# SPE Drilling & Completion Volume 20, Number 1, March 2005, pp. 24-36

SPE-84246-PA

### A Compendium of Directional Calculations Based on the Minimum Curvature Method

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DOI  10.2118/84246-PA http://dx.doi.org/10.2118/84246-PA

### Citation

• Sawaryn, S.J. and Thorogood, J.L. 2005. A Compendium of Directional Calculations Based on the Minimum Curvature Method. SPE Drill & Compl20 (1): 24-36. SPE-84246-PA.

### Discipline Categories

• 1.4.3 Downhole Operations (Casing, Cementing, Coring, Geosteering, Fishing)
• 1.2 Drilling Design and Analysis
• 1.1 Drilling Project Management

### 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.

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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