A Fu dk R CA cos F  k R sin F ˜ r  kT ˜ k R sin F ˜ r  , (6.24) ds A A m fr dr  w . ds C If the geometry of the borehole centreline is known, equations (6.21), (6.24) allow us to formulate the problem of contact interaction of the drill string with its wall during drilling and tripping in/out operations. The system of six equations (6.21), (6.24) contains only three functions of forces Fu - , Fv - , Fw - and function of elastic twisting angle F , through which moments (6.23) are calculated at given k R and kT . However, it is underdetermined as it also includes external distributed forces of contact f uc - , f vc - to be determined, the frictional force f wfr - , and the moment mwfr - of friction forces. The problems of

Modelling emergency situations in drilling deep boreholes

302

solid deformed bodies mechanics, in which the internal forces (or moments) are partially known, and some external forces (usually contact forces or friction forces) are to be determined, are called inverse. To formulate the inverse problem in this case, we reduce the first two equations of system (6.24) to the form: dk Fu  A R cos F  A k R kT sin F  C k R r sin F , ds (6.25) dk Fv A R sin F  A k R kT cos F  C k R r cos F . ds We differentiate both parts of this system with respect to s

dFu ds

dk dk d kT d 2 kR dF cos F  A R sin F  A R kT sin F  A k R sin F  ds ds ds ds ds 2 d kR dF dr dF r sin F  C k R  A k R kT cos F C sin F  C k R r cos F , ds ds ds ds A

dk dk d kT d 2 kR dF sin F  A R cos F  A R kT cos F  A k R cos F  2 ds ds ds ds ds d kR dF dr dF  A k R kT sin F C cos F  C k R r sin F . r cos F  C k R ds ds ds ds Then, equate their right parts to the right parts of the first two equalities of system (6.21). Taking into account the equalities of system (6.25), we obtain dk d § dk · f uc A ¨ R ¸ cos F  AkR kT2 cos F  Ck R kT r cos F  2 A R kT sin F  ds © ds ¹ ds dkT dk R  AkR sin F  C r sin F  mwfr k R sin F  Fw k R cos F  f ugr , ds ds (6.26) dk dk d § · f vc  A ¨ R ¸ sin F  AkR kT2 sin F  Ck R kT r sin F  2 A R kT cos F  ds © ds ¹ ds dkT dk R  AkR cos F  C r cos F  mwfr k R cos F  Fw k R sin F  f ugr . ds ds With the help of equations (6.26), we calculate the full force of the contact interaction of the drill string and the borehole wall dFv ds

A

fc

f  f c 2 u

c 2 v

.

(6.27)

As the drill string moves in the drilling liquid medium, it can be assumed that when the pipe surface of the drill string comes into contact with the borehole wall, Newton friction conditions arise, depending on the viscosity of the liquid. However, as shown by field experiments [29, 31], the presence of liquid does not lead to the essential change in the type of friction interaction, which remains Coulomb. It should be

Chapter 6. Modelling resistance forces and drill string sticking effects…

303

emphasised only that coefficient P of this friction varies in a wide range from 0.15 to 3.0, reaching the highest values in boreholes with rapidly changing geometry [7, 31]. It should be noted that the latter statement cannot be considered reasonable as the Coulomb coefficient of friction does not depend on the forces of normal pressure but is determined only by the mechanical properties of the materials of the contacting bodies and the quality of their surfaces. It can be assumed that the found inflated values P ≈ 3.0 are not real but are caused by the incorrect calculation of normal pressure forces in experiments, which (as shown in our studies) are very sensitive to geometric imperfections. Nevertheless in this work, it is accepted that P = 0.2 and 0.3. We assume that during drilling and round-trips the drill string simultaneously carries out an axial motion with speed w and rotates with angular velocity Z . Then, the full force of friction f components f wfr

rP ˜ f c

fr

P ˜ f c can be decomposed into two mutually perpendicular w

w  Z d / 2 2

2

,

fZfr

rP ˜ f c

Zd 2 2 w  Z d / 2 2

,

(6.28)

proportional to the respective velocity components w and Z d / 2 . Here, P is the coefficient of friction; d is the outer diameter of the drill string pipe. The first of these forces prevents the axial motion of the drill string, the second one is directed in the circumferential direction and leads to the appearance of a distributed moment of friction forces d Z d2 . (6.29) mwfr fZfr ˜ rP ˜ f c 2 2 4 w 2  Z d / 2 Signs ‘ r ’ in formulas (6.28), (6.29) are selected depending on the direction of motion and rotation of the drill string. In the expression for f wfr , the sign ‘–’ corresponds to the procedure of tripping out the drill string, the sign ‘+’ is its tripping in and the drilling process. The relations derived in this subsection allow us to formulate a system of equations of elastic bending of a drill string in a borehole with a given axial line. Let's bring it to the final form: dF r  kT , ds dr 1  mwfr , ds C dFw k R cos F ˜ Fu  k R sin F ˜ Fv  f wgr  f wfr , ds

Modelling emergency situations in drilling deep boreholes

dk R cos F  A k R kT sin F  C k R r sin F , ds dk Fv A R sin F  A k R kT cos F  C k R r cos F , ds 2 2 w f wfr r P ˜ f uc  f vc , 2 w 2  Z d / 2

Fu

A

 

mwfr

304

rP ˜

f  f c 2 u

c 2 v

Z d2 4 2 w 2  Z d / 2

(6.30)

,

d 2 kR dk cos F  A k R kT2 cos F  C k R kT r cos F  2 A R kT sin F  ds ds 2 d kT dk R  A kR sin F  C r sin F  mwfr k R sin F  Fw k R cos F  f ugr , ds ds d 2kR dk f vc  A sin F  A k R kT2 sin F  C k R kT r sin F  2 A R kT cos F  ds ds 2 d kT d kR  A kR cos F  C r cos F  mwfr k R cos F  Fw k R sin F  f ugr . ds ds

f uc

A

The components f ugr , f vgr , f wgr of distributed gravity forces entering here are known and determined as follows:

f ugr

f vgr

 F J st  J l g nz cos F  bz sin F ,

F J st  J l g nz sin F  bz cos F ,

(6.31)

 F J st  J l g W z , where F is the cross-sectional area of the drill string pipe; J st is the density of the rod material; J l is the density of the drilling fluid. Corresponding boundary conditions are added to the constructed system of equations (6.30), (6.31). f wgr

6.3.4. Technique of computer determination of internal and external forces acting on the drill string A characteristic feature of the problem is that, as a rule, for the system of equations (6.30), (6.31) the boundary conditions at one end of the drill string are fully known. Indeed, for example, when performing tripping in/out operations, longitudinal force Fw , torque and the angle of elastic twist F are zero at the lower end of the drill string. In this case, it is possible to formulate the Cauchy problem for system (6.30), (6.31) and to apply the methods of direct numerical integration.

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305

The required variables are calculated numerically. The Runge-Kutta method is used to integrate the first three equations of system (6.30). At each step of its implementation, functions F si , r si , Fw si are first calculated, and then other unknown Fu si , Fv si , f wfr si , mwfr si are calculated from their found values. After that, the next step of integration is performed. With this approach, the main difficulty is the task of calculating the geometric characteristics of the centreline of the drill string. As the well centreline with localised geometric imperfections is described by sufficiently complex analytical expressions, to calculate functions k R , kT and their derivatives d k R d s , d 2 k R d s 2 , d kT d s included in constitutive equations (6.30), (6.31), it is necessary to find derivatives up to the fourth order of functions x(s) , y (s) , z (s) and then calculate the determinants of matrices (6.9) of size 3u 3 . It is clear that it is impossible to analytically perform such transformations. Therefore, in this book, functions k R (s) , kT (s) and their derivatives are calculated numerically using the finite difference method x  xi 1 z z y  yi 1 dx dz dy | i 1 | i 1 i 1 ; (6.32) | i 1 , , ds i 2 ˜ 's ds i 2 ˜ 's ds i 2 ˜ 's x  2 xi  xi 1 d 2x | i 1 , 2 2 ˜ 's ds i y  2 yi  yi 1 d2y | i 1 , 2 ˜ 's ds 2 i

's i-2

i-1

i

i+1

(6.33)

z  2 zi  zi 1 d 2z | i 1 ; 2 ˜ 's ds 2 i

i+2

x  2 xi 1  2 xi 1  xi 2 d 3x | i2 , ds 3 i 2 ˜ ('s) 3 y  2 yi 1  2 yi 1  yi 2 d3y | i2 , ds 3 i 2 ˜ ('s) 3

(6.34)

z  2 zi 1  2 zi 1  zi 2 d 3z | i2 ; 3 ds i 2 ˜ ('s) 3

Using formulas (6.32)–(6.34) and substituting them into expressions (6.5), (6.9), we find functions k R (s) and kT (s) in each integration point

k R i

xicc 2   yicc 2 zicc 2 ,

(6.35)

Modelling emergency situations in drilling deep boreholes

kT i

1

k R i2

xic yic zic ˜ xicc yicc zicc xiccc yiccc ziccc

306

(6.36)

xic yiccziccc xicccyiczicc  xiccyiccczic  xicccyicczic  xic yiccczicc  xiccyicziccc k R i2 . As expressions (6.35), (6.36) for finding curvature k R (s) and torsion kT (s) of a given curve are rather complex, the derivatives of these functions are also calculated using the finite difference method k  k R i1 k  2 k R i  k R i1 d kR d 2 kR | R i 1 , ; (6.37) | R i 1 2 ds i 2 ˜ 's 2 ˜ 's ds i

k  kT i1 d kT | T i 1 . ds i 2 ˜ 's

(6.38)

As in the places of localisation of geometric imperfections, variables x(s) , y (s) , z (s) , k R (s) , kT (s) are rapidly changing functions, to achieve acceptable accuracy in their calculation, it is necessary to choose a sufficiently small step of differentiation and integration ' s . In calculations, as a rule, it is chosen to be equal to ∆s = S/16,000. Therefore, the mathematical model developed in this section of the numerical determination of the forces of contact and friction interaction between the drill string and the borehole wall makes it possible to determine the parameters of the stress state of the drill string and the resistance forces generated during its axial motion. As these force characteristics depend on the relation between the values of the axial and rotational speeds of motions, a relationship can be chosen between them where the distributed resistance forces and their moments are minimal. Rational regulation of these characteristics at each stage of the tripping in/out operations with the help of the developed model and the technique of numerical analysis enables us to not only reduce the resistance forces but also to minimise the total energy consumption at all stages of drilling. Numerical results related to the implementation of the proposed technique are presented below. 6.4.

Modelling resistance forces in a well with localised harmonic imperfections

The results of numerous studies of the mechanical effects associated with pulling (tripping in/out) the drill string in the channel of a curvilinear borehole with an axial line that does not have geometric imperfections show that the resistance forces that arise in this case, as a rule, are smooth functions with relatively small values. However, specific types of their geometric shapes have a certain influence on the forms of these functions. The following examples of elliptic, parabolic, and hyperbolic well paths

Chapter 6. Modelling resistance forces and drill string sticking effects…

307

show that the functions of internal forces and moments depend on the angles of the centreline of the well at this segment and on their curvature. In this regard, analysis of the influence of geometric imperfections of the centreline of the well on the resistance forces to the axial motion of the drill string is performed for two types of curves— hyperbolic and elliptic. 6.4.1

Results of calculations for a well with hyperbolic trajectory

Let's investigate the elastic bending of DSs in wells whose design outline is described by a hyperbolic curve (Fig. 6-3, a). As branches of hyperbolic curves at a distance from vortexes approach straight asymptotes, they become almost straight at the end segments of these wells. Such trajectories are typical for directional and horizontal wells, so in this book they are considered first. The equations of their trajectories have the form: H L1  H y 0, z ˜ sin - . ˜ cos- , x (6.39) 1  H cos1  H cosHere, L is the horizontal distance of the lower end of the well from the drilling rig, H is the well depth, H is the parameter that determines the eccentricity of the hyperbola, is the scalar parameter. The relationship between parameter - and natural parameter s is given by the integral dependence -

³ DT dT ,

s

3S / 2

where D(- ) is the metric factor (6.14) equal

L2 (1  H ) 2 sin 2 -  H 2 (cos-  H ) 2 . (6.40) (1  H cos- ) 2 Sometimes, it is convenient to use parameters - and s simultaneously. As a result of any reasons related to the drilling technology or heterogeneities of mechanical properties of rock, in plane xOz of the well centreline, geometric distortions in the shape of h cos(ks) with amplitude h and wave number k superimposed on curve (6.39) may occur. In addition, these imperfections have a localised character modelled by the form of the amplitude h(s) function (Fig. 6-3, b). It is accepted that it changes according to the following law: D(- )

x 2  y 2  z 2

h( s) hc e

§ s  sc · D 2 ¨ ¸ © S ¹

2

,

(6.41)

Modelling emergency situations in drilling deep boreholes

308

z

4000 m

x

а

8000 m z

x

b Fig. 6-3 Geometries of the borehole hyperbolic centreline: a) without imperfections b) with localised harmonic imperfections

O

c 2hc

sc

в

Fig. 6-4 Geometry of localised harmonic imperfection on a larger scale

Chapter 6. Modelling resistance forces and drill string sticking effects…

309

where constant hc sets the maximum value of amplitude h(s) (Fig. 6-4); S is the total length of the centreline; sc is the distance from the initial point s 0 - 3S 2 to the centre point of the imperfection; D is the coefficient that determines the rate of decline of the h(s) function and the representative range l of the h(s) change, beyond which the h(s) value can be ignored. Then, the components of radius vector ρ of the borehole centreline with imperfections can be calculated using the formulas: L1  H H (cos-  H ) , ˜ cos-  h( s) ˜ cosks ˜ x 2 2 1  H cosL (1  H ) sin 2 -  H 2 (cos-  H ) 2

y 0, H L1  H sin , ˜ sin -  h( s) ˜ cosks ˜ z 1  H cosL2 (1  H ) 2 sin 2 -  H 2 (cos-  H ) 2 where S

2S

2S

2

2

³ D- dT 3³S 3S

(6.42)

L2 (1  H ) 2 sin 2 -  H 2 (cos-  H ) 2 dT is the total length of (1  H cos- ) 2

the centreline. Using these equations according to formulas (6.3)–(6.6), (6.9), (6.12), all necessary coefficients in system (6.30) are calculated, and the first three equations are integrated by - using the Runge-Kutta method in the range - 3S 2 to- 2S . At each integration step, all functions presented in system (6.30) are calculated. The initial conditions at - 3S 2 for functions F , r , Fw are selected depending on the type of the w of linear modelled regime. For tripping in/out operations with a given ratio Q Zd 2 velocities of the axial and rotational motions, it is taken F 3S 2 0 , r 3S 2 0 , Fw 3S 2 0 . Analysis of elastic bending of the drill string pipe in the borehole with the selected imperfections is performed at the following values of the characteristic parameters: L = 8,000 m, H = 4,000 m, d = 0.1683 m is the outer diameter of the drill string pipe, G = 0.01 m is the pipe thickness, E = 2.1·1011 Pa is the modulus of elasticity, G = 0.8077·1011 Pa, J st = 7,850 kg/m3 is the density of steel, J l = 1,500 kg/m3 is the density of the drilling fluid, P = 0.2 is the coefficient of friction Q 100 . With this initial data, the length of the borehole centreline equalled S = 9,220 m. Parameter D that determines the rate of the h- decrease with variable change was set to 10. In this case, the representative interval of - change, in which there are significant values of h- , was l = 2,219 m. It should be noted that although outside this interval the curvature of the imperfection is so small that it can be

Modelling emergency situations in drilling deep boreholes

310

neglected, the distortion of the borehole centreline caused by imperfection was considered throughout the length of the segment. The step of integration of equations (6.30) using the Runge-Kutta method was chosen equal '- (2S  3S 2) 16,000 , reasoning from the condition of calculations convergence. First, the dependence of the resistance forces on the drill string motion in the borehole in the presence of imperfections for the lifting operation was considered. Table 6-1 presents the results of calculations for cases when the centres of imperfections with relatively small amplitude hc 5 m and with the pitch O 92.2 m are located in the middle of the first, second, third, and fourth quarters of the borehole centreline length (Fig. 6-5, a, b, c, d, respectively). It presents distance value sc from

initial point s 0 - 3S 2 to the central point of imperfection; the value of axial force Fw S at point s S of suspension of the drill string; the value of relation Fw S Pt of this magnitude to value Pt of the total gravity of the drill string found taking into account buoyant effect of the drilling liquid; total elastic elongation 'S of the drill string; the amount of torque M w S at the point of suspension and angle M S of elastic twist of the drill string. They indicate that when the imperfections zone is transferred to more curved segments of the borehole centreline, force Fw S and torque M w S significantly increase (positions 1–5 in Table 6-1 for tripping out the drill string). It is interesting to compare value Fw S with total gravity Pt acting on the drill string and calculated taking into account the action of the buoyant force of the drilling fluid. It is determined by the formula Pt 9.81S ˜ F J st  J l 2,856,303 N . It must be emphasised that in cases where the imperfections are located in the second, third, and fourth quarters, it is necessary to apply force Fw S and torque M w S in the tripping-out operation that is 35 and 100 times higher than their value in a borehole with ideal (design) geometry. Therefore, it can be assumed that in these cases the tripping-out operation is impossible to perform, and the situations in question will be abnormal. Table 6-1 also gives the value

'S

2S

1 Fw (- ) Dd- , EF 3S³ 2

(6.43)

which is equal to the total elastic elongation of the centreline of the drill string and the value

M S

2S

1 M w (- ) DdGI w 3S³ 2

(6.44)

Chapter 6. Modelling resistance forces and drill string sticking effects…

z

x

311

z

x

s S 2S

Localised imperfections

-

H s

0 3S 2

-

gas bearing bed

sc

L

а

b

z

x

z

x

c

d

Fig. 6-5 Geometries of the trajectories for different positions of the location of harmonic imperfections

Table 6-1 Values of forces, moments, and displacements at the suspension point of the borehole for different positions of harmonic imperfections (tripping-out operations) No.

sc

Fw S

(N)

1

(m) 0

1.870 ˜ 10 6

0.65

6.6

0.53 ˜103

M S (rad) 0.78

2

S/8 = 1,152

7.396 ˜ 10 6

2.59

39.3

5.18 ˜103

12.14

3

3S/8 = 3,457

34.710 ˜ 10

6

12.15

139.8

28.16 ˜10

3

47.08

4

5S/8 = 5,762

64.119 ˜ 10

6

22.45

150.9

52.90 ˜10

3

50.96

5

7S/8 = 8,067

71.200 ˜ 10 6

24.93

50.0

58.86 ˜103

15.84

Fw S Pt

'S (m)

M w S

(Nm)

Modelling emergency situations in drilling deep boreholes

312

equal to the total angle of the drill string twist. To analyse the effect of localised harmonic imperfections on the stress state of the drill string, Fig. 6-6, a, b give the graphs of longitudinal force Fw s and torque M w s , respectively, where sc 3S 8 . It can be seen that in the absence of imperfections (curves 1) functions Fw s and M w s also increase monotonically. However, the nature of changes in functions Fw s and M w s for boreholes with imperfections (curves 2) has changed significantly. Both graphs show that in the imperfection zone these functions rapidly increase, and the resulting geometric features are the main reason for the possible drill string sticking. This conclusion can also be confirmed by the graphs of the change of shearing forces Fu (s) , Fv (s) in Fig. 6-7, a, b, respectively. In the imperfection zone, these functions have the shape of oscillating curves with a significant amplitude, and they are smooth in the other part. The resulting shearing force FR

Fu 2  Fv 2

has a large spike in the

imperfections zone (Fig. 6-8). Outside the imperfections zone, this force is practically zero. It is interesting to trace the nature of changes to external forces that influence the deformation of the DS. Fig. 6-9 shows a graph of changes in distributed axial gravity

f wgr s (N/m). In the imperfections zone, this function has the shape of an oscillating curve, and it is smooth in the other part. However, distributed friction force f wfr s (N/m) caused by the contact interaction tangent to the centreline of the DS depends more on the imperfections, and its changes in the zone of distortion of the geometry occur with one big spike and small oscillations (Fig. 6-10, a). Similar changes occur with function graph mwfr s (N) (Fig. 6-10, b). Fig. 6-11 shows graphs of functions of

projections f ugr s (a), f vgr s (b) of the distributed gravity. They first take the form of smooth curves, but in the imperfections zone they begin to oscillate sharply with a significant frequency, then oscillations become less frequent. The influence of the imperfections of the centreline of the drill string on its geometric characteristics can also be noted. The function of changing the angle of elastic twisting F (rad) (Fig. 6-12) is smooth, after the appearance of the distortion of the centreline geometry it starts to increase monotonically. Twisting function r ( m 1 ) (Fig. 6-13) in the zone of imperfections increases dramatically. It is of interest to determine the impact of the geometric imperfections locations on the nature of changes in functions Fw s and M w s . To analyse it, Fig. 6-14 combines curves 1–5, the numbers of which correspond to the position numbers in Table 6-1. Curves 1 correspond to a borehole without geometric imperfections.

Chapter 6. Modelling resistance forces and drill string sticking effects…

Fw , MN

40

313

M w , Nm

30000

30

2

20000

2 20

10000

10

1

0 0

2000

4000

6000

8000

1 s, m

0

10000

0

2000

4000

6000

8000

s, m

10000

a b Fig. 6-6 Functions of longitudinal force Fw (a) and torque M w (b) (tripping-out operation): (1) = borehole without imperfections, (2) = borehole with imperfections (sc = 3,457 m)

Fu , N

4000

2000

2000

0

0

-2000

-2000

-4000 0

2000

Fv , N

4000

4000

6000

8000

s, m

-4000

10000

0

2000

4000

6000

8000

s, m

10000

a b Fig. 6-7 Functions of shearing forces Fu (a) and Fv (b) FR , N

4000

f wgr , N / m

0

3000

-100

2000

-200

1000

-300

0 0

2000

4000

6000

8000

s, m

10000

Fig. 6-8 Function of resulting shearing force FR

-400 0

2000

4000

6000

8000

s, m

10000

Fig. 6-9 Function of the distributed gr longitudinal force of gravity f w

Modelling emergency situations in drilling deep boreholes

f wfr , N / m

0

0

314

mwfr , N

-10000 -10

-20000 -20

-30000

s, m

-40000 0

2000

4000

6000

8000

s, m

-30 0

10000

2000

4000

а

f ugr , N / m

f wfr

0

0

-200

-200

s, m

-400 4000

f vgr , N / m

400

200

2000

6000

10000

fr (a) and moment mw of friction forces (b)

200

0

8000

b

Fig. 6-10 Functions of distributed friction force

400

6000

8000

s, m

-400 0

10000

2000

4000

6000

8000

10000

a b gr gr Fig. 6-11 Functions of components f u (a) and f v (b) of distributed gravity force

F , rad

50

r , m 1

0.012

40

0.008 30

20

0.004 10

s, m

0 0

2000

4000

6000

8000

10000

Fig. 6-12 Function of elastic twisting angle F

s, m

0 0

2000

4000

6000

8000

10000

Fig. 6-13 Twisting function r of centreline

Chapter 6. Modelling resistance forces and drill string sticking effects…

315

It should be noted that in all cases in the segments of the variable s change containing localised harmonic imperfections function Fw s increases rapidly (curves 2– 5 compared to curve 1 in Fig. 6-14, a. Moreover, if the imperfections are localised on an almost horizontal segment of the borehole (curve 2 in Fig. 6-14, a corresponds to Fig. 65, a), the function receives a relatively small increase. If they are located in the second quarter (Fig. 6-5, b), then on curve 3, in their localisation, there appears a segment of faster growth. With further transfer of imperfections (Fig. 6-5, c, d), the magnitude of the jumps of force function Fw becomes even larger (curves 4, 5 in Fig. 6-14, a). In these cases, the forces of resistance are the most sensitive to imperfections. Similar patterns can be traced for function M w s (N/m) presented in Fig. 6-14, b. FFww,,HN

80000000

M Mww,,Hм Nm

60000

4

60000000

4

40000

5

5 40000000

3

3 20000 20000000

2

2

1 0 0

2000

4000

6000

8000

s, m

10000

1 0 0

2000

4000

6000

8000

s, m

10000

a b Fig. 6-14 Functions of longitudinal force Fw (a) and torque M w (b) (tripping-out operations) (1) = without DS centreline imperfections (2) = with imperfections at point sc = S/8 = 1,152 m (3) = with imperfections at point sc = 3S/8 = 3,457 m (4) = with imperfections at point sc = 5S/8 = 5,762 m (5) = with imperfections at point sc = 7S/8 = 8,067 m

At first glance, it seems contradictory to reduce the deformability of the drill string if the imperfections are located in the fourth quarter (values 'S and M S for positions 5 in Table 6-1 compared with other cases) despite the sharp increase in values Fw S and M w S . This peculiarity, however, is explained by the shift of functions Fw S and M w S growth areas on curves 5 (Fig. 6-12, a, b) to the edge and the associated reduction of the zone of intense deformation of the drill string. It can be expected that the nature of the mechanical behaviour of the drill string undergoes significant changes in the transition from its tripping out to tripping in. This

Modelling emergency situations in drilling deep boreholes

316

feature can be explained by the redistribution of gravity and resistance forces acting on the string and changes in their balance. Indeed, if the tripping-out operation of the drill string is resisted by the forces of gravity acting on it, which are added to the resistance forces, the tripping-in operation has the opposite effect. In this case, according to the results of numerous studies, the gravitational forces contribute to the axial motion of the string and to some extent compensate for the resistance forces. Table 6-2 and Figs. 6-15–6-23 represent the values of the most characteristic integral values determining longitudinal force Fw and torque M w at point of drill string

suspension s S as well as total elastic elongation 'S and total angle of twist M S for Q 100 during the tripping in of the drill string in the borehole with harmonic imperfections with amplitude hc 5 m and pitch O 92.2 m . Table 6-2 Values of forces, moments, and displacements at the suspension point of the borehole for different positions of harmonic imperfections (tripping-in operation) No. 1 2 3 4 5

sc (m) 0 S/8 = 1,152 3S/8 = 3,457 5S/8 = 5,762 7S/8 = 8,067

Fw S (N)

0.726 ˜ 10 6 0.692 ˜ 10 6 0.607 ˜ 10 6 0.474 ˜ 10 6 0.176 ˜ 10 6

Fw S Pt 0.25 0.24 0.21 0.17 0.06

'S (m)

M w S (Nm)

2.09 1.80 1.35 1.11 1.30

0.431 ˜103 0.460 ˜103 0.532 ˜103 0.644 ˜103 0.894 ˜103

M S (rad) 0.80 0.90 1.06 1.14 1.08

We considered cases where there are no imperfections, and their central points are located in cross-sections s S / 8 , 3S / 8 , 5S / 8 , and 7S / 8 (Fig. 6-5, a, b, c, d, respectively, and positions 1–5 in Table 6-2) for boreholes with horizontal distance of L = 8,000 m and depth H = 4,000 m. Torque M w (S ) and angle of elastic twisting M (S ) with the introduction of imperfections also significantly increase. As it turned out, the added imperfections significantly affect value 'S , which essentially decreases with their complication. Fig. 6-15, a, b show the features of the change of values Fw and M w along length s for case sc = 3S/8 = 3,457 m during tripping in. For a borehole with ideal design geometry (curves 1), functions Fw (s) and M w (s) monotonically increase. The distortion of the centreline geometry causes the local maximum and then the local minimum of function Fw (s) (curve 2 Fig. 6-15, a), after which the function begins to increase again.

Chapter 6. Modelling resistance forces and drill string sticking effects…

Fw , N

700000 600000

M w , Nm

600

1

317

2

500000 400

400000

1

2

300000 200

200000 100000

s, m

0 0

2000

4000

6000

8000

s, m

0 0

10000

2000

4000

6000

8000

10000

a b Fig. 6-15 Functions of longitudinal force Fw (a) and torque M w (b) (tripping-in operation) (1) = without imperfections, (2) = with imperfections (sc = 3,457 m)

FF ,N u ,uN

4000

2000

400

0

0

-2000

-400

s, m

-4000 0

2000

Fv , N

800

4000

6000

8000

s, m

-800

10000

0

2000

4000

6000

8000

10000

a b Fig. 6-16 Functions of shearing forces Fu (a) and Fv (b)

FR , N

4000

3000

-100

2000

-200

1000

-300

s, m

0 0

2000

f wgr , N / m

0

4000

6000

8000

10000

Fig. 6-17 Function of resulting shearing force FR

s, m

-400 0

2000

4000

6000

8000

10000

Fig. 6-18 Function of the distributed gr longitudinal force of gravity f w

Modelling emergency situations in drilling deep boreholes

f wfr , N / m

400

318

mwfr , N

0

300 -0.1

200

-0.2

100

s, m

0 0

2000

4000

6000

8000

s, m

-0.3

10000

0

2000

4000

6000

8000

10000

a b fr Fig. 6-19 Functions of distributed friction force f w (a) and fr moment mw of friction forces (b)

f ugr , N / m

400

200

100

0

0

-200

-100

s, m

-400 0

2000

4000

f vgr , N / m

200

6000

8000

s, m

-200 0

10000

2000

4000

6000

8000

10000

a b gr gr Fig. 6-20 Functions of components f u (a) and f v (b) of distributed gravity force

F , rad

1.2

r , m 1

0.00025

0.0002 0.8

0.00015

0.0001 0.4

5E-005

s, m

0 0

2000

4000

6000

8000

10000

Fig. 6-21 Function of elastic twisting angle F

s, m

0 0

2000

4000

6000

8000

10000

Fig. 6-22 Twisting function r of the centreline

Chapter 6. Modelling resistance forces and drill string sticking effects…

319

Quite a different behaviour can be seen in function M w (s) (curve 2, Fig. 6-15, b). It increases on the entire length and increases even faster in the imperfections zone. Functions of shearing forces Fu and Fv are also sensitive to the imperfections (Fig. 6-16, a, b). In the zone where geometric distortions appear, these functions begin to oscillate intensively. Outside the imperfections zone, functions Fu (s) , Fv (s) almost equal zero. Resulting shearing force FR

Fu 2  Fv 2 (N) also has a large spike in

the imperfections zone (Fig. 6-17). These peculiar properties of changes in internal forces and moments are mainly due to the appearance of oscillations in the imperfections zones of distributed axial friction forces f wfr (s) and moments of friction forces mwfr (s) (Fig. 6-19), the distributed longitudinal force of gravity f wgr (s) (Fig. 6-18), and the components of gravity force f ugr (s) , f vgr (s) (Fig. 6-20 a, b). The influence of the imperfections of the borehole centreline is also noticeable for the deformation characteristics of the drill string. Therefore, the function of changing the angle of elastic twisting F (Fig. 6-21) is smooth, after the appearance of distortion of the centreline geometry it starts to increase monotonically. Twisting function r (m) (Fig. 6-22) in the imperfections zone also increases steeply. Fig. 6-23, a, b show the features of the change of values Fw and M w along length s for the considered case L = 8,000 m, H = 4,000 m, hc = 5 m, with pitch O 92.2 m , Q 100 , and localised imperfections located in the middle of the first, second, third, and fourth quarters of the length of the centreline of the borehole (Fig. 65, a, b, c, d, respectively). The numbers of curves 1–5 correspond to positions 1–5 in Table 6-2. Analysis of the presented results leads to the conclusion that in a borehole with an ideal geometry the function of longitudinal force Fw (s) increases monotonically with an increase of s (curve 1 in Fig. 6-23, a). However, when approaching the imperfection localisation zones, these functions reach a local maximum and begin to decrease rapidly, reaching local minima (curves 2– 5). Then, these curves increase again. Another trend has a place for torque function

M w (s) (Fig. 6-23, b). Here, all the curves 1–5 have outlines of increasing functions, and they increase most rapidly in segments containing imperfections. It is interesting to juxtapose the results of the analysis for the processes of the drill string tripping-in/out operations. By comparing Tables 6-1 and 6-2, we can see that a much greater force Fw and torque M w must be applied to perform the tripping-out operation.

Modelling emergency situations in drilling deep boreholes

Fw , N

800000

1

320

M w , Nm

1000

5 2 600000

800

3

4 600

3

4

400000

2 400

1 200000

200

5

s, m

0 0

2000

4000

6000

8000

10000

s, m

0 0

2000

4000

6000

8000

10000

a b Fig. 6-23 Functions of longitudinal force Fw (a) and torque M w (b) (tripping-in operation) (1) = without DS centreline imperfections

(2) = with imperfections at point sc = S/8 = 1,152 m (3) = with imperfections at point sc = 3S/8 = 3,457 m (4) = with imperfections at point sc = 5S/8 = 5,762 m (5) = with imperfections at point sc = 7S/8 = 8,067 m Therefore, it should be kept in mind that if the drilling and string tripping-in operations are performed without complications, it does not mean that they will be excluded when the string is tripped out. Therefore, when performing technological drilling operations, it is always necessary to check the possibility of performing the subsequent tripping out of the string. 6.4.2 Calculation results for a borehole with an elliptical trajectory Boreholes with the centreline geometry approaching the arc of an ellipse are less common in practice because even at their ends they have variable curvature. However, there are also boreholes that approach such trajectories in practice. Consider the case when the design outline of the borehole centreline has the shape of a quarter of an ellipse x L cos- , y 0 , z H sin - 3S 2 d - d 2S . (6.45) Here, semi-axes H and L are equal to the depth of the borehole and the horizontal distance of its end from the drilling rig (Fig. 6-24, a). For this geometry, the above problem of determining the internal and external force factors acting on the drill string during its motion is first solved. Then, the local distortion in the form of harmonic with pitch O and variable amplitude h(s) , which is given by expression (6.41), is introduced into the geometry of the borehole centreline.

Chapter 6. Modelling resistance forces and drill string sticking effects…

321

z

x

H

0

L а

z

x

0

b Fig. 6-24 Geometries of the borehole with elliptical centreline a) without imperfections b) with localised harmonic imperfections

In this case, the equations of the borehole centreline with harmonic imperfections acquire the final form (Fig. 6-24, b):

x L ˜ cos-  hce y

§ s  sc · D 2 ¨ ¸ © S ¹

-

³ D- dT

3S 2

˜ cosks ˜

H cosL sin -  H 2 cos2 2

2

,

(6.46)

0,

z H ˜ sin -  hce where s

2

-

³ 3S

§ s  sc · D 2 ¨ ¸ © S ¹

2

˜ cosks ˜

L sin L sin -  H 2 cos2 2

2

,

L2 sin 2 T  H 2 cos2 T ˜ dT , the total length of the centreline is

2 2S

S

³ 3S 2

L2 sin 2 T  H 2 cos2 T ˜ dT .

(6.47)

Modelling emergency situations in drilling deep boreholes

322

With these equations, the curvature and twisting are calculated by formulas (6.16), (6.17). Then, metric factor D is calculated by equality (6.14), which allows us to go to the natural parameterisation of curve (6.1), calculate unit vectors τ , n , b of Frenet trihedron, and move to the integration of system (6.30). When performing tripping in/out operations with a given ratio Q

w of Z d /2

linear velocities of the axial and rotational motions of the element of the outer surface of the drill string pipe, desired functions F , r , Fw equal zero at - 3S 2 , but gravity force (6.31) differs from zero. This allows us to calculate variables f uc , f vc , f wfr , mwfr ,

Fu , Fv at this point, make one step of the integration of the first three equations of system (6.30), and then follow the same scheme to continue the solution of the system in the next steps to point - 2S . Integration step '- is selected from the condition of convergence of calculations. Computer simulation of elastic bending of the drill string in an elliptical borehole with harmonic imperfections is performed based on the developed technique with the following values of the characteristic parameters: L = 4,000 m, H = 2,000 m are the semi-axes of the ellipse, d = 0.1683 m is the outer diameter of the drill string pipe, G = 0.01 m is the pipe thickness, E = 2.1·1011 Pa is the modulus of elasticity, G = 0.8077·1011 Pa, J st = 7,850 kg/m3 is the density of steel, J l = 1,500 kg/m3 is the density of the drilling liquid, P = 0.2 is the friction coefficient, Q 100 , 25, and 1. With this initial data, the length of the borehole centreline calculated by formula (6.47) equals S 4,844 m . Parameter D that determines the rate of decrease h- in variable - , as before, was set to 10. In this case, a representative range of s change, in which amplitude h- has a significant value, equals l = 1,593 m. It should be noted, however, that although outside this interval radius h- is negligibly small, nevertheless, as above, the distortion of the centreline of the borehole caused by imperfection was considered throughout the segment 3S 2 d - d 2S . The dependence of the resistance forces on coordinate sc of the imperfection centre is investigated. To do this, calculations of the tripping-out operation in the states in which point sc is located in the middle of the first, second, third, and fourth quarters of the ellipse (Fig. 6-25, a, b, c, d, respectively) were made.

Chapter 6. Modelling resistance forces and drill string sticking effects…

z

x

323

z

x

s S

-

Localised imperfection

2S

H s

0 3S 2

-

sc

gas bearing bed

L

а

z

b

x

c

z

x

d

Fig. 6-25 Geometries of the trajectories for different positions of harmonic imperfections locations

Cases when the harmonic wavelength of the imperfection was O = 96.88 and 48.44 m were considered. For the first case, the numerical integration increment was chosen equal '- 2S  3S 2 8,000 , for the second, '- 2S  3S 2 16,000 . Table

6-3 represents the found values of longitudinal force Fw S and torque M w S at the point of suspension of the drill string as well as values Fw S P t , 'S , M S for different values hc at O 96.88 m and the location of the centre of imperfection in the middle of the first quarter of the length of the drill string ( sc S / 8 606 m ; Fig. 6-25, a.

In relation Fw S P t , denominator P t is equal to the difference of gravity force of the entire DS and the buoyancy force of the drilling fluid Pt F J t  J l S 1,500,713 N . It is equal to the force that needs to be applied to the upper end of the drill string to keep it stationary in the channel of the corresponding vertical borehole filled with drilling fluid. Then, relation Fw S P t shows what part of force P t is axial force Fw S , if you start to lift the string placed in a curvilinear borehole. Value 'S presented in Table 6-3 is equal to the total elastic elongation of the drill string during lifting calculated by formula (6.43). Value M S is equal to full angle (6.44) of elastic twisting of the drill string.

Modelling emergency situations in drilling deep boreholes

324

Table 6-3 Values of forces, moments, and displacements at the point of drill string suspension (harmonic imperfections; tripping-out operation; L = 4,000 m, H = 2,000 m, sc S 8 606 m , O 96.88 m , Q 100 )

hc

Fw S

(m)

(N)

Fw S Pt

'S (m)

M w S

1

0.0

0.882 ˜10

6

0.59

1.43

221

M S (rad) 0.20

2

0.3

0.882 ˜106

0.59

1.43

221

0.20

0.5

0.882 ˜10

6

0.59

1.43

221

0.20

0.882 ˜10

6

0.59

1.43

221

0.20

0.891 ˜10

6

0.59

1.47

229

0.22

6

0.64

1.69

282

0.29

0.67

1.84

320

0.35

No.

3 4 5

1.0 2.0

6

4.0

0.954 ˜10

7

5.0

0.999 ˜106

(Nm)

Analysis of data in Table 6-3 allows us to conclude that with an increase in the amplitude of the imperfections to hc 5 m (position 7), force Fw S and torque M w S increase, but their values remain acceptable, and the operation for extracting the drill string from the well is feasible. For comparison, it can be noted that Fig. 6-25 shows harmonic imperfections with value hc 15 m because even with hc 5 m these imperfections on the charts are visually hardly noticeable. If the wavelength of the imperfections is reduced to O 48.44 m, their effect on the ability to perform the tripping-out operation increases (Table 6-4). For example, already at hc 2 m , sc S / 8 , force Fw S is almost 3 times greater than the value for the string in the borehole without imperfections, and torque M w S increases almost sevenfold (positions 5 and 1, respectively). In the case of hc=4 m (position 6), these force factors become so large that the tripping-out operation of the drill string becomes impossible. This state of the system can be considered abnormal. Fig. 6-26, a, b show the features of the change of values Fw and M w along length s for case sc S 8 606 m , Q 100 , hc 2 m , O 48.44 m that corresponds to position 5 in Table 6-4 during tripping-out operations. For a borehole with ideal design geometry (curves 1), functions Fw (s) and M w (s) monotonically increase.

Chapter 6. Modelling resistance forces and drill string sticking effects…

325

Table 6-4 Values of forces, moments, and displacements at the point of drill string suspension (harmonic imperfections; tripping-out operation; L = 4,000 m, H = 2,000 m, sc S 8 606 m , O 48.44 m , Q 100 ) No.

1 2 3 4

hc

Fw S

(m)

(N)

0.0

0.882 ˜106

0.3

0.890 ˜10

6

0.5

0.926 ˜10

6

1.121 ˜10

6

2.432 ˜10

6

1.0

5

2.0

6

4.0

7

Fw S Pt

5.0

12.341 ˜10

6

18.472 ˜10

6

0.59 0.59 0.62 0.75 1.62 8.23 12.31

Fw , N

2500000

M w S

'S (m)

M S (rad)

(Nm)

1.43

0.221 ˜103

0.20

1.46

0.228 ˜10

3

0.21

0.258 ˜10

3

0.26

0.424 ˜10

3

0.50

3

1.95

3

12.37

1.59 2.27 6.46 36.40 54.71

1.524 ˜10

9.867 ˜10

15.025 ˜10

3

18.74

M w , Nm

1600

2000000 1200

2

2

1500000 800

1000000

1

400

500000

1 s, m

0 0

1000

2000

3000

4000

5000

s, m

0 0

1000

2000

3000

4000

5000

a b Fig. 6-26 Functions of longitudinal force Fw (a) and torque M w (b) (tripping-out operation) (1) = without imperfections, (2) = with imperfections ( sc 606 m)

However, the nature of changes in functions Fw s and M w s for a borehole with imperfections (curves 2) changed significantly. Both graphs show that in the imperfections zone these functions rapidly increase, and the resulting geometric features are the main reason for the possible sticking of the drill string. They strongly oscillate in the imperfections zone. After that, they become almost zero again. Such regularities are also observed for the function of resulting shearing force FR

Fu 2  Fv 2 .

Modelling emergency situations in drilling deep boreholes

326

The above calculation results are obtained for motion modes in which the axial sliding of the drill string and its rotation are combined. In addition, the relation of the  Z d 100 . axial and rotational speeds is chosen equal Q 2w This conclusion can also be confirmed by the graphs of the change of shearing forces Fu (s) , Fv (s) in Fig. 6-27, a, b, respectively. Fu , N

15000

10000

400

5000

200

0

0

-5000

-200

-10000

-400

s, m

-15000 0

1000

Fv , N

600

2000

3000

4000

5000

s, m

-600 0

1000

2000

3000

4000

5000

a b Fig. 6-27 Functions of shearing forces Fu (a) and Fv (b)

Therefore, it is of interest to analyse the axial motion of the drill string in the borehole channel and select a magnitude Q minimising these values. In this regard, with the same values of the characteristic parameters of the system and the geometric characteristics of the imperfections, calculations were made for tripping out the drill string at values Q 100 , 25, and 1. These results are summarised in Tables 6-5–6-7, respectively. It is known that the choice of different values of magnitude Q can adjust the values of axial force Fw and torque M w performing this regime. The calculations showed that the forces of motion resistance of the drill string in the borehole to a considerable extent also depend on the position of the zones of imperfections on the borehole centreline. Table 6-5 presents the results of calculations for cases when at Q 100 the centres of imperfections with relatively small amplitude hc 2 m are located in the middle of the first, second, third, and fourth quarters of the borehole centreline (Fig. 6-25, a, b c, d, respectively). They show that with the transfer of the imperfections zone to more curved segments of the borehole centreline force Fw S and torque M w S significantly grow (positions 1–4 in Table 6-5 for O 96.88 m and positions 5–8 for O 48.44 m ).

Chapter 6. Modelling resistance forces and drill string sticking effects…

327

Table 6-5 Values of forces, moments, and displacements at the point of drill string suspension (harmonic imperfections; tripping-out operation; hc 2 m , Q 100 ) No. 1 2 3 4 5 6 7 8

O (m)

sc

Fw S

(m)

(N)

96.88

S8

96.88

3S 8

96.88

5S 8

96.88

7S 8

48.44

S8

48.44

3S 8

48.44

5S 8

48.44

7S 8

Fw S Pt

0.89 ˜10

6

1.05 ˜10

6

1.39 ˜10

6

1.65 ˜10

6

2.43 ˜10

6

6.47 ˜10

6

12.01 ˜10

6

15.92 ˜10

6

0.59 0.70 0.93 1.10 1.62 4.31 7.97 10.59

'S (m)

M w S

M S (rad)

(Nm)

1.47

0.23 ˜ 103

0.22

1.84

0.36 ˜ 10

3

0.35

0.65 ˜ 10

3

0.46

0.86 ˜ 10

3

0.34

1.52 ˜10

3

1.95

4.93 ˜10

3

4.56

9.54 ˜10

3

5.29

12.92 ˜10

3

2.12

2.17 1.83 6.46 13.94 16.04 6.96

Table 6-6 Values of forces, moments, and displacements at the point of drill string suspension (harmonic imperfections; tripping-out operation; hc 2 m , Q 25 )

O (m) 96.88

sc

Fw S

(m)

(N)

96.88

3S 8

96.88

5S 8

96.88

7S 8

5

48.44

6 7

No. 1 2 3 4

8

S8

Fw S Pt

0.89 ˜ 10

6

1.05 ˜ 10

6

1.39 ˜ 10

6

1.65 ˜ 10

6

S8

2.42 ˜ 10

6

1.62

48.44

3S 8

6.46 ˜ 106

48.44

5S 8

11.92 ˜ 106

48.44

7S 8

6

15.97 ˜ 10

'S (m)

M w S (Nm)

M S (rad)

1.47

0.92 ˜ 103

0.87

1.84

3

1.38

3

1.84

3

1.37

6.45

3

6.07 ˜ 10

7.79

4.30

13.91

19.64 ˜ 103

18.17

7.95

16.01

38.05 ˜ 103

21.09

6.95

3

8.48

0.59 0.70 0.93 1.10

10.57

2.17 1.83

1.43 ˜ 10 2.60 ˜ 10 3.45 ˜ 10

51.30 ˜ 10

Modelling emergency situations in drilling deep boreholes

328

Table 6-7 Values of forces, moments, and displacements at the point of drill string suspension (harmonic imperfections; tripping-out operation; hc 2 m , Q 1 ) No. 1 2 3 4 5 6 7 8

hc

O (m)

sc

Fw S

(m)

(N)

96.88

S8

96.88

3S 8

96.88

5S 8

96.88

7S 8

48.44

S8

48.44

3S 8

48.44

5S 8

48.44

7S 8

0.804 ˜ 10

6

0.875 ˜ 10

6

1.061 ˜ 10

6

1.232 ˜ 10

6

1.281 ˜ 10

6

2.415 ˜ 10

6

4.191 ˜ 10

6

5.880 ˜ 10

6

M w S

Fw S Pt

'S (m)

0.54

1.28

15.55 ˜ 103

14.95

1.46

3

21.21

3

28.08

3

22.67

3

73.65

3

149.74

3

181.07

3

88.84

0.58 0.71 0.82 0.85 1.61 2.79 3.92

1.65 1.50 2.96 5.15 6.05 3.40

M S (rad)

(Nm) 21.50 ˜ 10 37.16 ˜ 10 51.49 ˜ 10 55.67 ˜ 10 151.11 ˜ 10 300.53 ˜ 10 442.69 ˜ 10

Fig. 6-28,a shows dependencies Fw s for the imperfections with amplitude 2 m and wavelength O 96.88 m and with the ratio of the linear velocities of the

axial and rotational motions Q 100 at different sc . The numbers of curves 2–5 correspond to the numbers of positions 1–4 in Table 6-5. It should be noted that if the imperfections are localised on an almost horizontal part of the well (curve 2 in Fig. 6-28,a corresponds to Fig. 6-25,a), function Fw s is smooth. If they are located in the second quarter (Fig. 6-25, b), then on curve 3, in their localisation, there is a segment of rapid growth. With further transfer of imperfections (Fig. 6-25, c, d), the magnitude of the function Fw s jumps on them become even greater (curves 4, 5 in Fig. 6-28, a). Similar rules can be traced for function M w s (Fig. 6-28, b), only the values of the jumps of this function turn out to be larger. With a decrease in the wave pitch of imperfections to O 48.44 m , the marked features of the external and internal forces balance are more noticeable (positions 5–8 in Table 6-5). This can be seen on the graphs of longitudinal force Fw s (Fig. 6-29, a) and torque M w s (Fig. 6-29, b). Analysing the data in Tables 6-5–6-7, it can be concluded that with a decrease in the relation of the linear velocities of the axial and rotational motions Q =100, 25, and 1, respectively, the value of longitudinal force Fw at suspension point of the drill string s S decreases , and torque value M w at this point increases significantly.

Chapter 6. Modelling resistance forces and drill string sticking effects…

Fw , N

1800000

329

M w , Nm

900

5

5

4

1200000

600

4

3

3

2 1

600000

2

300

1 s, m

0 0

1000

2000

3000

4000

s, m

0 0

5000

1000

2000

3000

4000

5000

a b Fig. 6-28 Functions of longitudinal force Fw (a) and torque M w (b) (tripping-out operation; hc 2 m ; O 96.88 m ; Q 100 ) (1) = without DS centreline imperfections (2) = with imperfections at point sc = S/8 = 606 m (3) = with imperfections at point sc = 3S/8 = 1,817 m (4) = with imperfections at point sc = 5S/8 = 3,028 m (5) = with imperfections at point sc = 7S/8 = 4,239 m

Fw , N

16000000

M w , Nm

14000

5 12000000

4

12000

5

10000

4

8000 8000000

3 2 1

4000000

0 0

1000

2000

3000

4000

6000

2000

s, m

5000

3

4000

0 0

1000

2000

3000

4000

2 1 s, m

5000

a b Fig. 6-29 Functions of longitudinal force Fw (a) and torque M w (b) (tripping-out operation; hc 2 m ; O 48.44 m ; Q 100 ) (1) = without DS centreline imperfections (2) = with imperfections at point sc = S/8 = 606 m (3) = with imperfections at point sc = 3S/8 = 1,817 m (4) = with imperfections at point sc = 5S/8 = 3,028 m (5) = with imperfections at point sc = 7S/8 = 4,239 m

Modelling emergency situations in drilling deep boreholes

Fw , N

2000000

330

M w , Nm

8000

5

1600000

6000

5

4

1200000

3 800000

3

2 1

400000

0

1000

2000

3000

4000

2 1 s, m

2000

s, m

0

4

4000

0

5000

0

1000

2000

3000

4000

5000

a b Fig. 6-30 Functions of longitudinal force Fw (a) and torque M w (b) (tripping-out operation; hc 2 m ; O 96.88 m ; Q 25 ) (1) = without DS centreline imperfections

(2) = with imperfections at point sc = S/8 = 606 m (3) = with imperfections at point sc = 3S/8 = 1,817 m (4) = with imperfections at point sc = 5S/8 = 3,028 m (5) = with imperfections at point sc = 7S/8 = 4,239 m Fw , N

100000000

M w , Nm

60000

5

80000000

4

5 40000

4

60000000

40000000

3

3 20000

2

20000000

2 1 s, m

1 0 0

1000

2000

3000

4000

s, m

5000

0 0

1000

2000

3000

4000

a b Fig. 6-31 Functions of longitudinal force Fw (a) and torque M w (b) (tripping-out operation; hc 2 m ; O 48.44 m ; Q 25 ) (1) = without DS centreline imperfections

(2) = with imperfections at point sc = S/8 = 606 m (3) = with imperfections at point sc = 3S/8 = 1,817 m (4) = with imperfections at point sc = 5S/8 = 3,028 m (5) = with imperfections at point sc = 7S/8 = 4,239 m

5000

Chapter 6. Modelling resistance forces and drill string sticking effects…

Fw , N

1400000

M w , Nm

100000

5 4 3

1200000 1000000 800000

2 1

600000

331

80000

5 60000

4 3

40000

2

400000 20000 200000

s, m

0 0

1000

2000

3000

4000

1 s, m

0

5000

0

1000

2000

3000

4000

5000

a b Fig. 6-32 Functions of longitudinal force Fw (a) and torque M w (b) (tripping-out operation; hc 2 m ; O 96.88 m ; Q 1 ) (1) = without DS centreline imperfections

(2) = with imperfections at point sc = S/8 = 606 m (3) = with imperfections at point sc = 3S/8 = 1,817 m (4) = with imperfections at point sc = 5S/8 = 3,028 m (5) = with imperfections at point sc = 7S/8 = 4,239 m Fw , N

6000000

M w , Nm

500000

5

5

400000

4

4000000

300000

4

3

3

200000

2 1

2000000

s, m

0 0

1000

2000

a

3000

4000

5000

2 1 s, m

100000

0 0

1000

2000

3000

4000

5000

b

Fig. 6-33 Functions of longitudinal force Fw (a) and torque M w (b) (tripping-out operation; hc 2 m ; O 48.44 m ; Q 1 ) (1) = without DS centreline imperfections

(2) = with imperfections at point sc = S/8 = 606 m (3) = with imperfections at point sc = 3S/8 = 1,817 m (4) = with imperfections at point sc = 5S/8 = 3,028 m (5) = with imperfections at point sc = 7S/8 = 4,239 m

Modelling emergency situations in drilling deep boreholes

332

The graphs of functions Fw s and M w s for different positions of borehole centreline imperfections at wave pitch O 96.88 m and Q =100, 25, and 1 are shown in Figs. 6-28, 6-30, 6-32, respectively. On them, curves 2–5 correspond to positions 1–4 in Tables 6-5–6-7. Similar regularities are observed with decreasing of the wavelength to O 48.44 m . At Q =100, 25, and 1, the longitudinal force Fw and torque M w dependences are shown in Figs. 6-29, 6-31, and 6-33. Curves 2–5 match positions 5–8 in Tables 6-5–6-7. It can be concluded that, depending on Q , functions Fw s and

M w s practically do not change their outlines, but the values of functions Fw s are reduced with decreasing of Q , and the function M w s values, on the contrary, increase

several fold. We must emphasise again that not only do the developed methods of computer modelling allow us to analyse the influence of the parameters of geometric imperfections and their location on the forces of resistance to the displacement of drill strings, but they also establish their critical states in which the tripping-out operation becomes impossible. As noted previously, the drill string tripping-in operation is different from the tripping-out operation because the direction of gravitational forces in this case contributes to the process of the axial motion of the string and is generally accompanied by smaller values of forces Fw s and torques M w s . However, for this procedure, it is possible that the generated friction forces are so large that they balance the gravity forces, and the tripping-in operation becomes impossible. This happens when rated force Fw at suspension point s S becomes negative, and for the implementation of the tripping-in process it is necessary not to hold but rather push the string down. To perform computer simulation of the tripping-in process, it is necessary—as was done before for the hyperbolic well—to specify the appropriate directions and signs of friction forces and their moments. Table 6-8 presents the values of the most characteristic integral values determining longitudinal force Fw and torque M w at point of suspension s S of the drill string during tripping in as well as total elastic elongation 'S and total angle of twist M S for Q 100 when tripping in the drill string in an elliptical well described by formulas (6.46). Calculations are made for different values of hc at O 48.44 and location of the centre of imperfection in the middle of the first quarter of the length of the drill string (sc = S/8 = 1,211 m; Fig. 6-25, a), with horizontal extension L = 8,000 m and depth of H = 4,000 m. From the results, it follows that the amplitude hc values selected from 0 to 1 m (positions 1–4 in Table 6-8) have almost no effect on the values of the integral values presented in the table, and they remain almost unchanged.

Chapter 6. Modelling resistance forces and drill string sticking effects…

333

Table 6-8 Values of forces, moments, and displacements at the point of drill string suspension (harmonic imperfections; tripping-in operation; L = 8,000 m, H = 4,000 m, O 48.44 m , sc = S/8 = 1,211 m, Q Fw S Pt

100 )

No.

hc

Fw S (N)

1

(m) 0.0

M w S

0.74 ˜ 10 6

0.25

0.77

0.42 ˜103

M S (rad) 0.91

2

0.3

0.74 ˜ 10 6

0.25

0.77

0.42 ˜103

0.91

3

0.5

0.74 ˜ 10

6

0.25

0.77

0.42 ˜10

3

0.91

4

1.0

0.74 ˜ 10

6

0.25

0.42 ˜10

3

0.91

5

2.0

0.40 ˜ 10

6

0.13

-1.65

0.71 ˜10

3

1.75

6

4.0

-79.06 ˜ 10 6

-27.84

-493.91

71.38 ˜103

173.01

7

5.0

-292.90 ˜ 10 6

-97.59

-1712.53

248.05 ˜103

596.83

'S (m)

(Nm)

0.77

Table 6-9 Values of forces, moments, and displacements at the point of drill string suspension (harmonic imperfections; tripping-in operation; L = 8,000 m, H = 4,000 m, hc 5 m ,

O =96.88 m, sc = S/8 = 1,211 m) No.

Q

1 100 2 3

10

4

5

5

1

6

0.1

7

0

w Zd / 2 without imperfections with imperfections with imperfections with imperfections with imperfections with imperfections with imperfections

Fw S

'S (m)

M w S

(N)

Fw S Pt

0.740 ˜106

0.25

0.77

0.42 ˜103

0.314 ˜ 10

6

0.10

-2.28

0.332 ˜ 10 6

0.11

-2.17

0.381 ˜ 10 6

0.13

-1.85

0.864 ˜ 10 6

0.29

-1.38

1.188 ˜ 10 6

0.40

3.13

1.239 ˜ 10 6

0.41

3.38

M S (rad)

(Nm)

0.78 ˜10

3

0.91 1.97

7.63 ˜103

19.32

14.43 ˜103

36.43

31.56 ˜103

69.71

42.49 ˜103

88.04

44.05 ˜103

91.90

Modelling emergency situations in drilling deep boreholes

Fw , N

800000

334

M w , Nm Nm

16000

600000

1

12000

2

8000

400000 200000

2

1

0 -200000

4000

-400000

s, m

-600000 0

2000

4000

6000

8000

s, m

0 0

10000

2000

4000

6000

8000

10000

a b Fig. 6-34 Functions of longitudinal force Fw (a) and torque M w (b) (tripping-in operation; L = 8,000 m, H = 4,000 m, O 96.88 m ,Q 5 ) (1) = without imperfections; (2) = with imperfections sc = S/8 = 1,211 m)

Fu , N

4000

Fv , N

3000 2000

2000

1000 0

0 -1000

-2000

-2000

s, m

-4000 0

2000

4000

6000

8000

10000

s, m

-3000 0

2000

4000

6000

8000

10000

a b Fig. 6-35 Functions of shearing forces Fu (a) and Fv (b)

FR , N

4000

f wgr , N / m

100

0

3000

-100 2000

-200 1000

-300

0 0

2000

4000

6000

8000

s, m

10000

Fig. 6-36 Function of resulting shearing force FR

s, m

-400 0

2000

4000

6000

8000

10000

Fig. 6-37 Function of the distributed gr longitudinal force of gravity f w

Chapter 6. Modelling resistance forces and drill string sticking effects…

f wfr , N / m

mwfr , N

0

800

335

600 -4

400 -8 200

s, m

0 0

2000

4000

6000

8000

s, m

-12 0

10000

2000

4000

6000

8000

10000

a b fr Fig. 6-38 Functions of distributed friction force f w (a) and fr moment mw of friction forces (b)

F , rad

40

r , m 1

0.006

30

0.004 20

0.002 10

s, m

0 0

2000

4000

6000

8000

Fig. 6-39 Function of elastic twisting angle F

10000

s, m

0 0

2000

4000

6000

8000

10000

Fig. 6-40 Twisting function r of the centreline

But with growth in hc from 1 m (position 4) to 5 m (position 7), force Fw (S ) starts to decrease and, for values hc = 4 m (position 6) and 5 m (position 7), becomes generally negative. At the same time, torque M w (S ) begins to increase with an increase of hc . This means that the friction forces when the string is lowering have become so large that the string needs to be pushed with a great force. It can be concluded that in this case the tripping-in operation becomes unfeasible. It is interesting that the choice of different values of magnitude Q w Z d / 2 (the relation of linear velocities of the axial and rotational motions of the element of the

Modelling emergency situations in drilling deep boreholes

336

outer surface of the drill string pipe) can adjust axial force Fw and torque M w . Therefore, it is of interest to analyse the axial motion of the drill string in the borehole channel and select values Q that minimise these forces. In this regard, for the case L = 8,000 m, H = 4,000 m, hc 5 m with pitch O 96.88 m and localised imperfections with the centre at point sc = S/8 = 1,211 m, calculations were made for the tripping out the drill string at values Q 100 , 10, 5, 1, 0.1, and 0. These results are summarised in Table 6-9. It should be noted that with decreasing of value Q , longitudinal force Fw and torque M w at point of suspension S of the drill string become larger. The lowest value of Fw (S ) is reached at Q 100 . Fig. 6-34, a, b show the features of the change of values Fw and M w along length s for the case Q 5 (position 4 of Table 6-9). For a borehole with an ideal design geometry (curves 1), function Fw (s) declines monotonically, reaching its minimum value, and then begins to increase again. Function M w (s) (curve 1, Fig. 6-34, b) monotonically increases. The introduction of distortions in the geometry of the borehole centreline becomes the reason for the sharp drop of function Fw (s) in the area of the imperfections, after which the function increases again. This regime of tripping in can be considered unfavourable since the string is compressed in a large segment and can lose its stability. Quite a different behaviour can be seen in function M w (s) (curve 2, Fig. 6-34, b). It grows on the entire length and increases even faster in the imperfections zone. The functions of shearing forces Fu and Fv are sensitive to imperfections (Fig. 635, a, b). They strongly oscillate in the geometric distortions zone of the borehole centreline. Outside this segment, functions Fu (s) and Fv (s) almost equal zero. s

Resulting shearing force FR

Fu 2  Fv 2

also has a large spike in the imperfections

zone (Fig. 6-36). These properties of changes in internal forces and moments are mainly due to the appearance of oscillations in the imperfections zones of distributed longitudinal gravity force f wgr (s) (Fig. 6-37), distributed axial friction forces f wfr (s) (Fig. 6-38, a), and moments of friction forces mwfr (s) (Fig. 6-38, b). The influence of the tortuosities of the borehole centreline is also noticeable for the deformation characteristics of the drill string. Therefore, the function of changing angle of elastic twisting F (Fig. 6-39) is smooth, after the appearance of geometric distortions it starts to increase monotonically. Twisting function r (Fig. 6-40) also increases sharply in the imperfections zone.

Chapter 6. Modelling resistance forces and drill string sticking effects…

337

Fig. 6-41, a, b show the features of changes of functions Fw (s) and M w (s) for different values of Q , as before at L = 8,000 m, H = 4,000 m, hc 5 m with pitch O 96.88 m and localised imperfections with the centre at point sc = S/8 = 1,211 m. Curves 1–6 correspond to values Q 100 , 10, 5, 1, 0.1, and 0 (positions 2–7, Table 6-9). It should be noted that for function Fw (s) curves 1, 2 (Q 100 , 10) grow monotonically in contrast to curves 4–6 (Q 1, 0.1, 0) that in the zone reach the minimum negative values, then they grow. Graphs of function M w (s) monotonically increase. Fw , N

1200000

1 2

800000

3

50000

M w , Nm

1

40000

2 30000

3

400000

4 5 6

0

-400000

s, m 0

2000

4000

6000

8000

10000

4

20000

5 10000

6 s, m

0 0

2000

4000

6000

8000

10000

a b Fig. 6-41 Functions of longitudinal force Fw (a) and torque M w (b) (tripping-in operation; Q 100 (1), Q 10 (2), Q 5 (3), Q 1 (4), ν = 0.1 (5) (5), Q 0 (6))

The conducted numerical analysis of the influence of the geometric parameters of harmonic imperfections on the values of axial forces Fw (s) and torques M w (s) showed that their presence in the trajectories of the centrelines of the drill strings significantly affected the values of external and internal force factors both during drill string tripping out and tripping in. However, when tripping out the drill string, the scheme of action of gravity forces and friction (resistance) forces is such that they are oriented in the same direction and prevent the implementation of this technological operation. Moreover, the resistance forces are non-linearly dependent on the amplitude of the imperfections, and for some values of the geometric parameters, which set their shape, the resistance becomes so large that the drill string tripping-out operation becomes impossible. When the drill string is tripping in, the force of gravity and axial force resisting to displacement are oriented in opposite directions; therefore, to some extent, they neutralise each other. However, during this technological operation, abnormal modes are also possible, when the resistance forces are so large that the

Modelling emergency situations in drilling deep boreholes

338

gravity forces cannot overcome them, and the drill string gets stuck in the well. In practice, to overcome this emergency situation, it is necessary to load the top of the drill string with special heavy masses that lead the drill string away from the accident. In this regard, it can be noted that the theory and computer model of the drill string motion in the well channel developed in this work allows us to predict and avoid emergency drilling situations associated with a sharp increase in resistance forces in the areas of geometric imperfections. 6.5

Modelling resistance forces in a well with localised spiral imperfections

6.5.1. Model of spiral wavelets

4,000 m

If the technology of curvilinear borehole drilling processes is violated, situations arise associated with the appearance of localised geometric imperfections in the form of three-dimensional irregular spirals (spiral wavelets) on its trajectory. To analyse their impact on the possibility of fulfilling the design requirements, let's investigate the elastic deforming of the drill string in wells whose given shape is described by hyperbolic curve (6.39) (Fig. 6-42) L1  H H y 0, x ˜ cos- , z ˜ sin - . 1  H cos1  H cosAs before, when carrying out calculations together with parameterisation (6.39), it is also convenient to use natural parameterisation, which is set by parameter s equal to the length of the arc of the drill string centreline, measured from an initial point.

8,000 10,000 m 12,000 m Fig. 6-42 Geometries of curvilinear boreholes with spiral imperfections of the centreline

Fig. 6-43 Geometry of localised spiral imperfections

λ is the pitch of the spiral imperfections; 2hc is the largest diameter of the spiral; sc is the coordinate of the central cross section of the spiral imperfection; l is the representative length of spiral imperfections

sc

2hc

λ

Chapter 6. Modelling resistance forces and drill string sticking effects… 339

Modelling emergency situations in drilling deep boreholes

340

The relationship between parameters - and s is set by the integral dependence -

³ DT dT , where D(- )

s

L2 (1  H ) 2 sin 2 -  H 2 (cos-  H ) 2

is the metric factor, (1  H cos- ) 2 which is calculated, as above, using formula (6.40). We assume again that in the process of drilling, for a number of technological reasons, the borehole trajectory acquired additional distortions in the form of localised spirals superimposed on initial axial line (6.39) (Fig. 6-43). The pitch distance O 2S / k of spirals is determined by wave number k and is constant, amplitude h(s) has maximum value hc at some point s sc and decreases according to the law 3S / 2

h( s) hc ˜ e

§ s  sc · D 2 ¨ ¸ © S ¹

2

.

(6.48)

Here, sc is the distance from initial point s 0 ( - 3S / 2 ) to the centre point of the imperfection, S is the total length of the borehole centreline, D is the coefficient that determines the representative range of h(s) change, beyond which value h(s) can be ignored (though it is not neglected). Then, the components of radius vector ρ x i  y j  z k of the borehole centreline with localised spiral imperfections can be calculated using the formulas L1  H H (cos-  H ) x ˜ cos-  h( s) ˜ cosks ˜ , 2 2 1  H cosL (1  H ) sin 2 -  H 2 (cos-  H ) 2 y h(s) ˜ sin( ks) ,

z

H L1  H sin ˜ sin -  h( s) ˜ cosks ˜ . 2 2 1  H cosL (1  H ) sin 2 -  H 2 (cos-  H ) 2

(6.49)

As is customary in this work, when analysing the elastic bending of the drill string in the well cavity, we consider the fact that the centrelines of the drill string and the well coincide. This allows us to use equalities (6.3)–(6.6), (6.9), (6.12), calculate the necessary coefficients in system (6.30), and solve it numerically with the Runge-Kutta method. Desired functions F , r , Fw at - 3S 2 equal zero, but gravity forces (6.31) are different from zero. 6.5.2. Results of calculations for a well with hyperbolic trajectory When planning the geometry of a curvilinear borehole and its drivage technology, it is necessary to take into account a large number of different factors that characterise borehole depth H (more than 4 km), greatest distance L (more than 12 km) of the drill bit from the drilling rig, diameter d of the DS pipe (up to 0.4 m), the rod material (steel, titanium, aluminium, composite), the coefficient of friction ( P = 0.2–0.3), ratio Q

Chapter 6. Modelling resistance forces and drill string sticking effects…

341

between velocities of the DS axial motion and the circular motion of the external points of its pipe. In these studies of elastic bending of DS in boreholes with hyperbolic trajectories, cases are considered when their initial geometry is ideal and is set in the domain 3S / 2 d - d 2S of equations (6.39) at H = 4,000 m, L = 8,000 m, 10,000 m, 12,000 m, and H 3 (Fig. 6-42), and when in the middle of the second ( sc 3S / 8 ) or fourth ( sc

7S / 8 ) quarters of their length S there appeared spiral imperfections with pitch

O 91 m and greatest amplitude hc 2 m . Note that in Fig. 6-42 spiral imperfections are not visually noticeable for the selected geometric relations. Therefore, in Fig. 6-43, they are presented in a larger scale. The values of other parameters are taken to equal: d 0.1683 m , pipe wall thickness G 0.01 m , E 2.1 ˜1011 Pa , G 0.8077 ˜1011 Pa ; density of the rod material J st = 7,850 kg/m3; density of the drilling liquid J l = 1,500 kg/m3; P 0.2 , Q 0.01 , 1, 100. With the described approach, two drill string operations—tripping out and tripping in—were simulated. Calculations find all functions defined by correlations (6.30). Tables 6-10–6-12 show the most typical integral values that determine longitudinal force Fw and torque M w at point of suspension s S of the drill string during its tripping out as well as total elastic elongation 'S total angle of twist M S

1 2S Fw - Dd- and EF 3S³/ 2

2S

1 M w Dd- at Q GI w 3S³/ 2

100 , 1, 0.01, respectively.

As it turned out, torque M w (S ) and angle of the elastic twist 'M (S ) increase significantly with both an increase in L and the introduction of imperfections. Cases of the absence of imperfections and their location with central points in cross sections s 3S / 8 and 7S / 8 for wells with horizontal extension of L = 8,000 m, 10,000 m, and 12,000 m are considered. The tables also give values S of the total length of the DS and the total gravity force P F J st  J l S found taking into consideration the action of the buoyancy force of the drilling fluid. From the specified results, it follows that force Fw (S ) at the point of suspension of the drill string during the tripping-out operation increases with length S of DS and the introduction of imperfections in the geometry of the borehole centreline. The features of the change of values Fw and M w along length s at Q 1 for the cases L = 8,000 m, 10,000 m, and 12,000 m can be traced in Fig. 6-44, a, b, c. Curves 1–9 correspond to positions 1–9 in Table 6-11. It is clear that in segments free from

Modelling emergency situations in drilling deep boreholes

342

imperfections these functions change slowly, but when approaching segments with geometric distortions they increase sharply. Table 6-10 Values of forces, moments, and displacements at the point of drill string suspension (spiral imperfections; tripping-out operation; h c 2 m , O 91.13 m , Q 100 ) No.

sc , m

Fw (S ) ,

1 2 3 4 5 6 7 8 9

– 3S/8=3,417 7S/8=7,973 – 3S/8=4,103 7S/8=9,574 – 3S/8=4,807 7S/8=11,217

MN 1.923 12.176 23.141 2.091 22.198 43.520 2.258 41.761 82.703

L = 8,000 m S = 9,113 m P

2.82 MN

L = 10,000 m S = 10,943 m P

3.39 MN

L = 12,000 m S = 12,820 m P

3.97 MN

'S , m

6.89 48.89 22.30 8.82 104.44 41.62 11.02 225.02 79.52

M w (S ) , kNm 0.57 9.20 18.43 0.71 17.63 35.58 0.85 34.09 68.53

'M (S ) , rad 0.77 15.37 6.13 1.14 34.42 12.51 1.60 76.00 25.42

Table 6-11 Values of forces, moments, and displacements at the point of drill string suspension (spiral imperfections; tripping-out operation; h c 2 m , O 91.13 m , Q 1 ) No. L = 8,000 m S = 9,113 m P 2.82 MN L = 10,000 m S = 10,943 m P 3.39 MN L = 12,000 m S = 12,820 m P 3.97 MN

1 2 3 4 5 6 7 8 9

sc , m

Fw (S ) ,

– 3S/8=3,417 7S/8=7,973 – 3S/8=4,103 7S/8=9,574 – 3S/8=4,807 7S/8=11,217

MN 1.703 5.182 9.511 1.815 7.584 14.781 1.926 11.452 23.200

'S , m

M w (S ) ,

6.24 21.76 12.68 7.86 38.02 19.80 9.67 66.79 31.15

kNm 38.9 331.7 696.0 48.3 533.7 1,139.3 57.6 859.2 1,847.6

'M (S ) , rad 54.2 593.7 278.1 80.8 1,129.5 495.7 113.0 2,099.0 859.9

Chapter 6. Modelling resistance forces and drill string sticking effects…

343

Table 6-12 Values of forces, moments, and displacements at the point of drill string suspension (spiral imperfections; tripping-out operation; h c 2 m , O 91.13 m , Q 0.01 )

L = 8,000 m S = 9,113 m P = 2.82 MN L = 10,000 m S = 10,943 m P = 3.39 MN L = 12,000 m S = 12,820 m P = 3.97 MN

No.

sc , m

Fw (S ) ,

'S , m

M w (S ) ,

1 2 3 4 5 6 7 8 9

– 3S/8=3,417 7S/8=7,973 – 3S/8=4,103 7S/8=9,574 – 3S/8=4,807 7S/8=11,217

MN 1.245 1.255 1.257 1.247 1.259 1.279 1.248 1.262 1.285

4.70 4.76 4.74 5.56 5.64 5.61 6.45 6.57 6.53

kNm 50.3 137.8 277.1 61.8 165.0 332.5 73.1 193.0 388.8

'M (S ) , rad 76.8 269.6 181.0 114.6 387.5 262.4 160.4 531.4 361.4

Fig. 6-45, a, b show changes in values Fw and M w along length s for the case of sc = S/8 = 1,152 m, Q 100 , hc 5 m , H 1.5 during the tripping-out operation. For a borehole with ideal design geometry (curves 1) functions Fw (s) and M w (s)

monotonically increase. However, the nature of changes in functions Fw s and M w s for boreholes with imperfections (curves 2) has changed significantly. Both graphs show that in the imperfections zone these functions rapidly grow, and the resulting geometric features may be the main reason for the possible sticking of the drill string. This conclusion can also be confirmed by the graphs of the change of shearing forces Fu (s) , Fv (s) in Fig. 6-46, a, b, respectively. In the imperfections zone, these functions have the form of oscillating curves with a significant amplitude, and they are smooth in the other part. Resulting shearing force FR

Fu 2  Fv 2

is a smooth function, in the

imperfections zone it has a noticeable spike (Fig. 6-47). It is interesting to trace the nature of changes in external forces that influence the deformation of the drill string. So, in the imperfections zone, the function of distributed axial gravity force f wgr s has the shape of an oscillating curve, and it is smooth in the other part.

Modelling emergency situations in drilling deep boreholes

Fw , N

10000000

s, m 0

3000

6000

3 2 1

400000

1

0

M w , Nm

800000

3 2

5000000

344

s, m

0 0

9000

3000

6000

9000

а Fw , N

15000000

10000000

6

5000000

5 4

0 0

3000

M w , Nm

1200000

6000

9000

6

800000

5 4

400000

s, m

0

12000

0

3000

6000

9000

s, m

12000

b

Fw , N

25000000

M w , Nm

2000000

20000000

1600000

15000000

1200000

9

9

10000000

800000

8 7

5000000

8 400000

s, m

0 0

3000

6000

9000

12000

15000

7 s, m

0 0

3000

6000

9000

12000

15000

c Fig. 6-44 Functions of longitudinal force Fw (s) and torque M w (s) for the regime of DS tripping out in a well with spiral imperfections at hc 2 m , Q 1 a) L1 = 8,000 m, S1 = 9,113 m; b) L2 = 10,000 m, S2 = 10,943 m; c) L3 = 12,000 m, S3 = 12,820 m

Chapter 6. Modelling resistance forces and drill string sticking effects…

Fw , N

50000000

M w , Nm

40000

40000000

2

345

2

30000

30000000 20000 20000000 10000

1

10000000

s, m

0 0

2000

4000

6000

8000

1 s, m

0 0

10000

2000

4000

6000

8000

10000

a b Fig. 6-45 Functions of longitudinal force Fw (a) and torque M w (b) (spiral imperfections; tripping-out operation; hc 5 m , Q 100 ) (1) = without imperfections; (2) = with imperfections (sc = 1,152 m)

Fu , N

4000

2000

2000

0

0

-2000

-2000

s, m

-4000 0

Fv , N

4000

2000

4000

6000

8000

s, m

-4000 0

10000

2000

4000

6000

a b Fig. 6-46 Functions of shearing forces Fu (a) and Fv (b)

FR , N

4000

3000

2000

1000

s, m

0 0

2000

4000

6000

8000

10000

Fig. 6-47 Function of resulting shearing force FR

8000

10000

Modelling emergency situations in drilling deep boreholes

346

f wfr , N / m

0

mwfr , N 0

-10000 -10

-20000

-20

-30000

s, m

-40000 0

2000

4000

6000

8000

10000

s, m

-30 0

2000

a

4000

6000

8000

10000

b

fr Fig. 6-48 Functions of distributed friction force f w (a) and fr moment mw of friction forces (b)

However, distributed friction force f wfr s caused by the contact interaction tangent to the DS centreline is significantly dependent on the imperfections, and its changes in the zone of distortion of the geometry occur with one large spike (Fig. 6-48,

a). Similar changes occur with the graph of function mwfr s (Fig. 6-48, b). It is also interesting to note the influence of the imperfections of the drill string centreline on its geometric characteristics. For example, the function of changing the angle of elastic twist F (Fig. 6-49) is an almost linear polyline, and in the imperfections zone the angle of inclination is greater than in another segment. Twisting function r (Fig. 6-50) in the imperfections zone increases dramatically. F , rad 0.016 400 r , m 1 300

0.012

200

0.008

100

0.004

s, m

0 0

2000

4000

6000

8000

10000

Fig. 6-49 Function of elastic twisting angle F

s, m

0 0

2000

4000

6000

8000

10000

Fig. 6-50 Twisting function r of the centreline

Chapter 6. Modelling resistance forces and drill string sticking effects…

347

When performing the tripping-in operation, the scheme of forces and moments acting on the drill string undergoes qualitative changes in comparison with the procedure when it is tripped out. Table 6-13 and Fig. 6-51 show the results of modelling the major external and internal forces that accompany the tripping in of the drill string in wells with depths of L = 8,000 m, 10,000 m, and 12,000 m at Q 1. As in Table 6-11 (corresponding to the DS tripping out), Table 6-13 shows the most characteristic integral values that determine the values of longitudinal force Fw and torque M w at suspension point s S during tripping in as well as the values of total elastic elongation 'S and total angle of twist M S . Table 6-13 Values of forces, moments and, displacements at the point of suspension of the drill string (spiral imperfections; tripping-in operation; h c 2 m , O 91.13 m , Q 100 ) No. L = 8,000 m S = 9,113 m P = 2.82 MN L = 10,000 m S = 10,943 m P = 3.39 MN L = 12,000 m S = 12,820 m P = 3.97 MN

1 2 3 4 5 6 7 8 9

sc , m

Fw (S ) ,

– 3S/8=3,417 7S/8=7,973 – 3S/8=4,103 7S/8=9,574 – 3S/8=4,807 7S/8=11,217

MN 0.850 0.663 0.290 0.764 0.600 0.252 0.680 0.542 0.226

'S , m

M w (S ) ,

3.101 1.968 2.334 3.177 1.977 2.323 3.116 1.929 2.222

kNm 32.6 48.5 79.8 39.9 53.7 83.1 46.9 58.6 85.2

'M (S ) , rad 54.6 94.3 81.8 81.4 123.6 111.9 114.1 156.0 146.1

The table also gives values S of the total length of the DS and total gravity force

P

F J t  J l S found taking into account the action of the buoyant effect of the

drilling fluid. From the presented results, it follows that force Fw (S ) required to hold the drill string at the point of its suspension during the tripping-in operation decreases as length S increases, and imperfections are introduced into the geometry of the well trajectory. The first feature is due to the fact that force Fw (S ) is determined mainly by depth H of the well, and in the absence of friction forces between the drill string and the well wall it does not depend on L . Therefore, taking into account the contact interaction, with L increasing at constant H , the vertical component of the gravity force remains unchanged, while the friction forces along the centreline increase in the segments of the

Modelling emergency situations in drilling deep boreholes

900000

Fw , N

348

M w , Nm

90000

1

3

2

600000

60000

2 300000

30000

3

1 s, m

0 0

3000

6000

s, m

0

9000

0

3000

6000

9000

a 900000

90000

Fw , N

M w , Nm

6

4 60000

600000

5

5 4

30000

300000

6 s, m

0 0

3000

6000

9000

s, m

0 0

12000

3000

6000

9000

12000

b

Fw , N

900000

M w , Nm

90000

7

600000

9

60000

8

8

7 30000

300000

9 s, m

0 0

3000

6000

9000

12000

15000

s, m

0 0

3000

6000

9000

12000

15000

c Fig. 6-51 Functions of longitudinal force Fw (s) and torque M w (s) for the DS tripping-in regime in a well with spiral imperfections at hc 2 m , Q 1 a) L1 = 8,000 m, S1 = 9,113 m b) L2 = 10,000 m, S2 = 10,943 m c) L3 = 12,000 m, S3 = 12,820 m

Chapter 6. Modelling resistance forces and drill string sticking effects…

349

trajectory approaching the horizontal. This explains the nature of significant change Fw (S ) in Table 6-12. This effect is even more evident when the imperfections are localised in the most curved, upper part of the well (positions 3, 6, 9 in Table 6-13). As it turned out, elongation 'S of the string practically does not depend on L , although the introduced imperfections significantly affect its value. At the same time, torque M w (S ) and angle of elastic twist 'M (S ) increase noticeably both with an increase in L and with the introduction of imperfections. The features of the change of values Fw and M w along length s for cases L = 8,000 m, 10,000 m, and 12,000 m can be traced in Fig. 6-51. Curves 1 to 9 correspond to positions 1–9 in Table 6-13. Analysis of the results of calculations leads to the conclusion that in a well with ideal geometry longitudinal force function Fw (s) increases monotonically with an increase in s (curve 1 in Fig. 6-51, a). However, when approaching the imperfection localisation zones, these functions reach the local maxima and begin to decrease rapidly, approaching the local minima (curves 2, 3). Then, these curves grow again. Similar effects are typical for curves 4–9 of longitudinal force function Fw (s) . Another trend is for torque function M w (s) (Fig. 6-51). In this case, at the selected value Q 1, all curves 1–9 have the form of increasing functions, and they increase most rapidly in segments containing imperfections (curves 3, 6, 9). It is interesting to analyse the influence of spiral imperfections, the central points of which are located in cross sections s S / 8 , 3S / 8 , 5S / 8 , and 7S / 8 (Table 6-14, Fig. 6-52) for the tripping-in operation at hc 5 m , O 92.2 m , Q 100 . Table 6-14 Values of forces, moments, and displacements at the point of suspension of the drill string (spiral imperfections; tripping-in operation; h c 5 m , O 92.2 m , Q 100 ) No.

sc

Fw S

(N)

1

(m) 0

2

S/8 = 1,152

3

3S/8 = 3,457

4

5S/8 = 5,762

5

7S/8 = 8,067

M S

M w S

Fw S Pt

'S (m)

0.726 ˜ 10

6

0.25

2.09

0.431 ˜10

0.677 ˜ 10

6

0.24

1.68

0.473 ˜103

0.94

0.586 ˜ 10

6

0.21

1.21

0.549 ˜10

3

1.11

0.446 ˜ 10

6

0.16

0.97

0.667 ˜10

3

1.20

0.129 ˜ 10

6

0.04

1.18

0.934 ˜10

3

1.13

(Nm) 3

(rad) 0.80

Modelling emergency situations in drilling deep boreholes

Fw , N

800000

1

M w , Nm

1000

2 600000

350

5

800

3

4 600

400000

3

4

2 400

1 200000

200

5

s, m

0 0

2000

4000

6000

8000

10000

s, m

0 0

2000

4000

6000

8000

10000

a b Fig. 6-52 Functions of longitudinal force Fw (a) and torque M w (b) (spiral imperfections; tripping-in operation; h c 5 m , O 92.2 m Q 100 ) (1) = without DS centreline imperfections

(2) = with imperfections at point sc = S/8 = 1,152 m (3) = with imperfections at point sc = 3S/8 = 3,457 m (4) = with imperfections at point sc = 5S/8 = 5,762 m (5) = with imperfections at point sc = 7S/8 = 8,067 m From the results, it follows that force Fw (S ) that is needed to keep the drill string at its suspension point decreases with the introduction of imperfections in the geometry of the borehole trajectory. Table 6-14 and Fig. 6-52 present the values of longitudinal force Fw and torque M w at drill string suspension point s S during tripping in as well

as full elastic elongation 'S and full angle of twist 'M S . The nature of the Fw (S ) change in Table 6-14 appears when imperfections are localised in the most curved, upper part of the well (position 5 in Table 6-14). It should be noted that for the case under consideration the results of calculations coincide qualitatively with the data given in Table 6-13 and Fig. 6-51. To conclude this subsection, we should emphasise that in a well with a hyperbolic centreline, the structure of the force factors acting on the drill string during the tripping in/out operations has significant differences. At the same time, the introduction of localised spiral imperfections (as well as for harmonic imperfections) may be accompanied by a complication of the conditions of the corresponding operation, up to the occurrence of an abnormal situation when the considered technological operation is impossible.

Chapter 6. Modelling resistance forces and drill string sticking effects…

6.5.3.

351

Study of the functions of resistance forces during the motion of the drill string in a well with breaks of the centreline

Together with the forms of imperfections described above in the mode of localised harmonic and spiral wavelets, imperfections in the form of flat or threedimensional smoothed breaks (dog leg) occur quite often in drilling practice. Let us consider the influence of such imperfections on the example of an elliptic well (Fig. 653) described by the equations y 0, x L cos- , z H sin 3S 2 d - d 2S , (6.50) where, as above, H and L are the depth and horizontal distance of the lower end of the well from the drilling rig. We shall model the shapes of imperfections that are superimposed on this curve, by combinations of vertices of hyperbolas with different eccentricities and angles between their asymptotes (Fig. 6-54). If the asymptotes of the left branches of two similar hyperboles with different signs are oriented in the negative direction of axis Ox (Fig. 6-54, a, b), then their superposition with small gap ' z in plane xOz approximately forms a broken straight line with curvature radius Rh at the tops of the breaks (Fig. 6-54, c). If you then superimpose this line on the initial trajectory (6.50), the final configuration of the centreline of the well will take the shape shown in Fig. 655, a. The analytic expression for the hyperbola shown in Fig. 6-54, a, can be written in the form: 2 ­ ½ ° tg D x  x0 ° ª tg D  x  x0 º (6.51)  «  H z ® ¾, » 2 2 ¬ ¼ ° ° ¯ ¿ where D is the angle between the asymptotes of the hyperbola, x0 is the coordinate x of the hyperbola vertex, H is the parameter that characterises the hyperbola eccentricity. The hyperbola shown in Fig. 6-54, b is described by the equation 2 ­° tg D  x  x ½° ª tg D  x  x0 º 0 (6.52)  «  H ¾. z ® » 2 2 ¬ ¼ °¯ °¿ The considered breaks can also be oriented in plane xOz , then they are described by equations 2 ­ ½ ° tg D  x  x0 ° ª tg D  x  x0 º  «  H y r® (6.53) ¾. » 2 2 ¬ ¼ ° ° ¯ ¿

Modelling emergency situations in drilling deep boreholes

352

z

x

H

0

L Fig. 6-53 Diagram of a borehole with elliptical trajectory z

x

x0

0

D Rh

a

-

Rh

z

-

D x

0

x0 b z

's

si

0

'z

c Fig. 6-54 Modelling of a localised break

x -

Chapter 6. Modelling resistance forces and drill string sticking effects…

353

z

x

0

3S 2

а

z

x

0 4,000 m

-

- 2S

8,000 m

b Fig. 6-55 Well trajectories (a) = with one localised break (b) = with four localised flat breaks (Task 2)

Fig. 6-56 Geometry of a localised dog leg break

Modelling emergency situations in drilling deep boreholes

354

In general, flat and three-dimensional imperfections of the considered form can be localised in several places and can have different radii Rh and angles D , then they need to be modelled by a superposition of different breaks of form (6.51)–(6.53) on the design trajectory (6.50). The developed approach was used for computer simulation of the drill string motion in the well channel with selected geometric imperfections. In general, when planning drilling processes, there is a large number of different factors that affect the technological procedure. They differ in the diameter of wells (up to 40 cm), DS materials (steel, aluminium, titanium, composite), friction coefficients ( P 0.2  0.3 ), distance from the drilling rig (more than 12 km), the outlines of the well paths, and other parameters. In our studies, the following values of characteristic parameters were chosen: L = 8,000 m, H = 4,000 m, d 0.1683 m, column pipe thickness G 0.01 m,

E 2.1 ˜ 1011 Pa , G = 0.8077·1011 Pa, J st = 7,850 kg/m3, J l = 1,500 kg/m3, P 0.2 , Q 100 . Table 6-15 shows the values of curvature radii Rh for four cases of imperfections. First, a well without imperfections was considered (position 1 in Table 6-15), then four breaks were added to the trajectory, two of which (positions 2 and 3) were associated with plane breaks (equations (6.51), (6.52)), and in the other two cases (positions 4 and 5) they were oriented in plane xOz (equation (6.53)). Table 6-15 Radii of curvature and breaks of the well Task number 1 2 3 4 5

Imperfe ction shape – Flat 3D

Break 1 Rh,1 Rh, 2 (m) – 3,676 107 2,938 132

(m) – 2,363 101 2,762 129

Break 2 Rh,3 Rh, 4

Break 3 Rh,5 Rh,6

(m) (m) (m) – – – 257.5 260.3 63.7 143.8 147.8 38.4 223.9 217.4 91.7 91.7 90.0 70.4

(m) – 60.5 36.2 90.5 69.4

Break 4 Rh,8 Rh,7 (m) (m) – – 198.9 233.7 198.9 233.7 57.9 57.2 57.9 57.2

For Tasks 2, 4, break centres si were localised at points s1 S 8 , s2 3S 8 , s3 5S 8 , s4 7S 8 ; for Tasks 3, 5, they were located at points s1 4S 8 , s2 5S 8 , s3 6S 8 , s4 7S 8 . Vertices of conjugate hyperbolas were located at points si ± 50 m. Here, S is the total length of the centreline. For the considered case, it was

Chapter 6. Modelling resistance forces and drill string sticking effects… 2S

S

³ D(-)d-

355

9688 m .

3S 2

In all cases, the angles of hyperbolic opening were D 0.983 rad , while the radii of curvature were different, their values are shown in Table 6-15. As an example, Fig. 6-55, b shows the outline of the centreline of the well for Task 6. The first break on this curve is visually invisible, so it is not significant. The tripping-out operation of the drill string was considered with the selected values of the initial data. Integration of the constitutive equations in the range 3S 2 d - d 2S was performed using the Runge-Kutta method with the integration step '- 2S  3S 2 8000 . Table 6-16 shows the results of calculations. This gives the values of axial force Fw S at point s S of suspension; the relation value of this force to gravity force Pt F J t  J l S 3001427 N found taking into account the buoyant action of the drilling fluid; the value of the total elastic elongation of the drill string

'S

2S

1 Fw - Dd- ; torque M w at point s S of suspension, and the total angle of EF 3S³/ 2

DS elastic twisting M S

2S

1 M w - Dd- . Some results obtained for Task 2 are GI w 3S³/ 2

schematically presented in Figs. 6-57–6-61. For comparison, Fig. 6-57 represents curves 1 and 2 corresponding to the ideal geometry of the considered well and to the case of a well with imperfections. It can be seen that the introduction of even such minor imperfections leads to noticeable jumps in functions Fw (s) and M w (s) . Table 6-16 Values of DS stress-strain state parameters (imperfections in the shape of localised breaks; tripping-out operation) Imperfection Fw S Fw S Pt M w S Task M S 'S number

shape

1

–

2 3 4 5

Flat 3D

(N)

(m)

1.765 ˜ 10

6

– 0.59

5.74

(Nm) 442

(rad) 0.82

2.210 ˜ 10

6

0.74

6.95

803

1.22

2.791 ˜ 10

6

0.93

8.18

1,301

1.66

2.525 ˜ 10

6

0.84

7.20

1,082

1.32

3.191 ˜ 10

6

1.06

8.59

1,642

1.79

Significant peaks in the places of geometry discontinuity are achieved by

functions f c s , f wfr s , mwfr s , and M R (s) (see Fig. 6-58–6-61).

Modelling emergency situations in drilling deep boreholes 2500000

356

1000

Fw , N

M w , Nm

2

2000000

800

1500000

600

2 400

1000000

1 200

500000

s, m

0 0

2000

4000

6000

8000

1 s, m

0

10000

0

2000

4000

6000

8000

10000

a b Fig. 6-57 Functions of longitudinal force Fw (a) and torque M w (b) (imperfections in the shape of smoothed breaks; tripping-out operation) (1) = without imperfections; (2) = with imperfections (Q 100 )

f c,N /m

16000

12000

-1000

8000

-2000

4000

-3000

s, m

0 0

2000

4000

6000

8000

c

(Q

0

-1

40000

-2

20000

s, m 0

2000

4000

6000

8000

10000

Fig. 6-60 Function of distributed moment

mwfr

of friction forces (Q

100 )

4000

6000

8000

10000

M R , Nm

60000

-3

2000

Fig. 6-59 Function of friction fr force f w distribution (Q 100 )

100 )

mwfr , N

0

s, m

-4000

10000

Fig. 6-58 Function of distributed contact force f

f wfr , N / m

0

0 0

2000

4000

6000

8000

s, m

10000

Fig. 6-61 Function of resulting moment M R

Chapter 6. Modelling resistance forces and drill string sticking effects…

Fw , N

800000

357

M w , Nm

600

1

2

600000 400

1

400000

200

2

200000

s, m

0 0

2000

4000

6000

8000

10000

s, m

0 0

2000

4000

6000

8000

10000

a b Fig. 6-62 Functions of longitudinal force Fw (a) and torque M w (b) (tripping-in operation) (1) = without imperfections; (2) = with imperfections (Q 100 )

FR , N

2500

2000

0

1500

-100

1000

-200

500

-300

s, m

0 0

2000

f wgr , N / m

100

4000

6000

8000

10000

-400 0

2000

Fig. 6-63 Function of resulting shearing force FR

f wfr , N / m

1000

6000

8000

s, m

10000

Fig. 6-64 Function of the distributed gr longitudinal force of gravity f w

mwfr , N

0

800

4000

-0.2

600 -0.4 400 -0.6

200

0 0

2000

4000

6000

8000

s, m

10000

-0.8 0

2000

4000

6000

8000

a b fr Fig. 6-65 Functions of distributed friction force f w (a) and fr moment mw of friction force (b)

s, m

10000

Modelling emergency situations in drilling deep boreholes

358

Figs. 6-62–6-65 show the results of computer simulation of the drill string tripping-in operation in graphical form. In Fig. 6-62, curves 1 correspond to the string tripping into a well that has no geometric imperfections at Q 100 . Apparently, the distortion of the borehole centreline leads to a noticeable decrease in longitudinal force Fw (s) , while function M w (s) with the introduction of imperfections experiences an increase in its values (curves 2). The marked jumps of functions Fw (s) and M w (s) are caused by peak values of functions FR (s) (Fig. 6-63), f wgr (s) (Fig. 6-64), f wfr s , and

mwfr s (Fig. 6-65). The considered tasks for performing the technological operations of tripping in are characterised by the fact that at the lower end of the string ( s 0 ) the conditions of the absence of axial force Fw (0) and M w (0) are fulfilled. These conditions change if the drilling operation is performed. Then, on the edge, it is necessary to apply force Fw (0) and torque M w (0) defining the reaction of interaction of the drill bit with the well bottom and the moment of shearing forces. For other defining data, the functions must be specified with a condition corresponding to the string tripping-in. In this formulation of the task, the drilling operation is simulated with

Fw (0) 5 ˜ 10 4 N , M w (0) 105 Nm . Figs. 6-66 – 6-69 show the results of calculations for three-dimensional geometry of imperfection. They have some similarities with the results of modelling the drill string tripping-in shown in Figs. 6-62–6-65. However, their significant difference is that the drill string is compressed in its lower part. Therefore, the Eulerian stability of the drill string should be further investigated in this segment. To analyse the influence of the nature of geometric imperfections (planar or three-dimensional), Table 6-17 presents the results of modelling the processes of tripping out, tripping in, and drilling for instances where there are no imperfections (positions 1, 4, 7), when they are located in the well plane (positions 2, 5, 8), and when imperfections come out of its plane (positions 3, 6, 9). Comparison of the values of functions Fw (S ) and M w (S ) allows us to conclude that the type of imperfections significantly affects the parameters of the stress-strain state of the drill string. The represented results indicate that for the considered forms of geometric imperfections the functions of the stress-strain state at the contact and frictional force interaction of the drill string with the well wall increase sharply in the areas of localisation of geometric singularities. These effects manifest themselves differently for the technological tripping in/out operations of the string. Therefore, when performing the tripping-out operation, the distortions of the geometry of the borehole centreline lead to the need to increase axial force Fw (S ) and in the worst cases may be accompanied by emergency situations when,

Chapter 6. Modelling resistance forces and drill string sticking effects…

Fw , N

600000

359

M w , Nm

100600

400000

100400 200000

100200 0

s, m

-200000 0

2000

4000

6000

8000

s, m

100000 0

10000

2000

4000

6000

8000

10000

a b Fig. 6-66 Functions of longitudinal force Fw (a) and torque M w (b) (imperfections in the form of smoothed breaks; drilling operation, Q 100 )

k R , 11/m м

0.02

Nm M R , Hм

60000

4

0.016 40000

3

0.012

0.008 20000

2 0.004

1

s, мm

0 0

2000

4000

6000

8000

s, м m

0 0

10000

2000

6000

8000

10000

Fig. 6-68 Function of resulting moment M R

Fig. 6-67 Curvature function k R of the borehole centreline

f wfr , H /м N/m

1200

4000

N mwfr , H

0

-0.2

800 -0.4

-0.6

400 -0.8

s, м m

0 0

2000

4000

6000

8000

10000

-1 0

2000

4000

6000

8000

a b fr Fig. 6-69 Functions of distributed friction force f w (a) and fr moment mw of friction forces (b)

s, мm

10000

Modelling emergency situations in drilling deep boreholes

360

Table 6-17 Values of DS stress-strain state parameters (imperfections in the form of localised breaks; L = 8,000 m, P 0.2 , Q 100 ) Fw (S ) M w S Imperfection M S Operation 'S No. form type (m) (MN) (Nm) (rad) 4 1 – 1.765 5.74 0.82 0.044 ˜ 10 Tripping 4 2 2.210 6.95 1.22 Planar 0.080 ˜ 10 out 4 3 2.525 7.19 1.31 3D 0.108 ˜ 10 Tripping in

4

–

0.739

0.77

0.042 ˜ 10 4

0.90

5

Planar

0.716

0.74

0.045 ˜ 10 4

0.94

3D –

0.636

0.62

0.051 ˜ 10

4

0.97

0.682

0.26

10.042 ˜ 10

4

384.21

Planar

0.662

0.21

10.046 ˜ 10

4

387.52

3D

0.588

0.09

10.051 ˜ 10

4

386.20

6 7

Drilling

8 9

due to the sharp growth in the resistance forces, the tripping-out operation becomes impossible. For a drill string tripping-in operation, the opposite effect occurs. With the introduction of imperfections in the well geometry, axial force Fw (S ) required to hold the drill string decreases and can even become negative. This means that to carry out the operation in this case, it is necessary to overload the drill string at the upper end with an additional compressive force. The observed effects are most noticeable in situations where geometric imperfections are located in the upper (most curved) part of the drill string. 6.6

Simulation of energy-saving regimes of tripping in/out operations in a well with geometric imperfections

In the previous subsections, it is shown that geometric imperfections even with relatively small amplitudes can lead to a significant increase in resistance forces and be the reason for the drill string sticking. It is also found that the combination of axial and rotational motions of the drill string can significantly reduce the resistance forces as well as avoid sticking. In this subsection, the task of developing technological techniques for computer simulation is set to minimise energy consumption when moving the drill string in a well channel with geometric imperfections. It can be used to design the geometry of the borehole and to identify the requirements for its accuracy in designing bore drilling regimes and their implementation as well for releasing the drill string from sticking.

Chapter 6. Modelling resistance forces and drill string sticking effects…

361

Indeed, the integration of the system of equations (6.30) under given initial conditions

Fw (0) Fw0 , M w (0) M w0 (6.54) makes it possible to build functions Fw (s) , M w (s) ( 0 d s d Si ) and determine the values of axial force Fw ( Si ) and torque M w ( Si ) that must be applied at point of suspension s Si of the drill string, to implement a given process regime at given time ti , length Si of the drill string, and given relation K 1 Q of velocities of the circumferential ( Z d 2 ) and axial ( w ) motions. At the same time, as shown by theoretical studies in the previous subsections for wells with different geometric imperfections, the value of parameter K significantly affects the performance of technological drilling operations. So, in general, increasing parameter K leads to an increase in M w ( Si ) and a decrease in Fw ( Si ) , and by reducing K , you can reduce M w ( Si ) and increase Fw ( Si ) . However, there may be the cases when increasing rotation velocity Z , the resistance forces decrease to such an extent that the torque also begins to decrease. This possibility of changing the values Fw ( Si ) , M w ( Si ) and controlling the process at each stage by varying parameter K allows us not only to predict and avoid abnormal situations but also to select the least energy-consuming drilling modes. Indeed, let well length Si be reached in the process of the technological operation. We suggest that real geometry and imperfections formed on this length are established by logging sensing. Work d Wi on the implementation of the technological operation on elementary segment ds of this stage can be represented as (6.55) d Wi Fw (Si )ds  M w (Si )dM . Since Z d 2 (d 2) dM K , w ds K ds and present elementary work d Wi in the form you can replace dM d 2 (6.56) d Wi >Fw (Si )  K M w (Si )@ds Qi ds , where Qi Fw (Si )  K M w (Si ) is the summarised force corresponding to the displacement ds . By solving system (6.30), with allowance made for actual length Si and the geometric imperfections at different K available at this stage, we select such a value K i with which d Wi for the given mode at given length Si has a minimum. In this case, value Zi is selected based on the technical data and capabilities of the driving device of the drilling rig.

Modelling emergency situations in drilling deep boreholes

362

The found values Zi , Ki , Fw ( Si ) , M w ( Si ) are used to drill a well on its segment ' S i Si 1  Si , after which values Zi 1 , Ki 1 , Fw ( Si 1 ) , M w ( Si 1 ) are calculated by the described technique, and the described procedure is repeated on segment ' Si 1 . It should be noted that if the drill string lifting operation is performed, then ' Si  0 , and Si discretely varies from S1 S to S N S N , where N is the number of segments into which the centreline of the well is conditionally divided. During the tripping-in of the drill string ' Si ! 0 and Si changes from S1 S N to S N S . It should be noted that the proposed approach can be used both at the stage of the well design and when it is drilled. In the first case, the hypothetical parameters of the well trajectory and its imperfections can change within a wide range based on the technological capabilities of their implementation. In the second case, the real values of the parameters found as a result of the logging operations are set. With the help of the proposed approach, the problem of designing an energysaving mode of tripping out a drill string in a well with localised spiral imperfections is solved. The case was considered when the trajectory is a hyperbola (Fig. 6-1, a) in the domain 3S 2 d - d 2S of change of dimensionless parameter - with the value of maximum depth H = 4,000 m, horizontal distance from the drilling rig by the value L = 10,000 m, and eccentricity H 3 . On this trajectory, the spiral imperfections of form (6.48) were superimposed with amplitude value hc 2 m and pitch O 91 m . The centre of imperfection is at point sc 3S 8 . The values of all other parameters are given in subsection 6.4. At the same time, the total length of the well was S = 10,943 m, the total gravity force found taking into account the action of the buoyant force of the drilling liquid was Pt gF (J st  J l )S 3389972 N . To minimise the energy consumed during the drill string tripping-out operation, we use the item-by-item search method (bisection method) at each stage of the optimisation analysis. To do this, at the modelling stage, length S of the drill string was divided into 10 equal segments ' Si S 10 ( i 1, 2,,10 ), which were alternately removed from the bottom, simulating the tripping-out operation. The sequence of drill string configurations at these stages is shown in Fig. 6-70. It should be emphasised that on the scale these imperfections are indistinguishable in this drawing, so the scale of their amplitude in the diagrams has been increased 15-fold to improve their visualisation. In total, 10 tasks were solved, one for each configuration in Fig. 6-70. First, a drill string of full length S was considered (task 1). The values of the most characteristic parameters are presented in column 1 of Table 6-18. From these results, it

Chapter 6. Modelling resistance forces and drill string sticking effects…

363

z 0

x

(e) S 6

5S 10 , K | 0,01

9S 10 , K 1.171

(f) S 7

4S 10 , K | 0

(c) S3

8S 10 , K 1.458

(g) S8

3S 10 , K | 0

(г) S 4

7S 10 , K

0,782

(h) S9

2S 10 , K | 0

(d) S5

6S 10 , K

0.221

(i) S10

S 10 , K | 0

(а) S1

S, K

(b) S 2

2.033

Fig. 6-70 The sequence of configurations of the drill string in its tripping-out state

Modelling emergency situations in drilling deep boreholes

364

follows that the energy consumption (or the value of summarised force Q1 ) at this stage is minimal if coefficient value η = 2.033. Table 6-18 Values of task characteristic parameters Characteristic factors Si (m) Pt (MN) Ki Fw,i S (MN) M w,i S

(kN ˜ m ) Qi (MN)

Task number 1 1094 2 3.39

2

3

4

5

6

7

8

9

10

9848 8754

7660

6566 5472 4378

3284

2190

1096

3.05

2.37

2.03

2.71

1.70

1.36

1.02

0.68

0.34

2.033 1.721 1.458 0.782 0.221 0.01

|0

|0

|0

|0

3.17

2.73

2.30

2.02

1.49

0.93

0.73

0.52

0.29

330

229

154

71

22

11.13

7.41

4.97

2.67

1.38

1.12

0.16 0.062 0.022 0.006 0.001 1.12

0.93

0.73

0.52

0.29

Some numerical results obtained for this regime are schematically presented in Figs. 6-71–6-72. Fig. 6-71 shows the graphs of the change along longitudinal coordinate s of axial force functions Fw (s) (a) and torque M w (s) (b). Curves 1 on these graphs correspond to the case of an ideal well, curves 2 to a well with given imperfections. It can be seen that if the well has no geometric imperfections, functions Fw (s) and M w (s) are smooth and have relatively small values. However, the introduction of minor imperfections into the well geometry (which are even visually impossible to distinguish in the diagrams of Fig. 6-42) leads to a noticeable increase in these functions. The function of resulting shearing force FR is calculated by the formula

FR

Fu2  Fv2 .

Chapter 6. Modelling resistance forces and drill string sticking effects…

Fw , N

4000000

365

M w , Nm

400000

300000

3000000

2

2 200000

2000000

1

1000000

s, m

0 0

4000

8000

1

100000

s, m

0 0

12000

4000

8000

12000

a b Fig. 6-71 Functions of longitudinal force Fw (a) and torque M w (b) (spiral imperfections; tripping-out operation)

f wfr , N / m

0

mwfr , N

0

-200 -40

-400 -80

-600 -120

-800

s, m

-1000 0

4000

8000

12000

s, m

-160 0

4000

8000

12000

a b fr Fig. 6-72 Functions of distributed friction force f w (a) and fr moment mw of friction forces (b)

This is determined only by the values of the functions of total curvature k R and twisting kT and experiences a monotonous character of change with the introduction of the chosen imperfections, while acquiring an extremum in its area. Component f wgr (s) of the distributed force of gravity also acquires a special character. It has high-frequency oscillations corresponding to the oscillations of imperfection.

Modelling emergency situations in drilling deep boreholes 2,5

366

Ki

2

1,5

1

0,5

0 1

2

3

4

5

6

7

8

9

i

10

Fig. 6-73 Graph of parameter K i change when tripping out the drill string 3,5

Fw Si , МN

3 2,5 2 1,5 1 0,5 0 1

2

3

4

5

6

7

8

9

10

i

Fig. 6-74 Graph of longitudinal force Fw ( Si ) change at the point of suspension when tripping out the drill 350

M w ( Si ) , kN ˜ m

300 250 200 150 100 50 0 1

2

3

4

5

6

7

8

9

10

i

Fig. 6-75 Graph of torque M w ( Si ) change at the suspension point when tripping out the drill string

Chapter 6. Modelling resistance forces and drill string sticking effects…

367

The functions of distributed friction force f wfr (s) and torque mwfr (s) differ from each other only by factors (Fig. 6-72). They have extrema at the middle point of the spiral wavelet and small oscillations in its vicinity. With tripping out the drill string, its length Si and the values of the parameters Fw ( Si ) , M w ( Si ) are reduced, and the need to combine the operation of tripping out and rotation becomes smaller (see Figs. 6-73–6-75 and Table 6-18). This conclusion is confirmed by decreasing parameter K . When removing the drill string from imperfection zones ( i 6 ), values K are rapidly reduced, and the tripping-out operation can be performed without combining it with the rotation operation. A similar approach can be used in the tripping-in operation. It can also be useful in cases where DS sticking is caused by imperfections. In general, the developed technique can be used to monitor the drilling process. 6.7

Method for minimising resistance force in the interfacing areas of the borehole curvilinear segments

The problem of designing curvilinear trajectories in the form of a set of simple curvilinear segments is widely used in engineering. It can be assumed that the most important and representative issues in this area relate to transport technology—namely, to the design of the trajectory of a transport track. For example, if a vehicle is moving along a straight road, and it goes on a circular track contacting tangentially with a rectilinear one, it experiences instantaneous centripetal acceleration at the connection point between sections of different curvatures, leading to the appearance of an impact centrifugal force. With an increase in speeds on the road and rail transport, the shock effects at the points of connecting segments of trajectories of different curvature become more destructive, and it becomes obvious that to mitigate this effect, additional measures should be used to eliminate the curvature of the trajectory breaks and prevent the occurrence of impact phenomena. This goal is achieved by adding additional bridge segments at the trajectory connection points of different curvatures. It was recognised that from a geometric point of view the most suitable curve for this purpose is the curve in the shape of a clothoid (Cornu spiral). It can be assumed that such a method of smoother connection of two curves of different curvature may be effective in the drivage of deep curvilinear wells. The clothoid (Cornu spiral) is a curve whose curvature changes linearly with a change in its length s (Fig. 6-76). This fact characterises the transition geometry between the straight and circular sections of the curve—namely: x The clothoid curve begins with zero curvature on the straight-line segment (at the tangent point) and increases linearly as the length of the curve increases.

Modelling emergency situations in drilling deep boreholes

368

x At the point where the clothoid meets (touches) the arc of the circle, its curvature becomes the same as the latter. The shape of the clothoid is represented in an integral form s

>

@

³ sin >(as) s

2 ³ cos (as) ds ,

x( s )

y( s)

0

2

@ds

,

(6.57)

0

where x , y are the coordinates of the curve points; s is the natural parameter;

a 1 2 Rc sc is the constant; Rc and sc are the radius of curvature and the length of the curve at the considered point. y

x

Fig. 6-76 The trajectory of the clothoid used on railway transport for smooth connection of track segments

Let curvature k kc of the clothoid at point s sc ( sc z 0 ). Then, according to the condition a const , the curvature of the curve at any point can be calculated kc ˜ s . (6.58) k ( s) sc Similarly, taking into account that curvature k and angle T of inclination of the tangent to the curve are related by a differential relation (6.59) h ds dT , equality can be derived

T

(as) 2 (6.60) To complete the formulation of the problem on the geometric construction of smooth transition from a straight line to the arc of a circle without breaking the curvature of the trajectory, let's write the following formulas for the coordinates of the centre of the curve at point x(s) , y (s) : xc

x

>

@

y c ( xc) 2  ( y c) 2 , xc y c xcc y cc

yc

y

>

@

xc ( xc) 2  ( y c) 2 , xc y c xcc y cc

(6.61)

Chapter 6. Modelling resistance forces and drill string sticking effects…

369

where the symbol prime denotes differentiation in respect of s. These relations are usually presented in a simpler form (6.62) xc x  R d y d s , yc y  R d x d s The above formulas for bridge sections (though not simple) are widely used in railway and road transport for smooth transition between horizontal sections of roads of different curvature. Note that the above relations are not simple, so if the arc angle of the bridge segment is small, the clothoid curve can be replaced by a cubic parabola arc, whose curvature in small segments also varies linearly and begins with a zero value. Similar problems arise in the drilling of deep curvilinear boreholes. Typically, the trajectories of curvilinear boreholes are designed based on geometric analysis of shape without an exhaustive consideration of the mechanical aspects of the DS bending related to the drilling procedure. In practice, curvilinear boreholes are mainly designed using the minimum curvature method, according to which the trajectory of the borehole is represented as a combination of several straight sections connected to each other by arcs of circles so that the angle of inclination of the tangent to the axis of the borehole changes continuously (sections AB , BC , and CD , Fig. 6-77). However, the curvature (i.e., radius of curvature and elastic bending moment in the column) turns out to be a discontinuous function, which leads to an increase in the forces of contact between the string and the borehole wall.

D

Borehole channel Drill string

C B

C

A Fig. 6-77 Diagram of a multi-sectional borehole

B Fig. 6-78 Diagram of a drill string in a borehole channel

Field observations found that the geometry of the borehole has a significant impact on the friction forces acting on the drill string during its motion. As a rule, these forces determine the maximum horizontal borehole distance that can be achieved without the string sticking or transverse buckling. Therefore, the problem of reducing friction forces due to the choice of rational borehole geometry is given great attention. To evaluate these effects, mathematical models and software systems were developed that were used to calculate friction and resistance forces. However, these models were

Modelling emergency situations in drilling deep boreholes

370

created using simplified schemes based on hypotheses suggesting that the drill string is a flexible cable that does not have bending stiffness. Such hypotheses allow us to simplify the calculations, but they are acceptable only for smoothly varying trajectories with a small curvature and lead to significant errors when using the minimum curvature method for a real drill string in the connection areas of the sections of wells with geometry discontinuities. Therefore, the friction and resistance forces unaccounted for by such models lead to the following negative effects: x They lead to a deterioration in the mobility of the drill string in the well and reduce the conductivity of the torques from the upper drive device to the bit at the bottom of the well. x They lead to an increase in compressive axial forces in the drill string contributing to its local and global buckling. x They contribute to the wear of drill string pipes. x They lead to an increase in energy consumption for the drilling process. x They can lead to the drill string sticking. Therefore, the authors of this work have developed a more accurate software package based on the theory of elastic curvilinear rods, which allows us to more precisely simulate the friction and resistance forces in the connection sections of drill strings and develop measures to reduce their impact. Let's illustrate these effects on the example of multi-sectional string ABCD (Fig. 6-77). It consists of two straight segments AB , CD and one connecting section BC in the shape of a circular arc. To study the forces acting on the string, we introduce coordinate parameter s determined by the length of the well centreline from some initial point to the current one. It can be used to calculate the bending moment at all points of the string axis, in particular, in section BC (Fig. 6-78). As the geometry of the string is given, internal bending moment M and the external distributed (localised) contact forces can be easily calculated. In fact, bending moment M (s) is determined by the formula (6.63) M ( s) E I ˜ k E I R , where E is the elastic modulus of the string rod material, I is the moment of inertia of the string cross-section, k is its curvature, R is the curvature radius. Therefore, moment M (s) is zero on straight segments AB and CD and remains constant M (s) E I R inside the arc BC . The graph of this function is shown in Fig. 679.

Chapter 6. Modelling resistance forces and drill string sticking effects…

371

M(s) EI/R s A

B

D

C

Fig. 6-79 Graph of function M(s) in a sectional rod

With the formula for internal shearing force F (s) , it can be concluded that it is (6.64) F ( s) d M ( s) d s zero on the entire segment AD , except for the connection points B and C where it takes infinitely large values since function M (s) is discontinuous. And further using the equation of the elastic equilibrium of the pipe element,

d F ( s) d s  f k ( s) where f

k

(6.65)

is the external distributed contact force, it can again be concluded that

cont

f (s) is equal to zero everywhere, with the exception of the connection points B and C where it also turns to infinity since function F (s) is discontinuous. Using these simple differential calculations, it can be concluded that the configuration of the string inside the well channel, shown in Fig. 6-78, can only be formed by a system of force actions in the form of concentrated bending moments applied at points B and C (Fig. 6-80). But, in turn, each moment M can only be caused by a pair of external contact forces Q with arms h (Fig. 6-81) M Q˜h (6.66) D

D

Q

M M A

Q B

C A B

Fig. 6-80 The shape of a rod formed by the action of two external bending moments M applied at points B and C

Q

Q

h C

h

Fig. 6-81 Diagram of two pairs of forces Q constituting the bending moments at points B and C

Modelling emergency situations in drilling deep boreholes

372

In reality, each of these forces is not concentrated due to the elastic embeddability of the rock and can be represented as the resulting contact distributed forces f cont (Fig. 6-82). It is obvious that the smaller the gap between the well wall and the drill string, the smaller arm h and, according to equations (6.65), (6.66), the greater Q and force f cont . Due to their increase when the string in the well is pulled, the friction forces also increase; therefore, the force acting on the drill bit decreases, the wear rate of the pipe string increases, the string can buckle and stick, etc. Q

f

cont

C

f

Straight segment h

cont

C cc

Clothoid

Q

C Circular arch

Fig. 6-82 Diagram of concentrated (Q)

Cc

Fig. 6-83 Diagram of a rod with clothoidal insert C cC cc between straight and circular sections

cont and distributed ( f ) contact forces

M

A

D

s

D

s

Bc Bcc C c C cc a

F

C c C cc

A

Bc Bcc b Fig. 6-84 Diagram of the distribution of functions M (a) and F (b) in a rod with clothoidal inserts BcBcc and C cC cc

Chapter 6. Modelling resistance forces and drill string sticking effects…

373

Considering these arguments, it can be concluded that the connection of two segments of pipes with different curvatures can lead to an increase in the forces of contact and friction interaction and a decrease in the mobility of the string. To reduce these negative effects, it may be proposed to smooth the discontinuity of the well curvature by introducing at points B and C small bridge segments of the well with clothoid or cubic parabola geometry, where the radius of curvature changes almost linearly at a small length (Fig. 6-83). In this case, the graph shown in Fig. 6-79 is converted to the form given in Fig. 6-84, a. After that, function M (s) became continuous, but at points Bc , Bcc , C c , and C cc it is a polygon. Therefore, according to equality (6.64), internal shearing force F (s) differs from zero only in the bridge hyperbolic segments BcBcc and C cC cc (Fig. 6-84, b), while external contact force f cont (s) (according to equality (6.65)) is zero everywhere, except for points Bc , Bcc , C c , and C cc where it is discontinuous. This discontinuity can only be related to external distributed contact force f cont and resulting force Q . But in this situation, the distance between forces Q is approximately equal to the length of the clothoid or cubic parabola BcBcc , which is not small; therefore, f cont , the frictional force

f

fr

, and the shearing force are no longer large. A cubic parabola is the simplest curve whose curvature changes linearly on small segments. In fact, its equation in parametric form has the appearance

x bs 3 , z s Then, its curvature is calculated as follows k

>( xc)

2

@> >( xc)

(6.67)

@ @

 ( z c) 2 ˜ ( xcc) 2  ( z cc) 2   xcxcc  z cz cc 2

 ( z c) 2

2

3

,

(6.68)

where the prime mark denotes the differentiation with respect to parameter s . By substituting (6.67) into (6.68), we can get

k

>

6 b s 1  (3 b s 2 ) 2

@

32

.

(6.69)

If a short segment of the curve is selected, then (3b s 2 ) 2  1, and equality (6.69) takes a simple form: k 6b s . (6.70) In this case, the bending moment also changes linearly within the bridge segment (Fig. 6-83), and the effect of local increase of internal and external forces is mitigated in this zone. The presented arguments explain only the qualitative side of the problem because they do not take into account the influence of the gap between the string pipe and the well wall, which reduces the severity of these effects. However, they characterise the

Modelling emergency situations in drilling deep boreholes

374

main disadvantages introduced by the discontinuities of the curvature of the well trajectory. To confirm the effect of the trajectory curvature discontinuity on the values of the resistance friction forces generated by the motion of the drill string in a curved well, let's consider the model of the well (Fig. 6-85) with segments AB and CD of lesser curvature connected by the arc of circle BC . Their radii are R1 , R2 , and R3 , and the angles of coverage equal D1 15$ , D 2 60$ , and D 3 15$ , respectively. To perform a computer simulation of DS mechanical behaviour, the developed mathematical model of elastic curvilinear rod is used. According to it, the study of this construction is most convenient using the natural Frenet trihedron with unit vectors of principal normal n , binormal b , and tangent t and radius vector ρ(s) x i  y j  z k of the borehole element in the coordinate system Oxyz , where i , j , k are the unit vectors of this system; R is the curvature radius. Z

D1

s=S

x

D

R3

D3

D2

R1 A

R2 C B

s=0 Fig. 6-85 Geometry of the well trajectory outline

Assuming that the process of pulling the DS in the well channel occurs at constant speed v , we can represent vector f (s) of external distributed forces acting on the DS as follows:

f (s) f gr  f cont  f fr , where f

gr

is the force of gravity, f

the DS and the well, f

fr

cont

(6.71)

is the force of contact between the surfaces of

is the force of friction between these surfaces.

Chapter 6. Modelling resistance forces and drill string sticking effects…

375

Using the reference frame n , b , t , the equilibrium equations of the DS element can be presented in scalar form: d Fn k R Ft  kT Fb  f ngr  f ncont , ds d Fb kT Fn  f bgr  f bcont , ds d Ft k R Fn  f t gr  f t fr , ds (6.72) 0 k R M t  E I k R kT  Fb , d kR ds



1 Fn , EI

d Mt mtfr , ds where E I is the bending stiffness of the DS; Fn , Fb , Ft are the corresponding components of the internal forces vector; k R is the curvature of the DS; kT is its twisting; M t is the internal torque, mtfr is the external distributed torque. Four functions Fn (s) , Fb (s) , Ft (s) , and M t (s) are unknown in this system. External distributed forces f ncont (s) , f bcont (s) , f t fr (s) and torque mtfr (s) are also to be determined, while gravity forces f ngr (s) , f bgr (s) , and f t gr (s) are considered to be known. In this case, a plane system of forces is studied. Therefore, kT

f bcont

0 , Fb

0 , f bgr

0,

0 , and functions M b (s) , Fn (s) can be presented in the form:

Mb

E I kR ,

Distributed friction force

Fn

E I

dk R . ds

(6.73)

f t fr (s) and torque mtfr (s) are expressed through

distributed contact force f ncont (s) , friction coefficient P , and parameter K 100 that determines the relation between the axial and rotational speeds of the DS element. As a result, the state of quasistatic stretching of the DS with rotation is determined by the third and sixth equations of system (6.72), which take the form: d Ft k R Fn  f gr t z # f t fr , ds (6.74) d Mt fr mt . ds Here, f gr is the linear gravity force of the DS, f t fr P f ncont , P is the coefficient of friction, and the force of contact interaction is represented by the relation

Modelling emergency situations in drilling deep boreholes

f ncont

k R Ft  E I

The linear gravity force is calculated as



376

d 2kR  f ngr ds 2 .

(6.75)



f gr S J g r12  r22 , where r1 and r2 are the outer and inner radii of the cross section of the DS pipe; J is the density of its material, g = 9.81 m/s2 is the acceleration of gravity. As a result, system (6.74) contains only two unknown values ( Ft (s) and M t (s) ) and can be numerically integrated using the Runge-Kutta method. To demonstrate the effect of the influence of the well trajectory curvature function break on the forces of resistance to the motion of the DS in its channel, a computer simulation of this effect was made. First, the cases when the well centreline is composed of circular segments AB, BC, and CD without local smoothing of curvature at points B and C are considered (Fig. 6-83). Then, at these points short sections BcBcc and C cC cc were inserted in the form of cubic parabolas, and—as shown in Fig. 6-83— curvature discontinuities have been eliminated using cubic splines. Solutions of equations (6.74) were constructed using the Runge-Kutta method with integration step 's S /1330 , where S is the total length of the well centreline. It was calculated using the formula (6.76) S R1M1  R2M 2  R3M3 . In total, four tasks were considered. They differ in the values of radii R1 , R2 , R3 , which were R1 = 1,800 m, R2 = 48 m, R3 = 1,200 m (task 1); R1 = 3,600 m, R2 = 96 m, R3 = 2,400 m (task 2); R1 = 9,000 m, R2 = 240 m, R3 = 6,000 m (task 3), and R1 = 18,000 m, R2 = 480 m, R3 = 12,000 m (task 4). The values of well length S and the difference in the values of the curvatures at points B and C are given in Table 6-19. Table 6-19 Geometric parameters of the well trajectory Task No. 1 2 3 4

R1 (m)

1,800 3,600 9,000 18,000

R2 (m)

48 96 240 480

R3 (m)

1,200 2,400 6,000 12,000

S (m) 835.7 1,671.3 4,178.3 8,356.6

k 2  k1 (point B) (m-1) 0.02028 0.01014 0.00406 0.00203

k 2  k3 (point C) (m-1) 0.020 0.010 0.004 0.002

The following initial data on the DS rod material were selected: modulus of elasticity E=2.1·1011 Pa, density J = 7,800 kg/m3, g = 9.81 m·s-2. Two types of pipe cross section are considered. In the first case, its outer and inner radii are equal to r1 =

Chapter 6. Modelling resistance forces and drill string sticking effects…

377

0.08415 m and r2 = 0.07415. The thickness of the pipe was G = 0.01 m, the moment of inertia of its cross-section I=1.564·10–5m4, the linear force of gravity f gr = 380.5 N/m. In the second case, r1=0.1 m, r1 =0.088 m, δ = 0.012 m, I = 3.144·10–5m4 and f gr = 542.3 N/m. In addition, in computer simulation, the friction coefficient was set to P =0.3 and 0.4. For each combination of the calculated data, task 1 was solved at different lengths of the parabolic sections BcBcc and C cC cc . It was initially solved without spline interpolation (case 1). Then, the cases with parabolic inserts with lengths of two, five, and ten integration steps 's (cases 2–4, respectively) were investigated. The results of function calculations f t fr (s) for task 1 are shown in Fig. 6-86. Curves 1–4 (cases 1–4) correspond to the respective lengths of sections BcBcc and C cC cc . You can see that if circular arc sectors are connected without smoothing, the friction force f t fr (s) function has peak values at points B and C (curve 1). At the same time, the inclusion of transient cubic sections (curves 2–4 for lengths 2 's , 5 's , and 10 's , respectively) allows us to significantly reduce the effect of the curvature discontinuity. f t fr , kN/m

0

4

3

-10

2

-20

1

s, m

-30

0

200

400

600

800

1000

fr Fig. 6-86 Diagram of distributed friction force f t (task 1)

As shown on the graphs in Fig. 6-87, the local increase in function f t fr (s) is also due to the increase in the rate of increase in internal axial force Ft (s) at points B and C if the discontinuity of the curvature function is preserved (curve 1). However, the flattening of the trajectory curvature allows this force to be reduced (curves 2–4, respectively). The same conclusion can be made regarding the function of internal torque M t (s) (Fig. 6-88).

Modelling emergency situations in drilling deep boreholes

Ft , kN

250

1

Mt , N ˜ m

120

1 200

378

2

2

80

150

3

100

3 4

40

4 50

s, m

0 0

200

400

800

600

s, m

0 0

1000

Fig. 6-87 Diagram of axial force Ft (s) (task 1)

200

400

600

800

1000

Fig. 6-88 Diagram of torque M t (s) (task 1)

The found features become more obvious when increasing the bending stiffness of the DS. Figs. 6-89–6-91 illustrate similar results for strings with cross-section radii r1 = 0.1 m, r1 = 0.088 m, and moment of inertia I = 3.144·10–5m4. f t fr , kN/m

0

4

3 2

-20

-40

1 s, m

-60 0

200

400

600

800

1000

Fig. 6-89 Friction function diagram (task 1)

To estimate the dependence of the DS stress-strain states on the magnitude of the curvature functions discontinuity, the results of calculations for tasks 1–4 and cases 1, 4 at P = 0.3 are summarised in Table 6-20. They represent the peak values of functions

f t fr (B) , f t fr (C ) and axial force Ft (D) at top point D for trajectories with discontinuities of the trajectory (the top numbers) and trajectories with smoothing inserts with a length of 10 's (the bottom numbers). Similar results are given in Table 6-21 for value P 0.4 .

Chapter 6. Modelling resistance forces and drill string sticking effects…

Ft , kN

400

379

Mt , N ˜ m

250

1 300

200

2

1 2

150

200

3

100

3 4

4

100

50

s, m

0 0

200

400

600

800

0

1000

Fig. 6-90 Diagram of axial force function Ft (s) (task 1)

s, m

0 200

400

600

800

1000

Fig. 6-91 Diagram of torque function M t (s) (task 1)

Table 6-20 Extreme values of friction force f t

fr

and internal axial

force Ft at P =0.3 Task No.

1 2 3 4

I=1.564 ˜ 10 5 m4, fgr = 380.5 N/m fr F t (D) f t (B) f t fr (C ) (kN) (kN/m) (kN/m) -25.658 -25.638 247.649 -3.132 -3.331 205.552 -3.459 -3.687 401.334 -0.905 390.508 -0.696 -0.491 -0.739 971.978 -0.370 -0.581 970.193 -0.351 -0.562 1,940.215 -0.313 -0.561 1,939.680

I=3.144 ˜ 10 5 m4, fgr = 542.3 N/m fr F t (D) f t (B) f t fr (C ) (kN) (kN/m) (kN/m) -51.410 -51.426 384.351 -6.094 -6.371 299.987 -6.785 -7.098 579.811 -1.492 558.145 -1.195 -0.818 -1.171 1,386.458 -0.541 -0.841 1,382.896 -0.502 -0.816 2,765.394 -0.460 -0.802 2,764.370

Modelling emergency situations in drilling deep boreholes

380

Table 6-21 fr

Extreme values of friction force f t and internal axial force Ft at P =0.4 Task No.

1 2 3 4

I=1.564 ˜ 10 5 m4, fgr = 380.5 N/m

f t fr (B) (kN/m) -34.349 -4.319 -4.725 -1.061 -0.764 -0.627 -0.601 -0.527

f t fr (C ) (kN/m) -34.690 -4.705 -5.164 -1.444 -1.216 -1.010 -0.984 -0.980

f t fr (B) (kN/m) 305.485 244.250 474.172 458.481 1,139.702 1,137.360 2,273.933 2,273.712

I=3.144 ˜ 10 5 m4, fgr = 542.3 N/m

f t fr (C ) (kN/m) -68.762 -8.336 -9.210 -1.784 -1.247 -0.911 -0.858 -0.770

f t fr (B) f t fr (C ) (kN/m) (kN/m) -69.181 481.096 -8.888 358.344 -9.827 687.196 -2.330 654.745 -1.891 1626.067 -1.457 1621.301 -1.416 3,241.135 -1.405 3,240.435

From Tables 6-20, 6-21, it can be concluded that the effect of the curvature discontinuity on the external friction forces, internal axial force, and torque increases with an enlargement of the magnitude of the curvature discontinuity, the bending stiffness of the drill string, and the friction coefficient between the DS pipe and the well wall. Smoothing the curvature discontinuity has a beneficial effect on the drilling process. At the same time, if the difference in the curvature of the connected segments is small (task 4), the phenomenon of mitigating frictional effects becomes weakly noticeable. References to Chapter 6 1. Akgun F. A finite element model for analysing horizontal well BHA behaviour // Journal of Petroleum Science and Engineering. – April 2004. – V.42, No. 2 – 4. – P. 121 – 132. 2. Alexandrov M. M. Resistance Forces During the Movement of Pipes in a Well. – M.: Nedra. – 1978. – 208 p. (in Russian) 3. Analysis of wellbore instability in vertical, directional and horizontal wells using field data / [M.A. Mohiuddin, K. Khan, A. Abdulraheem, A. Al-Majed, M.R. Awall] // Journal of Petroleum Science & Engineering. – 2007. – V. 55. – pp. 83 – 92. 4. Basarygin Y. M., Bullatov A. I., Proselkov Yu. M. Complications and Accidents During Drilling of Oil and Gas Boreholes. – M.: Nedra. – 2000. – 680 p. (in Russian) 5. Bassov I.A. Deep sea drilling in the oceans // Soros Educational Magazine. – 2001. – Vol. 7, No 10. – P. 59 – 66. (in Russian)

Chapter 6. Modelling resistance forces and drill string sticking effects…

381

6. Bernt S. Aadnoy, Kenneth Larsen, Per C. Berg. Analysis of stuck pipe in deviated boreholes // Journal of Petroleum Science and Engineering. –March 2003. – V.37, No. 3 – 4. – pp. 195 – 212. 7. Bernt S. Aadnoy, Ketil Andersen. Design of oil wells using analytical friction models // Journal of Petroleum Science and Engineering. – 2001. – V.32, – pp. 53 – 71. 8. Birades Michel. Static and dynamic three-dimensional bottomhole assembly computer models // SPE Drilling Engineering. – June 1988. – pp. 160 – 166. 9. Chia C.R., Phillips W.J., Aklestad D.L. A new wellbore position calculation method // SPE Drilling & Completion. – September 2003. – V.18, No3. – pp. 209 – 213. 10. Choe Jonggeun, J.J. Schubert, Juvkam-Wold. Well-control analyses on extendedreach and multilateral trajectories// SPE Drilling and Completion. – June 2005. – pp. 101 – 108. 11. Completion and well-performance results, genesis field, deepwater Gulf of Mexico / [Robert D. Pourcian, Jill H. Fisk, Frank J. Descant, R. Bart Waltman] // SPE Drilling & Completion. – June 2005. – V.20, No 2. – pp. 147 – 155. 12. Cox R.J., Lupick G.S. Horizontal underbalanced drilling of gas wells with coiled tubing // SPE Drilling & Completion. – March 1999. – P. 3 –10. 13. Demarchos A.S., Porcu M.M., Economides M.J. Transversely multi - fractured horizontal wells: a recipe for success // SPE Annual Technical Conference and Exhibition, 24 – 27 September 2006, San Antonio, Texas, USA. – pp. 23 – 29. 14. Experience and problems of horizontal well construction / [V.P. Erokhin, N.L. Shchavelev, V. I. Naumov, E. A. Fadeyev] / / Borehole Drilling. - 1997. - No. 9. – P. 32 – 35 (in Russian) 15. Extended reach drilling (ERD) technology enables economical development of remote offshore field in Russia / [J.R. McDermott, R.A. Viktorin, J.H. Schamp et al] / SPE / IADC Drilling Conference, 23 – 25 February 2005, Amsterdam, Netherlands. – pp. 183 – 188. 16. Gullizade M. P. Turbine Drilling of Inclined Wells. – Baku: 1959. – 305 p. (in Russian) 17. Gulyayev V.I., Andrusenko E.N. Theoretical simulation of geometrical imperfections influence on drilling operations at drivage of curvilinear bore-holes // Journal of Petroleum Science and Engineering. – 2013. – V.112. – pp. 170 – 177. 18. Gulyayev V.I., Gaidaichuk V.V., Koshkin V.L. Elastic Deformation, Stability and Vibrations of Flexible Curvilinear Rods. – K. : Naukova Dumka. – 1992. – 344 p. (in Russian)

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19. Gulyayev V. I., Gorbunovich I.V., Glowach L.V. Elements of the Theory of Surfaces. – K.: Vidavnitstvo RVV NTU. – 2011. – 239 p. (in Russian) 20. Gulyayev V.I., Hudoly S.N., Glovach L.V. The computer simulation of drill column dragging in inclined bore-holes with geometrical imperfections // International Journal of Solids and Structures. – 2011. – V. 48. – P. 110 – 118. 21. Gulyayev V.I., Khudoliy S.N., Andrusenko E.N. Sensitivity of resistance forces to localised geometrical imperfections in movement of drill strings in inclined bore-holes // Interaction and Multiscale Mechanics. – March 2011. – V.4, No. 1. – P. 1 – 16. 22. Joseph H. Cho, Subhash N. Shan, Yeon-Tae Jeong. Development of statistical engineering process for analysing coiled tubing experimental data // SPE IcoTA Coiled Tubing Conference and Exhibition, 23-24 March 2004, Houston, Texas. – P. 35 – 41. 23. Kallinin A. G., Kulchinsky V. V. Natural and Artificial Well Deviation. – Izhevsk: IKI. – 2006. – 640 p. (in Russian) 24. Levitan M.M., Clay P.L., Gilchrist J.M. Do your horizontal wells deliver their expected rates // SPE Drilling & Completion. – March 2004. – V.19, No. 1. – P. 40 – 45. 25. Molchanov A. A., Lukyanov E. E, Rapin V. A. Geophysical Studies of Horizontal Oil and Gas Wells. – SPb: MANE. – 2001. –299 p. (in Russian) 26. Pogorelov A.V. Differential Geometry. – M.: Nauka, 1974. – 180 p. (in Russian) 27. Samuel, R.,. Systems and methods for modeling wellbore trajectories. ‒2008.‒US 886.2436 B2 Patent number US 12/145, 376. Publication date 14 Oct. 2014. 28. Samuel R., Gao D. Horizontal Drilling Engineering. Sigma Quadrant Publisher.‒2013.‒550 p. 29. Sawaryn, S.J., Thorogood J.L. A compendium of directional calculations based on the minimum curvature method // SPE Drilling & Completion, 2005, March, – pp. 24 – 36. 30. Selective placement of fractures in horizontal wells in offshore brazil demonstrates effectiveness of hydrajet stimulation process / [Jim B. Surjaatmadja, Ronald M. Willett, Billy W. McDaniel et al] // SPE Drilling & Completion. – June 2007. – V.22, No2. – pp. 137 – 147. 31. Sheppard M.C., Wick C.C., Burgess T. Designing well paths to reduce drag and torque // SPE Drilling Engineering. – December, 1987. – pp. 344 – 350. 32. Shepard S.F., Reiley R.H., Warren T.M. Casing drilling: an emerging technology // SPE Drilling & Completion. – March 2002. – V.17, No1. – pp. 4 – 14. 33. Suchkov B.M. Horizontal Wells. – M.: RKhD - 2006. – 424 p. (in Russian) 34. Sulakshin S. S. Directional Drilling. – M.: Nedra. – 1987. – 270 p. (in Russian)

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35. Technology of Oil and Gas Well Drilling / [A. N. Popov, A. I. Spivak, T. O. Akbulatov and others]. M.: Nedra. – 2004. – 524 p. (in Russian) 36. The application of rotary closed-loop drilling technology to meet the challenges of complex wellbore trajectories in the Janice field / [John Edmondson, Chris Abbott, Clive Dalton, John Johnstone] // SPE Drilling & Completion. – September 2002. – V.17, No 3. – pp. 151 – 158. 37. Tolstoy N. S. Vinogradov O. V. Horizontal drilling abroad // Geologiya Nefti i Gaza. – 1991. – No 12. – P. 30 – 32 (in Russian) 38. Tommy Warren, Bruce Houtchens, Garret Madell. Directional drilling with casing // SPE Drilling & Completion. – March 2005. – pp. 17 – 23. 39. Vicente R., Ertekin T. Modelling of coupled reservoir and multifractured horizontal well flow dynamics // SPE Annual Technical Conference and Exhibition, 24 – 27 September 2006, San Antonio, Texas, USA. – P. 33 – 39. 40. Yaremiychuk R.S., Semak G.G. Ensuring the Reliability and Quality of Deep Wells. – M.: Nedra. – 1982. – 264 p. (in Russian) 41. Zinchenko V.P. Directional Drilling. – M.: Nedra. – 1990. – 151 p. (in Russian)

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CHAPTER 7. CRITICAL STATES AND BUCKLING OF DRILL STRINGS IN CHANNELS OF DEEP CURVILINEAR WELLS 7.1. State of the problem of drill string stability in a curvilinear borehole 7.1.1.

Technical aspects of the phenomenon of drill string buckling in deep curvilinear wells

Over the past decades, the ‘shale gas revolution’ has been the greatest stimulus for the change in global energy market players. The North American energy balance and the positions of oil and gas sellers in Europe and the Pacific coast have been significantly affected. Even assuming that the ‘shale revolution’ is an exclusively regional phenomenon, it has nevertheless led to the redistribution of global energy flows throughout the world. The creation and development of directional (horizontal) drilling methods and fracturing procedures have made it possible to effectively extract shale gas and oil. However, as international practice has shown, the extremely rapid and successful development of technology for shale gas and shale oil exploration and production is ahead of the scientific understanding of the main features of their implementation. In this regard, there is an urgent need to intensify scientific work both in field research and computer modelling of all the critical aspects associated with drilling operations of extended oil and gas curvilinear wells. One of the most common emergency situations arising from curvilinear drilling is critical buckling of the drill string. This is caused by the loss of the ability of the drill string to carry an axial compressive force and is associated with a number of effects that negatively affect the entire technological process. They include: Increased distributed contact forces between the drill string and the borehole wall Increased distributed frictional forces between the string and the borehole wall Increased intensity of bending stresses in the drill string pipes Increased wear rate of drill string pipes Deterioration of the conductivity of axial force and torque from the drive mechanism to the bit. Increased total energy consumption for well construction As the accuracy of forecasting bifurcational buckling is relatively small, the buckling effect (especially in its initial phase) often goes unnoticed, and the process continues with abnormal conditions when the DS has partially lost its carrying capacity as a result of the buckling. Under these conditions, the listed negative effects are amplified, and an emergency situation (sticking) associated with a complete loss of string mobility may occur.

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Therefore, when studying the stability of an elastic DS in a borehole, the focus should be concentrated on determining the values of critical loads and modes of stability loss. For a string in a curvilinear well, such a study should be carried out using numerical analysis methods. They can be used to not only identify the onset of a critical state but also determine the pitch of the bifurcation harmonic, the width of the wave packet, and its localisation. After establishing the nature of the abnormal situation, measures are planned to deal with it. They can be associated with a change in the design of the string and the drilling tool, changing the drilling regime, and also redrilling the entire well or its individual sections. 7.1.2.

Mechanical influences leading to drill string buckling in deep curvilinear wells

Drilling and tripping in and out operations in deep curvilinear wells are accompanied by complex mechanical phenomena, the complexity of the drill stringwell system, the complexity of the forms of movement of its individual components, and the complex nature of the mechanical interaction between these components. Even if to assume that during bifurcational buckling of the drill string in the borehole, the influence of the drive mechanism and the type of drill bit design can be excluded from consideration, the remaining components of the entire structure, including the drill string, the borehole, and the drilling fluid, constitute a complex mechanical system. First of all, the string itself is a complicated element, as it consists of separate tubular pieces with different diameters and has additional supports (centralisers) in its lower part. In addition, it is inserted into the well channel with a clearance, whose axial line can have different geometric outlines. Finally, the drilling fluid moves inside the DS and the annular gap [46]. During operation, the drill string makes a forced rotation and moves in the axial direction. It is influenced by distributed gravity forces, contact forces of interaction with the borehole wall, circumferential and longitudinal frictional forces, and inertia forces from the drilling fluid flow, and at the upper and lower ends it is subject to axial forces and torque from the drive device and the bit. Under the influence of these disturbances, the drill string reaches states of complex stress and modes of oscillatory movements. With critical combinations of operating loads, these mechanical states may lose stability and be accompanied by string buckling. As noted above, the scientific literature indicates that string buckling can occur in twodimensional harmonic (sinusoidal) or three-dimensional spiral shapes. When discussing this feature, Cunha in article [10] comments that many experiments performed by Salies indicate that the string under the action of the axial force first protrudes in sinusoidal form, and then, if the axial force continues to increase, the

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amplitude of the sinusoidal shape increases to a certain limiting state in which the sinusoidal form snaps into spiral. Considering this circumstance, it is possible to connect the beginning of the unstable buckling of the drill string with its snapping along the sinusoidal form and, first of all, to investigate these critical states. Our book shows that this bifurcational mode is not just sinusoidal but has the appearance of a sinusoidal wavelet, so bifurcational analysis identifies the critical load and harmonic wavelet defined by the harmonic step, wavelet width, and the place of its location. It should be taken into account that these characteristics are determined by a complex combination of the variables of force and kinematic quantities, in turn, depending on the well geometry. The critical value of the axial compressive force in the string depends to a large extent on the balance of gravity forces acting on the elements of the system, frictional forces, and external forces applied to the lower and upper ends of the string. The horizontal and vertical components of these forces are expressed in terms of the inclination of the well trajectory, which can vary from zero to 90 . The contact force between the string and the borehole wall, which presses the string to the bottom of its surface and prevents the buckling process, depends strongly on this angle. And, finally, the system’s critical state is determined by the well centreline radius of curvature. All these factors should be taken into account when developing a mathematical model of bifurcational buckling effects in a curvilinear borehole. 7.1.3.

Mathematical aspects of stability problems of drill strings in curvilinear wells

7.1.3.1. Modified theory of flexible curvilinear rods and its features It is important to note that the task of investigating the stability of the drill string taking into account its contact interaction with the borehole wall has features that significantly complicate both the formulation of this task and the methodology for its solution. The first feature is that after buckling and coming into contact with the borehole wall, the DS continues to change its shape under the influence of increasing compressive force on the lower end, distributed contact forces, the forces of gravity, and torque. Therefore, simulation of this process can only be performed on the basis of the non-linear theory of flexible curvilinear rods. As a result, the process of critical and supercritical deformation of the DS in the well cavity can be compared with the motion of its elastic line along the corresponding channel surface. Second, the phenomenon of elastic bending DS in channels of curvilinear wells has one more specificity inherent in non-free mechanical systems, the movements of which are limited by constraints. The structure and complexity of the equations

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describing the motion of such systems are determined by the approach and coordinate systems used to model it. The Lagrange methods of the first and second kinds are commonly used in such cases. When the first approach is used, the system motion is considered in Cartesian coordinates, the equations of constraints are added to the constitutive equations, and the terms with additional unknowns (indefinite Lagrange multipliers) are added to the equilibrium equations (static or dynamic), which play the role of the constraint reactions. As the number of unknown variables in this case increases by the number of superimposed constraints, this method is only useful for systems with a small number of limiting conditions. At the same time, if we consider that each superimposed constraint reduces the number of its degrees of freedom by one, the advantage of the Lagrange method of the second kind (the method of generalised coordinates) becomes obvious. When applied, the problem is greatly simplified due to the possibility of preliminary consideration of some integral characteristics of system motion and further transition to the analysis of deformation and motion details and the calculation of constraint forces. Therefore, the difficulties introduced by the presence of constraints in the undetermined multipliers method become a source of advantages in the generalised coordinates method. In our book, these questions are treated specifically by using differential geometry methods and special reference frames (mobile trihedra) that allow us to separate the required variables and separately find them in a certain sequence. By excluding constraint forces (contact interaction forces) from the consideration, it was possible to construct a new (modified) theory of flexible curvilinear rods where only one unknown variable appears. We should remember that in the well-known theory [22, 45, 48] there are three of them (coordinates x(s) , y(s) , z(s) of the rod points). Therefore, the order of differential equations is reduced, and the general problem is essentially simplified. 7.1.3.2. Singularly perturbed equations in boundary-value problems of strings buckling in channels of curvilinear wells As noted above, the equations of drill string elastic bending in deep wells are singularly perturbed. In the general case in mathematical physics, this property of differential equations is associated with the presence of a small coefficient in front of the highest derivative [4, 52]. Then, the role of this term decreases in comparison with derivatives of a lower order, but it cannot be ignored since if it is rejected, not only the type of the differential equation but also the structure of its solution will change. Analysis shows that the presence of a small coefficient before the highest derivative leads to various losses of regularity in the structure of solutions, as

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singularities appear in them in the form of discontinuities, localised peaks or boundary effects. These features also occur in problems of drill string bending, but they appear in a somewhat different form. The point is that, for example, in the theory of rectilinear beams the leading (fourth and only) derivative usually has coefficient EI , where E is the modulus of elasticity of the beam material, I is the inertia moment of its cross section, and, as a rule, this coefficient is not small. In this case, if the beam is short, no specific features arise in the formulation of the problem and in its solution. However, if this beam is placed on an elastic foundation, an additional term appears in the equation of its bending in the form of the desired function multiplied by the coefficient of the bed (zero derivative). In this case, the role of the fourth derivative in the equation decreases, and the transverse load acting on the beam begins to be carried by an elastic foundation, whose influence increases with an increase in its bed coefficient. Finally, the effect under consideration is further strengthened as the beam length increases, as it becomes flexible, although its bending stiffness EI has not changed. The noted effects are even more pronounced in the problems of drill string buckling in channels of curvilinear wells since the bed coefficient (determined by the rigidity of the rock) and the length of the beam (string) are incommensurably large in comparison with cases typical of conventional technology. Therefore, it can be argued that the problems posed on curvilinear drill strings buckling are essentially singularly perturbed. Our results of modelling the effects of their loss of stability made it possible for the first time to construct buckling modes that are typical of them, which have the form of harmonic wavelets localised in the inner or nearboundary zones of the string. In this case, it was also possible to establish the basic characteristics of the wavelet, such as its pitch (the distance between two neighbouring zeros) and the width (a conditional value equal the distance between its beginning and end). Using the concept of singularly perturbed systems, it was also possible to answer for the first time a question of interest to specialists on buckling problems— namely, how boundary conditions affect critical loads and modes. It turned out that if a bifurcational wavelet is formed in the inner zone of the strings, the boundary conditions do not affect this process, if buckling occurs in the near boundary zone, the impact of provincial support is pretty weak since it applies only to one extreme spike in the wavelet. 7.1.3.3. Stuck and invariant (dead) states of drill strings in curvilinear wells One of the most unfavourable states that arise in drilling is when the drill string is stuck. This effect is usually associated with local caving of wall wells, pieces of

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rock filling the annular space, string misalignment, and loss of mobility (Fig.7-1). This situation is an emergency since it can be accompanied by a complete loss of the drill string and the well. However, in addition to these cases, similar states (but of a completely different nature) may arise during drilling. These are called dead lock states in the theory of machines and mechanisms.

Fig. 7-1 Diagram of a stuck drill string

They arise when the speed of the characteristic point of the leading link is perpendicular to the speed of the characteristic point of the driven link, and the system locks itself. For example, in a crank-connecting mechanism designed to convert the reciprocating motion of piston B in the rotational motion of crank OA (Fig.7-2, a), there are provisions where points O, A, and B lie on one straight line, and the system cannot be moved from this state by the movement of piston B along straight line OAB. This situation is dead if link B is the leading one and the system is insensitive to trying to move it. In more general cases in mechanics and in mathematics, when the system is insensitive in respect of any type of perturbation, it is said that the system is invariant in respect of this kind of perturbation.

A

B

O

a

O

B

A б

Fig. 7-2 Geometrically variable (a) and dead lock (b) positions of the two-link mechanism

In mechanics, invariant manifolds are certain functions that do not change during motion due to conservation laws: the law of conservation of energy, the law of

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conservation of momentum, the law of conservation of the angular momentum, the law of conservation of mass, etc. In dynamic systems, quantities that obey these laws are associated with integrals of motion that remain unchanged during the entire movement. The concept of invariant states is also widely encountered in automatic control systems implemented on the basis of the principle of invariance (independence, insensitivity) to any external influences. As already noted, in the theory of machines and mechanisms, the so-called limiting (dead lock) positions of spatial crank-and-rod mechanisms are known, which preserve the static geometry of their kinematic chain unchanged, with some combinations of forces acting on it, varying in proportion to a certain parameter. Usually, invariant states of variable systems are implemented based on the need to fulfil classical invariance conditions that require transformation to zero of the right-hand side of the differential equations describing the process under consideration. One of the most demonstrable examples of invariant systems encountered in mechanics of structures is the elastic beam, on which vertical force P acts as applied to one of its support points (Fig.7-3). P A

B

C

Fig. 7-3 Diagram of a three-support elastic beam

It is obvious that the stress-strain state of this beam is insensitive (invariant) in respect of force P , and, when it changes, only the support reaction at support B changes. f

Fig. 7-4 Diagram of an inextensible cable ring stressed by a uniformly distributed load

F M

F M

Fig. 7-5 Diagram of an elastic rod in a rigid holder

In this example, force P does not enter into the equation of elastic equilibrium of the beam itself. However, there are cases of invariant systems described by

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inhomogeneous equations. Fig.7-4 shows a diagram of a closed absolutely flexible inextensible thread stressed by distributed load f of equal intensity. The example of the invariant equilibrium of a flexible inextensible hose with an internal fluid flow is of important practical value. Its shape is invariant in respect of the flow velocity. The circular shape of a flexible rod placed in a rigid smooth cage (Fig.7-5) is also invariant in respect of axial force P . With an increase in force P , the circular shape of the rod does not change, and only distributed force f c of the contact interaction of its outer contour with the contour of the cage increases. From this example, we can proceed to a more complex case when the cage is a rigid smooth cylindrical cavity in which a long elastic rod is inserted in such a way that it is a regular cylindrical spiral conserved in equilibrium by axial force F , torque M , and distributed forces f c of contact interaction of the rod with the surface of the cylindrical cavity (Fig.7-6, a). It turns out that the cylindrical spiral created in this way is invariant in respect of force F and moment M , and, when they change, only contact distributed force f c changes (Fig.7-6, b). f

c

My

My

My

My

Fz

Fz

Fz

Fz

a

b Fig. 7-6 Invariant state of a flexible rod in a cylindrical cavity

An even greater generalisation of the examples considered is achieved when passing from a cylindrical surface to a constraining surface of a general form (differentiable required number of times), to the inner side of which an elastic rod is bent, curved along a geodesic curve. Since the length of the geodesic curve between its boundary points is minimal on a given surface, when the rod is loaded with axial forces and torque, it acquires the tendency to maintain its geometry unchanged (invariant). Such drill string states in curvilinear boreholes are similar to dead positions of mechanisms. To analyse the features of non-linear bending of an elastic rod contacting along its entire length with smooth rigid surface D , we will first distinguish some features of the state of its equilibrium. The simplest case occurs when the Gaussian curvature of surface D is positive, and absolutely flexible inextensible cable AB is pressed to

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it from the outside by two stretching axial forces Fc and Fcc equal in modulus and attached to the thread ends (Fig.7-7).

D A

B

Fcc

Fc

Fig. 7-7 Geodesic line AB on convex surface D

Under the action of these forces, the cable stretches and occupies a position in which its length is less than the length of any other curve on D , connecting points A and B . Such curve AB is called a geodesic one. It is the shortest on D between points A and B , and the curvature of its projection onto the plane tangent to D is equal to zero at each of its points. In this case, the cable is in equilibrium under the action of forces F c , Fcc and normal to D distributed force f c of its contact interaction with the surface, and this state does not change with a change of forces Fc and Fcc . It is said that in this case the system (cable) is invariant (insensitive) to forces Fc and Fcc and is stable under any their changes. A similar situation, but with some differences, occurs if a thin elastic rod is pressed against the inner side of surface D along geodesic curve AB by compressive axial forces F c and Fcc . Here, the rod will also be in equilibrium under the action of external forces F c , Fcc and distributed contact force f c , but at each point of its axis there is also an internal bending moment and a shearing force lying in the rectifying plane. These moments and forces are determined by the curvature of the normal section of surface D by rectifying plane and, therefore, do not depend on forces F c and Fcc . Bending moments and shear cutting forces in planes, tangent to surface D , are equal to zero since the geodesic curvature of curve AB is equal to zero. Therefore, the bending stress state of an elastic rod pressed by forces F c , Fcc to geodesic AB on surface D , does not depend on the values of forces F c , Fcc and is only determined by the geometry of the surface. In this case, with increasing Fc and Fcc in the rod, only internal axial force F increases, whose modulus is equal to the modulus of these forces, so does distributed force f c of contact interaction of the rod with the surface, that is, the rod is only pressed hard against the well wall. This force is normal to D

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and is determined from the equilibrium conditions of the external forces applied to the rod. In the result, as in the example with a cable pressed to a convex surface, the deformed state of a thin elastic rod pressed by two axial forces to a geodesic curve on a concave surface is insensitive (invariant) in respect of the magnitude of these forces, and the rod does not change its form when they vary. However, the situation takes a different form if curve AB is not geodesic. It is possible if not only axial forces F c , Fcc are applied to the rod but also other external influences (for example, gravity, friction, etc.). Then, with a change in external axial forces F c and Fcc the values of internal forces, moments, and deformations can evolve in an arbitrary way, and the axial line of the rod changes its geometry. The invariant state can only be implemented for it as a result of some special transformations. Both types of considered states can be implemented in drill strings. So, for a vertical well, surface D is cylindrical, and its geodesic lines are regular spirals, the number of which is infinite. Therefore, for such a system, an unlimited number of invariant states are possible, which manifest themselves as dead lock states. They almost do not react to the increase in axial forces and only press hard against the well walls. However, as the internal axial force F also increases, then, when Fc and Fcc increase, such invariant system state eventually comes that it becomes unstable, and, as a result of bifurcational buckling, jumps to another invariant condition or simply deforms with decreasing pitch of the spiral. Therefore, in the general case, when analysing drill string stability in a borehole, special attention should be paid to analysis of the stability of their invariant states. It should be noted that this is not an easy task even for cylindrical wells since with the curvature of their axial lines it is necessary to create special computational complexes for their analysis based on application of the theory of geodesic curves on channel surfaces of general form. 7.1.3.4. The role of frictional effects in the overall balance of force impacts on strings in curvilinear wells As a result of the forced curving of the drill string along the shape of the axial line of the borehole, bending moments arise in it. Under the action of these moments, the forces of contact and frictional interactions of the string with the well increase substantially. These forces can be calculated using our proposed modified theory of flexible curvilinear rods. The developed technique for calculating frictional forces and torque of friction makes it possible to determine these values with high accuracy in boreholes of almost any configuration taking into account the geometric imperfections of the well

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trajectory and local deviations of its centreline from the specified one. By using it on simpler examples, it is established that the effects of the formation of frictional forces in rectilinear and curvilinear wells are fundamentally different. In both cases when the string moves, the frictional force acting on it is equal to the product of the friction coefficient by the contact pressure force. However, in a rectilinear well, the force of the contact pressure depends only on the component of the external gravity normal to the axis of the well and does not depend on the internal axial force in the string. Therefore, it is believed that in this case the frictional force is additive. At the same time, for a string in a curvilinear channel, the situation is significantly complicated, as in this case the force of the contact pressure in the curved section depends not only on the gravity force but also on the internal axial force in the string and increases with the curvature of the channel. As a result, when the string moves in a curvilinear channel, additional frictional forces are generated. These forces lead to an increase in axial forces in the string, which increase the contact forces, thus leading to a growth of the frictional forces themselves. These forces further increase the axial forces, as a result of which the contact forces increase even more, etc. In connection with the frictional effects noted in this case, they are non-linear, and the described property of their formation is called multiplicative. Analysis of the results obtained from our methodology made it possible to draw the following conclusions: 1. As the direction of the friction forces has different orientations during tripping in and out operations, they interact differently with the forces of gravity. So, in the segment of increasing the DS depth, the frictional forces are added together with the forces of weight, when the string is tripped out, and they are subtracted when tripped in. Therefore, tripping out the drill string is associated with the application of greater force at its upper end than the descent. 2. The frictional forces on the curvilinear section depend on the bending stiffness of the drill string pipe. When it increases, these forces increase. They also increase in the places of rapid changes in the stiffness of the string. These effects occur at the locations of threaded joints of drill string pipes and in joints of pipes of different diameters. 3. The frictional forces depend on initial curvature κo of the drill string, and they increase when κo increases. 4. The frictional forces increase sharply in places where the initial curvature changes rapidly ( dκo / ds ). 5. The frictional forces increase in places where the curvature radius of the borehole decreases and in places where the curvature of the borehole rapidly changes (at the points of sharp bending and joining parts of different curvatures).

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6. The frictional forces increase due to the boundary effects of the appearance of a bending moment and a pair of cutting forces at the end of the drill string. The values of these forces (and friction forces) increase rapidly with increased bending stiffness EI of the drill string pipe as its diameter approaches the well diameter. With an increase in the borehole diameter relative to the diameter of the drill string and with decreasing EI , these forces fall. 7. Because the forces of normal pipe pressure on the borehole wall increase at the sites of well trajectory approaching horizontal, the friction forces at these sites also increase. In conclusion, we should emphasise that a more accurate calculation of the internal and external forces acting on a curvilinear DS allows us ‒ according to the experimental measurement of axial displacement and longitudinal force as well as the angle of twisting and torque moment‒to more accurately determine the positions of the stuck DS and take measures to release it. Obviously, the calculation of generated friction forces is also necessary in the study of drill string buckling. 7.1.3.5. The theory of bifurcations and the problem of string buckling The effect of the loss of stability of equilibrium and buckling of the drill string is a manifestation of one of the most general laws of philosophy‒the law of transition of quantitative changes to qualitative ones. In various spheres of reality, quantitative changes change into qualitative ones in different ways depending on specific conditions. Specific forms of transition, a jump from one state to another are the subject of study in many sciences. In mechanics, they are studied based on the theory of stability of equilibrium and bifurcations. The common and most typical forms of jumps are 1) the form of a relatively fast and sharp transformation of one quality into another when the object as an integral system immediately changes its structure; 2) the form of a gradual qualitative transition when the object does not change immediately but by the constant accumulation of qualitative changes. Both forms also exist in problems of the stability of mechanical systems. Depending on the type of qualitative changes with a loss of stability, the losses of stability of type I and type II are noted. When analysing the stability of the equilibrium state of a mechanical system, it is usually attempted to establish the limits of the change in load parameters under which the given system has a single equilibrium form. Euler, investigating the longitudinal bending of the rod, indicated the way of finding these limits on the basis of the transition to the eigenvalue problem (the Sturm-Liouville problem). This method subsequently became widely used and was strictly justified. The method

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adequately justified itself in respect of rods, rod systems and plates, which was facilitated by the fact that linearization in these cases takes place in the vicinity of the initial momentless stress state, which can be calculated by relatively simple means. However, attempts to use linearization to solve essentially non-linear stability problems often fail, because the usual linearization principle gives a distorted view of critical loads and modes. It turned out that it should be applied by linearizing the problem in the vicinity of a previously unknown solution or to reject linearization and proceed to a direct global investigation of the non-linear equations describing the deformation of the system. So, since these relations represent a complex system of partial differential equations containing load parameter λ , the problem is reduced to investigating the spectrum of some non-linear boundary value problem [18, 21]. As a rule, this research cannot be carried out strictly. Significant difficulties also arise when trying to solve this problem by approximate methods for drilling strings in channels of curvilinear wells since the axial line of the string assumes a buckling shape that has sections of smooth and rapid trajectory changes, which significantly complicates the task since in this case it is very difficult to approximate the shape of the centreline by simple approximating functions. The consequent change in the shape of the deformed axis during the development of the loading process leads to the need to investigate the string as a system with a large number of degrees of freedom. The difficulties mentioned can be overcome by moving to the Cauchy problem using numerical methods. In this case, the solution of the system of non-linear equations λb ,

F ( x)

(7.1)

describing the bending of the string is based on replacing non-linear operator F in the vicinity of the state by the generalised Taylor expression. Here, F is the nonlinear operator of the boundary value problem; x is the vector determining the deformed state of the string; λ is the parameter characterising the system perturbation. The restriction in the number of terms in the Taylor expansion makes it possible to introduce an analogue of the Newton-Kantorovich algorithms, tangential hyperbolas, and the like in the vicinity of state x( n ) : x( n1)

1

x( n)  ª¬ F( n) º¼ ( λ( n1)b  F( n) x( n) ) ,

(7.2)

where predicted element x(an ) is selected on the basis of elements x( m) (m d n) . Formula (7.2) describes the operation of finding a sequence of elements x(i ) that

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397

λ(i ) . It combines well the simplicity of

Newton's method and the high accuracy of higher-order algorithms such as Chebyshev and Aitken. Based on discretisation methods, operator equation (7.1) can be reduced to a system of non-linear algebraic equations, and Frechet derivative F c , to the Jacobi matrix. Analysis of the value of determinant J of the Jacobi matrix allows us to judge the appearance of branching (bifurcating) solutions and the stability of the system state. At the points of reference of the Jacobian J to zero, the problem of continuation of the solution by method (7.2) turns out to be incorrectly posed, as small changes in perturbation parameters can correspond to large changes in the solution of the problem. The notion of correctness and an example of an incorrect problem of mathematical physics were given at the beginning of the 21st century by Hadamard. A great contribution to the development of the theory and methods of solving problems of mathematical physics that are not correct in the classical sense was made by V. K. Ivanov, M. M. Lavrentyev, A. N. Tikhonov, and others. The limit value of parameter λ , under which the Jacobian vanishes, is critical. At these points, it is necessary to investigate the possibility of branching the solution ‒ that is, its bifurcation. We should recall that the bifurcation point of a solution is a point where one parameter value λ corresponds to two or more solutions in its small vicinity. Usually, a branching equation is used to study the trajectory of the loading of a mechanical system in the vicinity of a singular point based on the replacement of the non-linear equation by a certain number of terms of its expansion in a Taylor series in the vicinity of the bifurcation point. The solutions of the branching equation determine the directions of the branching solutions in a small vicinity of the degeneration point. In this case, multiple solutions should be considered separately. Continuation of the loading curve according to expression (7.2) for each of the directions found makes it possible to construct all branches of bifurcation solutions. The combination of the method of continuation by the parameter with branching theory methods makes it possible to find solutions and values of parameter λb at which branching takes place, determine the number of solutions branching off from xb , and investigate the behaviour of these solutions for different values λ . Special mathematical difficulties are associated with the study of the stability of mechanical systems with unilateral constraints ‒ that is, contact interactions capable of perceiving the efforts of only one sign. Such problems also arise when investigating the stability of a drill string in a cavity of a curvilinear well since in this case the borehole wall limits the movement of the string in only one direction.

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This class of problems is the least studied, it leads to solving a non-linear boundary value problem under additional conditions specified in the form of inequalities. To solve it, it is necessary to involve non-linear programming methods. In the general case, the solution of equation (7.1) can be constructed by continuing the solution in respect of the parameter. Let there be given a non-linear functional equation F ( x)

0.

(7.3)

The Frechet k ‒ times differentiable non-linear operator F translates elements of space X in space Y . Let G( x; λ) Φ( x)  λb ( x  X ; 0 d λ d 1 ) be an operator with values in Y , G( x; 1) Φ( x)  λb { F ( x)

(7.4)

G( x; 0) 0

(7.5)

G( x; λ) 0 .

(7.6)

and equation

has an obvious solution x0 . Consider the relation

Assume that equation (7.6) has a continuous solution x

x( λ) determined at

0 d λ d 1 and satisfying condition

x(0)

x0 .

(7.7)

If solution x( λ) is known, then formula x

(7.8)

x(1)

gives a solution to equation (7.3). We divide the interval [0; 1] with points λ(0)

0  λ(1)  ...  λ( m)

1.

(7.9)

Using the Taylor formula in the vicinity of point x( n ) for operator Φ , we have

Chapter 7. Critical states and buckling of drill strings in channels…

Φ( n )  Φc( n ) ( x( n 1)  x( n ) ) 

1 Φcc( n ) ( x( n 1)  x( n ) , x( n  2)  x( n) )  ...  2!

1  Φ(( rn)) ( x( n 1)  x( n ) ,..., x( n 1)  x( n ) ) | λ( n 1) b r!

399

(7.10)

Keeping in expansion (7.10) a certain number of terms and having element x( n ) , it is possible to find approximately x( n 1) . At r 1, the relation defining x( n 1) has the form x( n 1)

1

x( n )  ª¬Φ( n ) º¼ ( λ( n 1) b  Φ( n) x( n) ) .

This formula and equality x(0)

(7.11)

x0 describe a sequence of operations (Fig.7-8)

leading to element x( m ) approximately satisfying equation (7.3). Since each step of process (7.11) represents one step of the iterative Newton-Kantorovich method, for its convergence it is sufficient to satisfy the conditions following from the convergence conditions of the last algorithm [21].

λ(n+1)b-Φ(x(n))

λb

λ( n 1)b θ( n )

λ( n )b

λ(1)b

x( n 1)  xn Φ(x(n))

λ(2)b

Φ(x)

x(1) x(2)

x( n )

x( n 1)

x

Fig. 7-8 Geometrical representation of the algorithm for continuing the solution by tangent segments

When solving non-linear equation (7.3) using method (7.11), one must deal with singular points x(sn ) in which operator Φc( x(sn) ) degenerates, and continuation of the solution becomes impossible. If the defect of the linearized system of equations is equal to one, and the right-hand side is not compatible with the left one, the transition through the singular point is affected by a change of the leading parameter. In the general case, the solution of this equation at a singular point can turn out to be

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branching. To continue the solution, we need to use branching theory methods to find all the branches and continue the solution for each of the branches (Fig.7-9). λ H

A

G

D

F

C

B

R

S

E

x

Fig. 7-9 Types of branching solutions of non-linear equations

In mechanics of structures, the simplest effect associated with the branching (bifurcation) of the solution of the non-linear equilibrium equation is the loss of stability of the rectilinear shape of the hinged elastic rod under the action of axial compressive force λ (Fig.7-10). This problem was solved by L. Euler in 1744. As a result of analysis of the linearized bending equation

EI

d 4u d 2u  λ dx 4 dx 2

0,

(7.12)

where EI is the stiffness of the rod when bending; u( x) is the function of its small lateral displacement; x is the coordinate directed along its axis. λcr

λ

x

L

u ( x)

λcr

λ a

b

Fig. 7-10 Bifurcational buckling of a rectilinear elastic rod

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401

Equation (7.12) is valid for λ = λcr. When λ = λcr, deflections u( x) 0 , and with change of λ the loading curve shown in Fig.7-8 is represented by a vertical straight line (Fig.7-11). When λ = λcr, equation (7.12) degenerates, and in addition to unstable solution u( x, λ) 0 (λ = λcr) (shown in dashed lines on the vertical) it has a known non-zero solution in the form of a sinusoid that branches off from the vertical to the left ( u  0 ) and right ( u ! 0 ). Further investigation of the rod in supercritical states should be carried out on the basis of the corresponding non-linear equation. As its solution shows, as you move away from the vertical, the branching solutions rise. This indicates that in supercritical states the rod retains its carrying capacity, although it becomes very deformative.

λ λcr

O

u( L / 2)

Fig. 7-11 The load curve of a straight rod with axial force λ

In a similar statement, we also formulate the problem of bifurcation buckling of a long drill string lying on the bottom of a curvilinear well channel. It is believed that under subcritical conditions the string is stressed by axial force but retains its load-bearing capacity, and its transverse displacements are equal, and the stress state is calculated proceeding from this premise. With an increase in the axial force in the pre-critical stage proportional to parameter λ , loading of the string takes place according to the diagram shown in Fig.7-11 ‒ that is, parameter λ increases, but the transverse movements of the string remain zero. Then, when parameter λ reaches critical value λcr, the linearized string balance equation degenerates, and the solution branches, as is the case in Fig.7-11, and instead of one solution three arise, one of which (dashed) is unstable, and the sustainability of the other two (extending outward) should be tested further in the supercritical state. We should note that this investigation is not carried out in our case, and just as for Euler, there are only critical values λcr and the form of a branching (bifurcation) solution that determines the mode of loss of stability.

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7.1.4.

402

Review of the results of theoretical modelling of the bifurcational buckling phenomena of drill string in channels of curvilinear wells

The features of mathematical models that determine the process of drill string buckling in a curvilinear well given in paragraph 7.1.3 significantly complicate not only the structure of these models but also the algorithms of applied computer calculations. At an early stage of studying buckling processes, these difficulties were not explicitly formulated; nevertheless, they prevented a complete and reliable analysis of this phenomenon. Therefore, in the first stages, only some of its particular properties were studied with the help of very simplified models reflecting only the simplest characteristics of this process. Then, with the recognition of its main features, the computational models became more complicated and reliable. The problems of bifurcation buckling for free short rods were first formulated by Euler, but his solutions turn out to be practically unsuitable for extended curvilinear DSs. Attempts to formulate and solve this problem in the field of deep hole drilling were made by many authors, but because of its great complexity, as a rule, it was formulated in very simplified formulations and using hypotheses of a kinematic and static nature that reduced the value of the results obtained. For example, G. Woods and A. Lubinski (1960) studied in detail the longitudinal bending of a drill string in a vertical borehole describing it as a thirdorder differential equation, the general solution of which was given in the form of power series. Assuming that both ends of the string are pivotally supported, these authors obtained the first approximation formula for the critical length of drill pipes. They repeated some results first obtained by A. N. Dinnik and F. Willers. Furthermore, A. Lubinski [33] investigated the longitudinal bending of the lower section of the drill string with axial loads exceeding the value of the first-order critical force. A. Lubinski [33] was also one of the first to determine the effect of drilling mud on the longitudinal bending of the string. He showed that in stability calculations the effect of a liquid can be approximately taken into account when determining the weight of a unit length of pipes. M. Rothmann first obtained a rigorous solution to the problem of the stability of a heavy elastic rod of considerable length. The deformation of the axis of the rod during bending in the study of M. Rothmann was described by the same equation as in A. Lubinski, and its general solution was represented as a linear combination of integrals of Airy functions. Rothmann also found the value of the first-order critical length for a heavy elastic hinged support rod. D. Daring and B. Livesay (1968) and D. W. Daring and E. L. Radzimovsky (1965) investigated the longitudinal bending and lateral vibrations of a drill string on the basis of solving a fourth-order partial differential equation in respect of

Chapter 7. Critical states and buckling of drill strings in channels…

403

deflection. This equation was reduced to an ordinary differential equation whose general solution is represented in the form of a power series. The main drawback of this solution lies in the fact that the drill string stability is investigated under conditions that are not characteristic of rotary drilling (it is assumed that the lower end of the string is clamped, and the upper end is hinged, in addition, the spatial work of the string is not taken into account). Phenomena that have not yet received exhaustive analysis include, for example, bifurcation buckling of a rotating DS. Since a direct solution to this stability problem is difficult, the approach based on the formulation of the inverse problem connected with analysis of the supercritical state of the string has become widespread. It is assumed that the shape of the elastic rod in the supercritical state is a spiral, whose pitch is determined by the method of virtual displacements, and the longitudinal force that this spiral can withstand is calculated. The beginning of this approach was put by Arthur Lubinski in the 1950s, later this technique was covered in his monograph [33]. In the study of stability, Lubinski ignores rotation, torque, boundary conditions and assumes that the longitudinal compressive force is constant along the length of the infinite rod. Thanks to this significant simplification of the task, he managed to obtain its analytical solution. Then, the formulation of the task became more complicated. R. Mitchell in 1986 [36] took into account the frictional forces between the DS and the well walls. Then, he obtained an approximate solution of the problem taking into account gravity [35, 37, 38], the curvature of the borehole, and the presence of torque (1988). Kwon and W. Young as well as X. C. Tan et al. (1993) built a solution in the shape of a spiral with a variable pitch for a weighty pipe. De-Li Gao et al. [16, 17] studied the plastic deformation of spirals under conditions of high temperatures and the presence of frictional forces but without torque and rotation. S. Edwards, B. Matsutsuyu, and S. Willson (2004) studied the loss of wellbore stability in contact with the DS; Chao Sun and S. Lukasiewicz (2006) investigated the stability of an elongated rod in the well pump system. A review of the results of the stability study of DS is presented in J. C. Cunha's paper [10]. It shows that the found solutions concern the supercritical states of flexible rods. They are built, as a rule, without taking into account the specific length, boundary conditions, and the presence of centring devices ignoring the rotation and internal fluid flows and with various simplifying assumptions about the distribution of internal axial force and torque. Some general questions of the analysis of additional statistical and dynamic effects in drill strings, the determination of critical states and the elimination of accident consequences are discussed in publications [10, 11]. In [18, 49], a technique was proposed for determining the critical states and modes of buckling of vertical compressed-stretched twisted rotating DSs of large

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404

length taking into account all the factors used in constructing eighth-order constitutive equations. To integrate these equations, the initial parameter method was used in conjunction with the orthogonalisation procedure. The Cauchy matrix was constructed using the highly accurate Everhart method. Using this technique, critical values of characteristic parameters are found. It is shown that the shape of the buckling stability of the DS has a spiral outline predominantly in its lower part experiencing compression forces, this spiral being irregular and superimposed with smaller spiral waves (superposition of large-scale and small-scale spiral wavelets takes place (2007)). The technique was tested using examples of drill string lengths up to 10 km. It makes it possible to investigate the stability of the DS together with several sections of the BHA. The problem of linear buckling of the DS in the inclined well was first considered by Paslay and Bogy (1964) [42]. Dawson and Paslay (1984) [11] constructed an asymptotic method for solving the buckling problem of a long rod. Chen et al. (1990) [6] investigated the loss of stability in respect of the spiral shape of long rods in horizontal wells. Huang and Pattillo (2000) [30], using the Rayleigh-Ritz method, obtained solutions to the problem of spiral buckling of a pipe in a directional well. The loss of stability of the DS in directional wells with a large distance from the drilling rig was investigated by Mitchell (1988, 1997, 2002) [35, 36–38]. He proposed criteria for loss of stability in sinusoidal and spiral modes. However, all the above studies were conducted without taking into account the impact of frictional effects. At the same time, it is known that the effects of stuck DSs, in which they lose the ability to move due to frictional interaction with the borehole wall, limit the possibility of drivage elongated directional wells. As in these cases friction plays a decisive role in the occurrence of sticking effects (mainly in horizontal wells), it is very important to formulate the sticking conditions, especially in wells with small angles of inclination, in which the frictional force reaches the greatest values. It is known that the behaviour of the DS in the well is mainly influenced by two factors caused by friction. First, the frictional forces exert a decisive influence on the magnitude and distribution of the internal axial force along the length of the rod. Second, the frictional effect has a significant effect on the mobility of the string at the initial stage of loss of stability. One of the first, influences of friction forces on the stability of the DS was taken into account by Mitchell (1986). Then, to predict the sticking of a spirally protruded rod inside the cylindrical channel Wu and Juvkan-Wold (1993) investigated the question of its axial motion. Later, a more accurate analysis of the distribution of axial forces in a spirally curved column in a directional well was

Chapter 7. Critical states and buckling of drill strings in channels…

405

conducted by Miska et al. (1995). We should note, however, that this analysis was performed without taking into account boundary effects. An experimental study of the influence of frictional phenomena on the buckling process was carried out by McCann and Suryanatayana (1994) [34], Martinez et al. (2000), as well as Gao et al. (1995) [28, 29]. In the theoretical analysis of the results of the experiments, they confined themselves to a linear formulation. Further development of this problem was carried out using various methods and approaches. This has been reflected in a large number of publications in recent years. In 2008, Wicks et al. gave a detailed overview of the state of this problem for that period. It follows from this that the transition of the form of the loss of stability from a plane to a spiral harmonic is still poorly understood, and there is insufficient experimental data on the effect of DS bending in a cylindrical cavity. Moreover, there have been practically no attempts to study the effect of friction forces on bifurcation processes. Along with this, tests of stability loss in the laboratory with even short rods inside the cylindrical cavity were not fully studied. The mechanics of these phenomena have still not been fully analysed. As noted in the scientific literature, the significant complexity of the problems of analytical investigation of buckling of short rods in cylindrical cavities is due to the influence of gravity forces, frictional forces, and boundary conditions. Therefore, numerical methods, such as the Galerkin method (Mitchell, [37, 38]), the finite element method (Liu and Wang, 2004), and the method of difference quadratures (Hakimi and Moradi, 2010), were also used to solve them. They showed that the investigation of bifurcation effects taking into account the action of frictional and gravitational loads can only be carried out approximately, using local approaches, such as the finite element method. On the other hand, if a global approach is used, for example, the differential quadrature method, then a numerical instability can arise due to the large number of nodes of numerical sampling. In this regard, to overcome these difficulties, Gan et al. (2009) [15] proposed using the method of differential quadrature elements. To solve these problems, Wei (1999, 2001, 2002) proposed a discrete-singular convolution method with the property of increased accuracy inherent in global methods and the universality inherent in local methods. The proposed approach was subsequently used to analyse the stability of the DS (Civalek, 2007; Wei, et al., 2002; Xiang et al., 2002). An analysis of these results is given in the article by Wang and Yuan (2012). As noted in these works, the most negative effect associated with the buckling of the DS inside the well is the deterioration of the conditions of contact interaction between the DS and the well wall, thus leading to a significant increase in frictional forces.

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The most comprehensive analysis of research results in this area is presented in review articles (Cunha, 2004 [10], Mitchell, 2008 [38], Gao and Huang, 2015 [16]). It follows from them that, as a rule, the approaches used in bifurcation analysis problems are based on the approximation of eigen modes by regular sinusoids and spirals, and the values of critical loads and buckling patterns are mostly guessed. Therefore, as Cunha notes (2004), this circumstance seems to be the reason why the solutions to the problems of buckling stability in curvilinear wells obtained by different authors contradict each other and real observations. Mitchell in the review (2008) also emphasises that in the field of DS stability research there are still many challenges that need to be addressed and many difficult questions that need answering. They include unsolved fundamental problems related to the issues: How can we find the critical value of the load leading to DS buckling in a three-dimensional curvilinear well? What is the role of friction forces in the effect of initiating a loss of stability? What is the role of boundary effects on bifurcation phenomena? Recent advances in the analysis of bifurcation stability are discussed in a recent article by Gao and Huang (2015). They note that the fundamental unresolved issues in this area remain related to the establishment of the role played by friction forces in the phenomenon of loss of stability, and how they should be included in analytical and numerical models. Mitchell (2008) summarises that perhaps the most important force is the forces due to frictional effects, which is the least studied in the problems of DS buckling analysis. Usually it is not difficult to calculate the magnitude of the friction force, but at the same time the main difficulty is connected with determining the direction of the friction vector. We can agree that this vector cannot be determined in the presence of friction contacts in immobile elastic systems since this problem is statically indeterminate (Mitchell and Samuel 2009 [37]). However, as highlighted in our papers (Gulyayev, Shlyun 2016 [24]), if one of the contacting bodies slides over another, the vectors of slip velocity, and frictional forces are collinear, and then their magnitude and direction can be determined. This feature facilitates the analysis of frictional effects in bifurcational buckling of the DS. We should also emphasise that in all the cases under consideration the stability of small-length drill pipes is investigated. We have shown [19, 20, 24] that since the problems of studying the buckling of a drill string in deep wells are singularly perturbed, it is impossible to predict the localisation zone of the bifurcation form in advance. Therefore, it is necessary to formulate a boundary eigen value problem in the region of the entire length of the string. This approach, unlike those discussed above, is global.

Chapter 7. Critical states and buckling of drill strings in channels…

407

7.2. Buckling of a drill string in a directional channel of a rectilinear borehole 7.2.1. Specifics of the problem of drill string bifurcational buckling in a directional channel of a curvilinear borehole Since the phenomenon of the onset of the critical state and buckling of the drill string in the cavity of a curvilinear well is completely analogous to the effect of the Euler stability loss of a straight rod compressed by axial force, the approach used by Euler can be adopted for its analysis. However, if for a rectilinear rod up to critical value λcr in Fig.7-11 equation (7.12) is valid with constant coefficients and with trivial solution u( x) 0 , then for a curvilinear string it is necessary to construct a homogeneous linearized equation of neutral equilibrium with variable coefficients for a beam lying on a geodesic curve and in an invariant state, and when searching for critical value λcr each time to redefine contact and friction distributed forces, then recalculate the internal axial force and reassign the coefficients in the linearized bifurcation equation, and only then check whether parameter O is critical. Therefore, in this case, at each step of the bifurcation search algorithm, an additional independent problem has to be solved to determine the stress-strain state of the curvilinear string with selected O . Let us recall that it is discussed in Chapter 6. In this case, the first problem is solved by setting up the Cauchy problem, the second problem, by setting the boundary eigen value problem. Another feature of the stability problem is that, when buckling, the string continues to remain on the channel surface of the well. Therefore, we must also use an additional mobile trihedron, one of whose axes always remains orthogonal to the channel surface. Finally, because of the complexity of the geometry of the system, it is sometimes convenient to simultaneously use two parameterisations of the well centreline and the string using dimensionless and natural parameters. 7.2.2. General equations of elastic bending of a free curvilinear rod We shall assume that at the initial state the drill string tube does not undergo deformation and is rectilinear, and when driving in a long curved (directional and horizontal) drill wells it bends and takes the shape of the well trajectory (Fig.7-12). Since the structure of the DS undergoes considerable curvatures, it is inappropriate to use the theory of rectilinear beams to describe its elastic deformation, and it becomes necessary to describe its deformation using the theory of flexible curvilinear rods. Two approaches are possible. In the first approach, the centreline of the DS is identified with a curve, the position of an arbitrary point on which is given by the length from some initial point to the current one (natural parameterisation). In this case, we say that the metric of this curve is given by the length of its element. In

Modelling emergency situations in drilling deep boreholes

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this subsection, we derive the basic resolving equations of the theory of flexible curvilinear rods in this metric. Their advantages are that they have a simple structure; unfortunately, they are only convenient to use for curvilinear rods with the simplest outline of the centreline.

x

0

ds

Dd-

z Fig. 7-12 Diagram of elastic bending of a drill string in a cavity of a curvilinear borehole

Below, we consider the cases of DS bending in wells of complex geometry, in which it is rather difficult to use the metric of the length of the centreline. Therefore, another approach is used for the study, with the help of which the correlations of the elastic bending and equilibrium of the DS in the arbitrary metric of their centreline are derived using a dimensionless parameter. Based on these equations, the behaviour and stability of curvilinear DSs of a more complex shape is analysed. In the most complicated cases, both systems of these equations are jointly used [22]. We shall consider that straight or curved rods are called flexible rods if their cross-sectional dimensions are rather small in comparison with their linear dimensions or the curvature radius of the rod axis. An important property of flexible rods is that under the influence of external loads the elastic line of the rod can acquire a shape that differs significantly from its original one. In this case, however, the material of the rod obeys Hooke's law, and the rod operates in an elastic stage ‒ that is, the load at all its points is below the proportionality limit. The deformations of the rod will be quite small, the displacements of two close transverse cross-sections of the flexible rod are relatively small relative to each other and can be described by means of the basic equations of the theory of elasticity based on the Hooke linear law.

Chapter 7. Critical states and buckling of drill strings in channels…

409

The internal forces and moments arising in the curvilinear rod as well as the magnitude of the elastic displacements in each of its cross sections are non-linear functions of external loads. Therefore, theorems on reciprocity and the principle of superposition of stresses and displacements in the theory of flexible rods are not always appropriate. It also assumes that the transverse normal sections of the rod (flat before deformation) remain flat after deformation since the shifts are not taken into account. The total length of the rod remains unchanged; stretching or shortening of the elastic line of the rod during its deformation is ignored. In this case, internal force factors are determined from the equilibrium conditions. A significant change in the shape of the rod leads to the need to consider separately its internal and external geometry, using the Lagrange and Euler approaches for their determination. The internal geometry of the rod is given by coordinate s (Fig.7-13), which is conveniently measured by the length of the centreline from a certain starting point to the current one and by the moving right coordinate system (u, v, w) rigidly connected with the considered cross-section. Let the origin of this system lie at the centre of gravity of the cross-sectional area, axes u and v are directed along the principal central axes of inertia of the cross-sectional area, and axis w is along the tangent to the elastic line in the direction of increasing s . Coordinate s together with time forms Lagrangian variables. It should be noted that the convenience of using the longitudinal coordinate and the simplicity of the constitutive equations are determined by the metrics of longitudinal coordinate - of the well centreline in cases where it differs from its length s . This metrics is given by the square of the elementary arc (Fig.7-12) ds 2

D2 d- 2 ,

where D is the multiplier specifying the metrics [5]. If D = 1, the metric entity of coordinate - coincides with length s , and the geometric characteristics that determine the given curve acquire the simplest form. However, for curves with complex geometry, it is not always possible to select such a parameterisation, and we have to analyse the curve using coordinate - with variable metric factor D D(- ) . For the problem under consideration, sometimes it is convenient to use parameters s and - jointly.

Modelling emergency situations in drilling deep boreholes

410

u

z

F

n

τ w b

s

v

y

x Fig. 7-13 Axes u , v, w orientation and mobile trihedron diagram

We also introduce the natural trihedron of the elastic line of the rod with unit vectors of principal normal n , binormal b , and tangent τ . In this case, the w axis coincides with unit vector τ , and unit vector n and axis u (or b and v) in the general case may not coincide and form angle F , which varies from unit vector n up to axis u. In this case, F can vary along the elastic line. The main trihedron, which belongs to any elastic line point with a load changing during deformation of the rod, can be arbitrarily oriented in space (move progressively and rotate). The external geometry of the rod determines the position of each of its points and the entire elastic line in fixed coordinate system Oxyz . Geometric coordinates of space Oxyz are Euler variables. It is convenient to jointly apply the Lagrange and Euler approaches using the first one for the rod point individualisation, and the second one for describing its geometry. Then, the main problem of the theory of flexible rods is reduced to establishing a connection between the Lagrange and Euler variables x

x (s,- ) ,

y

y (s,- ) ,

z

z (s,- ) .

(7.13)

Chapter 7. Critical states and buckling of drill strings in channels…

411

We give the main relationships that determine the geometry of the elastic line of a curvilinear rod. Let point M with radius vector r(s,-) describe a certain line in space. Then, this line is the hodograph of radius vector r , and the derivative of vector dr ds is the unit tangent vector directed towards the s growth. So, τ

dr ds ,

τ

Using this equality, we calculate d 2r ds 2

1.

(7.14) dτ ds . Because τ is the unit

vector, its derivative dτ ds is a vector perpendicular to τ and equals lim 'M 's , 's o0

where 'M is the angle between two adjacent unit tangent vectors τ and τ  'τ , which is called the contiguous angle. The limit of the relation of the contiguous angle to element ' s is the curvature of the curve at a given point and is denoted as follows: lim 'M 's 1 R . 's o0

Since vector 'τ , equal to the difference of vectors τ  'τ and τ , lies in a plane that passes through the tangent and is parallel to the neighbouring tangent, vector lim 'τ 's dτ ds will be in the tangent plane. At the same time, vector 's o0

dτ ds is perpendicular to tangent τ at the point under consideration. It follows that

vector d 2r ds 2

dτ ds has value 1 R and is directed along the principal normal in

the direction of the concavity of the curve that is, it lies at the intersection of the tangent and normal planes. Similar to unit tangent vector τ , we introduce unit vector n directed along the principal normal in the same direction as dτ ds . Then, d 2r ds 2

So, we obtain

nx

dτ ds n R ,

R d 2 x ds 2 ,

ny

curvature radius R is determined from relation

1 R

2

n 1.

R d 2 y ds 2 ,

1 R2 2

(7.15) nz

R d 2 z ds 2 , where

d 2r d 2r , leading to the formula ds 2 ds 2 2

§ d 2x · § d 2 y · § d 2z · ¨ 2 ¸ ¨ 2 ¸ ¨ 2 ¸ . © ds ¹ © ds ¹ © ds ¹

(7.16)

Third unit vector b is directed to the binormal so that τ , n , and b constitute the right complete system of vectors. From the condition of their orthogonality, it follows that b τ u n , b 1 . Taking (7.15) into account, we obtain

Modelling emergency situations in drilling deep boreholes

db ds

d  τ u n ds

412

dτ dn un  τu ds ds

τu

dn . ds

As vector db ds is perpendicular both to b and τ , it is collinear with n ; therefore, it can be represented in the form:

db ds



n , T

(7.17)

where T is the radius of torsion of the curve at point M. db 1 '\ lim , where '\ is the angle Since b is the unit vector, then ' s o 0 ds T 's between two adjacent binormals. Considering the n change when moving along s and using formulas n b u τ , n u τ b , b u τ n , we obtain

dn ds

d (b u τ ) ds

db dτ u τ  bu ds ds

Given these relationships, we have

1 T

b τ  . T R

(7.18)

dr § d 2r d 3r · § d 2r d 2r · ˜ u ˜ or in ds ¨© ds 2 ds 3 ¸¹ ¨© ds 2 ds 2 ¸¹

the scalar form

1 T

xc R 2 xcc xccc

yc

zc

ycc zcc , yccc zccc

(7.19)

where the prime denotes differentiation in respect of s . It is convenient to determine the curve using equations (7.14), (7.15), (7.17), (7.18). We combine the last three equations in the system

dτ ds

n , R

dn ds



τ b  , R T

db ds



n , T

(7.20)

which is called the Frenet formulas. The system of three vector equations (7.20) is equivalent to nine scalar equations formulated in respect of the projections of vectors τ , n , b on the axes x, y , z

Chapter 7. Critical states and buckling of drill strings in channels…

413

dW z ds nz R ,

dW x ds nx R ,

dW y ds ny R ,

dnx ds W x R  bx T ,

dny ds W y R  by T ,

dbx ds  nx T ,

dby ds  ny T ,

dnz ds W z R  bz T , dbz ds

 nz T .

It is important to bear in mind that not all these equations are independent since they are related by the relations τ

1,

n 1,

τ ˜n

τun b,

0,

(7.21)

which follow from the orthonormality conditions of the Frenet basis. In scalar form, these equalities are reduced to the equalities

W x2  W y2  W z2 1, W y nz  W z ny bx ,

nx2  ny2  nz2 1 ,

W z nx  W x nz

by ,

W x nx  W y ny  W z nz

0,

W x ny  W y nx bz .

They are the first integrals of system (7.20) as can easily be verified by substitutions. Because knowledge k of the independent first integrals allows us to reduce the order of the system to k units, system (7.20) of ninth-order equations can be reduced with the help of first integrals (7.21) to a third-order system. Let us analyse the question of the mutual orientation of the main (u, v, w) and natural (n, b, τ) trihedra as they move along the elastic line of the rod and when it deforms. Axes u, v are oriented along the main bending axes of the rod, which are the lines of intersection of the plane of the cross section with the main planes of bending. Like unit vectors n, b , they lie in the normal plane and are rotated in respect of this pair of vectors by angle F, which is measured from unit vector n up to axis u. Axis w and unit vector τ are collinear. In the process of rod deformation and in proportion to the general motion of the main and natural trihedra along coordinate s, angle F changes. Let's consider vector of complete curvature (the Darboux vector) Ω b R  τ T , which is the vector of the angular velocity of rotation of the natural trihedron relative to its origin when the latter moves along the elastic line in the direction of increasing s with a unit linear velocity. Then, angular velocity vector ω F of the main trihedron (u, v, w) will be ωF

Ω  τ d F ds .

Projections of vector ω F on axes u, v, w are the curvatures of the projection of element ds on surfaces (v, w) and (u, w) and torsion of the axial line of the rod. We introduce notations p, q, r and calculate them using the formulas

Modelling emergency situations in drilling deep boreholes

1 sin F , R

p

q

1 cos F , R

414

r

1 dF  . T ds

(7.22)

Further, we mark values p, q, r, R, T, χ for the case of an undeformed state with index 0. If in this state F0 ( s) z const , then the rod is called naturally twisted. q0

r0

0 , R0

T0

f,

const . If the rod is straight and naturally twisted, then p0

q0

0 , R0

T0

f,

For the original rectilinear untwisted rod, we have p0

F0

but r0

0 , F0 z const .

We derive the differential equations of elastic equilibrium of a flexible rod under the action of external distributed forces with intensity f (s) and moments with intensity m(s) . For this, we consider the element of a deformed rod of length ds bounded by two cross sections. Let F be the principal vector of all internal forces (that act in the cross section) applied to the centre of gravity of the plane section, M be the principal moment of these forces relative to the same centre (Fig.7-14). In this case, the values that belong to the cross section with a large value of arc coordinate s will be considered positive and negative if they have a small value. Vector f ds determines the external distributed load acting on the rod element ds, and this load can be arbitrarily functionally distributed along the length, vector m ds is the external distributed moment.

m(s)

f ( s)

M F

F M

mds

fds

Mc

Fc

s

Fc

Mc

Fig. 7-14 Diagram of forces and moments acting on the rod element

At the end of element ds (Fig.7-14), we have F ' F  dF , M ' M  dM , where d is the total differential of a vector that varies along elastic line s in fixed d and the moment of force F coordinate system Oxyz . Then, vector D1D ' τ ds

relative to point D' will be > τ ds u F@

> τ u F@ ds . For load

f ds , the moment relative

to point D' is an infinitesimal second-order quantity. Proceeding from this, we obtain

Chapter 7. Critical states and buckling of drill strings in channels…

415

differential equilibrium equations for the elastic line of a flexible rod written in vector form:

dF ds

f ,

dM ds

τ u F  m ,

(7.23)

which are invariant relative to any coordinate system. If vector equations (7.23) are projected onto fixed axes Oxyz, then six scalar equilibrium equations of the element can be obtained. However, to clarify the nature of deformation of a flexible rod in the most general case, it is more convenient to project forces and moments on the movable axes of the main trihedron. To determine these projections, we first express in formulas (7.23) the derivatives of the vectors taken in the fixed coordinate system through the relative derivatives of these vectors in moving coordinate system (u, v, w). As is known, the absolute derivative of vector a is equal to the sum of the relative derivative of this vector da dt and the vector product of the angular velocity of the mobile coordinate system by this vector da dt

dF ds

ddF F  ωF u F , ds

dM ds

da dt  ω u a . Then,

dM d M  ωF u M . ds

Substituting the right-hand sides of these dependences into equation (7.23), we obtain the equilibrium equation for the rod element

ddF F ds

ω F u F  f ,

ddM M ds

ω F u M  τ u F  m .

(7.24)

Projecting the left and right sides of relations (7.24) on axes u, v, w, we obtain six scalar equilibrium equations. Let us write out three scalar equations of equilibrium of forces dFu ds qFw  rFv  f u , dFv ds rFu  pFw  f v , dFw ds  pFv  qFu  f w

and three scalar equilibrium equations of moments

(7.25)

Modelling emergency situations in drilling deep boreholes

416

dM u ds qM w  rM v  Fv  mu ,

dM v ds rM u  pM w  Fu  mv ,

(7.26)

dM w ds  pM v  qM u  mw .

Here, Fu and Fv are the shear forces; Fw is the longitudinal force; M u and M v are the bending moments; M w is the torque. The resulting differential equations are equations of equilibrium of the elastic line of a flexible rod in a scalar form (called the Kirchhoff equations). Equations of equilibrium (7.25) and (7.26) generally contain nine unknown functions: Fu, Fv, Fw, Mu, Mv, Мw, p, q, r. The six equations constructed are not sufficient to determine all these quantities. Forces Fu, Fv, Fw in the framework of the accepted hypothesis of the inextensibility of the axial line of the rod are purely static factors and are determined from equilibrium conditions (7.25). Taking into account that the axes of the main trihedron (u, v, w) are the main axes of the bending and torsion of the rod element, we represent projections M u , M v , M w of principal moment M of the internal forces in the form [22]: M u = A (р - р0),

M v = B (q - q0),

M w = C (r - r0),

(7.27)

where p, q, r are the functions of curvature and torsion of the rod in a deformed state; p0, q0, r0 are the same functions in the original undeformed state; A, B, C are the stiffnesses in bending and torsion. To calculate A, B, C, we write out equalities [22] A E Iu ,

B

E Iv ,

C

G Iw ,

where E is the modulus of elasticity of the rod material; G is the shear modulus; I u , I v are the moments of inertia of the cross-sectional area of the rod relative to axes u, v ; I w is the polar moment of inertia of the section. Studies of DS stability in the well channel will begin with the simplest case when the well is rectilinear but inclined at some angle to the horizontal. In practice, this model can be used to analyse the lower segment of an extended curvilinear well. 7.2.3. Geometrical prerequisites for analysis of drill string bending in a cylindrical cavity Let us assume that a rectilinear DS lies on the lower generatrix of the cylindrical surface of the directional well. Its axis is located in plane XOZ of fixed coordinate system OXYZ and is tilted at angle E in relation to vertical OZ (Fig.715). The internal geometry of the DS is determined by coordinate s measured by the length of the centreline from some initial to the current point (natural

Chapter 7. Critical states and buckling of drill strings in channels…

417

parameterisation). Consider that if the DS is immovable, then external distributed gravity forces ( f gr ) and contact forces ( f cont ) affect it, so do axial forces Fz (0) and Fz ( S ) applied to its ends. Here, S is the total length of the string. Surface of the borehole wall

Fz (0)

Y

O

X

s

f cont

Drill string

E Z

f fr f gr

Fz (S )

Fig. 7-15 Diagram of a drill string in a directional well

In the state of DS axial motion, distributed friction force f

fr

is added to these

forces, which is expressed through contact force with the help of the Coulomb law f fr ( s) r P f cont (s)

(7.28)

where P is the friction coefficient and signs ‘+’ and ‘-’ are selected depending on the direction of the DS movement. As a result of these factors, compressive axial forces are formed in some zones of the DS. When they reach critical values, the DS buckles, and its centreline L takes a new shape of equilibrium within the cylindrical cavity of radius а equal to the value of the annular gap. It is necessary to find the values of critical effects on the DS using theoretical modelling and to construct the mode of its bifurcation buckling. Since this investigation will be performed by non-linear analysis methods using the linearization procedure, we shall first construct non-linear equilibrium equations of the DS and linearize them in the vicinity of the stress-strain state under consideration. In this case, the external loads corresponding to the degeneracy of the linearized operator are bifurcational, and the eigen mode of this operator is a mode of the DS buckling. A characteristic feature of this problem is that if angle E of the incline of the well is not small, then the DS is pressed against the well wall by gravity f gr ( s) , which prevents its detachment from the surface of the borehole and stabilises its equilibrium. As a result, if stability is lost, the DS slides along the surface of the channel and remains in contact with it along its entire length during the buckling

Modelling emergency situations in drilling deep boreholes

418

process. In this regard, its geometry cannot be changed arbitrarily and is determined by the geometry of the cylindrical surface on which it lies. When analysing the geometric transformation of a buckling DS, this circumstance forces us to involve methods of the theory of surfaces and differential geometry [23, 44] in conjunction with the theory of flexible curvilinear rods [22, 45, 48]. To trace the geometrical transformation of the drill string inside the cylindrical channel of a directional rectilinear well, we introduce (in contrast to trihedron ( u, v, w ) adopted in the theory of free curvilinear rods) movable right-handed coordinate system oxyz with axis ox oriented along the inner normal to channel surface 6 and axis oz directed along the tangent to curve L in the direction of growth of parameter s . Unit vectors i , j , k are the unit vectors of this system (Fig.7-16). Along with this system, the Frenet trihedron is also used. Its unit tangent vector t , unit vector of principal normal n , and binormal unit vector b are calculated by formulas

dρ , ds

t

n

R

dt , ds

b

tun.

(7.29)

Here, ρ(s) is the radius vector of curve L in fixed coordinate system OXYZ , R is the curvature radius of the DS centreline.

O

a

X

s

v w Y

i

u

x

o j

k y

L z

Σ

Z

Fig. 7-16 Axial line L of the drill string when it is buckled in a cylindrical borehole

Chapter 7. Critical states and buckling of drill strings in channels…

419

Vectors (7.29) determine the orientation of the elements of the rod and the shape of curve L . By using them, the Darboux vector can be introduced Ω kRb  kT t ,

(7.30)

where k R 1 / R is the curvature of the curve, kT is its torsion. These parameters are calculated using formulas [23, 44]

kR

n

d 2ρ , ds 2

kT

dn · § t ¨ n u ¸. ds ¹ ©

(7.31)

However, it is more convenient to present parameters k R and kT in coordinate system oxyz and to study the bending of the DS, as its use makes it possible to transform the difficulties of the problem associated with the existence of limiting surface 6 into an advantage due to the fact that the geometry of curve L can be represented through surface parameters 6 . With this in mind, instead of the Darboux vector Ω (see (7.30)), we introduce the vector ω k xi  k y j  k zk ,

(7.32)

conditionally representing the angular velocity of system oxyz when it moves with unit velocity along curve L . The vector ω projections on axes x and y represent curvatures of the ds element projections on planes yoz and xoz , respectively; k z is the parameter characterising the torsion. They are expressed in terms of the known curvatures of cylindrical surface 6 parameterised by parameter u directed along the generatrix and parameter v that determines the position of the point on the circumferential direction of radius a (Fig.7-16). In this case, the 6 surface metrics is defined using the formula (ds)2

(du)2  a 2 (dv)2 .

(7.33)

With this in mind, we can conclude that curvature k x represents geodesic curvature k geod of curve L on surface 6 and can be expressed as follows [23, 44]:

kx

k geod

r EG  F 2 (uccvc  vccuc  Auc  Buc), ( Euc  2ucvc  Gvc2 )3/2 2

(7.34)

where the prime symbol denotes differentiation with respect to s ; A, B, E, F , G are the geometric quantities determined, in general, by the Christoffel symbols. If curve L and surface D are given in general form, the expressions for variables k x , k y , k z

Modelling emergency situations in drilling deep boreholes

420

have a rather cumbersome appearance. However, for the channel surfaces under consideration, they take a simpler form due to the orthogonality of coordinate curves u , v . In this case, geodesic curvature is determined by relation [9] kx

а11а22 [a11 (uc)2  a22 (vc)2 ]3/2 uccvc  vccuc  $vc  %uc ,

k geod

where a11 , a22 are the parameters of the first quadratic form of the channel surface; multipliers $ , % are expressed in terms of the Christoffel symbols of the second kind 2 $ *111  uc  2*112ucvc  *122  vc , % *11 uc  2*122 ucvc  *222  vc 2

2

2

2

.

As before, in these formulas the prime denotes differentiation in respect of s . For a cylindrical surface, these quantities are greatly simplified, and formula (7.34) can be reduced to the form: kx

a(uccvc  vccuc) .

k geod

(7.35)

Plane xoz is orthogonal to surface 6 ; therefore, curvature k y of curve L is equal to normal curvature k norm of this surface in the direction of unit vector t . Therefore, in accordance with the Euler theorem [23, 44], it is defined in terms of principal curvatures k1 , k 2 of the cylindrical surface ky

k norm

k1 cos2 T  k2 sin 2 T ,

(7.36)

where k1 0 , k2 1/ a , T is the angle between unit vector t and coordinate line u . Then, since sin T adv / ds , it follows that a(vc)2 .

ky

(7.37)

Torsion k z of curve L is defined by the formula kz

lim\ / 's .

's o0

Here, \ is the angle of rotation of axis ox towards axis oz at transition of system oxyz from point s to point s  's . In our case, this formula is reduced to the form: kz

ucvc .

(7.38)

Chapter 7. Critical states and buckling of drill strings in channels…

421

The obtained geometrical relations allow us to construct resolving non-linear equations of DS bending in a rectilinear well. 7.2.4. Equations of non-linear bending and buckling of a straight drill string (frictionless model) Using the constructed geometric relations, we can formulate the problem of elastic non-linear bending of the DS inside a cylindrical cavity of a directional rectilinear borehole. Let us assume that the DS is loaded by distributed gravitational forces f gr ( s) along axis OZ (Fig.7-15), throughout its entire length and at its lower end s S axial compressive force Fz ( S ) acts. The action of these forces leads to the generation of internal elastic force F( s) and moment M(s) as well as external distributed contact force f cont ( s) and frictional forces f fr ( s) . It has already been noted above that we should give special attention to the issue of the formation of friction forces. In our case, when the DS is immovable at the pre-critical state, and when it starts to buckle in the process of tripping in or out (mainly, tripping in), and when the unstable state is realised during drilling, it can be assumed that in the first case the string can be subject to various vibrations (for example, caused by non-stationary external and internal flows of the drilling fluid) that remove the frictional effects, and then we can put f fr 0 . At the same time, with longitudinal motion of the string and its rotation, these forces are determined by equality (7.28). Therefore, we can conclude that the question of establishing the main features of the influence of the parameters of the system and friction forces on the buckling process is of considerable interest. As a result of the action of these forces, the DS can be stretched in its upper zone and compressed within its lower segment. Thus, it can be concluded that critical buckling of the DS in a directional well predominates in its lower part where the well is almost rectilinear. In the process of critical deformation, the DS can take complex shapes, so for their analysis it is sensible to apply non-linear equilibrium equations for a curvilinear rod. Usually, equations (7.23) are formulated in a moving reference frame whose axes coincide with the main central axes of inertia of the cross sections of the flexible rod. Then, these equations are reduced to form (7.24). Using this formulation, the problems of simulating the downhole operations in curvilinear wells were considered in [18, 20]. In the analysed case, this need disappears since the DS section is circular, and all its central axes are the main ones. At the same time, the problem under consideration is associated with another complication due to the presence of a constraint in the form of a rigid surface of the well, which limits the motion of the

Modelling emergency situations in drilling deep boreholes

422

DS, and the appearance of additional unknown contact forces f cont ( s) increasing the number of required functions. However, if, as noted above, we consider the deformation of the DS in the accompanying frame of reference oxyz , it is possible not only to separate the variables and temporarily exclude from consideration unknown coupling reaction f cont (s) f cont (s)i but also to reduce the total number of unknown functions. Such an approach is analogous to the use of generalized coordinates in the method of second kind Lagrange equations in comparison with the application of first kind Lagrange equations. Taking this into account, we give absolute derivatives dF ds , dM ds in form (7.23): dF / ds dF / ds d  ωuF,

Here, d

dM / ds dM / ds d  ωuM .

(7.39)

ds is the local derivative. In frame of reference oxyz , equations

(7.23) become dF / ds ω u F  f ,

dM / ds ω u M  k u F .

(7.40)

We should emphasise that system (7.40) differs in principle from system (7.24) since it is considered in moving trihedron oxyz . Taking into account equality (7.32), we transform equations (7.40) to a scalar form separately for the relations of the force and moment groups

dFx / ds k y Fz  k z Fy  f xgr  f c dFy / ds k z Fx  k x Fz  f ygr dFz / ds k x Fy  k y Fx  f

(7.41)

gr z

dM x / ds

k y M z  k z M y  Fy

dM y / ds

k z M x  k x M z  Fx

dM z / ds

k x M y  k y M x

(7.42)

In equations (7.42), moments M x , M y are calculated using formulas [5] Mx

EI k x ,

My

EI k y

(7.43)

where, as above, E is the modulus of elasticity of the material of the DS pipe; I is the central axial moment of inertia of the pipe cross-section area. From these equalities, it follows that the right-hand side of the third equation of

Chapter 7. Critical states and buckling of drill strings in channels…

423

system (7.42) is zero; therefore, Mz

const ,

(7.44)

and its value is determined from the boundary conditions. Relations (7.32)–(7.44) together with the corresponding boundary conditions determine the non-linear deformation of the DS in the channel cavity of a curvilinear well. Let us consider a simpler problem of DS elastic bending in a cylindrical cavity. Since most of the directional wells end in their bottom part, typically, as rectilinear segments, it can be assumed that this represents the most practical interest. For cylindrical surface 6 , in accordance with equalities (7.35), (7.37), (7.38), the components of vector ω are determined by the formulas kx

a(uccvc  vccuc) ,

ky

a(vc)2 ,

kz

ucvc .

(7.45)

Using relations (7.43), (7.45), the first two equalities of system (7.42), and the first equation of system (7.41), we obtain

Fx

2 EIvcvcc  EIk x k z  M z k x

Fy

EIdk x / ds  M z k y  EIk y k z

fc

k y Fz  k z Fy  dFx / ds  f xgr

(7.46)

Using equations (7.41), (7.42), we can formulate a system of six first-order differential equations in respect of six unknown functions u , v , vc , k x , Fy , Fz

dFy / ds k z (2 EIvcvcc  EIk x k z  M z k x )  Fz k x  f gr avc, dFz / ds k x [ EI (k x )c  M z k y  EIa(vc) 2 k z ]  a(vc) 2 (2 EIvcvcc  EIk x k z  M z k x )  f z gr , dk x / ds 1 / EIa(vc) 2 M z  a (vc) 2 k z  1 / EIFy , dv / ds (vc),

(7.47)

d (vc) / ds 1 / ak x 1  a 2 (vc) 2 , du / ds

1  a 2 (vc)2 .

Here, f gr

g ( Ust  Ul )e , g = 9.81 m/s2, U st is the density of string material, Ul

is the density of the drilling fluid, e is the cross-sectional area of the string tube.

Modelling emergency situations in drilling deep boreholes

424

It is important to note that due to the approach used and the choice of orientation of the oxyz axes, variables Fx ( s) and f cont ( s) are excluded from system (7.47) and its structure is essentially simplified. The components of the gravitational forces entering into this system are calculated using the formulas f xgr

 f gr sin E cos v,

f ygr

f gr (cos E ˜ avc  sin E sin v ˜ uc),

gr z

f (cos E ˜ uc  sin E sin v ˜ avc).

f

(7.48)

gr

We should also emphasise that in addition to the unknown variables given in the left-hand side of system (7.47) this system includes also unknown variables vcc , (k x )c , and k z . To determine the first two, the right-hand sides of the fifth and third equations of the system are used, and torsion function k z is calculated using the third formula of system (7.45). Constructed sixth-order system (7.47) together with the corresponding boundary conditions determines the non-linear bending of the drill string in the cylindrical cavity of the rectilinear directional well. By using this, we can find all six of the required variables, then using equalities (7.46) to calculate shearing force Fx ( s) and contact interaction force f cont ( s) . Equations (7.47) are essentially non-linear. For numerical analysis, the solution method of continuation in respect of the parameter together with the NewtonKantorovich method [21] can be used according to scheme (7.11). At each step of the implementation of such an approach, system (7.47) and the corresponding boundary equations are linearized in the vicinity of the considered state. The states in which the linearized operator of the problem degenerates are bifurcations. In them, the bent drill string loses or restores equilibrium stability. Let the drill string with length L lie on the bottom of a cylindrical cavity whose axis makes angle E with the vertical. At the lower end, it is pivotally fixed, and longitudinal compressive force R and torque M z affect it, on the upper end it is rigidly clamped. These constraints correspond to the boundary conditions u

0, v

v

0 , kx

0 , vc 0 at s

0

(7.49) 0 , Fz

 R at s

L

At the initial state, the string is straight, so uc( s) 0 v(s) 0 , k x ( s) 0 , k y (s) 0 , k z ( s) 0 , Fz (s) g ( Ust  Ul )e( L  s)cos E  R . Let’s find value R , in

Chapter 7. Critical states and buckling of drill strings in channels…

425

which the string loses stability, and the mode of stability loss. To this end, we shall linearize equations (7.47) in the vicinity of the initial state

d G Fy ds

d G Fz ds

FzG k x  f gr aG vc ,

 f gr ,

d G kx ds

1 G Fy EI (7.50)

d G v G vc , ds

d G vc ds

1 G kx , a

d Gu 0 ds

Here, symbol G indicates small variations in the unknown functions. An important feature of the system is that it does not include torque M z . This means that if under the influence of gravity the string is pressed to the bottom v 0 of the rectilinear well as a result of critical buckling under the action of compressive force R , the L line continues to lie on the D surface and then the onset of the critical state does not depend on the M z torque. And, as follows from non-linear system (7.47), the M z influence begins to appear only when the L line becomes a curve, and quantities k x , k y , k z are different from zero. System (7.50) can acquire the usual form used in the theory of buckling of beams. To this end, we denote G y aG v . As in this system the second and sixth equations are not connected with the others, they can be considered separately, and the remaining fourth-order system can be reduced to the form used in the stability theory of rods. In this case, G k x aG vcc , G Fy EIaG vccc , and from the first equation of system (7.50) we obtain EIG y IV  [ f gr cos E ( L  s)  R]G ycc  f gr cos EG yc  ( f gr sin E / a)G y 0 . (7.51)

For a string in a horizontal well, E simplest form:

90o , and equation (7.51) takes the

EIG y IV  RG ycc  ( f gr / a)G y 0 .

(7.52)

It turns out to be analogous to the equation of buckling of a beam on an elastic foundation with bed coefficient f gr / a and for L f it determines the critical value ROcr

S 2 EI / O 2  O 2 f gr / (S 2a) ,

(7.53)

Modelling emergency situations in drilling deep boreholes

426

corresponding to the mode of stability loss

G yO G C sin(S s / O )

(7.54)

with length O of half-wave. Minimising ROcr by O , we obtain minimum value ROcr and the corresponding

O cr Rcr

2 EIf gr / a ,

O cr S 4 EIa / f gr

.

(7.55)

If the borehole axis is inclined at angle E (0  E  90o ) , equation (7.51), which determines the critical states, is also complicated, and because of the presence of a variable coefficient before G ycc , and the construction of a solution in a closed form becomes impossible. For these cases, the solution of the boundary eigen value problem for equation (7.51) was carried out using the method of initial parameters [2]. The corresponding solutions of the Cauchy problem for equation (7.51) were constructed using the Runge-Kutta method. As noted above, the problem posed is singularly perturbed, its particular solutions include rapidly increasing and rapidly decreasing functions. To overcome the computational difficulties associated with this factor, we also had to use the orthogonalisation procedure for particular solutions [2]. 7.2.5. Equations of non-linear bending and buckling of a straight drill string (friction model) We shall construct the equations of elastic bending and buckling of the DS for the case when it moves in the borehole. Then, all its elements are acted upon by distributed gravity forces f gr (s) , and axial forces are applied to the upper and lower ends Fz (0) and Fz ( S ) . Typically, force Fz (0) is tensile, force Fz ( S ) is compressive. The action of these forces is associated with the formation of internal forces Fz (s) and moments M z ( s) as well as external contact ( f cont ( s) ) and frictional ( f fr ( s) ) forces. In mobile system oxyz , the components of the forces of gravity are given by equalities (7.48), while in this system the contact and frictional forces have only one component f cont (s)

f xcont (s)i ,

f zfr ( s)

f fr (s)k .

Therefore, the scalar analogues of equations (7.40) take the form

(7.56)

Chapter 7. Critical states and buckling of drill strings in channels…

427

dFx / ds

k y Fz  k z Fy  f xgr  f xcont ,

dFy / ds

k z Fx  k x Fz  f ygr ,

dFz / ds

k x Fy  k y Fx  f

gr z

(7.57)

 fz ,

dM x / ds

k y M z  k z M y  Fy ,

dM y / ds

k z M x  k x M z  Fx ,

dM z / ds

k x M y  k y M x .

fr

(7.58)

As we can see, they differ from relations (7.41), (7.42) only in terms of the presence of summand f z fr in the third equation of system (7.52). This advantage was possible due to the use of movable axes oxyz . In this case, equalities (7.46) retain their form and equations (7.57) (7.58) are replaced by the system dFy

2 EIauc(vc) 2 vcc  ( M z  EIucvc)ucvck x  k x Fz  f ygr , ds dFz k x Fy  2 EIa 2 (vc)3 vcc  a ( M z  EIucvc)(vc) 2 k x  f zgr  f z fr , ds 1 1 dk x Fy ,  a (vc) 2 M z  auc(vc)3  ds EI EI dv vc, ds du 1  a 2 (vc) 2 . ds

(7.59)

To construct the equations of stability loss of the DS lying on the bottom of the well channel, it is necessary to linearize system (7.59) in the vicinity of state u(s) s , v(s) 0 , kx (s) k y (s) k z (s) 0 , M x (s) M y ( s) 0 . In this case, f xcont (s)  f xgr ,

(7.60)

and the frictional force is f z fr (s) r P f xcont (s)

P f xgr (s) .

(7.61)

As a result, by integrating the second equation of system (7.59), we can find axial force Fz ( s) that arises in the DS when it moves

Modelling emergency situations in drilling deep boreholes

428

s

Fz ( s)  Fz (0)  ³ ( f zgr  f z fr )ds ( f zgr  f z fr ) s  R .

(7.62)

0

If the lowering regime of the DS is considered, this equality implies Fz (s)

where R

f gr (cos E  P sin E )(S  s)  R

(7.63)

Fz (0) is the axial force acting on the bit and considered to be

known. Let us assume that when the DS is buckled from the considered state, its displacements, curvatures, forces, and moments take small increments. Then, they can be found from system (7.59) that is linearized in the vicinity of this state. It can be converted to the following form: dG Fy / ds

FzG k x  af gr cos EG vc,

dG Fz / ds

G f zgr

0,

dG k x / ds (1 / EI )G Fy ,

(7.64)

dG v / ds G vc, dG vc / ds (1 / a )G k x , dG u / ds 0.

We will take into account that G k x

aG vcc , G Fy

EIaG vccc , G f xgr ( s) 0 ,

g ( Ut  Um ) F , where g 9.81 m/s2; Ut , U m are the densities of the pipe material of the DS and the drilling fluid, respectively; F is the cross-sectional area of the DS pipe. In this case, the second and sixth equations of system (7.64) are satisfied identically and are not considered further, and the remaining four equations are reduced to one fourth-order linear differential equation f cont ( s)

f gr sin E , f gr

ª f gr (cos E  P sin E ) Rº ( S  s )  » G ycc  EI EI ¼ ¬ gr gr f (cos E  P sin E ) f sin E G yc  G y 0,  EI aEI

G y IV  « 

(7.65)

where G y(s) aG v(s) . We should note that this equation is analogous to the stability equation for a beam lying on an elastic foundation, and therefore, their solutions have the same properties. First, as equation (7.65) and stability equation of beams on elastic

Chapter 7. Critical states and buckling of drill strings in channels…

429

foundation are singularly perturbed, the mode of their stability loss has the appearance of edge effects or harmonic wavelets localised (perturbed) inside the domain occupied by the string. Second, as a rule, the harmonics of wavelets quickly decrease. An important property of equation (7.65) is that it does not contain torque function M z ( s) that is lost due to the linearization of the non-linear system in the vicinity of the state under consideration. This fact indicates that at this state the bifurcation effect of the system is insensitive to the effect of torque. 7.2.6. Analysis of buckling of strings in directional rectilinear wells We must again emphasise that equations (7.51) and (7.65) have variable coefficients; therefore, their eigen values and eigen modes cannot be found by analytical methods. So, they were investigated using finite difference methods. In addition, this problem is multi-parameter. Indeed, the buckling process depends on the length of the DS, the dimensions of the wells, the size of the annular gap, the absence or presence of friction effects and values of compressive force on the lower end, the type of boundary conditions, etc. Since a comprehensive study of such buckling processes is impossible, the results of calculations for some selected values of the characteristic parameters of the system are given below. They were: E = 2.1·1011 Pa, I = 2.7·10–4 m4, ρt = 7.8·103 kg/m3, ρm = 1.3·103 kg/m3, d1 = 0.2 m, d2 = 0.18 m, F S (d12  d22 ) / 4 5,97 ˜103 m2, μ = 0.2. It was believed that the upper and lower ends of the DS have hinged supports, so

G y(0) G y(S ) 0 ,

G ycc(0) G ycc(S ) 0 .

(7.66)

Initially, the stability was studied of a DS with lengths of S = 500 m and S = 1,000 m with the value of annular gap a = 0.166 m. Table 7-1 shows the results of computer studies for DS of S 500 m length. The two left columns of this table refer to the frictionless model, and the two right columns contain information related to analysis of the effect of friction forces on the bifurcation process. For each considered case, with the corresponding value of the angle of inclination, the critical values of axial force Rcr and the distribution diagrams along length 0 d s d S of internal axial force Fz ( s) and lateral movement wy(s) are shown. It can be seen that if angle E is not

very large ( E

45 and E

60 ), the tensile longitudinal component of gravity

gr z

f (s) leads to a significant decrease in axial compressive force Fz ( s) on the upper

end, and the mode of loss of stability becomes the form of the edge effect localised in the vicinity of the lower end (Fig.7-17). In this case, the frictional forces only slightly affect critical value Rcr and functions wy(s) (see items 1 and 2 in Table 7-1).

Modelling emergency situations in drilling deep boreholes

However, as angle E approaches value E f

gr

cos E and frictional forces f

fr

430

arcctg P (angle of friction), gravity

 f P sin E in equation (7.65) balance each gr

other, and longitudinal force Fz (s) R becomes constant. Then, this equation takes the form:

G y IV 

R f gr sin E G ycc  G y 0. EI aEI

(7.67)

β

Fig. 7-17 The boundary effect in the bifurcation mode of a drill string in a directional well

Eigen value Rcr and eigen mode wycr of this equation can be found analytically under boundary conditions (7.66). Indeed, we take

G y ( s) G C sin

S ns S

(7.68)

.

By substituting (7.68) into (7.67), we obtain 4

2

R §Sn · f gr sin E §Sn · ¨ ¸  ¨ ¸  aEI © S ¹ EI © S ¹

0.

(7.69)

This equality allows us to find Rn for selected number n 2

Rn

2

f gr sin E § S · §Sn · EI ¨ ¸  ¨ ¸ . a © S ¹ ©Sn ¹

(7.70)

In fact, the DS will lose stability with value n that provides minimal by modulus value Rcr . To determine it, we use condition dRn / dn 0 .

(7.71)

Chapter 7. Critical states and buckling of drill strings in channels…

431

Table 7-1 Critical values of axial force Rcr and loss-of-stability modes G y(s) for the cases of S 500 m and a = 0.166 m Frictionless model Friction model 1 Angle E 45 Rcr=203.772 kN

Rcr=202.129 kN

Gy

Fz , kN

200

Gy

Fz , k

200 0

0 -200

s,m 0

250

500

s, m 0

-200

E

2

s, m 0

500

250

250

500

s, m 0

250

500

60

Rcr=221.771 kN

Rcr=219.441 kN 200

200 0

0

-200

-200 0

250

500

0

E

3

0

500

250

250

500

0

250

500

78.495

Rcr=230.962 kN

Rcr=224.922 kN

200

200

0

0

-200

-200

0

250

500

0

500

250

0

E

4

250

500

0

500

250

78.69

Rcr=230.942 kN

Rcr=225.856 kN

200

200

0

0

-200

-200

0

250

500

0

500

250

E

5

0

250

500

0

500

250

79.06824

Rcr=230.997 kN

Rcr=227.518 kN 200

200 0

0

-200

-200 0

250

500

0

250

500

E

6

0

250

500

0

250

500

85

Rcr=230.962 kN

Rcr=229.450 kN

200

200

0

0

-200

-200 0

250

500

0

250

500

0

250

500

0

250

500

Modelling emergency situations in drilling deep boreholes

432

Then,

dRn dn

2

2 f gr sin E § S · §S · 2 EI ¨ ¸ n  ¨ ¸ an3 ©S¹ ©S ¹

2

(7.72)

0

and

ncr

S

S

4

f gr sin E aEI

(7.73)

.

Note that this force is independent on length

S.

In this case, 22.45 ,

E 90  arctgP 78 78.69 69 , and critical values of characteristic parameters ncr Rcr

225856 N. The case considered is represented by position 4 in Table. 7.1.

Fig. 7-18 Expansion of the zone of the boundary effect in the bifurcation mode

As can be seen from the results presented in the two left columns of Table 7-1, in the absence of frictional forces, critical force Rcr varies insignificantly with increasing E (from Rcr 203 kN up to Rcr 231 kN), the modes of stability loss do not change very much. This is because the maximum value of force Fz ( s) occurs at the lower end. However, if the process of the descent into the borehole is accompanied by forces of friction, then ‒ when the hole is approaching the horizontal ‒ the longitudinal force of friction causes longitudinal force Fz ( s) to grow when you remove it from the bottom end. Therefore, with increasing E , the zone of the boundary effect expands (Fig.7-18) and extends over the entire length of the string (positions 3–6 in the right column of Table 7-1). When the well is tilted at an angle of friction, the mode of loss of stability becomes purely harmonic (item 4 in Table 7-1 and Fig.7-19).

Chapter 7. Critical states and buckling of drill strings in channels…

433

Table 7-2 Critical values of axial force Rcr and loss-of-stability modes G y(s) for the cases of S = 1,000 m and a = 0.166 m Frictionless model Friction model 1 Angle E 45 Rcr=203.772 kN Fz , kN

200

Rcr=202.129 kN

Gy

Fz , kN

200

0

Gy

0

-200

s, m

0

500

1,000

-200

s, m 0

50

s, m

0

1,000

2

E

500

1,000

s, m 0

500

1,000

60 $

Rcr=221.771 kN

Rcr= 219.441 kN

200

200

0

0

-200

-200 0

500

1,000

0

500

1,000

E

3

0

500

1,000

0

500

1,000

78.495

Rcr=230.942 kN

Rcr=227.993 kN

200

200

0

0

-200

-200 0

500

1,000

0

500

0

1,000

4

E

500

1,000

0

500

1,000

78.69

Rcr=230.962 kN

Rcr=225.856 kN

200

200

0

0

-200

-200 0

500

1,000

0

500

1,000

E

5

0

500

1,000

0

500

1,000

79.06824

Rcr=230.995 kN

Rcr=227.684 kN

200

200

0

0

-200

-200 0

500

1,000

0

500

0

1,000

E

6

85

500

1,000

0

500

1,000

$

Rcr=230.622 kN

Rcr=228.841 kN 200

200 0

0

-200

-200 0

500

1,000

0

500

1,000

0

500

1,000

0

500

1,000

Modelling emergency situations in drilling deep boreholes

434

Fig. 7-19 Sinusoidal mode of stability loss of the drill string in a borehole tilted at the friction angle

If ncr is known, we can use equality (7.68) to find length Ocr of the bifurcation harmonic half-waves

Ocr

S / ncr

S 4 aEI / ( f gr sin E ) .

(7.74)

As the results of the numerical study and this formula show, the length of the half-wave of the harmonic pitch for all the examples presented in Table. 7.1 is equal to Ocr | 22.72 m. It has been pointed out above that the problem posed is singularly perturbed; therefore, its solutions, as a rule, have the mode of boundary effects localised in a small vicinity of the lower end of the DS. Problems of this type are also known as non-classical [12]. Therefore, one can expect that they are insensitive to an increase in the length of the DS. Indeed, this assumption is confirmed by the results of analysis of a 1,000 m long DS. As can be seen from Table 7-2, critical values of axial force Rcr retained their value for the frictionless model (two left columns) and in the presence of frictional forces (two right columns), although they are slightly increased at high values of angle E . In this case, the modes of loss of stability also did not change their form or their wavelengths. Of particular interest is the question of how the process of the DS buckling depends on annular gap a . Tables 7.3 and 7.4 show the results of stability calculations with gap a 0.08 m. From this data, it follows that when the value of a decreases, the Rcr module grows, and at the same time its values for strings with lengths of S 500 m and S = 1,000 m almost coincide. Of particular interest is the effect of the transformation of the bifurcation mode for a string of S 500 m length.

Chapter 7. Critical states and buckling of drill strings in channels…

435

Table 7-3 Critical values of axial force Rcr and loss-of-stability modes G y(s) for the cases of S 500 m and a 0.08 m Frictionless model Friction model 1 Angle E 45 Rcr=288.267 kN Rcr=286.618 kN F , kN 200 Fz , kN

Gy

Gy

z

200

0

0

-200

-200

s, m 0

250

s, m

s, m

500

0

500

250

2

E

0

250

500

s, m 0

250

500

60 $

Rcr=315.273 kN

Rcr=312.936 kN 200

200 0

0

-200

-200 0

250

500

0

500

250

0

250

500

0

E 78.495 Rcr=330.396 kN

3

250

500

Rcr=325.342 kN

200

200 0

0

-200

-200 0

250

500

0

250

0

500

E Rcr=330.450 kN

4

250

500

0

250

500

78.69 Rcr=325.342 kN 200

200 0

0

-200

-200 0

250

500

0

0

500

250

E

5

250

500

0

250

500

79.06824

Rcr=330.548 kN

Rcr=327.035 kN

200

200

0

0

-200

-200 0

250

500

0

250

500

E

6

0

85

250

500

0

Rcr=330.898 kN 200

250

500

$

Rcr=325.406 kN 200

0

0

-200

-200 0

250

500

0

250

500

0

250

500

0

250

500

Modelling emergency situations in drilling deep boreholes

436

Table 7-4 Critical values of axial force Rcr and loss-of-stability modes G y(s) for the cases of S = 1,000 m and a 0.08 m Frictionless model Frictional model Inclination angle E

1 Fz , kN

200

Gy

Rcr

288.267 kN

,

0

0 -200

s,m

s,m 500

1,000

0

500

E Rcr

200

0

0 -200 1,000

0

500

0

0

0 -200 1,00

0

500

Rcr

200

1,000

0

500

Rcr

200

-200 500

500

312.936 kN

1,000

78.495

200

0

1,000

$

330.390 kN

1000

0

E

$

4

500

Rcr

1,000

E Rcr

s,m 0

200

0

500

1,000

60 $

-200 0

500

315.273 kN

3

286.618 kN

s,m

1,000

2

Rcr

Gy

200 Fz kN

-200 0

45$

500

1000

0

327.364 kN

500

1000

78.69

330.450 kN

Rcr

325.342 kN

200

0

0

-200

-200 0

500

1,000

0

500

5

1,000

E Rcr

200

0

500

1,000

0

500

1,000

79.06824$

330.548 kN

Rcr

327.171kN

200

0

0

-200

-200 0

500

1,000

0

500

1,000

E

6

Rcr

200

0

0 -200 1,000

0

500

1,000

0

Rcr

200

0

500

1,000

330.898 kN

-200 0

500

500

1,000

85$

0

500

1,000

0

329.199 kN

500

1,000

Chapter 7. Critical states and buckling of drill strings in channels…

437

So, if the DS is immovable, and the frictional forces are absent, then forces Rcr grow with increasing angle E , although the modes of stability loss in this case vary little. However, in the presence of friction forces, the effect of the gradual transformation of the bifurcation mode with increasing E from the boundary effect to the disordered harmonics becomes more obvious. It is also interesting to note that with increasing E the distance between two neighbouring zeros in the mode of stability loss decreases from λ ≈ 40.5 m for

E

45 till O

31 m for E

85 .

The noted features of DS buckling are due to several factors. So, if angle E is not large, the DS is prestressed by tensile axial force Fz ( s) in the upper zone and compressive force in its lower part. In this case, the stress-strain state of the DS is essentially inhomogeneous. At the same time, the normal component of the forces of gravity f gr (s) pressing the DS to the well wall and preventing its buckling is small. Therefore, the DS almost extends freely in the zone of the lower end. With increasing angle E , the tensile forces decrease, the stress-strain state of the DS becomes more uniform, and the stabilising effect of the tensile forces decreases. In this case, the contact and frictional forces as well as the normal component of the forces of gravity increase. As a result, these features become responsible for the growth of the stabilisation effect of the string. At the same time, the interaction of these contradictory properties leads to a slight increase in critical force Rcr with increasing angle E . 7.3 Buckling of a drill string in a borehole channel with a circular outline 7.3.1. The main conceptual features of the problem of modelling bifurcations of a drill string in a curvilinear well The task of studying bifurcation buckling of a drill string in a directional well channel is significantly complicated if the well axis is curvilinear. In this case, the well wall is a channel surface of double curvature, and it is necessary to use such concepts as first and second quadratic forms to describe it. In addition, the influence of singular perturbation of the phenomena under consideration is also more noticeable, leading to the fact that the bifurcation modes of the string acquire the shape of harmonic wavelets whose localisation zones are unknown in advance. Therefore, even for curvilinear wells with the simplest trajectories in the form of circular arcs, the scientific literature only contains general arguments about methods

Modelling emergency situations in drilling deep boreholes

438

of modelling the effects of bifurcation buckling. And despite more than half a century of history, these tasks are still far from completion. Therefore, Cunha in his paper [10] notes: ‘A characteristic important point fixed in this literary review is that many results presented by many authors, as may be noted, are highly contradictory in terms of the values of critical forces and buckling patterns.’ In this section, the problem of drill string buckling in a circular well is solved using a new, global approach that allows one to analyse the drill string within its full length and to find the critical values of the acting forces as well as the zones and modes of the string buckling. Here are the main conceptual features of this problem. First, we once again focus attention on the fact that the problem under consideration is singularly perturbed; therefore, its solution has the mode of harmonic wavelets whose locations are unknown in advance. In this regard, the task must be solved in a global setting on the full length of the string. As a rule, the solution of these problems has poor convergence. The second characteristic feature of this approach, overwhelmingly used in the scientific literature, is connected with the assumption of the continuous movement of the column along the wall of the borehole during its buckling. Since the surface of the borehole wall with a circular cross section is a channel, it is described by complex analytical expressions, so the constraint limiting the movement of the column is essentially non-linear. Usually, when analysing mechanical systems with superimposed constraints, the first or second Lagrange methods are used. As noted in paragraph 7.1.3.1, when applying the Lagrange method of the first kind, constraint equations are added to the system of resolving equations, and the Lagrange multipliers play the role of additional unknown variables. At the same time, the Lagrange method of the second kind is based on the use of generalised coordinates. This method is more cumbersome when formulating the problem, but in our case, when applied, the order of the system of resolving equations decreases to four. Below, the Lagrange method of the second kind is used to solve this problem. The third aspect of our problem concerns the influence of torque on the stability of the DS and on the modes of stability loss. If the DS is in a vertical well, and at the initial stage of buckling it is not limited by its walls, then, as Greenhill established in 1883, the torque is the main cause of spiral buckling. If you impose a limiting constraint on the string, then, as shown in Section 7.2, torque at the initial stage of buckling does not affect the bifurcation process. Moreover, a similar conclusion for this phenomenon is made below for a curvilinear well. So, if the DS lies on the bottom of a flat curvilinear well, and only its lateral displacements are

Chapter 7. Critical states and buckling of drill strings in channels…

439

free, torque also does not affect the bifurcation process. However, if the DS is inside a well with a three-dimensional geometry, the action of the torque must be taken into account. The fourth question that awaits a solution is connected with taking into account the influence of friction forces on the critical state of the DS. As indicated above, if the drill string is in the channel of a directional rectilinear borehole, the frictional force function is additive, and it is easily calculated through the normal gravity component. However, if the well is curvilinear, the frictional forces depend not only on the direction of the gravity forces but also on the curvature radius of the channel centreline. In this case, as a rule, it increases with decreasing radius and depends on the axial and rotational motions of the DS. As noted above, the effect of the formation of the frictional force in this case is multiplicative. In this regard, the stability of the DS in a borehole with a circular outline is solved in two sets. In the first formulation, it is assumed that the DS is immovable, and there are no frictional forces between it and the well wall. In the second formulation, it is assumed that the DS is in the regime of axial motion, and axially distributed friction forces are formed between its surface and the borehole wall. The fifth feature of the problem under discussion is that the elastic deformation of the DS is described by essentially non-linear differential equations with variable coefficients. Therefore, its stability should be investigated on the basis of the application of the linearization method and the solution of the boundary eigen value problem for these linearized equations. In this section, special attention is paid to all selected issues, and it is shown that buckling DS with superimposed constraints bounding the channel surface can be described by an ordinary differential equation of the fourth order with a structure similar to an equation defining Eulerian stability of rectilinear beams. 7.3.2. Geometry of circular centreline channel surface Let us consider the deformation of an elastic drill string of radius r2 in the cavity of a curvilinear well during its tripping in or drilling. The drill string can buckle under the influence of internal and external forces; however, in the process of bending it remains in contact with the borehole wall over its entire length. To parameterise the geometry of this surface, we assume that it can be formed by the continuous motion of the circle generatrix along the borehole axis. In differential geometry, such surfaces are called channel surfaces.

Modelling emergency situations in drilling deep boreholes

440

X

O

Y

Z

T

Ma

v w u

M

6

T Fig. 7-20 Diagram of channel surface 6 that limits the movement of the DS centerline in a curvilinear well

Since in the deformed state centreline L of the DS continues to lie on channel surface 6 of radius a equal to the value of the annular gap, the geometry of line L depends on the geometry of surface 6 , so we parameterise this surface first. We introduce fixed coordinate system OXYZ and define surface 6 with parameter u , which specifies line T of centres of generating circles of radius a XT

X T (u ) ,

YT

YT (u ) ,

ZT

ZT (u ) ,

(7.75)

and parameter v , which determines the position of a point on the generatrix circle (Fig.7-20). In the general case, the T line can be both spatial and flat. For simplicity, we shall assume that it lies in plane X T (u ) 0 , then the equation of surface 6 can be represented in form (Fig.7-20): X (u, v) a sin v , Y (u, v) YT (u )  a cos v sin[M (u )] , Z (u, v) Z T (u )  a cos v cos[M (u )].

Here, M 20).

(7.76)

arctan(wYT / wZT ) is the angle of inclination of the generatrix circle (Fig.7-

Chapter 7. Critical states and buckling of drill strings in channels…

441

In this case, lines u const and v const form a system of curvilinear coordinates on channel surface 6 . Then, lines u const are circles of radius a , and line v 0 lies at the bottom of the channel. In the initial (undeformed) state, the position of curve L is defined by 0,

v( s )

(7.77)

where s is the natural parameter equal the length of line segment L from some initial point to the considered one. x

L

v

o

i

j

k

u

s

T

z y

6 Fig. 7-21 Coordinate system oxyz moving along the drill string centerline L

In the deformed state, the position of curve L on surface 6 is given by equalities (Fig.7-21): u u(s) , v v(s). Further, these variables will be used to determine the deformation of the DS. It is characteristic that their count is equal to two, although the problem in question is three-dimensional. The reduction of the dimension of the problem became possible due to the implicit consideration of constraints (7.76) and the use of variables u(s) and v(s) as generalised coordinates.

Modelling emergency situations in drilling deep boreholes

442

When studying the elastic bending of a free rod that is not subject to the action of constraints, a mobile reference system associated with the centreline is generally used. As a rule, this is a movable Frenet trihedron with basis vectors of normal n , binormal b , and tangent t . When it moves along an axial line with a unit velocity, this trihedron rotates with angular velocity ω6

knn  kt t

(7.78)

called the Darboux vector. Here, as usual, kn is the normal curvature of the curve, kt is its torsion. In this case, the DS is not free since centreline L lies and slides over surface 6 . Therefore, the geometry of curve L is determined by internal and external geometries of surface 6 , which are known and do not change with the transformation of curve L . For this reason, it is more convenient to introduce additional trihedron i, j, k moving along curve L over surface 6 (Fig.7-21). Its unit vector i is perpendicular to limiting surface 6 , and vector k is tangent to curve L . We can link axes ox , oy , oz with corresponding unit vectors of this trihedron. This technique allows us to separate the variables and simplify the problem in question. Then, we can use generalised Darboux vector ω in the form of

ω k x i  k y j  k z k.

(7.79)

Here, k x is the curvature of curve L in plane oyz , k y is its curvature in plane xoz , and k z is the torsion of the curve due to the motion of the trihedron along the curve. Since curve L lies on surface 6 , all these geometric quantities can be expressed in terms of the parameters of this surface. So, curvature k x is defined as geodesic curvature k geod of curve L , while curvature k y is normal curvature k norm of surface 6 calculated along tangent curve L [44]. These variables are evaluated using parameters aij and bij of the first and second quadratic forms )1 (u, v) a11 (u, v)du 2  2a12 (u, v)dudv  a22 (u, v)dv 2 , ) 2 (u, v) b11 (u, v)du 2  2b12 (u, v)dudv  b22 (u, v)dv 2

(7.80)

In general, when surface 6 is specified in any form, functions k x , k y , k z describe complex functions; however, if the surface is a channel, coordinate lines u const and v const are orthogonal and equalities a12 (u, v) 0 , b12 (u, v) 0 are valid. Then, the expression for the geodesic curvature simplifies and it takes form [44]:

Chapter 7. Critical states and buckling of drill strings in channels…

kx

k geod

>

a11a22 a11 (u c) 2  a22 (v1 ) 2

@

3

2

(u ccvc  vccu c  Avc  Bu c).

443

(7.81)

Here, multipliers A and B are expressed through Christoffel symbols *ijj using the formulas A *111 (uc) 2  *122 (vc) 2 ,

B

*112 (uc) 2  *222 (vc) 2 .

Normal curvature k y is expressed in terms of principal curvatures k1 and k 2 of surface 6 using Euler's theorem [44] ky

k norm

k1 cos 2 T  k2 sin 2 T ,

(7.82)

where T is the angle between the directions of curve L and coordinate line u . Value k z is determined by the rotation of system oxyz around vector k when moving along line L . It is calculated using the formula kz

lim '\ / 's.

's o0

7.3.3. Non-linear equations of bending of a drill string in the borehole with a circular centreline (frictionless model) As with the analysis of DS buckling in a rectilinear directional well (Fig.7-2), we will model DS buckling in a borehole with a circular centreline assuming that the length of DS and curvature radius of its centreline are much greater than the DS cross-section. Then, we can ascertain that the length of the DS does not change when it is bent, and the deformations are elastic. Therefore, the theory of flexible curvilinear rods can be used to describe them. Let, as in Section 7.2, the drill string be suspended from the upper end and axial force Fz ( S ) and torque M z ( S ) are applied to its lower end. In addition, distributed gravity forces ( f gr (s) ), contact interaction force ( f cont ( s) ), and friction force ( f fr (s) ) act on each of its elements. The well centreline geometry and the value a of annular gap are given, the DS does not rotate, and the influence of the drilling fluid flows is not considered. It is necessary to determine the critical values of force factors Fz ( S ) and M z ( S ) and to build a mode of DS buckling. This process is

analysed in four stages. At the first stage, non-linear equations of elastic equilibrium of the drill string are derived.

Modelling emergency situations in drilling deep boreholes

444

At the second stage, the equilibrium equations of the DS linearized in the vicinity of the initial state are constructed. At the third stage, the function of internal forces Fz ( s) is built, which is expressed through external distributed gravity forces f gr (s) and compressive force Fz ( S ) at the lower end.

At the fourth stage, the boundary eigen value problem for the constructed linearized system is solved, the critical loads are calculated, and the buckling mode is constructed. To derive resolving equations of DS deformation in the borehole with a circular trajectory, as in Section 7.2, we shall consider the DS element balance under the influence of resultants F(s) , M(s) of internal forces, internal moments and resultant f (s) of external distributed forces of gravity and contact interaction. Since equilibrium equations (7.40) are derived for a rod with arbitrary geometry, they are also valid in the case under consideration. Again, the advantages of the adopted approach become obvious when these equations are written in axes system oxyz of the selected mobile trihedron. Here again, the desired variables are separated and the force of the contact interaction can be temporarily excluded from consideration. Then, component Fx ( s) of internal force F( x) can be represented as follows:

Fx

 EIk cy  EIk x k z  M z k x

(7.83)

and system (7.41), (7.42) is transformed to the form:

dFy ds dFz ds dk x ds

EIk x k z2  M z k x k z  EIk cy k z  Fz k x  f ygr , EIk x k xc  EIk y k cy  f zgr , 

(7.84)

Mz 1 k y  k ykz  Fy . EI EI

This system is valid for the channel surface of general geometry; however, when considering a particular type of well, values k x , k y , k z must be given a concrete form since the derivation of the formulas for these quantities in the general case is associated with considerable difficulties. In a relatively accessible form, these values can be constructed for a well with a circular outline of its centreline. It should be noted that although in practice a drilling well with completely circular outline is unlikely to occur, nevertheless, the considered problem is of practical interest since

Chapter 7. Critical states and buckling of drill strings in channels…

445

their sections often have a circular shape. In addition, on the example of tasks for circular wells, some general patterns of buckling of curvilinear DSs are established. So, let centreline T of the well be a circle of radius R . Then, surface 6 is toroidal, and its equations (7.76) take the form: X

a sin v ,

Y

R(1  cos u) ,

Z

R sin u  a sin u cos v. (7.85)

On their basis, using the equalities 2

2

2

§ wX · § wY · § wZ · a11 ¨ ¸ ¨ ¸ ¨ ¸ , © wu ¹ © wu ¹ © wu ¹ wX wX wY wY wZ wZ a12 ˜  ˜  ˜ , wu wv wu wv wu wv 2

a22

2

(7.86)

2

§ wX · § wY · § wZ · ¨ ¸ ¨ ¸ ¨ ¸ , © wv ¹ © wv ¹ © wv ¹

parameters aij of first quadratic form )1 (u, v) (7.80) are calculated. They are equal to

a11

( R  a cos v) 2 ,

0,

a12

a2.

a22

(7.87)

Main curvatures k1 , k 2 of surface 6 are calculated in terms of quantities (7.86) and parameters bij of second quadratic form ) 2 (u, v) (7.80). Here, b12

0 , and

quantities b11 , b22 are calculated using formulas

1 a11a22

b11

w2 X wu 2 wX wu wX wv

w 2Y wu 2 wY wu wY wv

w2Z wu 2 wZ , wu wZ wv

b22

1 a11a22

w2 X wv 2 wX wu wX wv

w 2Y wv 2 wY wu wY wv

w2Z wv 2 wZ . (7.88) wu wZ wv

By substituting (7.85), (7.87), we obtain b11

cos v( R  a cos v) ,

b12

0,

b22

a.

It follows that the principal curvatures for surface 6 are equal k1

cos v , R  a cos v

k2

1 . a

(7.89)

Modelling emergency situations in drilling deep boreholes

446

In this case, geodesic curvature (7.81) is calculated using the formula

kx

k geod



a ( R  a cos v) ª¬(r  a cos v) 2 (uc) 2  a 2 (vc) 2 º¼

3/2

u

­ ª 2auc(vc) 2 ( R  a cos v)(uc)3 º ½ u ®uccvc  vccuc  sin v «  » ¾, a ¬ R  a cos v ¼¿ ¯

but normal curvature k y ky

(7.90)

k norm defined by (7.82) takes the form:

k norm

cos v( R  a cos v)3 (uc) 2  a 3 (vc) 2 .

(7.91)

Required variables uc , vc in equalities (7.90), (7.91) are not independent and can be expressed through each other using the relation

a11 (du) 2  a22 (dv) 2

(ds) 2 .

Then, taking (7.87) into account, we obtain

du / ds u c r

1  a 2 (vc) 2 . R  a cos v

(7.92)

To calculate components f xgr , f ygr , f zgr of distributed gravity f gr , we will project this vector on the corresponding axes

f xgr f

gr y

f zgr Here, f

gr

f

gr

sin u cos v,

f (sin u sin v cosT  cos u sin T ), gr

(7.93)

f gr ( sin u sin v sin T  cos u cosT ).

is the linear force of gravity calculated taking into account the

buoyancy force of the drilling fluid; sinT

avc , cosT

( R  a cos v)uc .

As a result, using relations (7.84), (7.90)–(7.93), it became possible to construct a system of resolving equations relative Fy (s) , Fz ( s) , kx (s) , v(s) , and u(s) . We shall represent it in the form of a system of first-order equations

Chapter 7. Critical states and buckling of drill strings in channels…

­ dFz 2 gr ° ds EIk x k z  M z k x k z  EIk cy k z  k x Fz  f (sin u sin v cosT  cos u sin T ), ° ° dFz EIk k c  EIk k c  f gr ( sin u sin v sin T  cos u sin T ), x x y y ° ds ° dk ° x  M z k y  k y k z  1 Fy , EI EI ° ds ° dv vc, ° ° ds ® 3/2 ª¬( R  a cos v) 2 (uc) 2  a 2 (vc) 2 º¼ ° d (vc) kx  ° a( R  a cos v)uc ° ds ° ª 2a(vc) 2 ( R  a cos v)(uc) 2 º uccvc ° sin v «  »  uc , a ° ¬ R  a cos v ¼ ° 2 2 1  a (vc) ° du ° ds ( R  a cos v) . ° ¯

447

(7.94)

At first glance, it seems that the general order of the system is six, and it exceeds the order of the equation of plane bending of the beam, which is four. However, it should be noted that the second equation of this system is used to determine axial force Fz ( s) , which is assumed to be given in rectilinear rods, and variable uc( s ) in the first five equations can be expressed in terms of the right-hand side of the sixth equation. If to exclude these two variables, the order of system (7.94) will be reduced to four, as in the theory of plane buckling of rectilinear rods. This system is homogeneous. It can be used for non-linear analysis of DS bending in a circular well under the action of axial forces and torques without frictional forces. The states in which the linearized operator of the system degenerates are bifurcation. In these cases, the eigen values and eigen modes of the linearized system determine the critical loads and buckling forms. Therefore, to analyse the stability loss of equilibrium and the buckling of a DS lying on the bottom of the borehole, it is necessary to linearize equations (7.94) in the vicinity of this state. Let us consider the problem of the buckling of a DS lying on the bottom of a concave section of a circular well (i.e., on the outer equator v 0 of toroidal surface 1 1 , u cc 0 , v 0 , vc 0 , vcc 0 , 6 (7.76)). At this state, u u0  s , uc Ra Ra 1 , k z 0 and after linearization, equations (7.94) take the form: kx 0, ky Ra

Modelling emergency situations in drilling deep boreholes

448

s · s · ­d § § gr gr ° ds GFy FzGk x  f sin¨ u 0  R  a ¸Gv  f a cos¨ u 0  R  a ¸Gvc, ¹ ¹ © © ° s · °d § gr ° ds GFz f sin¨ u 0  R  a ¸Gu, © ¹ ° °d (7.95) ® Gv G (vc), ° ds 1 1 °d ° ds G (vc)  a( R  a) Gv  a Gk x , ° °d ° ds (Gu ) 0. ¯ As follows from the sixth equation of this system, Gu(s) const . Then, starting from boundary condition Gu(0) 0 , we can obtain Gu(s) 0 . As a result, the second equation of the system reduces to form dGFz / ds

0 , and we also have GFz ( s)

0 . In

the remaining four equations, function Fz (s) is unknown. It is calculated using the second equation of system (7.94) that in the vicinity of state u(s) 0 becomes dFz ds

 f zgr

f

gr

s · § cos¨ u 0  ¸. R  a¹ ©

(7.96)

It has a solution s · §  f gr ( R  a ) sin¨ u 0  (7.97) ¸C, Ra¹ © in which constant C is found from the initial conditions. It is convenient to express this value through the known magnitude of compressive force Fz ( S ) at the lower end Fz ( s )

of the string. The remaining four first-order equations in system (7.95) are reduced to one ordinary homogeneous differential equation of the fourth order

F º 1 s · f gr § cos¨ u0   z »Gvcc  ¸Gvc  Ra¹ EI © ¬ a( R  a) EI ¼ ª

Gv IV  «

ª f gr º Fz s · § « sin¨ u0  Gv ¸ R  a ¹ aEI ( R  a) »¼ ¬ aEI ©

(7.98)

0.

It has a non-trivial solution with function Fz (s) in its coefficients, which turns it into degeneracy. In this case, the state corresponding to this form of Fz (s) is critical. Then, Fz (s) is the eigen function of equation (7.98) and its non-trivial solution wv(s)

Chapter 7. Critical states and buckling of drill strings in channels…

449

is its eigen mode representing the form of bifurcation buckling. We can see that by using the proposed approach associated with the introduction of new moving trihedron oxyz we managed to lead the three-dimensional deformation in curvilinear channel problem to the similar problem of the Euler planar buckling of straight rod. An important property of equation (7.98) is the absence in its coefficients of torque M z ( s) . This means that at the initial stage of DS buckling moment M z ( s) does not affect the buckling process (as well as for DS buckling in a directional rectilinear well, Section 7.2.). However, later, in a supercritical state, its effect becomes noticeable. As evidenced by the structure of equations (7.94), the effect of moment M z ( s) on the bifurcation process is also significant if the axial line of the well is three-dimensional. 7.3.4. Results of modelling the buckling of a drill string in a circular well (a frictionless model) It is clear that the study of solutions of equations (7.98) cannot be performed by analytical methods. Below are the results of its numerical simulation. It was carried out by the method of finite differences. Function Fzcr (s) of the internal axial force was formed as a result of the joint action of axial compressive force Fzcr ( S ) at lower end s S of the DS and distributed gravity f gr ( s) . The problem was solved by enumerating values Fz ( S ) using the half division method. Instances have been considered when the DS lies on the convex and concave segments of the well channel (on the outer and inner equators of torus surface 6 ). The DS is hinged at both ends. Studies of the influence of length S of the DS, radius R , value of annular gap a , and the positions of the DS in the channel on critical states and buckling patterns were carried out at the following values of the characteristic parameters: E 2.1˜1011 Pa, U st 7.8 ˜ 103 kg/m3, U m 1.3 ˜ 103 kg/m3, d1 0.1683 m, d 2 0.1483 m. Firstly, the case presented in Fig.7-22 was considered, when the DS laid on a concave section of a circular well, and at the lower end the tangent to its axis was parallel to the horizontal. Initial value of the u0 variable at the upper end of the DS was u 0 0 , and the angle of coverage of the arc of the well circumference was u(S ) S / 2 (90o ) . The calculations were carried out at values of annular gap a = 1, 0.5, 0.25, 0.1, 0.05, and 0.03 m, while radius R of the well centreline was fixed and was R 573 m. It should be noted that although the first two values of a are far from real, they are also included in the analysis to follow the trend of evolution of critical load values with changing values in the annular gap. In the calculations, segment 0 d s d S was divided into 500 finite-difference steps.

Modelling emergency situations in drilling deep boreholes

u0

450

0

us

Drill string pipe

cr

Fz ( S )

Fig. 7-22 Diagram of the drill string position inside the toroidal channel

Well wall Drill string in critical condition

Fig. 7-23 The mode of stability loss of the drill string in the lower zone of a curvilinear well

Chapter 7. Critical states and buckling of drill strings in channels…

451

Table 7-5 Critical forces F (S ) and buckling modes v(s) for case S = 900 m, u0 0 o , cr z

90o , u0  u S

uS No

a, m

90o .

Critical force Fzcr (S ) at end s S , kN

Axial force function Fz (s) , kN

The v(s) mode of buckling

400

1

1

-72.14

0 -400

s, m

s, m

0

400

800

0

400

800

0

400

800

0

400

800

0

400

800

0

400

800

0

400

800

0

400

800

800

400

2

0.5

-104.5

0 -400

400

3

0.25

-153.7

0 -400

400

4

0.1

-262.7

0 -400

400

5

0.05

-404.3

0 -400 0

400

800

0

400

0

400

800

0

400

400

6

0.03

-564.5

0 -400 800

Modelling emergency situations in drilling deep boreholes

452

The results of numerical analysis showed that if the length of the DS is relatively small (S < 20 m), and annular gap a is relatively large (a > 0.25 m), the DS loses its stability as a rectilinear rod with a half-sinusoidal buckling shape and an cr Euler critical force Fzcr (S ) | PEul

S

10 m and a

cr Eul

1 m, while P

S 2 EI / S 2 . So, Fzcr

327.4 kN in the case of

324.1 kN. This fact, to some extent, can serve as

evidence of the adequacy of the calculation model and the accuracy of the results. Increasing the length of the string and reducing the annular gap lead to an excess of cr Euler force PEul by value Fzcr (S ) and a significant complication of the bifurcation

form, which takes the shape of an edge effect (Fig.7-23). Table 7-5 shows the modelling results for a well of a quarter of the circumference ( u 0 0 , uS S / 2 ). This data includes the values of critical axial force Fzcr (s) at the lower end, the outline of function Fz ( S ) , and buckling shape wv(s) . They indicate that critical values Fzcr (S ) are relatively small when a ≥ 0.25 m and the upper end of the string is stretched. Therefore, the DS buckling prevails in the lower, compressed zone and takes the shape of the boundary effect typical of singularly perturbed systems. However, with a decrease in a , critical force Fzcr (S ) increases, and the DS compression zone begins to exceed the zone of its extension by gravity (graphs Fz ( s) in the table). In this case, the zone of the boundary effect expands, and the pitch of its sinusoidal harmonic decreases. If a = 0.03 m the boundary effect zone extends over the entire length of the string. u0 uS u0  u S

Fzcr (S )

Drill string Fig. 7-24 Diagram of the symmetric position of the drill string at the outer equator of a toroidal well

The case when the arch of the circular channel of the borehole is located symmetrically in respect of the vertical is of interest as well (Fig.7-24). Calculations are performed for R = 1,145 m and uS π (180°). The results of calculations for uS

S / 2 (90 ) , 2π/3 (120°), 5π/6 (150°), and

S / 2 (90 ) are presented in Table. 7.6. It

Chapter 7. Critical states and buckling of drill strings in channels…

453

contains an additional column of values of axial force Fzcr (S / 2) in middle segment s S / 2 that determines the maximum value of this force. It is established that, other things being equal, this force does not depend on the angle of the arch of axial line u S

since buckling occurs in the form of a harmonic wavelet localised in the central zone of the arch. It is also interesting to note that the half-step of a sinusoidal wavelet decreases from 27.2 m to 10 m with the reduction of the annular gap from a 1 m to a = 0.03 m. The three-dimensional shape of this wavelet is shown in Fig.7-25. In this case, a singular perturbation is manifested in the central zone of the string, while the critical values of force Fzcr (S / 2) and the buckling shape do not depend on the boundary conditions at the ends of the DS. Well wall Drill string in critical condition

Fig. 7-25 The mode of stability loss of the drill string in the central zone of a curvilinear well

If the DS lies on the convex part of the well trajectory (Fig.7-26), it is in contact with the inner equator of the torus channel. Then equality v S should be used in resolving equations (7.94). In this case, the buckling process acquires a different character. Cases are considered when, at the lower end, the tangent to the centreline is vertical ( u0  uS 2S ), and the angle of the circumferential arc is 1°, 2°, 10°, 20°, 40°, 60°, and 90°. The well centreline radius is R

285 m.

Modelling emergency situations in drilling deep boreholes

454

Table 7-6 Critical force F (S ) and buckling modes v(s) for the case S = 1,800 m, u0 45o , cr z

90o , u0  uS

uS

135o .

№

a, m

Critical force Fzcr (S ) at end s S, kN

1

1

37.7

Axial force extremum Fzext (S / 2) , kN

−68.73

Function of axial force Fz (s) , kN

Mode v(s) of stability loss

400 0

s, m

-400

s, m

0

800

1,600

0

800

1,600

0

800

1,600

0

800

1,600

0

800

1,600

0

800

1,600

0

800

1,600

0

800

1,600

0

800

1,600

0

800

1,600

0

800

1,600

400

2

0.5

8.120

−98.31

0 -400

3

0.25

−35.4

−141.8

400 0 -400 400

4

0.1

−127.5

−233.9

0 -400

400

5

0.05

−240.7

−347.1

0 -400 400

6

0.03

−362

−469.1

0 -400 0

800

1,600

A special property of the structural behaviour of these DSs is that if the length does not exceed 10 m, then with a loss of stability the Euler buckling mode is implemented, and the Euler value of critical load Fzcr (S ) S 2 EI / S 2 is achieved. But if uS

S / 2 (90 ) , the problem becomes singularly perturbed and buckling occurs in

the vicinity of the lower end. Moreover, critical load value Fzcr (S ) and the buckling shape are independent of length S and only slightly change with the a variation. This peculiarity can be explained by the fact that in almost vertical sections the force of compression (contact force) of the string to the wall of the borehole is negligible.

Chapter 7. Critical states and buckling of drill strings in channels…

455

u0  u S

Pipe of drill string uS

u0

Fzcr (S )

Fig. 7-26 Diagram of the asymmetric position of the drill string on the inner equator of the toroidal well

Well wall

Drill string in critical condition

Fig. 7-27 The mode of stability loss of the drill string in the convex segment of a curvilinear well

Table 7-7 presents the results of calculations for the case uS cr z

S / 2 (90 ) . As

can be seen, functions F (s) are almost identical in all cases, and the DSs are compressed in a narrow vertical zone near lower end s S . In this zone, contact forces f xc (s) are small, and they do not press the DS to the wall of the well; therefore, it buckles freely (Fig.7-27). Here again, it can be seen that the bulging effect of the DS does not depend on the boundary conditions at upper end s 0 .

Modelling emergency situations in drilling deep boreholes

456

Table 7-7 Critical force F (S ) and buckling shapes v(s) for the case S = 450 m, u0 270o , cr z

uS

90o , u0  u S No

a, m

360o .

Critical force Fzcr (S ) on edge s S, kN

Axial force function Fz (s) , kN

The v(s) mode of buckling

80

1

1

-22.88

0

s, m

-80

0

200

400

0

200

400

s, m 0

200

0

200

400

80

2

0.5

-23.99

0 -80

400

80

3

0.25

-24.72

0 -80

0

200

400

0

200

400

0

200

400

0

200

400

0

200

400

0

200

400

0

200

400

0

200

400

80

4

0.1

-25.23

0 -80 80

5

0.05

-25.42

0 -80 80

6

0.03

-25.49

0 -80

7.3.5. Friction model of drill string bifurcational buckling in a circular borehole The problem of mechanical instability of drill string bifurcational buckling is especially acute in the drilling technology of extended wells because it involves significant theoretical and technological difficulties. Modern experience shows that, as a rule, no deep wells are drilled without experiencing abnormal situations. They are caused by the complexity of the mechanical phenomena accompanying the

Chapter 7. Critical states and buckling of drill strings in channels…

457

drilling process and the lack of reliable methods of computer modelling that provide the ability to predict emergency situations and to eliminate them in advance. It has already been noted above that the problem of theoretical modelling of buckling in a curvilinear borehole channel acquires additional difficulties arising from the need to integrate differential equations with variable coefficients in the entire length of the DS. Together with this, the problem gets complication due to limiting constraint imposed by surface 6 of well walls, and the impact of contact, friction, and gravity forces, changing their orientation as the angle of the string element changes. These difficulties, as noted by Cunha [10], explain the lack of consistency in the results of the analysis of different authors and experimental data. Unresolved fundamental moments in this problem remain the questions about the role of friction forces in the phenomenon of loss of stability and how to use this factor in an analytical or numerical model. First, it is necessary to point out that each balanced stationary state of the DS is preceded by its stationary axial slip associated with the execution of downhole operations or drilling and accompanied by the appearance of kinetic friction (Fig.728). As a rule, the coefficients of kinetic friction exceed their values established at static equilibrium. Furthermore, these operations typically occur with certain finite longitudinal velocities, while DS buckling is implemented with very small transverse speeds. In this case, transverse buckling velocities (and the corresponding transverse frictional forces) can be assumed to be equal to zero in comparison with the forces of axial motion. Then, all frictional forces are axial and oriented in one direction (opposite the motion). In this case, the DS experiences the action of intense frictional forces, which adversely affect the stability of quasi-static equilibrium. Therefore, in this state, bifurcation buckling of the DS must be checked first. Second, due to the noted features of the formation of friction forces, they and all functions of the general stress-strain state of the DS can be determined by relatively simple calculations.

Modelling emergency situations in drilling deep boreholes

458

X

O

v

Y

u

Direction of drill string motion in the borehole

Z

Bore-hole wall

f cont

Drill string

f fr

f gr

Fig. 7-28 Diagram of a drill string in a borehole channel

Third, linearized homogeneous equations of critical equilibrium of the drill string can be constructed using these functions. Their eigen values and eigen modes determine the critical loads on the DS in the borehole and the modes of bifurcation buckling. To implement this approach, it is necessary to add the friction forces

f

fr

d P f cont

(7.99)

to the non-linear elastic bending equations (7.84) and (7.96). Following this relationship, static (between fixed bodies) and kinetic (between the surfaces of bodies sliding along one another) regimes of friction (Fig.7-29) are differentiated. Kinetic friction

Static friction

f fr

w Fig. 7-29 Coulomb friction diagram

Chapter 7. Critical states and buckling of drill strings in channels…

459

The static regime takes place when the driving force cannot overcome the resistance of the adhesion forces between the stationary bodies with limit (ultimate) value f ult P f cont . When limit value f ult is reached, the contacting bodies begin to slide over each other with the implementation of kinetic friction that does not depend on velocity w of relative motion and remains equal f ult . In addition, vector f fr of this force becomes collinear with this velocity w , and the following equality

f fr

 P f cont

w w

(7.100)

becomes effective. This relationship makes it possible to investigate the processes of DS buckling under axial motion. Indeed, if we assume that the DS moves along its axis at velocity wz without buckling, frictional forces f z fr (s) are directed along this velocity in the

opposite direction, and the transverse component of these forces is zero. Then, when critical axial force Fzcr (s) is reached and it slightly exceeds, the DS begins to buckle at low transverse velocity wy ( s) generating a small transverse frictional force f yfr (s)

f z fr (s) ˜ wy (s) / wz ,

(7.101)

preventing DS buckling at a velocity of wy . Assume that frictional force (7.101) that has arisen has stopped the buckling process, but then wy (s) 0 , and the transverse component also f yfr ( s) 0 . In this case, the critical value of the axial force is exceeded (albeit insignificantly), the DS remains unstable and again begins to buckle, but this time choosing absolutely low velocity wy ( s) in equation (7.101), which cannot stop the buckling process. Therefore, it can be assumed that if the DS slides along the centreline in the borehole, internal axial force Fz ( s) slightly exceeds found critical axial force Fzcr (s) , and the influence of resulting transverse frictional force f yfr (s) of the loss of equilibrium stability can be ignored. In this regard, we can assume that frictional force f z fr (s) is collinear to unit vector k at each point of the DS axis and its value is f z fr

r P f xcont ,

where signs ‘+’, ‘-’ are selected depending on the direction of motion of the DS. Then, in the system of equations (7.41), (7.42), it is necessary to use the equations of external and internal forces equilibrium

Modelling emergency situations in drilling deep boreholes

dFx / ds

460

k y Fz  k z Fy  f xgr  f xcont ,

dFy / ds k z Fx  k x Fz  f ygr , dFz / ds k x Fy  k y Fx  f

gr z

(7.102)

 fz , fr

and the equations of equilibrium of internal moments dM x / ds

k y M z  k z M y  Fy ,

dM y / ds

k z M x  k x M z  Fx ,

dM z / ds

k x M y  k y M x .

(7.103)

In this case, the remaining equations of the preceding section remain valid. Now, as a result of the introduction of proposed mobile trihedron oxyz representing the analogue of the Darboux vector, it becomes possible to rewrite the second equation of system (7.103) in the form:

Fx

 EIk cy  EIk x k z  M z k x

(7.104)

and represent the expression of contact force f xcont

EI (k ccy  kxc kz  kx kzc )  M z kxc  k y Fz  k z Fy  f xgr .

(7.105)

As a result, system (7.102), (7.103) is reduced to three equilibrium equations

dFy ds dFz ds dk x ds

EIk x k z2  M z k x k z  EIk cy k z  Fz k x  f ygr ,  EIk x k xc  EIk y k cy  f zgr  f z fr , 

(7.106)

1 Mz k y  k ykz  Fy . EI EI

It is complemented by surface 6 equations and the accompanying geometric relationships. We give the equations of the torus surface X

a sin v,

Y

R(1  cos u),

Z

R sin u  a sin u cos v .

(7.107)

They are used to calculate the parameters of the first and second quadratic forms and the principal curvatures

Chapter 7. Critical states and buckling of drill strings in channels…

a11

( R  cos v)2 ,

a12

0,

a22

a2 ,

b11

( R  a cos v)cos v,

b12

0,

b22

a,

k1

cos/ ( R  a cos v),

k2 1 / a.

Geodesic curvature k geod

k geod

461

kx

k x is calculated in this way

ª 1  a 2 (vc) 2 º a( R  a cos v) «uccvc  vccuc  sin vuc , a( R  a cos v) »¼ ¬

but for normal curvature k norm

k norm

(7.108)

(7.109)

k y we use the formula

ky

cos v

1  a 2 (vc)2  a(vc)2 . R  a cos v

(7.110)

Angle T between the directions of the tangents to lines L and u is given by trigonometric functions sinT

avc ,

cosT

( R  a cos v)uc .

Here, uc is given by

uc du / ds r

(1  a(vc) 2 . R  a cos v

(7.111)

Now, it seems possible to form non-linear equations for the elastic bending of the DS in the cavity of the toroidal channel ­ dFy 2 gr ° ds EIk x k z  M z k x k z  EIk cy k z  k x Fz  f (sin u sin v cosT  cos u sin T ), ° ° dFz  EIk k c  EIk k c  f gr ( sin u sin v sin T  cos u sin T )  f fr , x x y y ° ds ° ° dk x  M z k  k k  1 F , y y z y °° ds EI EI ® dv vc, ° ° ds ° d (vc) uccvc 1 1  a 2 (vc) 2 k x  sin v  , ° c ds a R a v u a R a v ( cos ) ( cos )   uc ° ° du 1  a 2 (vc) 2 ° . °¯ ds ( R  a cos v)

(7.112)

Modelling emergency situations in drilling deep boreholes

462

This system, with appropriate boundary conditions specifying the external axial force and torque, can be used to simulate non-linear elastic deformation of the DS in the circular channel of the well. It differs from system (7.94) by friction force presence in the second equation. We consider that in general the DS can bend and assume new deformed shapes remaining in contact with the surface of the well along its entire length. If during this deforming the small increments of external force perturbations correspond to small elastic displacements of the DS, the equilibrium state under consideration is stable. In the vicinity of this state, linear differential equations derived from non-linear system (7.112) by using linearization are degenerated and have only one solution. However, if external loads continue to grow, the coefficients of linearized equations continue to change, and a state can also be achieved where these equations become degenerate and acquire an additional (bifurcation) solution, together with the original. Therefore, the achieved state is critical (unstable), and the bifurcation solution represents the buckling mode. The feature of non-linear problems, set for a buckling DS in a circular channel is that in a pre-critical state the pipe does not change its shape, and, therefore, the coefficients of linearized equilibrium equations change as a result of the step-by-step increase in axial force Fz ( s) with an increase in external load and frictional forces. In this case, the problem posed is analogous to the problem of the Euler loss of stability of a rectilinear rod and turns out to be connected with the search for eigen values and the construction of eigen modes. To find the critical state of a drill string moving along the bottom of the borehole, system (1.112) must be linearized in the vicinity of the considered state, and its eigen values must be analysed. For a given axial motion of the drill string, the following quasi-statics and kinematics conditions are fulfilled: u

u0  S /(R  a) ,

k y 1/ ( R  a) , k z

uc 1/ ( R  a) ,

ucc 0 ,

v

0,

vc 0 ,

Fz ( S )

vcc 0 ,

kx

Fz ,

0,

0 . With bifurcation deformation, the system parameters acquire

small variations G v , G k x , G Fy . They can be calculated using a homogeneous system of linear equations

Chapter 7. Critical states and buckling of drill strings in channels…

463

­d s · s · § § gr gr ° ds G Fy FzG k x  f sin ¨ u0  R  a ¸ G v  f a cos ¨ u0  R  a ¸ G vc, © ¹ © ¹ ° °d °° ds G v G (vc), (7.113) ® 1 1 ° d G (vc)  G v  G kx , ° ds a( R  a) a ° 1 ° d Gk GF , °¯ ds x EI y which follows from system (7.112) after taking into account equalities G u 0 , G k y 0 , G Fz 0 . Coefficient Fz ( s) in the first equation is determined using the second equation of system (7.112). It is reduced to the form:

dFz ds

 f zgr  f z fr

 f gr r P f cont .

(7.114)

Here, signs r are selected depending on the type of DS tripping in/out operation, and contact force f cont is established based on the first equation of system (7.112) in the form:

f cont



1 Fz  f xgr Ra



1 Fz  f gr sin u . Ra

(7.115)

Substituting relations (7.115) into (7.114), we obtain a linear ordinary firstorder differential equation

dFz r P Fz du

( R  a) f gr cos u r P ( R  a) f gr sin u .

(7.116)

Let u U , s S , then

Fz (u )

( R  a) f gr ª r2P (cos u  cosU )  (1  P 2 )(sin u  sin U ) º¼  Fz (U ) , 1 P2 ¬

(7.117)

where Fz (U ) is the compressive force applied to lower end u U . Here, U is the coordinate value u at end s S . Four first-order differential equations (7.113) are equivalent to one homogeneous fourth-order differential equation

Modelling emergency situations in drilling deep boreholes

1 F º f gr s · §  z » G vcc  cos ¨ u0  ¸ G vc    a ( R a ) EI EI R a¹ © ¬ ¼

464

ª

G v IV  «

ª f gr º s · Fz § « aEI sin ¨ u0  R  a ¸  aEI ( R  a) » G v 0. © ¹ ¬ ¼

(7.118)

It was obtained on the basis of the assumption that when buckling frictional force f

fr

is directed along the DS centreline. To establish the state of bifurcation

buckling of a DS, it is necessary to formulate the Sturm-Liouville problem (eigen value problem) for equation (7.118). Formulation of this problem is based on the application of the finite differences method and the construction of the corresponding matrix of coefficients of algebraic equations. Elements of this matrix depend on factor Fz ( s) before required value G v in equation (7.118), which in turn is determined by the value of force Fz (U ) at edge u U and the distributed frictional force in equation (7.117). Then, the critical (bifurcational or eigen) value of external force Fz (U ) applied to edge u U (i.e., s S ) is found using the trial and error method. It is worth noting that torque M z ( s) is not present in equation (7.118). This means that the critical states of the drill string in wells of a circular outline do not depend on magnitude M z ( s) (just like in the case of rectilinear boreholes, Section 7.2). 7.3.6. Results of buckling analysis of a drill string moving in a circular borehole channel The main purpose of this section is to investigate the effect of frictional forces on the stability of drill strings moving within curvilinear boreholes. Despite the fact that the curvilinear borehole chosen for analysis has the simplest curvilinear configuration, this example allows us to analyse the main feature of the buckling process in curvilinear channels. It consists in the possibility of forming localised bifurcation wavelets in the most unexpected locations of the well channel. Additional uncertainty is introduced into the situation under consideration by the influence of frictional forces arising during tripping in/out operations. It is assumed that the motion is slow, and the forces of inertia can be ignored. Then, the DS can be partially compressed, partially stretched by variable distributed gravity forces f gr (s) , variable frictional forces f fr (s) , and axial force Fz (U ) applied to edge s S with angular

Chapter 7. Critical states and buckling of drill strings in channels…

465

coordinate u U . In this case, equation (7.118) cannot be solved using analytical methods. To find the DS bifurcation state, equation (7.118) was algebraised using the method of finite differences for different values of boundary force Fz (U ) , and the states in which the matrix of linear algebraic equations became degenerate were considered to be critical. In these states, we constructed eigen functions Fzcr (s) and eigen modes G v(s) representing the forms of DS buckling. In the above model, range 0 d s d S was divided into 500 finite-difference sections. Buckling results were tested with doubling the number of finite-difference divisions. Verification confirmed the sufficient accuracy of the calculations. To follow the influence of friction forces on the buckling process, computations in each considered example were carried out using frictionless and frictional statements of problems. Fig.7-30 shows the geometric diagram of a DS lying inside the lower quarter of the circular channel. At its lower end, the axis of the DS is tangent to the horizontal, the angle of the well arc is M U  u0 , and the position of upper end s

0 varied. The DS is hinged at both ends. The influence of

DS length S , radius R , annular gap a , and friction force f

fr

on critical values

cr z

F (U ) at the following values of the system parameters was studied:

E 2.1˜1011 Pa, d2

0.1483 m, P

J st

7.8 ˜ 103 kg/m3,

J mud 1.3 ˜103 kg/m3,

0.2 . Each example was investigated for cases f

d1 0.1683 m, fr

0 and f

fr

z0

and with values of annular gap a = 0.5, 0.1, 0.05, 0.03 m. Although the first value of a used here does not occur in drilling practice, it is included in the analysis to follow the trend of evolution of critical states with variation of gap a . The results of the calculations show that if the DS has a short length, gap a is not small, and radius R is large, the DS buckles under the scenario of the Euler beam under the action of axial critical force Pcr

S 2 EI / S 2 , and the friction force does not affect the bifurcation

process. However, the situation undergoes significant changes with increasing length S and angle M in connection with the fact that the DS becomes longer, and the properties of singularly perturbed structures begin to appear in the system. For this zone, buckling is localised in unpredictable ways in the form of short waves in the DS boundary zone (Fig.7-31 a, for the frictionless model) or in the DS inner zone, in which compressive axial force is at the maximum value (Fig.7-31 b, for accounting for forces of friction). In wave theory, such forms are called wave packets, and in applied mathematics they are called harmonic wavelets.

Modelling emergency situations in drilling deep boreholes

Fz (0)

s=0

Distributed friction forces

466

φ Directions of DS motion

Fz ( S )

s=S

Fig. 7-30 DS circular segment diagram

a

b

Fig.7-31 Deformation of drill string in a circular channel of a directional well: a = frictionless model of a stationary DS b = frictional model of a mobile DS

Chapter 7. Critical states and buckling of drill strings in channels…

467

Table 7-8 Functions of critical force Fzcr (s) and buckling modes G v(s) of drill strings for the case S =1,200 m, R= 1,146 m, φ= 60º No

a, m

Friction force

Critical axial force function Fzcr (s) , kN Fzcr

400

f

fr

0

-400

1

0.5

40

400

f

fr

z0

v

Fzcr (s) = -98.33

0

0

Mode of stability loss G v(s)

800

-93.97

s,m 1,200

0

400

800

s,m 1,200

Fzcr (s) = -66.02

0 -400 0

400

800

400

f

fr

0

1,200

0

400

800

1,200

0

400

800

1,200

0

400

800

1,200

0

400

800

1,200

0

400

800

1,200

Fzcr (s) = -234.0

0 -400

2

0

0.1

400

800

400

f

fr

z0

-224.9

0

1,200

Fzcr (s) = -196.9

-400 0

400

800

400

f

fr

0

1,200

Fzcr (s) = -347.1

0 -400 0

3

400

800

1,200

0.05 400

f

fr

z0

Fzcr (s) = -303.6

-331.6

0 -400 0

400

800

400

f

fr

0

1,200

Fzcr (s) = -469.2

0 -400 0

4

800

400

1,200

0

400

0

400

800

1,200

0.03 400

f

fr

z0

-445.2

0

Fzcr (s) = -417.26

-400 0

400

800

1,200

800

1,200

Modelling emergency situations in drilling deep boreholes

468

Table 7-8 shows the results for the 1,200 m long DS inserted in a circular well with radius R = 1,146 m. In this example, the angle of the well arc is M 60 . We can see that the character of DS buckling depends on the distribution functions of force Fz ( s) and the position of its extreme value. So, if a = 0.5 m, and f fr 0 , the DS is stretched in the upper zone in the vicinity S 0 m and is compressed in its lower zone in the vicinity s S . In this case, critical value Fzcr (S ) 98,33 kN formed at edge s S is the greatest compressive force for whole interval 0 d s d S . Therefore, the bifurcation wavelet is localised in the edge zone, thus confirming the properties of singularly perturbed systems [4, 18, 19, 52]. However, the situation changes if f fr z 0 (item 1 in Table 7-8). Then, maximum compressive value Fz fr 93,97 kN of the axial force is shifted to the inner zone of the DS, and the system becomes singularly perturbed within its length. In this case, the bifurcation wavelet also shifts inside the length of the DS. If a ≤ 0.1 m (positions 2–7 in Table 7-8), the DS buckles at large values of axial forces. In these cases, the smaller a , the more complex the shapes of stability loss. In the presence of frictional forces, it becomes more difficult to predict the localisation zone of bifurcation wavelets since it extends over a large DS length zone, and when a = 0.03 m, the boundary effect shifts from the right edge of the diagram to its left edge. In such situations, the step of buckling shape O becomes smaller and smaller. So, the length of the wavelet pitch O | 26 m at a = 0.03 m. Let us now consider the case when the arc of the well channel is located symmetrically in respect of the vertical (Fig.7-32). For this option, calculations are performed at values of S = 2,400 m, R = 1,146 m, M 120 . It is established that if friction effects are not taken into account, then representative functions determining DS buckling can be obtained by a simple symmetrical extension in respect of the vertical of the corresponding functions presented in Table 7-8 for an asymmetric arc (compare Fig.7-31a, and Fig.7-32a ). In this case, the critical values of edge forces Fzcr ( S ) of an asymmetric well become equal to critical values Fzcr (S / 2) for the corresponding columns in Table 7-9. As noted above, the presence of friction effects leads to a shift in the positions of the bifurcation wavelets in both cases (Figures 7.31 and 7.32), while the maximum values of corresponding critical functions Fzcr (s) (localised in the zones of wavelets) and the steps of wavelets remain approximately the same, although the shapes of their bifurcation modes acquire some differences.

Chapter 7. Critical states and buckling of drill strings in channels…

469

Table 7-9 Functions of critical force F (s) and buckling modes G v(s) of drill strings for cr z

the case S =2,400 m, R= 1,146 m, φ 120 No

a, m

Critical axial force function Fzcr (s) , kN

Friction force

cr 400 Fz

f

fr

0

0

0.5

s, m 2,400

fr

z0

1,600

800

0

1,600

800

400

f

0

800

0

800

1,600

0

800

1,600

0

800

1,600

2,400

0

800

1,600

2,400

0

800

1,600

0

800

1,600

2,400

0

800

1,600

2,400

0 -400

fr

1,600

s, m 2,400

0

Fzcr (S ) = 222.65

−94.11

400

f

v

Fzcr (S ) = 83.40

−98.29

0 -400

1

Mode of stability loss G v(s)

−233.9

2,400

2,400

Fzcr (S ) = −52.21

0 -400

2

0

0.1 400

f

fr

z0

800

1,600

2,400

2,400

Fzcr (S ) = 94.46

−222.3

0 -400 0

800

1,600

400

f

fr

0

Fzcr (S ) = −165.3

−347.0

0

2,400

-400 0

3

800

1,600

2,400

0.05 400

f

fr

z0

−332.8

0

Fzcr (S ) = −16.09

-400 0

800

400

f

fr

0

−468.9

0

1,600

2,400

2,400

Fzcr (S ) = −287.3

-400

4

0

0.03 400

f

fr

z0

800

−446.5

0

1,600

2,400

Fzcr (S ) = −129.8

-400 0

800

1,600

2,400

Modelling emergency situations in drilling deep boreholes

470

Table 7-10 Functions of critical force F (s) and buckling modes G v(s) of drill strings for the case S =3,600 m, R= 1,146 m, φ= 180º cr z

No

a , Friction m force

Critical axial force function Fzcr (s) , kN 400

f

fr

0

Fzcr

−98.18

0

0.5

0

fr

z0

900

1,800

−96.95

400

f

2,700

s, m 3,600

900

400

0

1,800

−233.7

2,700

3,600

0.1

0

fr

z0

900

1,800

−225,9

400

f

1,800

2,700

s, m 3,600

0

900

1,800

2,700

3,600

0

900

1,800

2,700

3,600

0

900

1,800

2,700

3,600

1,800

2,700

3,600

Fzcr (S ) = 129.7

0 -400

2

900

0 0

fr

0

Fzcr (S ) = 409.2

-400

f

v

Fzcr (S ) = 265.2

-400

1

Mode of stability loss G v(s)

2,700

3,600

Fzcr (S ) = 277,2

0 -400 0

900

400

f

fr

0

1,800

−346.5

2,700

3,600

Fzcr (S ) = 16.85

0 -400

3

0

0.05 400

f

fr

z0

900

1,800

−335.9

2,700

3,600

0

900

Fzcr (S ) = 167.2

0 -400 0

900

400

f

fr

0

1,800

−473.5

2,700

3,600

0

900

1,800

2,700

3,600

0

900

1,800

2,700

3,600

0

900

1,800

2,700

3,600

Fzcr (S ) = −110.7

0 -400

4

0.03

0

900

400

f

fr

z0

1,800

456.1

2,700

3,600

Fzcr (S ) = 47.05

0 -400 0

900

1,800

2,700

3,600

Chapter 7. Critical states and buckling of drill strings in channels…

471

a

b Fig. 7-32 Deformation of a drill string in a symmetric circular channel of a directional well: a = frictionless model of a stationary DS b = frictional model of a mobile DS

It is also interesting to note that if the length of the DS exceeds the width of the wavelet zone, then critical value Fzcr (s) does not depend on the DS length, and with an increase in coverage angle M these features become more obvious. These conclusions are supported by the data in Table 7-10 at M 180 , R = 1,146 m, S = 3,600 m. In this case, comparison of the analysis results with the initial data presented in Table 7-10 for M 180 allows us to conclude that in the absence of frictional forces bifurcational wavelets are localised in the central zone of the DS length, they are the same for each gap value a and are implemented at the same values of axial force Fzcr reached at midpoint s

S / 2 . It also becomes obvious that in this case the

buckling effect does not depend on the boundary conditions at ends s 0 and s S . In addition, step O of its eigen mode does not undergo significant changes. Their values for lengths S = 2,400 m and 3,600 m ( M 120 and M 180 ) are given in Table 7-11. As can be seen, they remain unchanged if the bifurcation buckling zone is small, but they decrease with a decrease in gap a . In cases of frictionless contact of the DS with the well wall, the values of steps O are identical for both lengths S with gaps a = 0.5 m and a = 0.1 m. However, they differ when a = 0.05 m and a = 0.03 m. The effect of frictional forces on the modes of stability loss is more pronounced, they lead not only to a shift in the positions of the bifurcation zones but also to their expansion (see Table 7-10). In addition, within each buckling zone, the pitches of the conditional harmonics also become variable (Table 7-11 for f

fr

z 0 ). It is interesting

Modelling emergency situations in drilling deep boreholes

472

to compare the results based on the data from analytical calculations obtained in limiting cases when curvature radius R tends to f in the absence of frictional effects. Then, it can be assumed that a curved well is horizontal, and an infinitely long column is prestressed by axial force Fz ( s) , which remains unchanged along its entire length. In this case, the critical value of force Fzcr (s) and step O are determined by equalities [19]: Fzcr

2 EIf gr / a ,

O cr

S 4 EIa / f gr .

Their bracketed values for the corresponding states are given in Table 7-11. It follows from this that the results of calculations closely coincide for large values of the gap, but the difference between them increases with decreasing a . Table 7-11 Extreme values of critical force F (s) and wavelet pitches O at critical states of the cr z

DS No. a , m Friction force

1

2

3

4

f

fr

0

f

fr

z0

f

fr

0

f

fr

z0

f

fr

0

f

fr

z0

f

fr

0

f

fr

z0

0.5

0.1

0.05

0.03

M 120 , S = 2,400 m M 180 , S = 3,600 m Fzext , kN Fzext , kN O,m O,m −98.29 (-91.26)

26 (26.65)

−98.18 (-91.26)

26 (26.65)

−94.11

26

−96.95

23.5

−233.9 (−204.1)

15.5 (17.8)

−233.7 (−204.1)

15.5 (17.8)

−222.3

15−15.5

−225.9

13.6–14

−347.0 (−288.6)

13 (14.9)

−346.5 (−288.6)

12 (14.9)

−332.8

11−12.5

−335.9

9.5–10.5

−468.9 (−372.6)

10.5 (13.1)

−469.2 (−372.6)

9.3 (13.1)

−446.5

10−10.5

−456.1

8.6–9

Chapter 7. Critical states and buckling of drill strings in channels…

473

Summarising the obtained results, it can be concluded that the established patterns of the realisation of critical states and the evolution of modes of stability loss are largely related to the geometry of the circular well and its radius R value. Therefore, it can be expected that the phenomena under consideration will be much more difficult for wells with variable curvature, both in cases of frictionless contact and in frictional interaction between moving elements of the system. It is also important to note the need to consider the influence of the presence of geometric imperfections of the well on the buckling process. Geometrical imperfections, as a rule, lead to an increase in the distributed forces of friction and axial force Fz ( S ) . On the other hand, geometric imperfections are associated with a change in curvature radius R . It can be concluded that these factors have a significant effect on the buckling process and should be specifically studied. 7.4. Global analysis of drill string buckling in a curvilinear well The phenomenon of DS buckling becomes further complicated if it occurs in the channel of a deep well of variable curvature. It seems that at present it can only be modelled theoretically with some significant simplifying assumptions concerning the well geometry and the conditions for realising the bifurcation effect. Since the study of the supercritical non-linear behaviour of the DS in the curvilinear channel is associated with considerable difficulties, the issue of establishing the critical state of the DS and predicting its deformation at the initial stage of buckling is of considerable urgency. Formulation of this problem is also based on the application of Euler's approach to the problem of stability of the initial form of equilibrium, but the study of DS bending on its entire length becomes difficult. This is because the curvature and the angle of inclination of the DS centreline vary along the longitudinal coordinate, and the movements are limited by the well wall. Therefore, the axial internal force in the DS as well as the contact force, frictional force, and orientation of the force of gravity pressing the DS to the well wall and preventing its buckling become essentially variable. However, as already noted, the most significant complication of the problem is associated with the long length of the DS and its relatively low stiffness. As shown earlier in the example of rectilinear wells, the differential equations describing these structures belong to the so-called singularly perturbed type. Chang and Howes [4] investigated this type of equation from the standpoint of applied mathematics. Analysing the two-point boundary value problem for ordinary differential equations, they showed that singularly perturbed tasks are poorly justified as their solutions have the forms of boundary effects and do not converge well. Here, it is appropriate to

Modelling emergency situations in drilling deep boreholes

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recall the monograph of Elishakoff et al. [12] devoted to the solution of non-classical problems of buckling of elastic rods. These singularities of the solutions of singularly perturbed problems were confirmed by us in modelling DS stability in deep directional and circular outline wells. In Subsections 7.2 and 7.3, it was shown that ‒ depending on the values of the angle of inclination and frictional forces ‒ the buckling effect appears in the form of short wave wavelets localised in the inner part of the DS length or in its edge zone. Nevertheless, these features have been identified for directional wells with simple geometry, whereas in practice the centrelines of the wells have variable curvature and inclination. In these cases, it is not possible to predict the localisation zone of buckling; therefore, the buckling effect must be analysed using an integrated approach along the full length of the string. In this section, to study DS stability in a channel with variable geometry, the non-linear theory of flexible curvilinear rods, the basic foundations of channel surfaces, and methods of differential geometry are used, except that for parameterisation of the centreline of the borehole and the drill string dimensionless and natural parameters are used together. DS critical states are established based on the double point boundary value Sturm-Liouville solution for the fourth-order equation, the bifurcational process models are considered: frictionless and friction. The problem is solved in two stages. At the first stage, external and internal forces acting on the string (gravity forces, contact forces, frictional forces, and internal axial force) are determined, and the bifurcation equilibrium equations are constructed. At the second stage, the eigen value problem is solved, critical loads are found, and modes of stability loss are constructed. 7.4.1. Theoretical analysis assumptions To formulate the problem, consider a DS in a deep directional well of variable curvature. Let us single out the case when the well trajectory can be represented by a hyperbolic curve (Fig.7-33) reflecting the most typical properties of its geometry. Indeed, it has a vertical tangent at the upper point, and its lower end approaches a straight line approximating the asymptote of the hyperbola. In fixed coordinate system OXYZ , it is convenient to present well axial line T in the general analytical form: X*

ε ch t ,

Y*

0,

Z*

λ sh t ,

(7.119)

where t is the dimensionless parameter, H is the semiaxis of the hyperbola, O is the value characterising the well depth.

Chapter 7. Critical states and buckling of drill strings in channels…

Z

ε

l

475

Y X O

T

h

β t

t0 Fig. 7-33 Geometric diagram of a directional well

We shall introduce channel surface 6 with axial line T and generating circles of radius a equal to annular gap a (d1  d2 ) / 2 , where d1 and d 2 are the diameters of the well and DS, respectively. We shall parameterise its parameter u associated with line T of the well and parameter v that determines the position of the point on the circle generatrix (Fig.7-34), where values v 0 correspond to the lower positions of points on these circles. Let us assume that at the initial state the DS lies on the well bottom and for its centreline L condition v 0 is true. When buckling, the DS slips over the well surface without separation, and then the axial line of the string moves along channel surface 6 . To describe the geometry of the DS centreline, it is necessary to set values h , l , and H that determine the well dimensions and express coefficient O in system (7.119) and initial value t0 of parameter t through the original data. From Fig.7-33 and equalities (7.119), it follows that X * (t0 ) H ch(t0 ) (l  H ), Z * (t0 ) O sh(t0 ) h.

From this, it follows: t0

arcch ª¬H  l / H º¼  0,

O h / sh t0 ! 0 .

(7.120)

Using these quantities, we find angle E of the curve T slope (Fig.7-33). For this, we use equality

tg E Since

dZ * / dX * .

Modelling emergency situations in drilling deep boreholes

dX *

X *dt ,

476

dZ *

Z *dt ,

we have

E

arctg(Z * / X * ) arctg( arctg O ch t / H sh t ),

(7.121)

where the dot above the symbol denotes differentiation with respect to t .

s

z u

L

x

v

Σ Arc of the generating circle

y

f gr

Fig. 7-34 Geometric diagram of the orientation of centerline L on channel surface Σ

The problem of DS loss of stability is solved on the basis of the theory of flexible curvilinear rods. In this way, we represent the non-linear equations of flexible rods in general form (7.40), and the states at which the linearized operator of these equations degenerates will be considered critical (bifurcation). Using this approach in Subsections 7.2 and 7.3, the stability of the DS in rectilinear directional and circular wells was investigated. However, this technique proved to be acceptable only for wells with the simplest geometry when the curvature of their axial lines is unchanged in length. In the case of wells with varying geometry of their lines, the non-linear equations of DS bending become so complex and cumbersome that it is difficult to imagine them in an explicit form. Therefore, it becomes more rational to immediately formulate a linearized equation without resorting to a non-linear one. An additional difficulty of the problem is caused by the fact that one must simultaneously apply two parameters. One of them ( t ) is used to parameterise well centreline T , and the second one ( s ), for parameterisation of the centreline of the DS. Parameter t has in fact already been used in equations (7.119) – (7.121), while parameter s is used in the theory of flexible rods to form the equilibrium equations of a rod element

Chapter 7. Critical states and buckling of drill strings in channels…

dF ds

f ,

dM ds

-τ u F ,

477

(7.122)

where, as before, F( s) is the resultant of internal forces, M(s) is the resultant of internal moments, f (s) is the vector of external distributed forces, τ is the unit vector tangent to line L , s is the natural parameter. The relationship between parameters t and s can be expressed by a differential relation ds

(dX * )2  (dZ * )2

( X * )2  ( Z * )2 dt

(ε sh t )2  ( λ ch t )2 dt .

(7.123)

Then, we have t

s(t )

³

(ε sh ξ ) 2  ( λ ch ξ ) 2 dξ .

(7.124)

t0

In the calculations, these parameters are used simultaneously. To determine the geometry of the DS centreline, it is convenient to use unit vectors of normal n , binormal b , and tangent τ (relations (7.14) – (7.18)). In doing so, we use the above assumption that when transforming line L , it does not come out of contact with surface 6 , thus limiting its movement. Therefore, it is also expedient to use reference system oxyz with unit vectors i , j , k (Fig.7-34). In this case, unit vector i is directed along the inner normal to surface 6 , vector k is tangent to curve L , and vector j complements the system to the right three. Further, we introduce an analogue of the Darboux vector

ω k xi  k y j  k zk ,

(7.125)

where k x and k y are the corresponding components of curvature vector K b / R , k z determines the rotation of trihedron oxyz around vector k when it moves along line L. The entered values determine the geometric characteristics of surface 6 . Its equations can be represented in parametric form:

X X *  a sin β cos v ε ch t  a sin β cos v, Y a sin v, Z

Z  a cos β cos v *

(7.126)

λ sh t  a cos β cos v.

In the case under consideration, radius a of the channel surface is small in comparison with its other parameters; therefore, variables t and u can still be combined; nevertheless, resolving non-linear equations of DS bending remain very

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complicated. Therefore, as noted above, critical states and small bifurcation displacements of the DS lying on bottom v 0 of surface 6 are studied. Then, there is no need to first construct non-linear equations of DS bending in a general form and then linearize them, as was done in Sections 7.2 and 7.3, but it is possible to build linearized stability equations in the vicinity of equilibrium state v 0 . They acquire the simplest form, if they are formed in mobile reference frame oxyz with unit vectors i , j , k . In this case, the non-linear equations of equilibrium of forces dFx / ds

k y Fz  k z Fy  f x ,

dFy / ds

k z Fx  k x Fz  f y ,

dFz / ds

k x Fy  k y Fx  f z

(7.127)

and the equilibrium of moments dM x / ds

k y M z  k z M y  Fy ,

dM y / ds

k z M x  k x M z  Fx ,

dM z / ds

k x M y  k y M x

(7.128)

are presented in standard, universal form. They can be investigated at any stage of integration using a two-step computational process. At the first step, the pre-critical state of the DS is analysed, and internal axial force Fz ( s) is calculated. At the second step, equations (7.127), (7.128) are linearized in the vicinity of this state, and the boundary value problem for eigen values is solved. As a result, the values of the critical loads are determined, and the buckling modes are constructed. We assume that in the case under consideration the DS is descending, it does not rotate, and, therefore, the frictional torque is zero. Let the DS motion occur at a low velocity, the inertia forces can be ignored, and only gravity forces ( f gr ), contact forces ( f cont ), and frictional forces ( f fr ) affect the DS. In this case, due to the chosen orientation of system oxyz , forces f gr ( s) are directed along axis OZ , force f cont ( s) acts along the normal to surface 6 , and frictional force f fr ( s) is directed in the axial direction against the velocity vector of the DS element. Therefore, these forces can be represented in system oxyz as follows: f gr

f cont

f xgr i  f ygr j  f zgr k ,

f xcont i ,

f fr

 f z fr k .

(7.129)

Since the DS slides along the channel cavity, frictional force f z fr (s) is determined using the Coulomb formula

f z fr

μf xcont ,

(7.130)

Chapter 7. Critical states and buckling of drill strings in channels…

479

and the components of the total external distributed forces are calculated in this way

fx

f xgr  f xcont ,

fy

f ygr ,

fz

f zgr  μf xcont .

(7.131)

In the pre-critical state, the DS lies at the bottom of the channel, and its axis coincides with line v 0 on surface 6 . If we take into account that at this state k x 0 , k z 0 , M x 0 , M z 0 , Fy 0 , then from the third equation of system (7.128) we can obtain or

dM z / ds 0

M z (s) const .

Here, M z ( s) is determined from the initial conditions. Let's take M z ( s) 0 . In this case, it follows from systems (7.127), (7.128) that dFx / ds k y Fz  f xgr  f xcont , dFz / ds k y Fx  f zgr  μ f xcont ,

(7.132)

 EIdk y / ds.

Fx

These relationships allow us to derive an equation for determining axial force Fz ( s) dFz / ds

 EIk y dk y / ds  f zgr  μ EId 2k y / ds 2  k y Fz  f xgr .

(7.133)

It contains only one unknown quantity Fz ( s) , so it can easily be integrated using the Runge-Kutta method. To implement the second step in the study of the buckling process, equations (7.127), (7.128) must be linearized in the vicinity v 0 dδFx / ds

δk y Fz  k y δFz  δk z Fy  k z δFy  δf x ,

dδFy / ds

δk z Fx  k z δFx  δk x Fz  k xδFz  δf y ,

dδFz / ds

δk x Fy  k xδFy  δk y Fx  k y δFx  δf z ,

dδM x / ds

δk y M z  k y δM z  δk z M y  k z δM y  δFy ,

dδM y / ds

δk z M x  k z δM x  δk x M z  k xδM z  δFx ,

dδM z / ds 0.

(7.134)

(7.135)

Here, symbol G ... indicates a small increment of the corresponding quantity. Then, variables δFx and δFy can be represented using equations (7.135) δFx

 EIdδk y / ds 0,

δFy

EIdδk x / ds  EIk y δk z .

(7.136)

Modelling emergency situations in drilling deep boreholes

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In the Eulerian theory of stability of rectilinear rods, the hypothesis is accepted that the axial force in the rod does not change with loss of stability. By using it, the third equation of system (7.134) can be discarded, and the other two are reduced to the form:

EId 2δk y / ds 2

Fz δk y  δf x ,

EIdk y / dsδk x  EIk y dδk z / ds  EId 2δk z / ds 2

 EIdk y / dsδk z  Fz δk x  δf y .

(7.137)

These include unknown functions G k x , k y , δk y , and δk z . To derive them, it is necessary to determine the curvatures of curve L lying on channel surface 6 . Therefore, they are defined in terms of its geometric characteristics. The internal geometry of this surface is given by parameters aij (u, v) of its first quadratic form

Φ1 (u, v) a11 (u, v)du 2  2a12 (u, v)dudv  a22 (u, v)dv 2 .

(7.138)

External geometry is defined by parameters bij of second quadratic form

Φ2 (u, v) b11 (u, v)du 2  2b12 (u, v)dudv  b22 (u, v)dv 2 .

(7.139)

In these equalities, aij (i, j 1,2) are components of a metric tensor connected with surface 6 . They characterise curvilinear coordinates u , v and the geometric objects lying in it, including curvature k x of curve L identified by geodesic curvature k geod . In the vicinity of line v 0 of the channel surface, lines u and v are orthogonal, so a12 (u,0) 0 , and curvature k z can be represented by formula (7.34)

[44] kx

k geod

a11a22 ª¬a11 (uc)2  a22 (vc)2 º¼

3/2

uccvc  vccuc  Avc  Buc ,

(7.140)

where A and B are represented through the Christoffel symbols in Subsection 7.2.3. Parameters of the first quadratic form a11 (u, v) and a22 (u, v) are calculated using formulas (7.86), and quantity a12 in the vicinity of curve v 0 is equal to zero. If we take into account that a 1, O >>1, and curvature k y is small. Then, in the vicinity of the pre-critical state, kz

sin ς cos ς / a 0 .

(7.152)

For small variations of G v , we have δk z

dδv / ds .

(7.153)

To determine the components of vector f gr , we represent it in reference frame oxyz (Fig.7-34)

f gr

 f gr cos β cos vi  f gr ( sin β sin ς  cos β sin v cos ς ) j 

 f gr ( sin β cos ς  cos β sin v sin ς )k , where f gr

(7.154)

π (r12  r22 )(γt  γm ) g ; r1 , r2 are the outer and inner radii of the DS pipe,

respectively; γt , γm are the densities of the material and the drilling fluid; 2 g 9.81 m/s is the acceleration of gravity. In the case under consideration, v 0 , ς 0 , so

f xgr δς

 f gr cos β ,

f ygr

0,

f zgr

 f gr sin β .

(7.155)

In the perturbed state, quantities v and ς acquire small perturbations δv and adδv / ds , so

Chapter 7. Critical states and buckling of drill strings in channels…

δf xgr

0,

f gr (a sin βdδv / ds  cos βδv) ,

δf ygr

483

δf zgr

0.

(7.156)

Taking into account equations (7.136), (7.137), (7.140), (7.144), (7.145), (7.149), (7.150), (7.152), (7.153), and (7.156), the second equation of system (7.134) and the first equation of system (7.135) can be transformed to the form:

­° ª β º ½° dδFy / ds dδv / ds  Fz ®aδvcc  «  λ ch t sin β  ε sh t cos β δv » ¾  ¬ a11 ¼ ¿° ¯°  f gr  a sin βdδv / ds  cos βδv , dδk x / ds

(7.157)

1 δFy  k y dδv / ds, EI

while the left-hand sides of the remaining equations of systems (7.134) and (7.135) are identically zero and must be discarded. Equations (7.157) are supplemented by equation (7.145) in the form:

dδvc / ds

1 β δk x  ( λ ch t sin β  ε sh t cos β )δv a aa11

(7.158)

and identity dδv / ds δ (vc) .

(7.159)

After the corresponding substitutions, the system of equations (7.157)–(7.159) is transformed to one homogeneous fourth-order differential equation ½ d 4δv EI d 2 ­ β ª λ ch t sin β  ε sh t cos β  aβ  β ¼º δv δ ¾ 4 2 ® ¬ ds a ds ¯ a11 ¿ 2 2 2 2 ­ ½ d sh t ch t dδv d δv Fz (ε  λ )sh t ch t dδv   EI (ε 2  λ2 ) 2 ® ¾  Fz ds ¯ (a11 )3/2 ds ¿ ds 2 ds (a11 )3/2

EI

(7.160)

β ª λ ch t sin β  ε sh t cos β  aβ  Fz β º¼ δv δ  aa11 ¬ cos β · §  f gr ¨  sin βdδv / ds  δv ¸ 0 a © ¹

0 d s d S.

A characteristic feature and complexity of this equation are that it includes ordinary derivatives with respect to variable t (marked with a dot) and s simultaneously. In the process of implementing the solution, this difficulty can be overcome by using variable s for the formulation of equation (7.160), while the coefficients of this equation must be expressed in the corresponding points of variable s through variable t . The transition from one parameter to another is done with the help of equality (7.124).

Modelling emergency situations in drilling deep boreholes

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The second difficulty is due to the need to formulate equation (7.160) in the vicinity of a given function of axial force Fz ( s) , which is unknown in advance. Therefore, the problem is solved in two stages. At the first stage, function Fz ( s) is constructed for given value Fz (0) using equation (7.133). Then, in the second stage, this function is inserted into equation (7.160), and its degeneracy is checked. If this homogeneous equation is degenerate, it has non-zero solutions (eigen modes), and the corresponding state is critical (bifurcation). Then, the corresponding eigen form is a mode of DS buckling. Special attention should be paid to the numerical implementation of the described approach. This is mainly because the problem is singularly perturbed [19], and, therefore, its solutions have the form of localised boundary effects or internal concentrated perturbations. The positions of these perturbations are unknown in advance, so the problem cannot be reduced to a local analysis and should be solved based on a global approach with a solution search in the whole region 0 d s d S of the DS length. As underlined above, the greatest difficulty of the problem posed for equation (7.160) is associated with large DS length S and its small bending stiffness. To comment on this assertion, we use the approach adopted in [4, 19] to justify singularly perturbed systems and analyse equation (7.160) with chosen parameter values E = 2.1•1011 Pa, I = 1.564•10–5 m4. For the sake of simplicity, value S = 10,000 m is selected, then equation (7.160) must be integrated over s in area 0 ≤ s ≤ 10,000. However, it is unusual to integrate a differential equation in this range, so by changing the scale of parameter s the transition to new independent variable T will be done. For example, it is possible to measure the length of the DS in kilometres (although we shall leave all other characteristics of the DS, including its bending stiffness EI , remain unchanged). Then, it should be written s = 1,000θ, where 0 d T d 4 and 4 10 . In this case, equation (7.160) can be transformed as follows:

½ EI d 4δv EI d 2 ­ β ª¬ λ ch t sin β  ε sh t cos β  aββ º¼ δv δ ¾  6 12 4 2 ® 10 dθ 10 a dθ ¯ a11 ¿ EI (ε 2  λ2 )

d 2 ­ sh t ch t dδv ½ d 2δv ¾  Fz 6 2  6 2 ® 3/2 10 dθ ¯ (a11 ) 10dθ ¿ 10 dθ

Fz (ε 2  λ2 )sh t ch t dδv dδv β  f gr sin β 3  Fz u (a11 )3/2 103 dθ 10 dθ aa11 u ª¬ λ ch t sin β  ε sh t cos β  aβ º¼ δv  f gr

cos β δv 0 a

(7.161)

0 d θ d Θ.

Chapter 7. Critical states and buckling of drill strings in channels…

485

Now, as can be seen the coefficient before the fourth derivative is negligible, while its last two terms remain unchanged. For this reason, these two terms play a much more significant role than the first one. However, this first term cannot be discarded since in this case the structure of equation (7.161) will change, and it will begin to describe a completely different phenomenon. But equation (7.161) in domain ( 0 d T d 10 ) and equation (7.160) in domain (0 ≤ s ≤ 10,000) are completely equivalent. Therefore, the importance of the first term in equation (7.160) is negligibly small in comparison with the other terms. However, this time the term cannot be neglected either, because if it is discarded, the remaining part in equation (7.160) will begin to describe the balance of the cable that cannot buckle. In practice, as a rule, DSs behave like ropes, so in English they are called ‘drill strings’ rather than drill rods. At the same time, if the string begins to lose stability, it is equivalent to an elastic rod in the buckling zone. In applied mathematics, equations with a small coefficient in front of the highest derivative are called singularly perturbed [4]. As a rule, tasks of this kind are poorly justified and have irregular divergent solutions. In this book, the finite difference method is used to analyse it. When implemented, the length of the drill string is broken down into n finite-difference steps 's S / n , and at each nodal point si derivatives of G v with respect to s are replaced by their finite-difference analogues. So, equation (7.160) sampled at each point si (1 d i d n ) allows us to construct a system of linear algebraic equations. The solution of this system determines the approximate values of function G v(s) at points si . Special mathematical software has been developed to implement this numerical modelling. It consists of programme blocks that prepare the initial data, perform geometric modelling of the details of the well, establish a correspondence between parameters s and t , set the initial loads, calculate internal axial force Fz ( si ) in all nodes of the finite-difference region, construct finite-difference equations for given loads and forces Fz ( si ) , solve the system of algebraic equations, count the determinant of the matrix, and then find a state in which the determinant changes its sign passing through the zero value. This state is critical, and the corresponding eigen mode of the matrix is a buckling shape. Next, the programme processes the results of the calculations and builds the graphs of the functions found. In the calculations, domain 0 d s d S is split into 1,000 finite-difference steps 's . Numerical results were verified with a doubled number of finite-difference divisions. Testing confirmed the sufficient accuracy of calculations.

Modelling emergency situations in drilling deep boreholes

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7.4.2. Analysis of the computer modelling results To determine the general pattern of DS buckling in curvilinear wells, their trajectories with different outlines and sizes were chosen. The examples considered make it possible to reveal the general features of the phenomena of buckling in curvilinear boreholes. One of them is the possibility of losing stability with the formation of harmonic wavelets in the most unexpected places of the DS length. This is due to the initial uncertainty of axial friction force f z fr (s) generated during string round trips. When it acts together with distributed gravity f zgr (s) and edge axial force Fz (0) at s 0 , the string can be partially compressed, partially stretched, which also complicates its analysis. To study these nuances for the study, wells with different inclines of the trajectories were chosen. In accordance with equations (7.119) and selected parameter values H , l , and h , the well axis is a square hyperbola with a vertical tangent at upper point s S and a tangent that tends to a rectilinear asymptote at its lower end s 0 (Fig.7-33). The DS is hinged at both ends. The influence of parameters l , h and angle β (0) of the well incline at its bottom point on critical values Fzcr (0) of axial compressive force Fz (0) was studied at the following values of the system parameters E

2.1˜1011 Pa, γt

7.8 ˜103 kg/m3,

γm 1.3 ˜103 kg/m3, d1 = 0.1683 m, d2 = 0.1483 m, a = 0.08 m, μ = 0 and μ = 0.3. The results of the study indicate that if μ 0 , DS buckling occurs in the vicinity of edge s 0 similar to the critical behaviour of a string in a straight borehole with incline angle E equal to angle β (0) of the original curvilinear well. In this case, the critical loads and the modes of stability loss of both systems almost coincide, and with bifurcation analysis it becomes possible to replace the curvilinear DS buckling study with the corresponding analysis of a rectilinear directional borehole. However, this phenomenon undergoes changes if the effect of frictional forces is taken into account. As it turned out, frictional effects contribute to a significant redistribution of internal axial force Fz ( s) , the evolution of their eigen functions and modes of stability loss, and the shift of their origin. Nevertheless, as shown below, there are some patterns associated with angle β (0) β0 and coefficient of friction μ = 0.3 as well as angle of friction βfr = arctgμ = 0.29145679 ≈16.699°. The calculation results show that the buckling phenomenon takes a sharply expressed form of the boundary effect, if β0 ! β fr . Table 7-12 presents the computational data for DS buckling in a borehole channel with an uplifted trajectory. The four types of geometry considered refer to wells of same depth h = 4,000 m and differ in the values of horizontal distance parameter l , which is 1 km (task 1), 2 km

Chapter 7. Critical states and buckling of drill strings in channels…

487

Table 7-12 Characteristics of DS stability loss in uplifted wells Task No. -6 km

0

Mode

δv( s) of buckling p curv = 10.87 m ( p rect = 12.2 m)

Critical force Fzcr ( s ) functions

Well centreline

Fzcr (0) = −152.337 kN

-4

( Fzrect (0) = −151.185 kN) Fzcr , kN

1,200

δv

-1

1

-2

400

-3

0 β 0 = 1.1760

-4

s, km

-400

0

-6

-4

2

0

4

6

s, km 0

2

4

Fzcr (0) = −185.471 kN

p curv = 16.30 m

( Fzrect (0) = −184.877 kN)

( p rect = 16 m)

6

1,200 -1

2

-2

400

-3

0 β 0 = 0.9273

-400

-4

0

-6

2

0

4

6

0

F (0) = −201.075 kN

-4

2

4

6

p = 17.39 m ( p rect = 17.5 m)

cr z

curv

( Fzrect (0) = −200.970 kN)

1,200

-1

3

400

-2

0

-3 β 0 = 0.7610

-4

0

-6

-400

2

0

4

6

0

F (0) = −209.565 kN

-4

2

4

6

p = 17.39 m ( p rect = 16.4 m)

cr z

curv

( Fzrect (0) = −209.774 kN)

1,200 -1

4

-2

400

-3 -4

0 β 0 = 0.6435

-400 0

2

4

6

0

2

4

6

Modelling emergency situations in drilling deep boreholes

488

(task 2), 3 km (task 3), and 4 km (task 4). The left column of the table shows the outline of the well. Charts of critical axial functions Fzcr (s) are shown in the middle column. As follows from the diagrams, these forces are positive (tensile) in the upper parts of the DS and negative (compressive) in their lower zones. The corresponding values of critical (extreme) compressive force Fzcr (0) at lower end s

0 of the string

are shown in the upper right corner of the corresponding column. It can be seen that the smaller inclination angle β (0) , the greater the value of critical force Fzcr (0) . This effect can be explained by the fact that when this angle decreases, the contact force pressing the DS to the well wall increases and prevents buckling of the string. It is of interest to confirm the obtained results by comparing them with other theoretical data. Considering that the problem of global stability of the DS in a curvilinear well is not yet complete, it is more convenient to consider some individual cases. In particular, as noted above, the outline of hyperbolic line L of the string approaches a straight line in the lower part of the well where function Fz (0) is extreme. Then, it is quite natural that in this case DS buckling should occur to a large extent similar to its loss of stability in a rectilinear directional channel. To test this assumption, calculations are made for DS buckling in directional rectilinear channels using the equation given in Section 7.2 ª f gr (cos β  μ sin β ) Rº δy IV  «  ( S  s )  » δycc  EI EI ¬ ¼ . f gr (cos β  μ sin β ) f gr sin β δyc  δy 0 EI aEI

(7.162)

Here, δy(s) aδv(s) . Critical values of edge axial forces Fzrect (0) for straight-line DSs, if appropriate, are given in the middle column of Table 7-12. As follows from the comparison, the results of the calculations agree well. As for the shapes of DS buckling, they exhibit the property of singularly perturbed structures experiencing short-wave bifurcations in the border zones (the right-hand column of Table 7-12). To analyse these modes in more detail, Fig.7-35 shows a fragment of function G v(s) for problem 1 in a greater scale. In domain 0 d s d 0.25 km, it is a decreasing harmonic with pitch p .

Chapter 7. Critical states and buckling of drill strings in channels…

489

δv

p

P 0

50

s, m 100

150

200

250

Fig. 7-35 The scaled up boundary effect of DS stability loss (task 1 Table 7-12)

Z X

Y O

Channel surface

Harmonic wavelet

Fig. 7-36 Schematic view of the DS buckling in a steep bore-hole

Modelling emergency situations in drilling deep boreholes

490

To get a more detailed idea of the properties of the buckling effect, the right column of Table 7-12 shows the values of steps p curv and p rect (in parentheses) for curvilinear and rectilinear channels, respectively. Their proximity allows us to confirm once again the conclusion that the critical buckling of a DS in an uplifted curvilinear channel occurs in the same way as it takes place in a rectilinear channel with same inclination angle E 0 . Fig.7-36 is a schematic representation of the shape of DS buckling in uplifted wells. This illustration makes it easier to understand the features of the bifurcation loss of stability. δv p

P

s, m 900

1,100

1,300

1,500

1,700

1,900

Fig. 7-37 Harmonic wavelet of DS stability loss mode (Task 6 in Table 7-13)

However, this regularity changes essentially with a decrease in angle β (0) when it becomes less than angle of friction βfr = arctgμ = 0.29145679 ≈16.699°. Table 7-13 shows the basic characteristics of buckling in shallow boreholes. Tasks 5– 8 relate to cases where the depth of the well has the same value (h = 0.5 km), while their horizontal removal varied from l 1 km (task 5) to l 4 km (task 8). For Task 5, angle E 0 still exceeds β fr , and the minimum value of function Fz ( s) occurs when s 0 . Therefore, the phenomenon of singular perturbation is implemented in the form of a boundary effect. If β0  β fr (tasks 6–8), the role of the frictional force exceeds the value of the forces of gravity, and the maximum value of compressive axial force Fz ( s) is achieved inside the length of the DS (the middle column of Table 7-13).

Chapter 7. Critical states and buckling of drill strings in channels…

491

Table 7-13 Characteristics of DS stability loss in shallow wells Task No.

Critical force Fzcr ( s ) functions

Mode δv(s) of buckling

Fzcr (0) = −227.340 kN

p curv = 16.6 m

( F rect (0) = − 225.229 kN)

( p rect = 16.3 m)

Centreline of the well

-6 km 0

z

-4

0

5

β0 =0.2941

-1 -2

cr z

F , kN

δv

-100

-300

s, m 0

-6

6

1,500 3,000 4,500

Fzcr (0) = − 183.999 kN

p curv = 15.9 m

( F rect (0) = − 229.981 kN)

( p rect = 15.3 m)

z

-4

0

s, m 1,500 3,000 4,500

0

0

232.452

-100

-1

β0 = 0.1655 -300

-2

0

1,500 3,000 4,500

0

Fzcr (0) = − 104.552 kN

p curv = 17.05

( F rect (0) = − 230.799 kN)

( p rect = 16.3 m)

z

-6 0

7

-4 0

235.976

-100 -1

β0 = 0.11849 -300

-2

0

0

-4

1,500 3,000 4,500

1,500 3,000 4,500

0

-6

1,500 3,000 4,500

Fzcr (0) = − 16.894 kN

p curv = 15 m

( F rect (0) = − 227.954 kN)

( p rect = 14.7 m)

z

0

8

-1 -2

β0 = 0.09348

238.401

-100

-300 0 0

1,500 3,000 4,500

1,500 3,000 4,500

Modelling emergency situationss in drilling deep bo boreholes

492

Z X

Y O

a

b

c

Fig. 7-38 Buckling wavelet shift depending on friction intensity in shallow bore-hole: μ =0 (a); μ = 0.3 (b); μ = 0.4 (c).

Therefore, the systems turn out to be internally singularly perturbed, and the buckling modes take the form of a harmonic wavelet with pitch p and width P (Fig.7-37 for task 6 in Table 7-13), which is shifted to the top edges of the drill string (the right column of Table 7-13). Fig.7-38 schematically illustrates the variations in the modes of stability loss as a function of the frictional effect. So, if the force of friction is negligible (for example, when the DS is fixed, and moving fluid flows eliminate frictional effects), the shape of buckling is a boundary effect (Fig.7-38). However, when the DS is tripping in, frictional forces are formed, and bifurcational wavelets are shifted into the interior of the well (Fig.7-38, b). If the friction interaction becomes more intense, the wavelet shifts higher (Fig.7-38, c). However, maximum values of Fz fr (s) are implemented at the midpoints of the bifurcation wavelets, and their values do not change significantly. From the obtained results, it can be concluded that if the well has an uphill trajectory, and the angle of its inclination in the lower zone exceeds the friction angle, the DS buckling takes place in the form of a boundary effect similar to the case in a rectilinear well with the same angle of inclination. In this case, bifurcation analysis

Chapter 7. Critical states and buckling of drill strings in channels…

493

can be performed using the DS theory in a rectilinear directional well. However, the buckling effect is localised inside the DS when the angle of inclination of the well at its lower portion is less than the angle of friction. To analyse the loss of stability and DS buckling in this situation, the approach proposed to the problem of DS bending in curvilinear channels should be used. References to Chapter 7 1. Analysis of wellbore instability in vertical, directional and horizontal wells using field data / [M.A. Mohiuddin, K. Khan, A. Abdulraheem, A. Al-Majed, M.R. Awall] // Journal of Petroleum Science & Engineering. – 2007. – V. 55. – pp. 83 – 92. 2. Beederman V.L. Mechanics of Thin-walled Structures. – Moscow, Mashinostroyenie. – 1977. – 488 p.(in Russian) 3. Challamel N. Rock destruction effect on the stability of a drilling structure. – Journal of Sound and Vibration. – 2000. – V.233, №2.– P. 235–254. 4. Chang K.W., Howes F.A. Non-linear Singular Perturbation Phenomena. – Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 1984. 5. Cheatham JB Jr. Helical postbuckling configuration of a weightless column under the action of m axial load. – Soc Pet Eng J. – 1984. – 24(4). – P.467– 472. 6. Chen Y., Lin Y., Cheatham J.B. Tubing and casing buckling in horizontal wells (includes associated papers 21257 and 21308). – SPE J Pet Technol. – 1990. – 42(2). – P.140–191 7. Chui C. Introduction to Wavelets. – M .: Mir. – 2001. – 412 p. 8. Critical buckling load assessment of drill strings in different wellbores using the explicit finite element method. / [J.S. Daily, L. Ring M. Hajianmaleki et al.]// SPE offshore Europe oil and gas conference and exhibition, Aberdeen. – 2013.http://dx.doi.org/10.2118/166592-MS 9. Critical buckling of drill strings in curvilinear channels of directed bore-holes. / [V.I. Gulyayev, V.V. Gaidaichuk, E.N.Andrusenko, N.V.Shlyun] // Journal of Petroleum Science and Engineering. – 2015. − Vol. 129, No 1/ − P. 168−177. (USA). 10. Cunha J.C. Buckling of tubulars inside wellbores: a review on recent theoretical and experimental works. – SPE Drilling and Completion. – 2004. – V. 19, № 1. – P. 13 – 19. 11. Dawson R., Paslay P.R. Drill pipe buckling in inclined holes. - J Pet Technol. 1984. - V.36 (10). - P.1734-1738. 12. Elishakoff I., Li Y., Starnes J.H. Non-Classical Problems in the Theory of Elastic Stability. − 2001. − Cambridge: Cambridge University Press.

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13. Feodosiev V.I. Selected Problems and Questions on Strength of Materials. M.: Nauka. – 1967. – 237 p. (in Russian) 14. Foster B. ‘Network Graphics’ improving the performance of drilling wells with horizontal displacement of the face. – Oil and Gas Technologies. – 2005. – No. 3. – P. 19 – 24. (in Russian) 15. Gan L., Wang X., Tan M. Non-linear buckling analysis of tubular in a deviated well with differential quadrature element incremental iterative method. – Chin. J. Comput. Phys. −2009. −V.26. − P.129–134 (in Chinese). 16. Gao G.H., Huang W.J. A review of down-hole tubular string buckling in well engineering. – Petr. Sci.– 2015. –12(3). – P. 443−457. 17. Gao G.H., Miska S.Z. Effects of friction on post–buckling behaviour and axial load transfer in a horizontal well. – SPE J. – 2010a. – 15(4). –P.1104 – 1118. 18. Gulyayev V.I., Andrusenko O.M., Shlyun N.V. Critical states of drill strings in channels of rectilinear directional wells. Mechanics of a Solid Body. – 2016. – No. 1. – P. 174 – 185. (in Russian) 19. Gulyayev V.I., Andrusenko E.N., Shlyun N.V. Theoretical modelling of post – buckling contact interaction of a drill string with inclined bore-hole surface. – Structural Engineering and Mechanics. − 2014. −Vol. 49, No 4. − P. 427−448. 20. Gulyayev V.I., Andrusenko E.N. Theoretical simulation of geometrical imperfections influence on drilling operations at drivage of curvilinear boreholes. - J. of Petroleum Science and Engineering. – 2013. – V.112. – P. 170 – 177. 21. Gulyayev V.I., Bazhenov V.A. Gotsulyak E.A. Stability of Non-linear Mechanical Systems. – Lviv: Vishcha shkola. – Publ. House at Lviv University. – 1982. – 255 p. (in Russian) 22. Gulyayev V.I., Gaydaychuk V.V., Koshkin V.L. Elastic Deformation, Stability and Oscillations of Flexible Curvilinear Rods. – K.: Naukova Dumka, 1992. – 344 p. (in Russian) 23. Gulyayev V.I., Gorbunovich I.V., Glovach L.V. Elements of Surface Theory. – К .: Vidavnitstvo RVV NTU. – 2011. – 239 p. (in Ukrainian) 24. Gulyayev V.I., Shlyun N.V. Influence of friction on buckling of a drill string in the circular channel of a bore hole. - Petroleum Science. - 2016. - V. 13. - P. 698 - 711. 25. Gulyayev V.I., Shlyun N.V. Invariant states of drill strings in channels of curvilinear boreholes. – Proceedings of NTU. – 2013. – No. 28. Part 2. – P. 116 –123. (in Ukrainian) 26. Hajianmaleki M., Daily J.S. Critical-buckling-load assessment of drillstrings in different wellbores by use of the explicit finite-element method. – SPE Drill Complet. – 2014. –V. 29(2). − P. –256–264.

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495

27. He X., Halsey G.W. Interactions between torque and helical buckling in drilling. – SPE annual technical conference and exhibition. – 22–25 October, Dallas. 1995.http://dx.doi.org/10.2118/166592-MS 28. He X., Kyllingstad A. Helical buckling and lock–up conditions for coiled tubing in curved wells. – SPE Drill Complet. – 1995. –V.10(1). –P.10–15. 29. How drillstring rotation affects critical buckling load? / [S. Menand, H Sellami, Akowanou J, et al.] // IADC/SPE drilling conference, 4–6 March, Orlando, 2008.http://dx.doi.org/10.2118/112571-MS. 30. Huang N.C., Pattillo P.D. Helical buckling of a tube in an inclined well bore. – Int. J. Non-Linear Mech. – 2000. – V.35. – P. 911 – 923. 31. Huang W.J., Gao D.L. Helical buckling of a thin rod with connectors constrained in a torus. – Int J Mech Sci. – 2015. –V.98. –P.14–28.456 Pet. Sci. – 2015. –V.12. –P.443–457 32. Kyllingstad A. Buckling of tubular strings in curved wells. – J Pet Sci. Eng. – 1995. –V.12(3). –P.209–218 33. Lubinski A. Developments in Petroleum Engineering, V.1. – Houston, TX, USA: Gulf Publishing Company, 1987. – 438 p. 34. McCann R.C., Surayanarayana P.V.R. Experimental study of curvature and frictional effects on buckling. – OTC 7568. Presented at the 26th Annual Offshore Technology Conference, 2-5 May, Houston, TX 35. Mitchell R.F. Comprehensive analysis of buckling with friction. – SPE Drill Complet. –1996. – V. 11(3). – P.178–184. 36. Mitchell R.F. Simple frictional analysis of helical buckling of tubing. – SPE Drilling Engineering. – 1986. – V.1, No. 6. – P. 457 – 465. 37. Mitchell R.F., Samuel R. How good is the torque / drag model? – SPE Drilling & Completion. – 2009. – V. 24 (1). – P. 62–71. 38. Mitchell R.F. Tubing buckling − The state of art. – SPE Drilling & Completion. – 2008. – 23 (4) . – P.361−370. 39. Musa N., Gulyayev V., Shlyun N., Aldabas H. Сritical buckling of drill strings in cylindrical cavities of inclined bore-holes. – Journal of Mechanics Engineering and Automation. – 2016. V. 6. – P. 25 – 38. USA 40. Newland D. E. Wavelet analysis of vibration – part I: Theory. – Department of Engineering University of Cambridge, Cambridge CB2 1PZ, UK. – P. 1–23. 41. O’Donnell M.A. Boundary and corner layer behavior in singularly perturbed semilinear systems of boundary value problems. – SIAM J. Math. Anal. – 1984. – 15.- p. 317-332. 42. Paslay P.R., Bogy D.B. The stability of a circular rod laterally constrained to be in contact with an inclined circular cylinder. – J Appl Mech. –1964. – V. 31(4). – P.605–610.

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43. Perelmuter A. V., Slivker V. I. Stability of Structural Equilibrium and Related Problems. – M.: Publishing House SKAD SOFT, 2010. – T.1. – 704 p. 44. Pogorelov A.V. Differential Geometry. – M.: Nauka, 1974.–180 p. 45. Popov E.P. Non-Linear Problems of Statics of Thin Rods. – Moscow: OGIZ, 1948. – 178 p. (in Russian) 46. Samuel R., Gao D. Horizontal Drilling Engineering. Sigma Quadrant Publisher.‒2013.‒550 p. 47. Shlyun N.V. Computer modelling of dragging of the drill strings in channels of curvilinear bore-holes. – Proceedings of NTU. – 2015. – No. 1 (31). – P. 574 580. 48. Svetlitsky V.A. Mechanics of Rods. – M.: High School, 1987. – Part 2: Dynamics. – 304 p. (in Russian) 49. The buckling of elongated rotating drill strings / [V.I. Gulyayev, V.V. Gaidaichuk, I.L. Solovjov, I.V. Gorbunovich] // J. of Petroleum Science and Engineering. – 2009. – V.67. – P. 140 – 148. 50. Timoshenko S.P., Gere J.M. Theory of Elastic Stability. 2nd ed. – New York: Tata McGraw-Hill Education. – 1963. 51. Tsigler G. Fundamentals of the Theory of Structural Stability. – M., Mir, 1971. – 192 p. (in Russian) 52. Vasilyeva A.B., Butuzov V.F. Singularly Perturbed Equations in Critical Cases. – M.: Ed. Moscow State University. - 1978. – 286 p.(in Russian) 53. Walker J.S. A Primer on Wavelets and their Scientific Applications. – University of Wisconsin, Eau Claire, USA. – 2007.– 320 p. 54. Zhao Y.B., Wei G.W., Xiang Y. Discrete singular convolution for the prediction of high frequency vibration of plates. – Int. J. Solids Struct. – 2002. – V.39. – P. 65–88.

Index

497

INDEX

A Aadnoy, B. S. 293, 294, 381 Aarrestad, T.V. 104, 110, 214 Abbott, C. 383 Abdel Magid, Y.L. 290 Abdullayev, F.Y. 64, 105 Abdulraheem, A. 293, 380, 493 Aitken A.C. 397 Akbulatov, T. O. 383 Akgun, F. 294, 380 Aklestad, D.L. 381 Akowanou, J 495 Aldabas, H. xiii, xiv, 495 Alexandrov, M. M. 380 Alfutov, N. A. 63 Al-Majed, A. 293, 380, 493 Andersen, K. 293, 381 Andronov, A.A. 8,107, 111, 112, 115, 212 Andrusenko, E.N. xii, xiii, xiv, 381, 382, 493, 494 Ashley, H. 104 Astarita, J. 212 Awall, M. R. 293, 380 B Babakov, I.M. Bailey, J.J. Balthazar, P. Barakat, Elie R. Basarygin, Y. M. Bassov, I.A. Bazhenov, V.A. Beederman, V. L. Behr, S.M. Berg, P.C. Besaisow, A.A. Blackwood, K. Bogy, D.B. Borisov, A.V. Borshch, E.I. Borshch, O.I. Brett, F.J. Brooks, A.G. Bullatov, A. I.

104 110, 212, 213 18 63, 104 380 380 494 63, 104, 493 289 293, 381 110, 212 289 404, 495 289 104, 105, 289 xiii 105, 110, 212, 289 104 380

Burgess, T. Butenin, N.V. Butuzov, V.F.

293, 382 212 65, 496

C Cauchy, A.L.

21, 22, 24, 26, 29, 31, 41, 53,60, 93,396, 404, 407, 426 Challamel, N. 110, 212, 493 Chang K.W 10, 63, 104, 212, 474, 493 Chao, S. 403 Cheatham, J.B. 493 Chebyshev 397 Chen, David C.-K. 104 Chen, S.L. 289 Chen, Y. 404, 493 Chenevez, E. 110, 212 Chia, C.R. 381 Cho, J. H. 63, 382 Christoffel 419, 420, 443, 480 Christoforou, A.P. 104, 110, 212, 290 Chui, C. 63, 493 Civalek 405 Clay, P.L. 382 Coriolis, G.-G. 4, 14, 16, 67 Coulomb 418, 479 Cox, R.J. 381 Crumrine, M.J. 10, 64, 105, 214 Cunha J.C. 385, 403, 406, 438 457, 493 D Dahl, T. Daily, J.S. d'Alembert, J.R. Dalton, C. Dankers, S. Darboux Dareing, D.W. Daring, D. W. Dashevskiy, D. Dawson, R. Demarchos, A. S. Den-Hartog, J. P. Descant, F.J. Dinnik, A. N. Doppler, Ch.

104 493, 494 13, 234 383 104 413, 419, 442, 460, 477 110, 212 402 104 110, 213, 404,493 294, 381 104 293, 381 402 88

Modelling emergency situations in drilling deep boreholes E Economides, M.J. 381 Edmondson, J. 383 Edwards, S. 403 Elishakoff, I. 104, 474, 493 Erokhin, V.P. 381 Ertekin, T. 383 Euler, L. 33, 279, 400, 401, 439, 481 Everhart 22, 41, 60, 93 F Fadeyev, E. A. Feodosyev, V.I. Filippov, A.P. Finnie, I. Fisk, J. H. Foster B. Fourier J.-B.J. Friedrichs, K. Fufayev, N. A.

381 63, 104, 495 63 110, 212, 213 293, 381 494 38, 39 19 212, 290

G Gaidaichuk, V. V. 213, 381,493 Galerkin 405 Gan L. 405, 495 Gao, D.L. 382, 405, 406, 495, 496 Gao, G.H. 495 Gaydaychuk, V.V. xii, xiii, 64, 105, 495 Gere J.M. 496 Gewinnyan, G.M. 214 Gilchrist, J.M. 382 Glazunov, S.N. xii, xiii, xiv,105, 213, 289 Glovach, L.V. xiii, 64, 382, 494 Glowach, L.V. 382 Glushakova, O.V. xiii, xiv, 213, 289 Godunov, S.K. 7, 20, 27, 32, 51, 91 Golovin, A.A. 65, 105 Gorbunovich, I.V. xiii, 64, 382, 496 Gossuin, L. 110, 212 Gotsulyak, E.A. 495 Gradshteyn, I.S. 19 Gudkov, V. V. 64 Gullizade, M. P. 381 Gulyaev, V. I. 213, 289 Gulyayev, V.I. xii, xiii, xiv, 104, 64, 105,214, 295, 381, 382, 493-495 Gutenmacher, L.I. 19 H Haar Hajianmaleki, M. Hakimi

37 493, 496 405

498

Halsey, G.W. Hans, C. Haviland, G. Hayashi, T. He, X. Hopf, E. Houtchens, B. Howes F.A. Huang, N.C. Huang,W.J. Hudoliy, S.N. Hudoly, S.N.

110, 214, 495 10, 63, 105 104 214 495 8 383 10, 63, 104, 212, 473, 493 64, 404, 406, 494 494, 495 213 xiii, 382

I Ivanov, V. K. Iyoho, A.W.

397 10, 64, 105, 111, 214

J Jacobi Jansen, J.D. Jeong, Y. T. Jerome, J.S. Johnstone, J. Jonggeun, C. Juvkam-Wold, H. C.

397 105, 110, 214, 217 382 10, 105 383 10, 105, 293, 381 10, 63, 293,404

K Kallinin, A. G. Kantorovich Kaplunov, J. Karki, H. Karkoub, M. Kerimov, E.G. Kerr, R.A. Keultjes, W.J.G. Khaikin, S.E. Khan, K. Kharkevich, A.A. Khaykin, S.E. Khudoley, S.N. Khudolii, S.N. Kilin, A. A. Kirchhoff Klokov, Yu. A. Kononenko, V.O. Koshkin ,V.L. Kovalyshen, Y. Kulchinsky, V. V. Kutta, M.W. Kwon Kyllingstad, A.

382 396, 399, 424 104 290 290 105 10, 64 290 212 293, 380, 493 214 19 xiv, 64 213, 289, 382 289 417 64 108, 214 64, 381, 495 218, 290 382 94, 426, 479 403 104, 110, 214, 495

Index

499

L Lagrange, J. 221,387 Lamine, E. 289 Landa, P.S. 214 Larsen, K. 293, 381 Lavrentyev, M. M. 397 Leine, R.I. 110, 214, 218, 290 Lepin, V. D. 64 Levinson, N. 19 Levitan, M.M. 382 Li, Y. 493 Lin, Y.Q. 110, 213, 214, 493 Lindberg, R.E. 290 Liouville, J. 6, 7, 9, 20, 21, 22, 53, 91, 93, 395, 464, 474 Liu 405 Livesay, B.J. 110, 212, 402 Lofts, J.C. 104 Longman, R.W. 290 Lubinski, A. 65, 402, 403, 495 Lugovoi, P.Z. 213 Lukasiewicz, S. 403 Lukyanov, E. E. 382 Lupick, G.S. 381 Lurie, A.I. 64 Lyapunov, A.M. 17 Lysne, D. 110, 214 Lyusternik, L.A. 19 M Madell, G. Majeed, F. Abdul Mamaev, I.S. Markeev, A. P. Marrucci, J. Marsden, G. Martinez Matsutsuyu, B. Maugeri, L. McCann, R.C. McColpin, G. McCracken, M. McDaniel, B. W. McDermott, J.R. Meize, R.A. Menand, S. Michel, B. Mihajlović, N. Millheim, K.K. Minett-Smith, D. Mirzadzhanzade, A.H. Mirzoyan, A.A.

383 290 289 290 212 214 405 403 11, 64 405, 495 293 214 382 381 10, 64, 105, 214 495 294, 381 290 10, 64, 105, 214 290 214 214

Mishchenko, E.F. Miska, S.Z. Mitchell, R.F. Mohiuddin, M. A. Molchanov, A. A. Moradi Musa, N. W. Myshkis, A.D. Myslyuk, M.A. N Naumov, V. I. Newland, D. E. Newton, I. Neymark, Yu. I. Nicholson, J.W. Nijmeijer, H. Nodle, E. O O’Donnell, M.A. Ostrogradskiy, M.V.

215 63,104, 405, 494 403,404, 405, 406, 495 293, 380, 493 382 405 xiii, xiv, 290,495 214 11, 64, 105 381 65, 495 396, 399 212, 290 215 290 104 19, 65, 495 221

P Pagett, J. Pascal, M. Paslay, P.R. Pattillo, P.D. Payne, M.L. Perelmuter, A. V. Phillips, W.J. Pogorelov, A.V. Poincare, H. Ponomarev, A. Ya. Popov, A. N. Popov, E.P. Porcu, M.M. Pourcian, R. D. Prandtl, L. Proselkov, Yu. M

290 290 404, 494, 495 64, 404, 495 212 496 381 382, 496 8, 17 64 383 496 381 293, 381 18 380

R Radchenko, V.N. Radzimovsky, E. L. Rapin, V. A. Rayleigh Reiley, R. H. Ribchich, I.I. Ring ,L. Ritz Rosielle, P.C.J.N. Rothmann, M.

65 402 382 404 293, 382 64 493 404 290 402

Modelling emergency situations in drilling deep boreholes Rozov, N.H. Rubanik, V.P. Runge, K. Ryabikhina, S.M. Rybchich, I.I.

214 214 94, 426, 479 65, 105 11, 105

S Sairdza, M.K. Samuel, R. Sandstrom, B. Sawaryn, S. J. Schamp, J.H. Schubert, J.J. Scott LaPierre Sellami, E. Sellami, H Semak, G.G. Shan, S. N. Shchavelev, N.L. Shepard, S. F. Sheppard, M. C. Shevchuk, L.V. Shlyun N.V.

215 290, 382, 406, 495, 496 293 293, 294, 382 381 63, 293, 381 104 110, 212 495 383 382 381 294, 382 293, 294, 382 xiii, xiv, 289, 290 xii, xiii, xiv, 64,406, 94, 493,495,496 Shyu, R.-J. 215 Skrypnik, S.G. 65, 105 Slivker, V. I. 496 Smith, M. 104 Soloviev, I. L. 64 Solovjov, I.L. xiii, 496 Spanos, P.D. 110, 213 Spivak, A. I. 383 Starnes, J.H. 493 Stroud, D. 219, 290 Sturm 395, 464, 474 Suchkov, B.M. 382 Sulakshin, S. S. 382 Surayanarayana P.V.R. 405, 495 Surjaatmadja, J. B. 382 Svetlitsky ,V.A. 496 T Takach, N. Tan, M. Tan, X. C. Teodorchik, K.F. Tikhonov, A. N. Timoshenko, S.P. Tolbatov, E.Yu.

63, 104 494 403 214 19, 397 496 xiii, 64, 105

Tsigler, G. Tucker, R.W. V Van Campen, D.H. Van de Wouw, N. Van den Steen, L. Van Der Pol. Vandiver, K.J. Vashchilina, E.V. Vasilyeva, A. B. Vazov, V. Vicente, R. Viktorin, R.A. Vinogradov, O. V. Vishik, M.I. Vitt, A.A. W Walker, J.S. Waltman, R. B. Wang, C. Wang, X. Wang, Y.H. Warren, T.M. Wei, G.W. Wick, C. Willers, F.. Willett, R. M. Willson, S. Woods, G. Wu, J. X Xiang, Y. Y Yaremiychuk, R.S. Yigit, A.S. Young, W. Yuan, z Z Zhao, Y.B. Zigler, G. Zurcher, D. Zinchenko, V.P.

500 105, 496 110, 111, 215 110, 214, 290 290 110, 214 107, 108, 112, 113, 215 215 xii, 104, 105, 289 19, 65, 496 19 383 381 383 19 112, 212 65, 290, 496 293, 381 215 405, 494 110, 214 289, 293, 383 405, 496 293, 382 402 382 403 65 404 405, 496 11, 64, 105, 383 104, 212 ,290 403 405 496 65 104 383