An Improved Method for Computing Wellbore Position Uncertainty and its Application to Collision and Target Intersection Probability Analysis
Authors: Brooks Andrew G. , Wilson Harry 
Published: December 2007

Abstract
As modern drilling projects continue to include increased use of extended-reach wellbores directed to smaller targets, the need for an accurate assessment of uncertainty in bottom-hole location is becoming increasingly critical.
Incorrect assessment of the probability of intersecting the target can lead to an equally incorrect assessment of the viability of the project.
Most published methods for computing wellbore position uncertainty are based on the analysis of systematic errors in inclination, azimuth and measured depth. The underpinning of such analyses is that these various error terms are uncorrelated constants, but this assumption may not always be justified. The technique has therefore been generalized to make use of more fundamental input error terms. and to take account of the probabilistic nature of such terms, thereby calculating an ellipsoidal probability field for each point along the well. Examples are presented which illustrate the advantages of this method, including the ability to take into account the rotation of an electronic magnetic survey instrument such as an MWD tool.
When one survey tool is followed in the hole by another, some error terms may behave in a systematic fashion across the tie-on point while others combine randomly. Since the computed uncertainty in bottom-hole location depends on the degree of correlation between one set of errors and the next, a flexible means is suggested for accommodating such tie-ons.
Since a probability can be assigned to the hypothesis that a point in a planned or drilled wellbore occupies a given volume of space, this method can also be used to determine the probability that a particular point in the well might intersect an adjacent wellbore or an arbitrary target volume. This makes possible improved computation of the probabilities of wellbore collision or target penetration.

Introduction
The calculation of wellbore position uncertainty has been addressed by Walstrom et al and by Wolff and deWardt. It is commonly accepted today that such uncertainty is dominated by systematic errors, thus most popular models are based on the systematic analysis outlined by Wolff and deWardt. With extensive field usage of these models over the years, some limitations have become apparent. These include the deterministic nature of the systematic model as originally described, the definition of most instrument errors in terms of fixed inclination and azimuth uncertainties, failure to include random errors as well as systematic, and the inability to assign a probability to borehole position. A modified treatment of uncertainty which overcomes these limitations is described here. Although several of these improvements have been used for many years by a number of companies, until now there has been little documentation of them in the literature.

Calculation of Wellbore Position Uncertainty
Wellbore surveys are typically performed at a number of discrete survey stations along the course of the well by measuring components of the earth's gravity field and either the local magnetic field or a rotation vector. An instrument performance model is used to convert these raw measurements to inclination I, and azimuth A, which together with the along-hole depth L, make up a three-component measurement vector p. A wellbore trajectory model is then employed to convert the set of measurement vectors into a position vector r in a north, east, and vertical (N, E, V) coordinate system.
An instrument performance model includes potential sources of error or uncertainty in the measurement. P. 411