SPE RU. Copyright 2014, Society of Petroleum Engineers

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1 SPE RU Development of General-Purpose Software for String Dynamics Simulation V.S. Tikhonov, Weatherford, Moscow, Russia; A.I. Safronov, Krasnaya Zvezda OJSC, Moscow, Russia; Kh.R. Valiullin, SPE, Weatherford, Houston, USA; O.S. Bukashkina, Weatherford, Saint Petersburg, Russia Copyright 2014, Society of Petroleum Engineers This paper was prepared for presentation at the SPE Russian Technical Oil and Gas Exploration and Production Conference and Exhibition held on October, 2014, in Moscow, Russia. This paper was selected for presentation by an SPE Program Committee following review of information contained in an abstract submitted by the authors. Contents of this paper, as presented, have not been reviewed by the Society of Petroleum Engineers and are subject to correction by the authors. The material, as presented, does not necessarily reflect any position of the Society of Petroleum Engineers, its officers, or members. Electronic reproduction, distribution, or storage of any part of this paper for commercial purposes without the written consent of SPE is prohibited. Permission to reproduce in print is restricted to an abstract of not more than 300 words; illustrations may not be copied. The abstract must contain conspicuous acknowledgement of SPE copyright. Abstract One of the most topical problems of drilling mechanics is development of a software program for 3D string dynamic simulation in a 3D well on various operation regimes with maximum account of all external factors. The paper describes the results of long-term development of general-purpose DYNTUB software designed for string dynamic simulation. The software calculates: String torque and drag; Drillstring lateral and torsional vibrations; Drillstring buckling and whirling; Well path while drilling in sliding regime and in rotary regime with Rotary Steerable Systems (RSS) or systems controlled by downhole motor angular velocity modulation; Standoffs between the casing and the wellbore wall with various centralizers. The software is based on the stiff string dynamic model. It accounts for deformation of the wellbore walls contacted by the string, initial drill pipe bending, PDC bit interaction with the bottomhole under downhole and pore pressure, rock anisotropy, refined multicomponent string friction model, etc. The software proved applicability of the soft drill string model to calculate torque and drag, enabled study of multicomponent friction effects on string buckling, well path calculation while RSS drilling, and solving other complex problems of drilling mechanics. Software efficiency is demonstrated based on case studies; the paper also includes results of comparative analysis using data from other references. The software is designed for expert analysis of drilling, casing, coil tubing and similar object parameters while well drilling and completion. 1. Introduction The growing complexity of BHA and well designs necessitates development of an advanced engineering tool for mathematical simulation of various downhole operation regimes. The advantages of mathematical models for string dynamic analysis at the respective operation design stage are obvious. They enable to play emergency scenarios and select the optimal configuration ensuring accident-free operation under extreme static and dynamic loads. The idea of development of a general-purpose model and software to analyze various drilling operations appeared about 10 years ago (Tikhonov, et al, 2006). Originally the developed 3D drill string dynamic model in a 3D well was supposed to be used for drill string buckling studies only. However, while model development it became evident that is had a rather general nature. Changing boundary conditions it is possible to model the drill string behavior virtually for any downhole operation. This was how the first DYNTUB software version came to life. Two years later the model and a new software version were developed for string dynamic stability: snaking and whirling (Tikhonov & Safronov, 2008). Effects of bulges on the pipe body, i.e. tool joints, stabilizers, stiff centralizers on BHA buckling and whirling were studied in DYNTUB software and described by Tikhonov & Safronov (2011). The model was developed to calculate the well path while drilling in sliding and rotary regimes with RSS (Tikhonov, et al, 2012a) and systems controlled by amplitude modulation of downhole motor angular velocity. For a long time experts in drilling mechanics were concerned with the issue of accuracy of the soft model of a string with various bulges not accounting for string bending stiffness. DYNTUB software based on the stiff pipe model proved the classical soft model adequacy for drag force and torque calculations (Tikhonov, et al, 2013).

2 2 SPE RU The software was successfully used to simulate casing with bow-spring centralizers while running in hole with account of hole diameter variations with depth (Tikhonov, et al, 2014a). The mathematical model including boundary conditions characteristic of one or another string operation regime in the well and the computational method for solving string motion equations are described in Section 2. Further sections give examples of various string operation regimes studies in the DYNTUB software, comparative analysis of results calculated using this software with the results described in other papers, as well as analyze new results. 2. Mathematical Model String mathematical model used in DYNTUB software is described in detail by Tikhonov, et al (2006). Only unique features of the model will be discussed herein Equations of Motion The principal idea underlying the developed string model is the wellbore wall stiffness. It would seem at first sight that such insignificant factor can t be critical for model formulation. However, this allows interaction of the contact force with the string axis lateral offset relative to the wellbore axis. Thus, complicated and not always having unambiguous solution problem of defining contact forces (Fedorenko, et. al 2000) acting on the string in the well with absolutely rigid walls was reduced to formal solving of equations of motion in relation to deformation. The developed equations use a quadratic-elastic model of pipe wall lateral deformation (Tikhonov, et al, 2006) that enables improvement of numeric scheme stability. The proposed approach considerably simplified the problem of defining contact zones and contact forces not only within smooth string sections but also at points of local bulges, tool joints, stabilizers, centralizers, etc. As for the rest, the approach to derivation of equations of motion is not much different from the conventional one. Partial differential equations by length and time describing the dynamics of lateral (in two orthogonal directions), torsional and axial motion were presented as projections on the coordinate axes aligned with a 3D well. The string was considered to be elastic for bending, torsion and tension. Added fluid (mud) weight was considered as the inertia terms of equations of lateral motion, and the string buoyant weight as external forces. To describe the friction force and torque acting on the string at the point of contact with the wellbore wall, a new model was developed accounting for the effect of transition from sliding friction to rolling friction common to rotary drilling. In order to develop this model, a special experiment was carried out with real drill pipe specimens that allowed localizing and describing the transition zone and identified the effect not described in any investigations (Tikhonov, et al, 2012b) Boundary Conditions Boundary conditions characterize specific features of BHA design and string operation modes. During analysis of buckling at the string top end, a longitudinal motion or compressive axial stress is given, and at the bottom end an equation correlating the weight on bit with the well bottom elastic deformation (see subsection 2.1). Other conditions characterizing string ends attachment are defined by specific features of the given mode. However, these conditions have effect only near end points and do not have any effect on buckling development in the string middle part. To study string buckling during rotation, the same parameters of longitudinal and lateral forms are given as for buckling analysis. The string angular velocity is defined at the top, and the weight on bit at the bottom. Boundary conditions for well path calculation are more complex. In case of well curving with the help of a bent sub without drill string rotation at the string axis kink point, the conditions of bending moment and axial force to the left and right of the kink point are specified, as well as the conditions which define the string kink angle. For RSS drilling y g and z g coordinates of shaft offset in the system non-rotating unit (Fig. 1) are specified, as well as the equation of lateral forces acting on the unit and conditions of matching the lateral shaft and unit offset at the inlet and outlet of the latter. a) b) Figure 1. Rotary Steerable System (a) and the scheme of shaft offset while well path steering (b)

3 SPE RU 3 In case of well curving with the help of a bent sub and drill string rotation, the law of amplitude modulation of downhole motor angular velocity is specified at the set phase as well as kinematic equations defining bit coordinates while its rotation in the wellbore cross section (the hole diameter drilled using this drilling method will be larger than the bit diameter). In all options of well path calculations, boundary conditions include equations of cutting and friction force and torque acting at the cutting edges. The cutting model is based on the equations developed by Detourney & Defourny (1992) to consider forces acting on the cutter vs the cutting depth, specific intrinsic energy of the rock, width of blunt edges of cutters and friction factor. The model also includes downhole and pore pressure effects on the specific intrinsic energy (Detourney & Tan, 2002; Detourney & Atkinson, 1991). The developed curving model accounts for the lateral bit milling capacity and anisotropy of inclined stratified formations, which affects well path vector deviation from the specified direction Numerical Method Numerical method of solution for the equations of string motion together with boundary conditions is described in more detail by Tikhonov, et al (2006). It is based on the method of lines. At first, four partial differential equations, each of the 2 nd order by time, describing lateral (two degrees of freedom), torsional and axial (one degree of freedom) string motion modes, are reduced to eight first order equations. Then the derivatives of the string length are approximated by finite differences that allows for reducing the partial differential equation system to the finite dimensional system of ordinary differential equations. The latter is supplemented by conditions at external and internal boundary points (see subsection 2.2). The system of ordinary differential equations is integrated by time using one of conventional methods. The Runge-Kutta's explicit integration scheme was selected for the purposes hereof. Implicit schemes (Newmark's, θ-wilson's methods, etc.) allow integration with a larger time increment than explicit schemes. However, this requires solution of the high order nonlinear algebraic equation set at each time step. The computational experiment demonstrated that the numerical solution resulting from implicit schemes does not always converge. This is especially indicative of jump of system parameters, for example, in case of buckling. In this case the integration step has to be reduced, which brings advantages of implicit schemes to nought. At the same time, numerical solution of explicit scheme is much simpler than that of implicit one. Tikhonov, et al (2012a) found the required condition for selection of the integration step by time to ensure numerical convergence of the explicit scheme depending on the minimum increment of string subdividing by length and the maximum stiffness. In case of subdividing 4,000 m string with a mesh step equal at 2-3 m, the maximum integration step by time is sec. The calculating time of 100 sec transient process for such string on a personal computer with 2 GHz processor ranges from several dozens of minutes to 2-3 hours depending on the operation regime. While analyzing 100-m BHA transient processes, which define behavior of the most critical string components, the calculating time is reduced to several minutes. 3. Buckling study String buckling occurs at rather high axial compression load and torque. Buckling is common for drilling of straight hole section, running of coiled tubing, and setting of packers in the wellbore. There are two kinds of buckling: sinusoidal buckling (sinusoid lying on the low wall of the hole), and helical buckling (string helically curved inside the hole). The latter is especially hazardous because it features jump increase of contact force preventing string motion up to lock-up (Wu & Juvkum- Wold, 1995). Quite many studies, both theoretical and experimental, were devoted to buckling (Hajianmaleki, et al, 2012). However, there is still no well defined understanding of the process. To a large extent this is related to inconsistency of testing results and misjudgment of such important factors as string length, friction, initial pipe bending, etc. DYNTUB software was used to analyze the results of laboratory experiments performed by McCann & Suryanarayna (1994) and differing from other experiments by high precision. String loading was studied as defining the dependence of the weight on bit on the specified displacement of the string top end. Figure 2 demonstrates loading of the rod, which simulates the string in the horizontal pipe-well obtained from McCann & Suryanarayna (1994) tests and with the help of DYNTUB software. The curve differences do not exceed 15%. Load characteristics defined by DYNTUB software in Figs. 3-5 demonstrate the effects of rod length, friction factor and its initial bending (Tikhonov & Bukashkina, 2013). The first basic peak of the curves defines the sinusoidal buckling load. The following minor peaks correspond to forming of the 1 st (B1, C1), 2 nd (B2, C2), etc. turns of the helical buckling. According to the curves, common formulas of buckling loads (Dawson & Paslay) may lead to erroneous results, and experimental results may be misinterpreted. The obtained results have been recently confirmed by independent theoretical investigations and laboratory studies of Gao & Miska (2009, 2010).

4 4 SPE RU Figure 2. Experimental (black) and calculated (red) rod load characteristics at buckling Figure 3. Effect of string length on load characteristic Figure 4. Effect of friction factor on load characteristic Figure 5. Effect of Initial rod bending on load characteristic 3. Studies of Lateral String Vibrations Dynamics of the drillstring while rotation and associated well cleanup problem are the most important factors while extended reach drilling. At rather low rotation speeds, drillstring sections lie on the low side of the bore. However, when the first threshold is reached, the string starts snaking. Upon further increase of the rotation speed, vibration becomes chaotic being accompanied with shocks and bounces of the pipe elements against the wellbore wall. Finally, reaching the second threshold, string vibrations are starting to feature eccentric motion of its axis around the well axis in the direction opposite to the string rotation around own axis (backward whirling). This mode may lead to serious damages of the string: accelerated wear of thread connections, premature damage of bit blades, MWD failure due to impact loads at the wellbore wall, well deviation. One of the best software for analysis of string dynamics in hole has been developed by Heisig & Neubert (2000) based on FEM. But its major disadvantage is long computing time necessary for transient analysis. So, to calculate lateral BHA vibrations with the mesh divided by length into 155 elements in 75 seconds of real time, this software program needs 75 hours of running time. With DYNTUB, this time is reduced by almost two orders (Tikhonov & Safronov, 2011). Figure 6 shows the scheme and parameters of BHA taken from Heisig & Neubert (2000) and used in present paper for demonstration of string lateral vibration calculations made with DYNTUB software. The results of calculation of contact force vibrations and angle of string axis rotation in relation to the well axis at 100 kn WOB and 6,500 kn m TOB are presented in Fig. 7. String rotation started from the 50 th second: the angular velocity increased by the damped exponential with the time constant of 10 sec. As seen, chaotic vibration with string bouncing from the wellbore wall and eccentric motion depends on the string rotation speed and well inclination. String snaking up to 90 is the most characteristic motion in the deviated well in normal conditions. Contact force distribution by the drill pipe joint length upon completion of the transient analysis (100 seconds) is shown in Fig. 8.

5 SPE RU 5 Section Outer Dia., mm Inner Dia., mm Note PDC Bit Bit is modeled as a point Near-Bit Section st stabilizer, D = mm Downhole Motor nd stabilizer, D = mm FE-Sub th stabilizer, D = мм MWD Sub Drill Pipe Box, pin, wear knots, D = 172 мм Figure 6. Bottom Hole Assembly Figure 7. Effect of well inclination (α), protector setting interval (h) and friction factor (k) on the of BHA midpoint contract force and rotation angle m m m m Figure 8. Contact force distribution by drill pipe length at various well inclinations, protector setting intervals and friction factors upon completion of calculation process (see the nomenclature in Fig. 7)

6 6 SPE RU 4. Torque and Drag Analysis Drillstring torque and drag analysis of drilling and tripping operations is a key drillstring design tool for a drilling engineers. The majority of known software designed for this purpose is based on a static soft string model first introduced by Johancsik, et al (1984). This model is assuming zero bending stiffness of the string, what inevitably leads to continues string contact with wellbore wall, including spans between string bulge elements (tool joints, centralizers, stabilizers, etc.) throughout its length. It should be recognized that notwithstanding such strict restriction, the soft model allows to obtain torque and drag predictions that are quite acceptable for majority of practical applications. Although, soft model predictions accuracy becomes questionable for the high dogleg wells, cases with tight string clearance radii, and practically useless for the analysis of local contact forces and bending moments in the area of string bulge elements and in sections with high DLS. Analysis of string stiffness effects on torque and drag as well as distribution of contact forces was performed by Tikhonov, et al (2013). Equilibrium (quasi-static) position of drillstring within well bore has been defined as a result of a steady process achievement by means of solution of equations with rather arbitrary initial conditions. For string lengths about 4,000 m and mesh lengths approximately equal 2 m the required computing time was not exceeding several minutes. One of the particular field cases analyses was a short radius build section well, with dog leg severities up to 50 deg/30 m (Fig. 9). The drillstring design is shown in Fig. 10. Figure 11 shows distribution of contact force along the well length. Figure 12 represents a fragment of this distribution in a high DLS section while drilling with drillstring rotation. Three models were compared: soft, rigid" without bulge elements (tool joints) and stiff model with tool joints. As seen, stiffness of the string without tool joints has insignificant effect on the contact force value. However, in case of tool joints, the contact force is increasing sharply. Thus, bending moment in the string sharply increases around the contact points. Nevertheless, notwithstanding significant changes of local contact force distribution, integral distribution of torque and drag along the well length obtained with the help of soft and stiff models appear to be closely approximated (Figs. 13, 14). Thus, soft model is quite suitable for string torque and drag analysis, however, a more detailed string design based on the stiff model is required while contact forces and bending moment analysis, which is critical for fatigue strength estimation. Drill string assembly 4¾ inch bit 3¾ inch BHA, m 2⅞ inch drill pipes to surface BHA 1 - PDC bit, 2 - Drilling motor, 3 - X-over sub, 4 - UBHO sub, 5,6 Flex NMDC, 7 - X-over sub Figure 9. Well profile, m Figure 10. Drillstring assembly Figure 11. Contact force distribution by MD for T&D analysis using different models

7 SPE RU 7 Figure 12. Contact force distribution by MD in high DLS section (Fragment of Fig. 11) Figure 13. Drag force vs MD Figure 14. Torque vs MD 5. Flexible Centralizers Calculation One of the key requirements to centralizers is ensuring minimum necessary standoff between the wellbore wall and casing for uniform annulus cementing. Rigid centralizers can be considered as common bulge elements. Standoff calculation in this case can be performed with DYNTUB software module described in Section 4. Flexible centralizers are more challenging. The majority of developed up to date spring centralizer design software is based on the paper of Juvkum-Wold and Wu (1992). However, this paper does not consider the possibility of string axis turn at the point of spring centralizer setting. Gorokhova et al (2013) provide casing string calculation in the well with significant size variations of hole diameter while considering the above effect. However, the authors failed to find any papers where all types of actual string contact is clearly defined and considered: Absence of contact; Contact along the whole centralizer perimeter upon offset of its axis towards the well axis when the centralizer diameter is greater than the hole diameter; Contact along a part of the centralizer perimeter when the centralizer diameter is greater than the hole diameter; Contact along a part of the centralizer perimeter when the centralizer diameter is less than the hole diameter. All the above specific features of spring centralizer contact were taken into account while development of DYNTUB software (Tikhonov, et al, 2014a). The results of sample calculations taken from Gorokhova et al (2013) for two 168 mm and 178 mm casing strings with bow-spring centralizers in the well with 60 inclination and variable diameter of hole diameter (Fig. 15) are shown in Fig. 16. Here the standoff ratio is defined as the ratio of the standoff between the wellbore wall and the string when their axes match to the minimum standoff in case of axes displacement.

8 8 SPE RU Figure 15. Hole diameter vs MD Figure 16. Standoff ratio vs MD at different casing diameters 6. Well path calculations DYNTUB software was adopted for well path calculations (Tikhonov, et al, 2012a): prediction of directional tendencies, inclination and directional angle rate dependencies while drilling. Considering real rates of penetration, which are considerably low as compared to string lateral vibrations, the problem was solved in quasi-static statement at each given increment of trajectory formation. The following directional tools were considered: mud motor with bent sub for slide drilling, RSS and mud motor with bent sub for conventional rotary drilling at amplitude modulation of downhole motor angular velocity (Farley, et al, 2011). The model includes bit aggressiveness effects, or lateral milling capacity, and anisotropy of formation inclined beddings (Griguletsky and Lukyanov, 1990). Figure 16 presents modeling results of well trajectory formed by simulation of directional drilling with RSS, with and without consideration of bit lateral milling capacity, for the assembly shown in Fig. 10 (bent angle settled to 1.5 ) (Tikhonov, et al, 2012a). Figure 17 shows well trajectories predicted for slide drilling with bent sub and rotary drilling at 60 rpm, and with contemporary downhole motor amplitude modulation of rotation speed equal to 300±200 rpm. Figure 18 demonstrates effect of drilling tendency alternation from its initial vertical direction while drilling through boundary of inclined formation bedding having different anisotropy indices (Tikhonov, et al, 2012a). Figure 16. Profiles of RSS drilled wells with and without account for lateral bit milling capacity Figure 17. Profiles of wells drilled with bent subs with and without drillstring rotation with various lateral bit milling capacity coefficients (k f)

9 SPE RU 9 Figure 18. Rock anisotropy effect on the vertical well deviation from the specified drilling orientation 7. Conclusion 1) The proposed dynamic string 3D model enables simulation of virtually any well operation related to string mechanics analysis. 2) DYNTUB software developed based on the proposed model allows simulation of buckling and lateral vibrations of string, carrying out T&D analysis and transient processes of any arbitrary string assembly taking into account tool joints, stabilizers, centralizers, etc., calculating the well path with various steering tools. 3) At present, DYNTUB software is used as an expert program allowing simulation and analysis of complicated string operation modes. However, it can become a general use program subject to further development of computer engineering enabling run time minimization. 4) Buckling calculation results were verified based on the experimental lab results comparisons, torque and drag analysis prediction validity on the over hand has been confirmed on several case histories, direct field testing. The future major goal is to continue further validations of the rest of the rig operation modeling features available within DUNTUB. 5) Further DYNTUB improvements are associated with extension and enhancement of its functionality. Further development efforts are to be focused in modeling of coupled torsional, longitudinal and lateral modes of string oscillations including stick-slip vibrations (Tikhonov, et al, 2014b), string torque and drag in noncircular, key-seat wellbore, dynamic/time domain analysis of tripping operations, etc. References Dawson, R. and Paslay, P.R Drillpipe buckling in inclined holes. Journal of Petroleum Technology, October: Detourney, E., and Atkinson, C Influence of pore pressure on the drilling response of PDC bits. Rock Mechanics as a Multidisciplinary Science, Ed. Roegiers, Balkema, Rotterdam: Detournay, E., and Defourny, P A phenomenological model of the drilling action of drag bits. Int. J. Rock Mech. Min. Sci. & Geomech. Abstr., 29 (1): Detournay, E., and Tan, C.P Dependence of Drilling Specific Energy on Bottom-Hole Pressure in Shales. Paper SPE/ISRM presented at the SPE/ISRM Rock Mechanics Conference, Dallas, TX, October. Farley, S.R., Maranuk, C.A., Hronek, B.J., Jasper, C Controlling Well Path Trajectory with Rate of Penetration Modulation. Paper SPE/IADC presented at the SPE/IADC drilling Conference and Exhibition, Amsterdam, The Netherlands, 1-3 March. Fedorenko, R.P., Tikhonov, V.S., R.P., and Fedorina, O.Yu Numerical study of the buckling of an elastic rod in a tube. Comput. Math. Math. Phys. 40 (3): Gao, G., and Miska, S Effects of Boundary Conditions and Friction on Static Buckling of Pipe in a Horizontal Well. Paper IADC/SPE presented at the 2008 IADC/SPE drilling Conference, Orlando, FL, 4-6 March. Gao, G., and Miska, S Effects of friction on post-buckling behavior and axial load transfer in a horizontal well. SPE Journal December: Gorokhova, L., Parry, A., & Flamant, N Comparing Soft-String and Stiff-String Methods Used to Compute Casing Centralization. Paper SPE/IADC presented at the SPE/IADC Drilling Conference & Exhibition, Amsterdam, The Netherlands, 5-7 March. Griguletsky, V.G., and Lukyanov, V.T Design of bottom-hole assemblies of drillstrings (in Russian), Nedra, Moscow, 302 p. Hajianmaleki, M., Daily, J.S., Ring, L., and Gandikota, R Buckling analysis of drill strings in inclined wellbores using the explicit finite element method. Paper IMECE З presented at the ASME 2012 Intern. Mechanical Congress & Exposition, Houston, TX, 9-15 November. Heisig, G., and Neubert, H Lateral Drillstring Vibrations in Extended Reach Wells, Paper IADC/SPE presented at the IADC/SPE Drilling Conference, New Orleans, LA, February.

10 10 SPE RU Johancsik, C.A., Friesen, D.B., and Dawson, R Torque and Drag in Directional Wells Prediction and Measurement. Journal of Petroleum Technology 36 (6): McCann, R.C., and Suryanarayana, P.V.R Experimental study of curvature and frictional effects on buckling. Proc. Offshore Technology Conf., OTC 7568, Houston, TX, 2-5 May. Tikhonov, V.S., Safronov, A.I., and Gelfgat, M.Ya Method of Dynamic Analysis of Rod-in-Pipe Buckling. Paper ESDA presented at the 8 th Biennial ASME Conference on Engineering Systems Design and Analysis, Torino, Italy, 4-7 July. Tikhonov, V.S., and Safronov, A.I Numerical Simulation of the Drillstring-in-Hole Dynamics at Rotary Drilling. Paper ESDA presented at the 9 th Biennial ASME Conference on Engineering Systems Design and Analysis, Haifa, Israel, 7-9 July. Tikhonov, V.S., and Safronov, A.I Analysis of Postbuckling Drillstring Vibrations in Rotary Drilling of Extended-Reach Wells. Transactions of the ASME, JERT 133 (December): / /8. Tikhonov, V., Valiullin, Kh., and Ring, L. 2012a. Numerical Simulation of Well Trajectory while Drilling of Isotropic and Anisotropic Formations. Paper ESDA presented at the 11 th Biennial ASME Conference on Engineering Systems Design and Analysis, Nantes, France, 2-4 July. Tikhonov, V., Giers, M., Yakhimovich, V., Shemyakinsky, B. et al. 2012b. Multi-component friction testing of full scale drill pipe specimen. Tribology and Design II. 4 th International Conference on Tribology and Design, Ed. M. Hadfield & C.A. Brebbia, WIT Transactions: Tikhonov, V.S., and Bukashkina, O.S Rod-in-tube buckling comparative analysis of numerical and test results. Paper SEMC presented at the 5 th International conference on Structural Engineering, Mechanics and Computation, Cape Town, South Africa, 2-4 September. Tikhonov, V., Valiullin, Kh., Nurgaleev, A., Ring, L. et al Dynamic Model for Stiff String Torque and Drag. Paper SPE/IADC presented at the SPE/IADC Drilling Conference and Exhibition, Amsterdam, The Netherlands, 5-7 March. Tikhonov, V.S., Bukashkina, O.S., and Gandikota, R. 2014a. Numerical Simulation of Casing Centralization. Paper 1297 presented at the 11 th World Congress on the Computational Mechanics, Barcelona, Spain, July. Tikhonov, V.S., Bukashkina, O.S., and Gandikota, R. 2014b. Stick-Slip Model for PDC Bits Accounting for Coupled Torsional and Axial Oscillations. Paper ESDA presented at the 12th Biennial Conference on Engineering Systems Design and Analysis ESDA, Copenhagen, Denmark, June Wu, J., and Juvkum-Wold, H.C The effect of wellbore curvature on tubular buckling and lockup. Trans. ASME. Journal of Energy Resources Technology 117 (September):

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