Computing Directional Surveys With a Helical Method (includes associated papers 6409 and 6410 )
- N.P. Callas
- Document ID
- Society of Petroleum Engineers
- Society of Petroleum Engineers Journal
- Publication Date
- December 1976
- Document Type
- Journal Paper
- 327 - 336
- 1976. Society of Petroleum Engineers
- 1.5.1 Surveying and survey programs, 1.6 Drilling Operations, 1.6.6 Directional Drilling, 1.9.4 Survey Tools
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- 223 since 2007
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Callas, N.P., Colorado School of Mines, Golden, Colo.
Before the introduction of the radius of curvature method in 1968, departures were computed along straight-line segments with the so-called tangential method. Since 1968 many variations of the curvature method have been developed. This paper extends available departure methods to a helical method that uses torsional as well as curvature information in the raw directional-survey data. Fortran routines have been written implementing this method. The algorithms presented here were used to implement these routines.
The raw directional-survey data collected at a drilling site consists of measurements of the drift declination angles from the vertical direction and the drift directions of the borehole in the horizontal plane, using a plumb and compass (or gyro) at selected measurement stations. Further, the amount of drilling pipe in the hole gives the accumulated length of the borehole at each station. Translated geometrically, this information represents a collection of unit vectors, where (Ui, i = 1, 2, ..., n, in three-dimensional space; each datum indicates the direction of the borehole at a measured position, where n represents the total number of station points. The direction vectors are represented as
The vector components or so-called direction cosines are given by Cx = COS beta sin delta, Cy = sin beta sin delta, and Cz = cos delta....................(2)
where the angle beta is the drift direction measured from due east in a counterclockwise direction and delta is the drift declination angle measured from the downward vertical direction. From these data the course of the borehole is to be integrated as accurately as possible to obtain the horizontal and vertical departures of intermediate or bottom-hole positions. If station readings are taken sufficiently close together, it is evident that even the tangential approach for computing directional surveys would be satisfactory to give acceptable departure results. Since the economics of the situation place a limit on the amount of data that may be collected, it is of interest to consider mathematical models for the borehole that account for its curvature and/or torsional properties, thus better simulating its true shape. Wilson's radius of curvature method provided the first published example of such a curvature model. Since 1968 many variations of this basic method have been developed and used. In 1973, Zaremba developed an explicit circular arcs method that pieces together circular arcs that fit the raw data at station points.
CIRCULAR ARCS MODEL -> -> -> Let the set U1, U2, ...., Un represent the survey data as a collection of consecutive borehole direction vectors. Suppose that we know the x, y, and z coordinates of the kth station point, Pk, in standard rectangular coordinates, where the index k assumes any of the values k=1, 2,...,n - 1. The origin of the coordinate system may conveniently be taken as the starting position of the borehole at ground level. Further, we shall consider that the z axis of the coordinate system points straight down, so that the z values of departures will be synonymous with the vertical departure of the borehole. The first objective of this paper is to model the kth segment of the borehole with a circular arc segment spanning the points Pk and Pk+ 1 in such a manner that the direction vectors -> -> Uk and Uk + 1 are tangent to the circular arc at its respective end-points. The existence of such a space arc is quite clear. To compute the coordinates 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 a canonical or vertical frame of reference such that -> the vector Uk is transformed into the vector (0, 0, 1). (See Appendix A for T's functional representation.
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