Analysis of Helical Buckling

Abstract

Summary. A new solution to helical buckling is presented yielding real helical shapes of buckled pipes with a varying helix pitch. This solution has been developed with the beam/column equation. The change in length and the bending stress can be computed from the helical-buckling solution. In addition, a new equation for tool clearance in helically buckled pipes is provided. Introduction Critical well conditions-e.g., deep, high pressure, or lethal gas service-require a reliable design and analysis of tubing and casing strings because such wells are often designed to the designated design-factor limits. Helical buckling is an important parameter in such a design analysis. This phenomenon was first investigated by Lubinski et al. and led to the derivation of the well-known helix-pitch/force relationship, which was subsequently used extensively by others. The underlying assumption for the Lubinski et al. equation is that the helix pitch is constant. That assumption is correct for a weightless uniform pipe inside a uniform, concentrically surrounding pipe. All pipes have weight, so "weightless" really means that the weight of the pipe is negligible compared with externally applied forces. The total pipe length in oil or gas wells, however, can be very long. Consequently, its total weight, unlike external load, cannot be neglected, especially in wells requiring thick-walled tubes. In this paper a new helical-buckling equation is derived from the beam/column equation. Because the length of the column is substantially larger than the cross-sectional dimension, the thin-beam theory was used. For this assumption, the deflection caused by shear force is neglected. The resultant helical-buckling equation is a fourth-order nonlinear differential equation. It may be solved nu-merically, but its solution is tedious and time-consuming. Consequently, a semianalytic solution was developed with a series-solution method. This approach yields helix pitches varying along the length of pipe. This approach also computes the change in column length and bending moments resulting from buckling. Application of this solution to combination completions is also discussed. In addition to solving for the forces and stresses in helically buckled pipes, it is often desirable to compute tool clearance in buckled pipes. The tool clearance formula was derived by Lindsey et al.; however, in that derivation, the helix was approximated by a torus. This ap-proximation resulted in an overestimation for tool clearance. Thuse of a real helical shape needs to be considered for a conservative estimation of tool clearance. A new solution for tool clearance has been derived with a real helical shape. Helical Buckling of Weightless Pipes An external force applied on the bottom end of a vertically suspended pipe will remain constant along the length of the pipe if the pipe is weightless. This implies that the helically buckled shape will be the same along a weightless pipe if helical buckling occurs. Con-sequently, helix pitch is constant through the buckled portion of the pipe. Because of this simplicity, the helical-buckling equation of a weightless pipe was derived from a simple helix geometry with energy methods. The buckling equation expresses helix pitch in terms of an external force: 8 2 EIP2 = ----------------.......................................(1)Fe and 4 2 EIP2 = ------------------,.................................... (2)Fe where P = helix pitch, E = Young's modulus, I = second moment of inertia of cross section, and Fe = force causing helical buckling. These equations were obtained by Lubinski et al and Cheatham and Pattilo. Eq. 1 was derived by taking a derivative of total potential energy with respect to P with a constant radial clearance, r, while Eq. 2 was derived by taking a derivative of total potential energy with respect to r with a constant P. Once a helically buckled pipe contacts the wall of a concentrically surrounding pipe, r is constant and P is the only variable. Thus, Eq. 1 may be more reasonable for finding a physically stable helix pitch. The helical-buckling equation for a weightless pipe can also be obtained from the beam/column equation in space. Eq. 1 is also found from this derivation. Appendix A lists the derivation of the helical-buckling equation from the beam/column equation. Helical Buckling of Pipe With Weight When the weight of a pipe is not negligible compared with the force applied on it, the force does not remain constant along the length of the pipe. This implies that helix pitch is not constant, but varies along the pipe. Thus, a helical-buckling equation cannot be found from a simple helix geometry under these circumstances. To overcome this difficulty, the generalized beam/column equation can be used to find a helical-buckling equation for a pipe with weight. The resultant equation is a fourth-order nonlinear differential equation, which can be solved with numerical methods. Numerical methods are time-consuming, however, and may yield incorrect, diverging solutions for nonlinear problems. Consequently, a semianalytic method, such as a series solution approximation, is applied. The solutions are derived in Appendix B and provide Eq. 3. 2 d--- = ---- =, .............................(3)P dz Fe 1/2a = -----, .................................................(4)2EI a =-----[1/2+2/3 a L2-(6 1/4+2 2/3 L2+4/9 L4)1/2],...(5)3L a1 31a2 -- ----- + -------, ................................ (6)2L 2ao SPEDE P. 211^