Movement, Forces, and Stresses Associated With Combination Tubing Strings Sealed in Packers
Authors: Hammerlindl D.J. 
Published: December 2007

Changes in temperature and pressure during pump-in and production usually result in changes in tubing length or tubing and packer forces, depending on whether the tubing is free to move. An earlier paper presented means for calculating these effects in completions with constant tubing and casing sizes. This paper summarizes the earlier paper and generalizes it to completions with varying tubing and/or casing sizes.

Introduction
Lubiriski et al.1 presented a means for calculating the effects of changes in pressure and temperature on tubing sealed in packers. These tubing changes are either changes in length if the packer permits free tubing movement or changes in force if the packer permits no tubing movement.
Lubinski et al. primarily treat uniform completion (that is, a single tubing and casing size). Means for solving problems in combination completions (varying tubing and/or casing sizes) are only briefly outlined for packers permitting tubing movement. There is no treatment for combination completions where tubing movement is not permitted.
This paper presents a method for solving problems in uniform completions as well as combination completions. The theoretical aspects of uniform completions must be obtained from the Lubinski et al. paper; to avoid redundancy, parts of Lubinski et al. 's paper are recommended for additional information and clarity. Numerical values in both papers follow the same pattern, but the ones given here pertain to combination completions.
This paper presents two items not covered by Lubinski et al.:a direct mathematical method for calculating forces in uniform completions where tubing movement is not permitted, anda method for handling hydraulic packers set with the wellhead in place.
Problems in combination completions are solved as though each section is a separate uniform completion and the tubing is initially free to move. The analysis begins with the lower section and the algebraic sum of each section's length change is the total length change. Where no tubing movement is permitted, the force required to return the tubing to its starting position is then determined.

Field Applications
The text of Ref. 1 for uniform completions holds entirely true for combination completions.

Applied Forces

Uniform Completions
A uniform completion is one with a single-sized tubing string containing only one fluid, and with a constant-ID casing string where the annulus contains only one fluid. The annular fluid may or may not be the same as that contained in the tubing.
Consider the effect of an axially induced force on a freely suspended tubing string in a cased well containing no fluid (see Fig. 1A). One of two types of forces may be applied at the lower end—a tensile force that elongates the tubing or a compressive force that shortens and buckles the tubing.

Uniform Completions
A uniform completion is one with a single-sized tubing string containing only one fluid, and with a constant-ID casing string where the annulus contains only one fluid. The annular fluid may or may not be the same as that contained in the tubing.
Consider the effect of an axially induced force on a freely suspended tubing string in a cased well containing no fluid (see Fig. 1A). One of two types of forces may be applied at the lower end—a tensile force that elongates the tubing or a compressive force that shortens and buckles the tubing.