A Compendium of Directional Calculations Based on the Minimum Curvature Method

Abstract

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.

Wellbore-Trajectory Control by Use of Minimum Well-Profile-Energy Criterion for Drilling Automation

“…When the actual well-trajectory data (MD A , a A , / A ) are received, the actual coordinates (X A , Y A , Z A ) will be calculated by use of wellbore-trajectory-estimation models; examples include the industrystandard minimum-curvature model (Zaremba 1973;Sawaryn and Thorogood 2005) or the simple balanced tangential model (Walstrom et al 1972).…”

“…With fuzzy logic, a controller can be designed to use experience and common-sense knowledge because the rule-of-thumb method exists primarily in the form of linguistic statements that are usually, but not always, true. Stoner (1997) developed a fuzzy-logic drilling-direction controller (FDDC) for a closed-loop rotary-directional-drilling system. The controllable variables of the drilling system are the eccentricity settings of a nonrotating, near-bit downhole adjustable stabilizer.…”

“…Compared with Stoner's (1997) eight input parameters, their model seems to be much simpler because of significantly fewer input parameters. Xue et al (2012) compared the simulation results of step response of fuzzy control with that of traditional PID control.…”

Kick-Off Point (KOP) and End of Buildup (EOB) Data Analysis in Trajectory Design

“…If the points are deviated, borehole will be directed to the point that already set before. There are many methods that can be used to determine suface point coordinates, tangential, balance tangential, average angle, radius of curvature, minimum of curvature, and mercury method (Inglis, 2005) (Zaremba, 1973) (Wilson, 1968 (Sawaryn and Thorogood, 2005). The calculation of each methods based on measurement of 3 kinds of parameter.…”

“…The first reference to the minimum curvature directional survey calculation method is credited to Mason andTaylor in 1972. (Sawaryn andThorogood, 2005) (Taylor and Mason, 2013) Minimum of curvature method seperate 2 intervals where the upper interval and lower interval. For the upper interval, inclination angle and azimuth is being used at the initial point.…”