# SPE Drilling & Completion Volume 24, Number 2, June 2009, pp. 311-325

SPE-110014-PA

### A Compendium of Directional Calculations Based on the Minimum Curvature Method--Part 2: Extension to Steering and Landing Applications

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DOI  10.2118/110014-PA http://dx.doi.org/10.2118/110014-PA

### Citation

• Sawaryn, S.J. and Tulceanu, M.A. 2009. A Compendium of Directional Calculations Based on the Minimum-Curvature Method--Part 2: Extension to Steering and Landing Applications. SPE Drill & Compl  24 (2): 311-325. SPE-110014-PA. doi: 10.2118/110014-PA.

### Discipline Categories

• 1 Drilling and Completions

### Keywords

• directional, trajectory, steering, landing, geometry

### 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.

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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