Exact Analytic Solutions for Pipe Buckling in Vertical and Horizontal Wells

Summary The first buckling solution was developed by Lubinski and Woods. When the differential equation describing buckling was derived, this solution was found to be an exact solution for vertical wells. Since these results were published, no other exact analytic solution has been discovered. Many numerical results have been obtained, however, suggesting that other solutions did exist. Because the buckling differential equation is nonlinear, it is not surprising that no other analytic solutions have been discovered. This paper presents three new analytic solutions for the vertical well problem and two new analytic solutions for the horizontal well problem. These analytic solutions are valuable both for predicting previously unanticipated buckling behavior and for providing guidance in further numerical evaluations of this problem. In this paper, these five solutions are described, including techniques for evaluating the analytic functions. Buckling length change calculations are determined analytically, and pipe curvature, bending moment, and bending stresses are evaluated. The contact loads between tubing and wellbore are determined, then used to limit the range of validity of the solutions. The critical force for helical buckling is determined for horizontal wells. Possible applications for these solutions include the analysis of bottomhole assemblies, drillpipe, casing, and tubing. The solutions are simple formulas, but require computer evaluation of the analytic functions.