Tubing Buckling - The Rest of the Story
- Robert F. Mitchell (Halliburton)
- Document ID
- Society of Petroleum Engineers
- SPE Drilling & Completion
- Publication Date
- June 2008
- Document Type
- Journal Paper
- 112 - 122
- 2008. Society of Petroleum Engineers
- 2 Well Completion, 4.1.5 Processing Equipment, 1.10.1 Drill string components and drilling tools (tubulars, jars, subs, stabilisers, reamers, etc), 1.8 Formation Damage, 4.1.2 Separation and Treating, 1.6 Drilling Operations
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The need to improve completion reliability has prompted a second, deeper look at the analysis of tubing buckling. The conventional-buckling analysis is assumed to hold "far from the packer", but what happens "near" the packer?
The original development by Lubinski and Woods used the method of virtual work to determine a constant pitch helix. All attempts to connect a constant pitch helix to boundary conditions, such as a packer, have failed. The conventional wisdom says that the virtual work solution applies "far from the packer". How far is this, and what happens between the packer and this point? Even worse is the case of tapered strings. The generally accepted answer to the tapered string problem is a "stacked" set of Lubinski solutions, with each solution far from the interconnection of the different sized tubulars (1962).
Perhaps there are nonconstant pitch solutions? Because the buckling-differential Eq. is nonlinear, it is not surprising that no other analytic solutions have been discovered until recently. Once these solutions are known, the initial value problem can be posed and solved. Two possible boundary conditions are considered in this study: a cantilever boundary condition to model a packer, and a "pinned" boundary condition to model a centralizer. For the first time, analytic solutions were found for both problems, and interestingly, both solutions converged rapidly to Lubinski's constant pitch solution at a specified distance from the boundary. For this reason, many of the original Lubinski calculations remain essentially correct, (such as length change and contact forces). Alternatively, the bending moment near the boundary exceeds the bending moment calculated from the Lubinski solution by approximately 20%. For this reason, conventional pipe-buckling stress analysis must be revised. Further applications of these solutions allow different sized tubulars to be interconnected, solving the tapered-string problem for the first time. An additional, useful result is the solution to the helix "reversal" problem.
The principal applications for these solutions are the stress analysis and design of tubing and casing strings. The boundary solutions are simple formulas, suitable for spreadsheet or hand calculations.
To improve completion reliability, there has been increasing use of "fixed string" completions across the industry, where the tubing is fixed at the tubing hanger and packer, rather than incorporating expansion joints that may prove vulnerable to leakage causing reduced-completion service life. Furthermore, there has been increasing uptake of hydrostatic-set packers that eliminate the requirement for wireline or coiled tubing to set a tailpipe plug. Although the hydrostatic packer has proven popular, it creates additional compressive loads in the tubing above the packer. This compares with a hydraulic-set packer that reduces tubing compressive loads, in that tubing pressure is applied against a tailpipe plug to set the packer.
Wider awareness of the buckling condition in recent years has provided impetus for tubing manufacturers to develop completion connections with a better service rating in compression. Most vendors now offer 100% connection-compression efficiency, where the connection has been qualified to operate with compressive loads as high as the rated-tensile capacity of the pipe.
The need to improve completion reliability has prompted a second, deeper look at the analysis of tubing buckling. When solving a mechanical equilibrium problem, the engineer is usually confronted with a system of differential Eq.s and a set of boundary conditions. The tubing buckling problem is seldom posed this way. Why should this be the case?
This is partly because of historical reasons. The first generally accepted buckling solution was developed by Lubinski and Woods in 1962. They posed a virtual-work problem to determine the pitch of the helically buckled pipe. Others followed the same path, and this is generally the way tubing-buckling problems are solved today. Examples of this style of analysis are given in Cheatham et al. (1984); He and Kyllingstad (1993); and Miska et al. (1995ab). The second reason is that the differential Eq.s describing tubing buckling are nonlinear, and are therefore difficult to solve analytically. Only recently have general solutions to this problem become available [see Appendix A (Mitchell 1982)]. The final reason, as will be shown, is that posing and solving these boundary-value problems are often tedious and difficult.
In the Lubinski method, the boundary conditions are not considered explicitly, but are embedded in the formulation in a subtle way. Examination of the formulation clearly shows that the boundary conditions are not consistent with typical packers. The conventional wisdom is that the buckling solution applies far from the packer (1962).
Few authors were aware that a solution near the packer was needed. However, a comprehensive analysis of tubing buckling consistent with boundary conditions at a packer or a centralizer was never done in a completely satisfactory way. Attempts were made to connect the constant pitch-Lubinski solution to a beam-column solution that brings the pipe from the wellbore wall to the packer (Mitchell 1982). The following conditions must be met where the two solutions join
- wellbore contact
- wellbore tangency
- continuity of curvature
- continuity of shear tangent to wellbore
- positive contact force between pipe and wellbore
- all pipe displacements within the borehole
These six conditions cannot all be satisfied for a constant-pitch helix joined to a single beam-column section. Sorenson was able to connect a two-section beam-column solution to a numerical solution of the full-contact buckling problem (1986). He chose one boundary condition to be Lubinski's solution for the extreme end of the pipe. This solution only approximated the constant-pitch solution in the limit of a long pipe, and only at the extreme end of the pipe. Did the use of this boundary condition impose Lubinski's solution on this problem? Application of this model has been extremely limited because of the complexity of the solution.
A major roadblock to understanding the effect of the packer on buckling was the seeming incompatibility of the constant-pitch helix solution with the beam-column solution. The constant-pitch solution has only a single unknown constant, the helix pitch. Together with the constants in the beam-column solution, there are not enough degrees of freedom to join the two solutions. Fortunately, recent work has found a family of analytic solutions to the full-contact buckling problem that allow the solution of general initial-value problems or a solution to the beam-column connection problem (Mitchell 1982).
This paper discusses the general solutions to the pipe-buckling problem, explicitly states the constraints on the solution imposed by several models for pipe-packer interaction, and solves for the full-contact solution for each case. The properties of each solution are examined in depth. The effects produced by cantilever packers and centralizers, (such as contact forces and bending stresses) are developed in detail. The location and magnitude of maximum bending stresses are determined.
This paper highlights that existing models underestimate buckling-bending stresses directly above the packer by some 20%. Once this 20% error is recognized, it is not surprising to learn that there are field examples of tubing leaks and early completion failures caused by buckling above the packer.
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Cheatham, J.B. Jr. and Pattillo, P.D. 1984. Helical Postbuckling Configuration ofa Weightless Column Under the Action of an Axial Load. SPEJ24 (4): 467-472. SPE-10854-PA doi: 10.2118/10854-PA
Gradshteyn, I.S. and Ryzhik, I.M. 2000. Table of Integrals, Series, andProducts. London: Academic Press.
Hammerlindl, D.J. 1977. Movement, Forces, and StressesAssociated with Combination Tubing Strings Sealed in Packers. JPT29 (2): 195-208; Trans., AIME, 263. SPE-5143-PA doi:10.2118/5143-PA
He, X. and Kyllingstad, A. 1995. Helical Buckling and Lock-UpConditions for Coiled Tubing in Curved Wells. SPEDC 10 (1):10-15. SPE-25370-PA doi: 10.2118/25370-PA
Lubinski, A., Althouse, W.S., and Logan, J.L. 1962. Helical Buckling of Tubing Sealed inPackers. JPT 14 (6): 655-670; Trans., AIME,225. SPE-178-PA doi: 10.2118/178-PA
Miska, S. and Cunha, J.C. 1995a. An Analysis of Helical Buckling ofTubulars Subjected to Axial and Torsional Loading in Inclined Wellbores.Paper SPE 29460 presented at the SPE Production Operations Symposium, OklahomaCity, Oklahoma, 2-4 April. doi: 10.2118/29460-MS
Miska, S. and Cunha, J.C. 1995b. Helical Buckling of Long Weightless StringsSubjected to Axial and Torsional Loads. Drilling Symposium of the ASME Energyand Environment, Houston, 29 January-1 February.
Mitchell, R.F. 1982. BucklingBehavior of Well Tubing: The Packer Effect. SPEJ 22 (5):616-624. SPE-9264-PA doi: 10.2118/9264-PA
Mitchell, R.F. 2002. ExactAnalytical Solutions for Pipe Buckling in Vertical and Horizontal Wells.SPEJ 7 (4): 373-390. SPE-72079-PA doi: 10.2118/72079-PA
Mitchell, R.F. 2005. The Pitchof Helically Buckled Pipe. Paper SPE 92212 presented at the SPE/IADCDrilling Conference, Amsterdam, 23-25 February. doi: 10.2118/92212-MS
Press, W.H., Flannery, B.P., Teukolsky, S.A. and Vetterling, W.T. 1999.Numerical Recipes in Fortran 77: The Art of Scientific Computing. NewYork City: Cambridge University Press 382-386.
Sorenson, K.G. and Cheatham, J.B. Jr. 1986. Post-Buckling Behavior of aCircular Rod Constrained Within a Circular Cylinder. J. AppliedMechanics 53: 929-934.
Timoshenko, S.P. and Gere, J.M. 1961. Theory of Elastic Stability.New York City: McGraw-Hill Publishing Co., Inc.