Summary
The point-to-target calculations contained in Part 1 of the compendium
(Sawaryn and Thorogood 2005) were restricted to those cases in which the 3D
target coordinates were stated explicitly. However, another class of problems
exists where the target's structural position is determined indirectly from
other constraints defining the arc's orientation. This new paper builds on the
earlier work and contains complete details of twelve explicit algorithms,
including the calculation of a target on the basis of the toolface at the start
or end of an arc and for the landing of a well path parallel to a formation
bedding plane.
These cases find practical application in computing trajectories of motor
runs and in whipstock operations. The curvature of the trajectory landing on a
bedding plane varies with direction and has a distinct minimum and maximum.
These values may be used either to minimize the distance to the target or to
limit the dogleg severity (DLS) to avoid drillpipe fatigue, a concern in
short-radius drilling applications.
The algorithms add to the compendium and are useful extensions to the
engineer’s computational toolkit. The paper shows that the trajectories,
computed using the minimum-curvature calculation, are contained within the
geometric surfaces defined by a torus or cyclide. These geometries explain why
the explicit solutions exist, opening up possibilities for obtaining solutions
to the more-complex cases.
Introduction
The purpose of the compendium is to provide a consistent set of algorithms
related to the minimum-curvature calculation method, providing explicit
expressions where possible. The advantage of explicit expressions is that they
provide a ready means of calculating the critical values delimiting the regions
where solutions may be found. A graphical presentation of the geometries
involved is an effective means of identifying the mathematical behavior and
multiple solutions inherent in these calculations (Gray 1999). Several examples
of these techniques are provided.
It is assumed that the reader is familiar with the algorithms described in
Part 1 of the compendium, and the reference to the equations it contains is
made by enclosing them in square brackets, for example [Eq. A-74]. The angle
α subtending the arc may assume values such that 0 ≤ α <
π and it is also assumed that the start points and endpoints of the arc
are not coincident. For convenience, the main vector operations and common
methods are summarized in Appendix A.
© 2009. Society of Petroleum Engineers
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History
- Original manuscript received:
27 July 2007
- Meeting paper published:
11 November 2007
- Revised manuscript received:
22 April 2008
- Manuscript approved:
28 April 2008
- Published online:
1 June 2009
- Version of record:
1 June 2009
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