An Analysis of Uncertainty in Directional Surveying

Abstract

Knowledge concerning the position of the bottom of a directionally drilled hole is uncertain. This uncertainty can be measured quantitatively either by analysis based on principles of probability and statistics, or by Monte Carlo simulation using a computer that yields a pattern of possible bottomhole locations. Introduction and Model Description In view of the large number of wells directionally drilled from offshore platforms and urban drillsites, greater interest is being focused on the directional survey and on the uncertainty inherent in calculating the position of the bottom of the hole, or other benchmarks in the wellbore. Of the two methods presented here for evaluating this uncertainty, the first is analytical and uses the principles of probability and statistics. The second method uses Monte Carlo simulation to generate a pattern of the scatter of possible bottom-hole locations. possible bottom-hole locations. Both methods are based on a model for which the usual assumption is made that the curvilinear axis of the wellbore is adequately represented by a large number, M, of linear segments joined end to end from the top to the bottom of the hole and closely following the course of the hole. The errors due to this approximation are assumed to be negligible and are not studied here. One form of the model is to consider that for a given hole there exists a perfect set of data describing it, of the above form. Each linear segment has an angle of inclination with the vertical Ii in radians, an azimuthal angle from true north Ai in radians, and a length Si, in feet. The subscript variable, i, extends from 1 to M. Each of the M linear segments representing the axis of the wellbore has three components in the X, Y, and Z directions. These components are calculated from the equations:....................(1) The bottom-hole coordinates as determined from the reference set of data for the wellbore are given by the following for i = 1 to M:..........(2) A directional survey instrument, lowered into the wellbore to the M segments, or stations, measures the values of the Ii and the Ti. In general, because of errors from various sources, the resulting angular measurements will not be precisely equal to the I, and Ai. Random variables Ii precisely equal to the I, and Ai. Random variables Ii and Ai are therefore considered with associated probability density functions fi(Ii) and gi(Ai), probability density functions fi(Ii) and gi(Ai), respectively, which represent the distributions of potential survey measurements for this hole. Since it is assumed that the values Si are accurately measured they are treated deterministically. The random variables corresponding to the surveyed bottom-hole coordinates are given, for i = 1 to M, by:............(3) P. 515