Summary.

An approximate analytic solution for the helical buckling of tubing with weight has been determined. This solution has good accuracy except near the neutral point. The solution used by Lubinski to solve the helical-buckling problem is shown to be a special case of this new analytic solution. It is generally accepted that Lubinski's solution is a good approximate solution, but the range of applicability has been unknown because the approximations used in its derivation were not well understood In this new solution, the approximations are explicit and an accuracy criterion is established. Initial conditions at the packer are investigated and their effects on the solution are described. Finally, the effect of this solution on tapered strings is described, and a new approximate buckled solution for tapered strings is presented. This analysis puts the Lubinski solution in a technical context that allows such further developments as inclined wellbores and friction.

Introduction

Buckling analysis of well tubing and its effect on packer selection and installation are basic parts of tubing design. Lubinski et al.'s analysis of this problem is commonly used and generally accepted to be correct. Later analyses, such as Hammerlandl's have applied the same basic techniques developed in Ref. 1 to other problemse.g., tapered strings. Lubinski et al. presented a mechanical model of well buckling behavior that predicted the buckled well configuration as a function of applied loads. The principal results from this model were the motion of the tubing at the packer and the stresses developed in the tubing because of buckling. The major features of the mechanical model are summarized here.1. The slender-beam theory is used to relate bending moment to curvature.2. The tubing is assumed to buckle into a helical shape.3. The wellbore is assumed to be straight and vertical.4. The pitch of the helix is related to the applied buckling load by the principle of virtual work.5. Friction between the buckled tubing and restraining case is neglected. The geometry of the helix (see Fig. 1) is described by the equations

,...........................................(1)

,.......................................(2)

and

,..........................(3)

whereu1, u2 = tubing centerline locations in x, y coordinatedirections, respectively,0 = angular coordinate, r = tubing/casing radial clearance, and P = pitch of helix.

The principle of virtual work relates the pitch, P, to the buckling force, Ff, through the following equation:

..........................................(4)

Cheatham and Patillo provide a lucid description of virtual work applied to this problem. Several points need to be made about Eqs. 1 through 4. First, the analysis applies to a buckling force that does not vary with depthi.e., the tubing is weightless and fluid densities are zero. Second, the pitch of the helix does not vary with depth. Finally, the assumed helical shape is not consistent with the end constraints imposed by the packer. The first attempt to reconcile the helix with the end constraints is given in Ref. 6. Sorenson proposed a more complete solution to the end-constraint problem based on analysis and experimental results. Sorenson also showed that, near the end constraints, the pitch of the helix was not constant, but did approach the result in Eq. 4 at the center of the buckled rod. Variable pitch was introduced to the tubing buckling problem in Ref. 1. The assumption was made that Eq. 4 still applied when the constants Ff and P were replaced by variables. An unpublished virtual-work analysis of tubing with weight completed by the author produced results very near to Lubinski's results, suggesting that this assumption may be reasonable in some cases. Nevertheless, the exact nature of the approximations in Lubinski's analysis and the range of their applicability remained unknown. Kwon recently discussed a virtual-work analysis of tubing buckling with weight. This paper solves the equations of equilibrium for a buckled tubing string directly to determine the exact nature of the assumptions used by Lubinski. A new relationship between pitch and buckling load is determined to be consistent with that solution and clearly states the conditions that result in Eq. 4. The generalized load/pitch relation is analyzed and found to predict instabilities in numerical analysis of helical buckling. A new definition of the neutral point is derived from a contact-force criterion. The boundary conditions proposed by Sorenson are shown to be consistent with the new relation between pitch and buckling load. An approximate solution to the tapered-string buckling problem is proposed and compared with Hammerlindl's tapered-string solution.

Equilibrium of a Helix With Variable Pitch

The equations that describe a slender beam with axial load were derived in Ref. 6:

,.........................(5)

whereEI = bending stiffness of tubing, Ff = buckling force, FHi = lateral loads exerted on tubing, and' = derivative with respect to Coordinate z.

The buckling force, Ff, is given by

...........................................(6)

whereFfo = applied buckling force, W = buckling weight per unit length of tubing, as givenby Eq. 5 of Ref. 1, and= angle of inclination from vertical of wellbore.

SPEDE

P. 303^

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