On dynamic stability of drill strings in a supersonic gas flow
DOI:
https://doi.org/10.26577/jmmcs-2018-1-488Keywords:
drill string, stability, nonlinearity, gas flowAbstract
In this work stability of the drill string nonlinear dynamics, complicated by the effect of an external
axial load, initial curvature of the drill string, geometric nonlinearity and the influence of a
supersonic gas flow as a circulating medium is studied. The drill string is modelled as a rotating
elastic isotropic rod with constant cross-section. Pressure of the gas flow used to clean the borehole
from drill cuttings and to transport them from the bottom to the surface is determined by the
nonlinear dependences of the piston theory in the third approximation. Utilization of the Galerkin
method allows to reduce the drill string mathematical model to an ordinary differential equation
for the generalized time function, containing an asymmetric nonlinear characteristic, which is further
eliminated by introducing the corresponding substitution. Considering a small perturbation
to the system and applying the harmonic balance method, characteristic determinants are constructed.
Equations describing boundaries of instability zones of basic resonance, which allow to
determine the range of dangerous frequency regimes and to increase safety of the drilling process,
are obtained.
References
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[4] Chen, J., Q.S. Li. “Analysis of Flutter and Nonlinear Dynamics of a Composite Laminated Plate.” International Journal
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Engineering Science 227(5) (2013): 946-960. https://doi.org/10.1177/0954406212455126.
[7] Gulyaev, V.I., P.Z. Lugovoi, M.A. Belova, I.L. Solov’ev. “Stability of the Equilibrium of Rotating Drillstrings.” International
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Novgorod, Russia, September 26-29, 2016.
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Paper presented at International Conference on Mathematical Methods & Computational Techniques in Science
& Engineering (MMCTSE 2017), Cambridge, UK, February 24-26, 2017.
[14] Khajiyeva, L., Askar Kudaibergenov, Askat Kudaibergenov. “The effect of gas and fluid flows on nonlinear lateral vibrations
of rotating drill strings.” Communications in Nonlinear Science and Numerical Simulation 59 (2018): 565-579.
https://doi.org/10.1016/j.cnsns.2017.12.008.
[15] Kudaibergenov, Askhat, Askar Kudaibergenov, L. Khajiyeva. “Stability Analysis of Drill Rods as Shells in the Gas Stream.”
Journal of Applied Mechanics and Materials 665 (2014): 593-596.
[16] Lian, Zh., Q. Zhang, T. Lin, F. Wang. “Experimental and Numerical Study of Drill String Dynamics in
Gas Drilling of Horizontal Wells.” Journal of Natural Gas Science and Engineering 27 (2015): 1412-1420.
https://doi.org/10.1016/j.jngse.2015.10.005.
[17] Librescu, L., P. Marzocca. “Supersonic/Hypersonic Flutter and Postflutter of Geometrically Imperfect Circular Cylindrical
Panels.” Journal of Spacecraft and Rockets 39(5) (2002): 802-812.
[18] Liu, X., N. Vlajic, X. Long, G. Meng, B. Balachandran. “State-Dependent Delay Influenced Drill-String Oscillations and
Stability Analysis.” Journal of Vibration and Acoustics – Transactions of the ASME 136 (2014): 051008-1-9.
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[20] Meng, Y., H. Li, G. Li, L. Zhu, N. Wei, N. Lin. “Investigation on Propagation Characteristics of the Pressure Wave in
Gas Flow through Pipes and its Application in Gas Drilling.” Journal of Natural Gas Science and Engineering 22 (2015):
163-171. https://doi.org/10.1016/j.jngse.2014.11.026.
[21] Nandakumar, K., M. Wiercigroch. “Stability Analysis of a State Dependent Delayed, Coupled Two DOF Model of Drillstring
Vibration.” Journal of Sound and Vibration 332 (2013): 2575-2592. https://doi.org/10.1016/j.jsv.2012.12.020.
[22] Sahebkar, S.M., M.R. Ghazavi, S.E. Khadem, M.H. Ghayesh. “Nonlinear Vibration Analysis of an Axially Moving Drillstring
System with Time Dependent Axial Load and Axial Velocity in Inclined Well.” Mechanism and Machine Theory
46 (2011): 743-760. https://doi.org/10.1016/j.mechmachtheory.2010.12.003.
[23] Szemplinska-Stupnicka, W. “Higher Harmonic Oscillations in Heteronomous Non-linear Systems with One Degree Of
Freedom.” International Journal of Non-Linear Mechanics 3 (1968): 17 30.
[24] Szemplinska-Stupnicka, W. “Non-Linear Normal Modes” and the Generalized Ritz Method in the Problems of Vibrations
of Non-Linear Elastic Continuous Systems.” International Journal of Non-Linear Mechanics 18(2) (1983): 149-165.
https://doi.org/10.1016/0020-7462(83)90042-2.
[25] Vaz, M.A., M.H. Patel. “Analysis of Drill Strings in Vertical and Deviated Holes Using the Galerkin techniques.” Engineering
Structures 17(6) (1995): 437-442.
[26] Volmir A.S. Ustojchivost’ deformiruemyh sistem [Stability of Deformable Systems]. Moscow: Nauka, 1967.
at ENOC 2017, Budapest, Hungary, June 25-30, 2017.
[2] Al-Hiddabi, S.A., B. Samanta, A. Seibi. “Non-linear Control of Torsional and Bending Vibrations of Oilwell Drillstrings.”
Journal of Sound and Vibration 265 (2003): 401-415. https://doi.org/10.1016/S0022-460X(02)01456-6.
[3] Bhattiprolu, U., A.K. Bajaj, P. Davies. “Periodic Response Predictions of Beams on Nonlinear and Viscoelastic Unilateral
Foundations Using Incremental Harmonic Balance Method.” International Journal of Solids and Structures 99 (2016): 28-39.
[4] Chen, J., Q.S. Li. “Analysis of Flutter and Nonlinear Dynamics of a Composite Laminated Plate.” International Journal
of Structural Stability and Dynamics 16 (2016): 1550019-1-20.
[5] Christoforou, A.P., A.S. Yigit. “Dynamic Modeling of Rotating Drillstrings with Borehole Interactions.” Journal of Sound
and Vibration 206(2) (1997): 243-260. https://doi.org/10.1006/jsvi.1997.1091.
[6] Ghasemloonia, A., D.G. Rideout, S.D. Butt. “Coupled Transverse Vibration Modeling of Drillstrings Subjected to Torque
and Spatially Varying Axial Load.” Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical
Engineering Science 227(5) (2013): 946-960. https://doi.org/10.1177/0954406212455126.
[7] Gulyaev, V.I., P.Z. Lugovoi, M.A. Belova, I.L. Solov’ev. “Stability of the Equilibrium of Rotating Drillstrings.” International
Applied Mechanics 42(6) (2006): 692-698.
[8] Hayashi, Ch. Nonlinear Oscillations in Physical Systems. New Jersey: Princeton University Press, 1986.
[9] Heisig, G, M. Neubert. “Lateral Drillstring Vibrations in Extended-Reach Wells.” Paper presented at the 2000 IADC/SPE
Drilling Conference, New Orleans, Louisiana, February 23-25, 2000.
[10] Jansen, J.D. Nonlinear Dynamics of Oilwell Drillstrings. Delft University Press, 1993.
[11] Kiiko, I.A., V.V. Pokazeev. “On the Formulation of the Problem of Strip Oscillations and Stability in Supersonic Gas
Flow.” Fluid Dynamics 44(1) (2009): 135-140.
[12] Kudaibergenov, Askat K. “Ob ustojchivosti nelinejnyh kolebanij buril’nyh kolonn [On Stability of Nonlinear Vibrations of
Drill Strings].” Paper presented at X All-Russian Conference on “Nonlinear Vibrations of Mechanical Systems”, Nizhny
Novgorod, Russia, September 26-29, 2016.
[13] Kudaibergenov, Askat K., Askar K. Kudaibergenov, L.A. Khajiyeva. “On resonance regimes of drill string nonlinear dynamics.”
Paper presented at International Conference on Mathematical Methods & Computational Techniques in Science
& Engineering (MMCTSE 2017), Cambridge, UK, February 24-26, 2017.
[14] Khajiyeva, L., Askar Kudaibergenov, Askat Kudaibergenov. “The effect of gas and fluid flows on nonlinear lateral vibrations
of rotating drill strings.” Communications in Nonlinear Science and Numerical Simulation 59 (2018): 565-579.
https://doi.org/10.1016/j.cnsns.2017.12.008.
[15] Kudaibergenov, Askhat, Askar Kudaibergenov, L. Khajiyeva. “Stability Analysis of Drill Rods as Shells in the Gas Stream.”
Journal of Applied Mechanics and Materials 665 (2014): 593-596.
[16] Lian, Zh., Q. Zhang, T. Lin, F. Wang. “Experimental and Numerical Study of Drill String Dynamics in
Gas Drilling of Horizontal Wells.” Journal of Natural Gas Science and Engineering 27 (2015): 1412-1420.
https://doi.org/10.1016/j.jngse.2015.10.005.
[17] Librescu, L., P. Marzocca. “Supersonic/Hypersonic Flutter and Postflutter of Geometrically Imperfect Circular Cylindrical
Panels.” Journal of Spacecraft and Rockets 39(5) (2002): 802-812.
[18] Liu, X., N. Vlajic, X. Long, G. Meng, B. Balachandran. “State-Dependent Delay Influenced Drill-String Oscillations and
Stability Analysis.” Journal of Vibration and Acoustics – Transactions of the ASME 136 (2014): 051008-1-9.
[19] Novozhilov, V.V. Foundations of the Nonlinear Theory of Elasticity. New York: Dover Publications, 1999.
[20] Meng, Y., H. Li, G. Li, L. Zhu, N. Wei, N. Lin. “Investigation on Propagation Characteristics of the Pressure Wave in
Gas Flow through Pipes and its Application in Gas Drilling.” Journal of Natural Gas Science and Engineering 22 (2015):
163-171. https://doi.org/10.1016/j.jngse.2014.11.026.
[21] Nandakumar, K., M. Wiercigroch. “Stability Analysis of a State Dependent Delayed, Coupled Two DOF Model of Drillstring
Vibration.” Journal of Sound and Vibration 332 (2013): 2575-2592. https://doi.org/10.1016/j.jsv.2012.12.020.
[22] Sahebkar, S.M., M.R. Ghazavi, S.E. Khadem, M.H. Ghayesh. “Nonlinear Vibration Analysis of an Axially Moving Drillstring
System with Time Dependent Axial Load and Axial Velocity in Inclined Well.” Mechanism and Machine Theory
46 (2011): 743-760. https://doi.org/10.1016/j.mechmachtheory.2010.12.003.
[23] Szemplinska-Stupnicka, W. “Higher Harmonic Oscillations in Heteronomous Non-linear Systems with One Degree Of
Freedom.” International Journal of Non-Linear Mechanics 3 (1968): 17 30.
[24] Szemplinska-Stupnicka, W. “Non-Linear Normal Modes” and the Generalized Ritz Method in the Problems of Vibrations
of Non-Linear Elastic Continuous Systems.” International Journal of Non-Linear Mechanics 18(2) (1983): 149-165.
https://doi.org/10.1016/0020-7462(83)90042-2.
[25] Vaz, M.A., M.H. Patel. “Analysis of Drill Strings in Vertical and Deviated Holes Using the Galerkin techniques.” Engineering
Structures 17(6) (1995): 437-442.
[26] Volmir A.S. Ustojchivost’ deformiruemyh sistem [Stability of Deformable Systems]. Moscow: Nauka, 1967.
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How to Cite
Kudaibergenov, A. K., & Khajiyeva, L. A. (2018). On dynamic stability of drill strings in a supersonic gas flow. Journal of Mathematics, Mechanics and Computer Science, 97(1), 101–110. https://doi.org/10.26577/jmmcs-2018-1-488
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