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.

The most generally accepted method for the analysis of buckling, tubing movement, and packer selection is the method developed by Lubinski *et al.*^{1} 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 included the following features:

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.^{2} Helical buckling in a deviated well, in this formulation, is described by a fourth-order nonlinear differential equation. The solution discovered by Lubinski and Woods was found to be an exact algebraic solution for a pipe with constant axial force and no lateral forces. This solution is not valid for deviated or horizontal wells with lateral gravity forces. Mitchell solved the general deviated well equation using numerical methods.^{3} The purpose of this paper is to find and develop analytic solutions to these equations.

Accurate solution of the buckling equations is important for several reasons. Bending stresses caused by tubing buckling may be underestimated for deviated wells using Lubinski's formula. However, Lubinski's solution applied to deviated wells may also overpredict tubing movement. For a fixed packer, this solution will overestimate tubing compliance, which may greatly underestimate the axial loads, resulting in a nonconservative 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 five new analytic solutions of the buckling differential equation. Results of interest are developed, including buckling length change, tubing contact forces, tubing bending moment, bending stresses, and dogleg angle.

In this formulation of the beam-column equations, the lateral displacements are given by

Equations 1 and 2

where ?=the helix angle and *r*=the tubing/casing radial clearance. This configuration is illustrated in **Fig. 1**. The differential equation for the helix angle ? is given by

Equation 3

where *w*=the lateral tubing weight per unit length, *EI*=the bending stiffness, *F*=the axial buckling force, and ' denotes *d/dz*. For details on the derivation of this equation, see Ref. 3. The axial buckling force *F* and the lateral tubing weight are both strongly influenced by fluid pressures and must be formulated accordingly.^{4} Note also that *F* is the value of the axial force in the buckled state. If we define the following dimensionless length ?

Equation 4

and if *Np*=the Paslay number, and *Fp*=the Paslay force,^{5} then

Equation 5

and Eq. 3 now takes the form:

Equation 6