The point to target calculations contained in part one of the compendium1 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 based on 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 whip-stocking 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 to either minimise the distance to the target or limit the dog leg severity to avoid drill-pipe 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 on 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.