Analytical approximate solution for Initial post-buckling behavior of pipes in oil and gas wells

  • Yu, Yongping (College of Construction Engineering, Jilin University) ;
  • Sun, Youhong (College of Construction Engineering, Jilin University) ;
  • Han, Yucen (School of Mathematical Sciences, Dalian University of Technology)
  • Received : 2012.04.13
  • Accepted : 2012.06.10
  • Published : 2012.06.25


This paper presents analytical approximate solutions for the initial post-buckling deformation of the pipes in oil and gas wells. The governing differential equation with sinusoidal nonlinearity can be reduced to form a third-order-polynomial nonlinear equation, by coupling of the well-known Maclaurin series expansion and orthogonal Chebyshev polynomials. Analytical approximations to the resulting boundary condition problem are established by combining the Newton's method with the method of harmonic balance. The linearization is performed prior to proceeding with harmonic balancing thus resulting in a set of linear algebraic equations instead of one of non-linear algebraic equations, unlike the classical method of harmonic balance. We are hence able to establish analytical approximate solutions. The approximate formulae for load along axis, and periodic solution are established for derivative of the helix angle at the end of the pipe. Illustrative examples are selected and compared to "reference" solution obtained by the shooting method to substantiate the accuracy and correctness of the approximate analytical approach.


  1. Belendez, A., Alvarez, M.L., Fernandez, E. and Pascual, I. (2009), "Linearization of conservative nonlinear oscillators", Eur. J. Phys., 30(2), 259-270.
  2. Benecke, S. and van Vuuren, J.H. (2005), "Modelling torsion in an elastic cable in space", Appl. Math. Model, 29(2), 117-136.
  3. Cheatham, J.B. and Pattillo, P.D. (1984), "Helical postbuckling configuration of a weightless column under the action of an axial load", Soc. Petroleum Eng. J. AIME, 24(4), 467-472.
  4. Denman, H.H. (1969), "An approximate equivalent linearization technique for nonlinear oscillations", J. Appl. Mech.-T. ASME, 36, 358-360.
  5. Gao, D.L., Liu, F.W. and Xu, B.Y. (2002), "Buckling behavior of pipes in oil and gas wells", Progress in Natural Science, 12(2), 126-130.
  6. Gulyayev, V.I., Gaidaichuk, V.V., Solovjov, I.L. and Gorbunovich, I.V., (2009), "The buckling of elongated rotating drill strings", J. Petrol. Sci. Eng., 67(3-4), 140-148.
  7. He, X. and Kyllingstad, A. (1993), "Helical buckling and lock-up conditions for coiled tubing in curved wells", Proceedings of the SPE Asia Pacific Oil and Gas Conference & Exhibition, Singapore, 8-10 February.
  8. Jonckheere, R.E. (1971). "Determination of the period of nonlinear oscillations by means of Chebyshev polynomials", ZAMM -Z. Angew. Math. Mech., 51(5), 389-393.
  9. Li, P.S., Sun, W.P. and Wu, B.S. (2008). "Analytical approximate solutions to large amplitude oscillation of a simple pendulum", J. Vib. Shock, 27, 72-74. (In Chinese)
  10. Lubinski, A., Althouse, W.S. and Logan, J.L. (1962), "Helical buckling of tubing sealed in packers", J. Petrol. Technol., 14(6), 655-670.
  11. McCann, R.C. and Suryanarayana, P.V.R. (1994), "Experimental study of curvature and frictional effects on buckling", Proceedings of the 26th Annual O!shore Technology Conference, Houston, TX, 2-5 May.
  12. Miska, S. and Cunha, J.C. (1995), "An analysis of helical buckling of tubulars subjected to axial and torsional loading in inclined wellbores", Proceedings of the the SPE Production Operations Symposium, Oklahoma City, OK, 2-4 April.
  13. Paslay, P.R. and Bogy, D.B. (1964), "The stability of a circular rod laterally constrained to be in contact with an inclined circular cylinder", J. Appl. Mech. T.-ASME, 31, 605-610.
  14. Qiu, W., Miska, S. and Volk, L. (1998), "Drill pipe/coiled tubing buckling analysis in a hole of constant curvature", SPEJ, 39795(3), 385-395.
  15. Seydel, R. (1994), Practical bifurcation and stability analysis-from equilibrium to chaos, 2nd Ed., Springer Verlag.
  16. Tan, M.L. and Gan, L.F. (2009), "Equilibrium equations for nonlinear buckling analysis of drill-strings in 3D curved well-bores", Science in China Series E: Technol. Sci., 52(3), 590-595.
  17. Tan, M.L. Wang, X.W. and Gan, L.F. (2006), "Buckling of a spatial elastic thin rod under torque", Chin. J. Comput. Phys., 23(4), 447-450.
  18. Timoshenko, S. and Gere, J. (1961), Theory of elastic stability, New York, McGRAW-HILL.
  19. Wicks, N., Wardle, B.L. and Pafitis, D. (2008), "Horizontal cylinder-in-cylinder buckling under compression and torsion: Review and application to composite drill pipe", Int. J. Mech. Sci., 50(3), 538-549.
  20. Wu, J. (1997), "Torsional load effect on drill-string buckling", SPEJ, 37477(6), 703-710.
  21. Wu, B.S. Sun, W.P. and Lim, C.W. (2006), "An analytical approximate technique for a class of strong nonlinear oscillators", Int. J. Nonlinear Mech., 41, 766-774.
  22. Wu, B.S. Yu, Y.P. and Li, Z.G. (2007), "Analytical approximations to large post-buckling deformation of elastic rings under uniform hydrostatic pressure", Int. J. Mech. Sci., 49(6), 661-668.
  23. Yu, Y.P. Lim, C.W. and Wu B.S. (2008), "Analytical approximations to large hygrothermal buckling deformation of a beam", J. Struct. Eng.- ASCE, 134, 602-607.