Summary
The need to improve completion reliability has prompted a second, deeper
look at the analysis of tubing buckling. The conventional-buckling analysis is
assumed to hold "far from the packer", but what happens "near"
the packer?
The original development by Lubinski and Woods used the method of virtual
work to determine a constant pitch helix. All attempts to connect a constant
pitch helix to boundary conditions, such as a packer, have failed. The
conventional wisdom says that the virtual work solution applies "far from
the packer". How far is this, and what happens between the packer and this
point? Even worse is the case of tapered strings. The generally accepted answer
to the tapered string problem is a "stacked" set of Lubinski solutions,
with each solution far from the interconnection of the different sized tubulars
(1962).
Perhaps there are nonconstant pitch solutions? Because the
buckling-differential Eq. is nonlinear, it is not surprising that no other
analytic solutions have been discovered until recently. Once these solutions
are known, the initial value problem can be posed and solved. Two possible
boundary conditions are considered in this study: a cantilever boundary
condition to model a packer, and a "pinned" boundary condition to model
a centralizer. For the first time, analytic solutions were found for both
problems, and interestingly, both solutions converged rapidly to Lubinski's
constant pitch solution at a specified distance from the boundary. For this
reason, many of the original Lubinski calculations remain essentially correct,
(such as length change and contact forces). Alternatively, the bending moment
near the boundary exceeds the bending moment calculated from the Lubinski
solution by approximately 20%. For this reason, conventional pipe-buckling
stress analysis must be revised. Further applications of these solutions allow
different sized tubulars to be interconnected, solving the tapered-string
problem for the first time. An additional, useful result is the solution to the
helix "reversal" problem.
The principal applications for these solutions are the stress analysis and
design of tubing and casing strings. The boundary solutions are simple
formulas, suitable for spreadsheet or hand calculations.
Introduction
To improve completion reliability, there has been increasing use of
"fixed string" completions across the industry, where the tubing is
fixed at the tubing hanger and packer, rather than incorporating expansion
joints that may prove vulnerable to leakage causing reduced-completion service
life. Furthermore, there has been increasing uptake of hydrostatic-set packers
that eliminate the requirement for wireline or coiled tubing to set a tailpipe
plug. Although the hydrostatic packer has proven popular, it creates additional
compressive loads in the tubing above the packer. This compares with a
hydraulic-set packer that reduces tubing compressive loads, in that tubing
pressure is applied against a tailpipe plug to set the packer.
Wider awareness of the buckling condition in recent years has provided
impetus for tubing manufacturers to develop completion connections with a
better service rating in compression. Most vendors now offer 100%
connection-compression efficiency, where the connection has been qualified to
operate with compressive loads as high as the rated-tensile capacity of the
pipe.
The need to improve completion reliability has prompted a second, deeper
look at the analysis of tubing buckling. When solving a mechanical equilibrium
problem, the engineer is usually confronted with a system of differential Eq.s
and a set of boundary conditions. The tubing buckling problem is seldom posed
this way. Why should this be the case?
This is partly because of historical reasons. The first generally accepted
buckling solution was developed by Lubinski and Woods in 1962. They posed a
virtual-work problem to determine the pitch of the helically buckled pipe.
Others followed the same path, and this is generally the way tubing-buckling
problems are solved today. Examples of this style of analysis are given in
Cheatham et al. (1984); He and Kyllingstad (1993); and Miska et al. (1995ab).
The second reason is that the differential Eq.s describing tubing buckling are
nonlinear, and are therefore difficult to solve analytically. Only recently
have general solutions to this problem become available [see Appendix A
(Mitchell 1982)]. The final reason, as will be shown, is that posing and
solving these boundary-value problems are often tedious and difficult.
In the Lubinski method, the boundary conditions are not considered
explicitly, but are embedded in the formulation in a subtle way. Examination of
the formulation clearly shows that the boundary conditions are not consistent
with typical packers. The conventional wisdom is that the buckling solution
applies far from the packer (1962).
Few authors were aware that a solution near the packer was needed. However,
a comprehensive analysis of tubing buckling consistent with boundary conditions
at a packer or a centralizer was never done in a completely satisfactory way.
Attempts were made to connect the constant pitch-Lubinski solution to a
beam-column solution that brings the pipe from the wellbore wall to the packer
(Mitchell 1982). The following conditions must be met where the two solutions
join
- wellbore contact
- wellbore tangency
- continuity of curvature
- continuity of shear tangent to wellbore
- positive contact force between pipe and wellbore
- all pipe displacements within the borehole
These six conditions cannot all be satisfied for a constant-pitch helix
joined to a single beam-column section. Sorenson was able to connect a
two-section beam-column solution to a numerical solution of the full-contact
buckling problem (1986). He chose one boundary condition to be Lubinski's
solution for the extreme end of the pipe. This solution only approximated the
constant-pitch solution in the limit of a long pipe, and only at the extreme
end of the pipe. Did the use of this boundary condition impose Lubinski's
solution on this problem? Application of this model has been extremely limited
because of the complexity of the solution.
A major roadblock to understanding the effect of the packer on buckling was
the seeming incompatibility of the constant-pitch helix solution with the
beam-column solution. The constant-pitch solution has only a single unknown
constant, the helix pitch. Together with the constants in the beam-column
solution, there are not enough degrees of freedom to join the two solutions.
Fortunately, recent work has found a family of analytic solutions to the
full-contact buckling problem that allow the solution of general initial-value
problems or a solution to the beam-column connection problem (Mitchell
1982).
This paper discusses the general solutions to the pipe-buckling problem,
explicitly states the constraints on the solution imposed by several models for
pipe-packer interaction, and solves for the full-contact solution for each
case. The properties of each solution are examined in depth. The effects
produced by cantilever packers and centralizers, (such as contact forces and
bending stresses) are developed in detail. The location and magnitude of
maximum bending stresses are determined.
This paper highlights that existing models underestimate buckling-bending
stresses directly above the packer by some 20%. Once this 20% error is
recognized, it is not surprising to learn that there are field examples of
tubing leaks and early completion failures caused by buckling above the
packer.
© 2008. Society of Petroleum Engineers
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History
- Original manuscript received:
12 July 2005
- Meeting paper published:
9 October 2005
- Revised manuscript received:
7 January 2008
- Manuscript approved:
10 January 2008
- Version of record:
20 June 2008
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