Determining Stress In Tubing Using Triaxial Evaluation
Authors: Greenip John F. 
Published: December 2007

Abstract
The tubing string is subjected to various load conditions throughout the productive period of the well. In addition to the load period of the well. In addition to the load conditions imposed while hanging off the tubing during the well completion, operations such as setting hydraulic packers, pressure testing the tubing, producing the well, shutting the well in, hydraulic fracturing, acidizing, and remedial squeeze cementing create stress in the tubing. The stress created by these operations is often greater than the stress created during the running and hanging off of the tubing. A triaxial stress analysis is made for some of these load conditions, using thick-walled cylinder theory and conventional axial stress theory to develop the radial, tangential, and axial principal stresses at specific points in the string. The distortion energy theory of failure is then applied to calculate the equivalent stress for that load condition and point in the tubing string.

Introduction
With the deep, high pressure wells that are being completed, the cost to workover a well to repair damaged tubing becomes increasingly high. In wells with production of hydrogen sulfide, the potential safety hazard of a leaking tubing string magnifies the importance of maintaining the pressure integrity of the tubing string. In these deep or critical wells, the tubing string must be designed with maximum attention given to the stresses put on the steel. With increased depth and pressure differentials and limited yield strength steels, it becomes more difficult to design the tubing string to stay within the maximum safe stress limits of the steel. The stress in the tubing must be evaluated under each anticipated loading condition and compared to the yield strength of the steel and the resulting design factor compared to the design standards set.
With each specific loading condition, certain parameters can be established for any point in parameters can be established for any point in the tubing string; internal pressure, external pressure, axial loads due to the pipe weight, pressure, axial loads due to the pipe weight, axial loads due to pressure and packer forces, and bending loads. In tubing design and analysis, these parameters are often analyzed uniaxially or, as with combined tension and external pressure loading, biaxially, rather than triaxially. Under some conditions of loading, the uniaxial or biaxial evaluation predicts a stress that is higher than actually exists, resulting in a margin of safety that is higher than predicted, and thus a more conservative design. Under other load conditions, the uniaxial or biaxial evaluation predicts a stress that is lower than actually exists, resulting in a margin of safety that is lower than predicted.

TRIAXIAL THEORY OF FAILURE
For a triaxial evaluation, the failure predicting criteria applied to the tubing is the predicting criteria applied to the tubing is the distortion energy theory. In cylindrical coordinates, the equivalent stress is given by:
(1)
Failure is defined as when the equivalent stress exceeds the yield strength of the steel.
The Lame formulae for thick-walled cylinders are used to determine the radial stress and the tangential stress.
(2)
(3)
The principal stress in the axial direction, the applied axial stress, is calculated from the axial load conditions and constraints using conventional theory, to be discussed.