Summary
The minimum curvature method has emerged as the accepted industry standard
for the calculation of 3D directional surveys. Using this model, the well’s
trajectory is represented by a series of circular arcs and straight lines.
Collections of other points, lines, and planes can be used to represent
features such as adjacent wells, lease lines, geological targets, and faults.
The relationships between these objects have simple geometrical
interpretations, making them amenable to mathematical treatment. The
calculations are now used extensively in 3D imaging and directional collision
scans, making them critical for both business and safety. However, references
for the calculations are incomplete, scattered in the literature, and have no
systematic mathematical treatment. These features make programming a consistent
and reliable set of algorithms more difficult. Increased standardization is
needed.
Investigation shows that iterative schemes have been used in situations in
which explicit solutions are possible. Explicit calculations are preferred
because they confer numerical predictability and stability. Though vector
methods were frequently adopted in the early stages of the published
derivations, opportunities for simplification were missed because of premature
translation to Cartesian coordinates.
This paper contains a compendium of algorithms based on the minimum
curvature method (includes coordinate reference frames, toolface,
interpolation, intersection with a target plane, minimum and maximum true
vertical depth (TVD) in a horizontal section, point closest to a circular arc,
survey station to a target position with and without the direction defined,
nudges, and steering runs). Consistent vector methods have been used throughout
with improvements in mathematical efficiency, stability, and predictability of
behavior. The resulting algorithms are also simpler and more cost effective to
code and test. This paper describes the practical context in which each of the
algorithms is applied and enumerates some key tests that need to be
performed.
Introduction
The first reference to the minimum curvature method is credited to Mason and
Taylor1 in 1971. In the same year, Zaremba2 submitted an identical algorithm
that he termed “the circular arc” method. In the minimum curvature method, two
adjacent survey points are assumed to lie on a circular arc. The arc is located
in a plane, the orientation of which is defined by the known inclination and
direction angles at the ends. By 1985, the minimum curvature method was
recognized by the industry as one of the most accurate methods, but was
regarded as cumbersome for hand calculation.3,4 The emergence of
well-trajectory planning packages to help manage directional work in dense well
clusters increased its popularity. It was natural to use the same model for
both the surveys and the segments of the well-plan trajectories. Today, with
the widespread use of computers, computational power is no longer an issue, and
the method has emerged as the accepted industry standard.
© 2005. Society of Petroleum Engineers
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History
- Original manuscript received:
27 January 2004
- Revised manuscript received:
15 January 2005
- Manuscript approved:
29 January 2005
- Version of record:
15 March 2005
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