Simple Frictional Analysis of Helical Buckling of Tubing

Abstract

Summary Previous analyses of helical buckling of tubing have not considered frictional forces. In this paper, differential equations are derived and solved for two simplified cases of interest: downward motion of the tubing--e.g., when buckling occurs during the landing of the tubing-and upward motion of the tubing--e.g., when buckling occurs as a result of thermal and differential pressure loading subsequent to landing. While somewhat more complicated than the conventional frictionless buckling equations, these solutions are still suitable for hand calculations. These solutions, however, do not represent general solutions to buckling with friction. Load reversals and lateral frictional forces add complications that would require computer analysis. Several examples are examined to evaluate the relative importance of friction, which has a significant impact on tubing length change for loaded cases. For instance, the choice of a conservative value for the friction coefficient may allow the solution of a difficult seal-design problem by reducing a large predicted length change. Friction also has an important effect on set-down loads. Frictionless buckling calculations do not give conservative results for this problem. Introduction The buckling behavior of well tubing has an important impact on well design and production operations. Lubinski et al. first analyzed tubing buckling comprehensively. Hammerlindl applied the same basic buckling model to more complicated situations, including combination strings and intermediate packers. The mechanical basis for this buckling model consists of the following features. 1. Slender-beam theory is used to relate bending moment to curvature. The tubing must remain elastic for the analysis to remain valid. 2. The tubing is assumed to buckle into a helical shape. This assumption is reasonable for a vertical wellbore but might not be valid for a deviated wellbore. 3. The principle of virtual work is used to relate the buckling load to the pitch of the helix. 4. Certain conditions on bending moment at the packer are implied by the formulation. It has been shown that packer are implied by the formulation. It has been shown that these boundary conditions influence the solution of the buckling problem. 5.Friction between the buckled tubing and the constraining casing is neglected. Recent work showed that the virtual work analysis is not necessary to derive useful approximate solutions to buckling equations that satisfy Item 4. These results are summarized in Appendix A. The friction assumption is often mentioned in the buckling literature, and the importance of friction is often stated. In Fig. 14 of Ref. 2, for instance, the 50% deviation of the measured buckling length change from the predicted length change is attributed to friction. In this paper, the buckling model is modified to include the effects of friction for two special cases. The elementary theory of friction is discussed and the history-dependence of friction forces is described. Two simple load histories are described that give analytical solutions: tubing loaded at the packer and tubing slacked off at the surface. Sample packer and tubing slacked off at the surface. Sample problems based on the cases presented by Lubinski et al. are problems based on the cases presented by Lubinski et al. are calculated to illustrate the importance of friction. The actual technical development of the buckling equations is presented in three appendices. In Appendix A, the presented in three appendices. In Appendix A, the buckling-force/pitch relation and the contact forces between the buckled tubing and the constraining casing are determined by an approximate solution to the slender-beam equations. In Appendix B, the differential equations governing the buckling force when friction is present are derived and solved for the cases of loading and landing. In Appendix C, the buckling and "piston-effect" length changes are determined for these special friction cases.