Buckling Analysis in Deviated Wells: A Practical Method
Authors: Mitchell R.F. 
Published: December 2007

Abstract
Current helical buckling models are valid for vertical wells, but provide only approximate solutions for horizontal wells. Solutions of the non-linear buckling equations for arbitrary well deviation have been developed, but are too complex for practical use. This paper presents a set of correlations that match the exact solutions extremely well, but are simple to use. These correlations show the effects of well deviation on buckling shape, tubing length change, contact force and bending stress.

Introduction
The most generally accepted method for the analysis of buckling, tubing movement, and packer selection is the method developed by Lubinski et. al. in reference 1. Analyses following Lubinski's basic approach have been developed for more complicated tubing configurations, e.g. tapered strings. Henry Woods, in the appendix to Lubinski et. al., developed a mechanical model of well buckling behavior that predicted the buckled configuration as a function of well loads. This model featured:The slender beam theory is used to relate bending moment to curvature.The tubing is assumed to buckle into a helical shape.The wellbore is assumed to be straight and vertical.The pitch of the helix is related to the buckling load through the principle of virtual work.Friction between the tubing and casing is neglected.
Mitchell developed a more general approach that replaced the virtual work relations with the full set of beam-column equations constrained to be in contact with the casing. Helical buckling in a deviated well, in this formulation, is described by a fourth order non-linear differential equation. For a vertical well, the solution to this equation can be accurately approximated by the simple algebraic equation discovered by Lubinski and Woods. This solution is not valid for deviated or horizontal wells because of the lateral gravity forces. The full deviated well equation was solved by Mitchell using numerical methods. The purpose of this paper is to put these results in a more usable form.
Accurate solution of the buckling equations is important for several reasons. Bending stresses due to tubing buckling will be overestimated for deviated wells using Lubinski's formula. However, Lubinski's solution applied to deviated wells will also overpredict tubing movement. For a fixed packer, this solution will overestimate tubing compliance, which may greatly underestimate the axial loads, resulting in a non-conservative design. For a free packer or PBR, exaggerated tubing motion will require excessive seal length. Further, because tubing incremental motion will control the friction load direction, errors in overall tubing displacement will generate further errors in friction loads.
This paper presents correlations to the numerical solution of the buckling differential equation. Calculation of results, including buckling length change, tubing contact forces, bending stresses and dogleg angle are developed. An application problem was solved and the effects of well deviation on stability, length change, and maximum bending stress were examined. Well deviation is shown to have significant impact on buckling results and tubing stress analysis.
At the end of this paper is a complete nomenclature and reference list.
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