Computing Directional Surveys With a Helical Method (includes associated papers 6409 and 6410 )
Authors: Callas N.P. 
Published: December 2007

Callas, N.P., Colorado School ofMines, Golden, Colo.

Abstract
Before the introduction of the radius of curvaturemethod in 1968, departures were computed alongstraight-line segments with the so-called tangentialmethod. Since 1968 many variations of the curvaturemethod have been developed. This paper extendsavailable departure methods to a helical methodthat uses torsional as well as curvature informationin the raw directional-survey data. Fortran routineshave been written implementing this method. Thealgorithms presented here were used to implementthese routines.

Introduction
The raw directional-survey data collected at adrilling site consists of measurements of the driftdeclination angles from the vertical direction andthe drift directions of the borehole in the horizontalplane, using a plumb and compass (or gyro) atselected measurement stations. Further, the amountof drilling pipe in the hole gives the accumulatedlength of the borehole at each station. Translatedgeometrically, this information represents acollection of unit vectors, where (Ui, i = 1, 2,..., n, in three-dimensional space; each datumindicates the direction of the borehole at a measuredposition, where n represents the total number of station points. The direction vectors are represented as
(1)
The vector components or so-called direction cosinesare given byCx = COS beta sin delta, Cy = sin beta sin delta, andCz = cos delta....................(2)
where the angle beta is the drift direction measuredfrom due east in a counterclockwise direction anddelta is the drift declination angle measured from thedownward vertical direction. From these data thecourse of the borehole is to be integrated asaccurately as possible to obtain the horizontal andvertical departures of intermediate or bottom-hole positions.If station readings are taken sufficiently closetogether, it is evident that even the tangentialapproach for computing directional surveys wouldbe satisfactory to give acceptable departure results.Since the economics of the situation place a limiton the amount of data that may be collected, it isof interest to consider mathematical models for theborehole that account for its curvature and/ortorsional properties, thus better simulating its trueshape.Wilson's radius of curvature method provided thefirst published example of such a curvature model.Since 1968 many variations of this basic methodhave been developed and used. In 1973, Zarembadeveloped an explicit circular arcs method thatpieces together circular arcs that fit the raw dataat station points.
CIRCULAR ARCS MODEL-&gt; -&gt; -&gt;Let the set U1, U2, ...., Un represent the surveydata as a collection of consecutive boreholedirection vectors. Suppose that we know the x, y, and z coordinates of the kth station point, Pk, instandard rectangular coordinates, where the index kassumes any of the values k=1, 2,..., n - 1. Theorigin of the coordinate system may convenientlybe taken as the starting position of the borehole atground level. Further, we shall consider that thez axis of the coordinate system points straightdown, so that the z values of departures will besynonymous with the vertical departure of theborehole.The first objective of this paper is to modelthe kth segment of the borehole with a circulararc segment spanning the points Pk and Pk+ 1 insuch a manner that the direction vectors-&gt; -&gt; Uk and Uk + 1 are tangent to the circular arc atits respective end-points. The existence of sucha space arc is quite clear. To compute thecoordinates of the (k + 1) station point, Pk + 1, a rigid motion, three-dimensional transformation, T, will be performed on this arc to place it in acanonical or vertical frame of reference such that-&gt; the vector Uk is transformed into the vector (0, 0,1). (See Appendix A for T's functional representation.
SPEJ
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