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 both business and safety critical. 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 standardisation is needed.

Investigation shows that iterative schemes have been used where 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 co-ordinate reference frames, toolface, interpolation, intersection with a target plane, minimum and maximum 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 behaviour. 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.

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