Directional Survey by the Circular Arc Method
Authors: Zaremba W.A. 
Published: December 2007

Abstract
A method for computing directional surveys is presented that eliminates the abrupt changes in presented that eliminates the abrupt changes in direction of the tangent to wellbore. For any well course not located in a single plane the twist of the representing curve will still experience discontinuities, but the effect may be expected to be small in practical cases. Ease of interpolation between the computed location of station points is retained, since the coordinates of any intermediate point are explicitly expressible in terms of the point are explicitly expressible in terms of the measured depth increment to such point, while the in-hole depth needed to reach any given horizon is uniquely determined, provided that the latter is at an intermediate depth to that of some two consecutive station points. The formulas are somewhat tedious for any extended routine hand calculation, but a digital computer program can be constructed to carry it out, and a copy of one written in ALGOL 60 is included here.

Introduction
Accurate directional surveys are required for the correlation of bottom-hole location with the geological data, for avoiding interference with existing wells while drilling, and for possible assessment of property rights in jurisdictional disputes. Though the well course is ultimately determined on the basis of simultaneous measurements of hole direction and depth at a number of survey points, the true shape between those "stations" is unknown; hence, some reasonable assumptions must be made as to its nature. Thus, various interpretations of the same data may lead to different computed locations of the same point, and the question of which one to choose becomes important. Unfortunately, detailed comparisons of the effects of various possible assumptions on the predicted positions vs those known from closely predicted positions vs those known from closely spaced and accurate surveys seem to be lacking; hence, no definite answer can be given.
Quite apart from shape uncertainty, both the angle and the depth data are also subject to measurement inaccuracies due to small instrumental errors; thus, other things being equal, the method minimizing the effect of those errors is preferable provided that we can be reasonably sure it makes provided that we can be reasonably sure it makes no unrealistic assumptions and, as a practical tool, is not too costly to use.
The commonly used and simplest solution is to take the well course between consecutive stations as a series of straight segments-each one inclined and directed as measured at its deeper end. This so-called "tangent projection" suffers from an obvious defect that all changes of direction are assumed to be concentrated at the survey points, whereas we can expect from physical considerations such as drill pipe rigidity, etc., that those changes are actually distributed over a considerable portion of each segment.
It has been proposed by Wilson to inarpolate between consecutive stations by segments of a space curve formed from a circular arc wrapped around on a vertical right circular cylinder, the two radii (arc and cylinder) being chosen so as to match the measured direction of the hole at two respective survey points, while the length of curve is equal to the difference of measured depths. This "radius of curvature" method removes abrupt tangent vector changes, but the intrinsic shape of the interpolating curve is, in general, a rather complex space helix with the total curvature varying from point to point.
In this article, we want to point out that it is possible to retain the tangent continuity across possible to retain the tangent continuity across survey points more simply by fitting true circular arcs between them. Each arc is, in general, located on a skew plane whose orientation can be easily determined from the known inclination and direction angles at each end, while the radius of arc follows from the requirement that the developed length of curve be the same as the measured separation of relevant station points. For any well located in a vertical plane, both Wilson's approach and the current method give identical results; but if there is an out-of-plane component the computed locations will diverge.
It must be stressed here that neither method removes discontinuities in the twist of the constructed space curve, while both retain ease of interpolating for the locations between station points, since the measured and the vertical depths points, since the measured and the vertical depths are explicitly solvable functions of each other.