A Comparison of Various Directional Survey Models and an Approach To Model Error Analysis
Authors: Walstrom J.E. , Harvey R.P. , Eddy H.D. 
Published: December 2007

The classic terminal angle method of directional survey calculation is grossly susceptible to error, and the recommendation is that it be abandoned. There are too many alternatives that are better. One of them is the balanced tangential method, which has proved highly efficient in more than three years of use.

Introduction
It is often desirable to know the position of specified points on the axis of a wellbore relative to the points on the axis of a wellbore relative to the tophole location. The current interest in directional survey calculations is related to the increased number of highly deviated holes drilled from offshore platforms and Arctic or urban drillsites. Positional platforms and Arctic or urban drillsites. Positional accuracy is important in the following situations:
when two wellbores approach each other closely and it is important to avoid an intersection; when a relief well is drilled to a wild well and it is important to achieve intersection, or near intersection, at some specified depth in the wild well; when the bottom of a well, or perhaps some intermediate point, is close to a property boundary; and when making equity participation calculations. participation calculations.
Suppose, as in the conventional manner, readings of inclination angle, I, with the vertical, and azimuthal angle, A, with the north, are taken at a number of specified depths in the wellbore. Each pair of measurements defines the direction of the tangent to the axis of the wellbore at the corresponding point or station. Suppose there are n + 1 stations along the wellbore, including the top of the hole, so that there are n station intervals. Suppose such data are provided in triads of readings Ii, Ai and Li, where Ii provided in triads of readings Ii, Ai and Li, where Ii and Ai are in radians, Li is the distance in feet along the axis of the wellbore from the top of the hole to station i, and where i = 0, 1, 2.... n, so that Lo = 0. The distance between two stations measured along the axis of the wellbore is given by Si =Li - Li -1 with So = zero.
The position of points along the axis of the wellbore is calculated with respect to a coordinate reference frame that has its origin at the top of the wellbore. The positive X direction is east, the positive Y direction is north, and Z is measured vertically downward. The computations used in all directional survey methods are based on the following three differential equations:
dX = sin I sin A dLdY = sin I cos A dLdZ = cos I dL
where L is the distance along the axis of the wellbore.
There is no calculational procedure that can be expected to determine the bottom-hole position relative to the top-hole position of a wellbore with complete certainty. Errors arise in two ways:
there are errors inherent in the calculational procedure used to determine the bottom-hole position, principally because the original data set is incomplete in terms of defining the hole; and random errors occur in the readings of I, A, and L.
In this paper we use the terms "model", "method", and "calculational procedure" interchangeably. procedure" interchangeably. The present paper is confined mainly to a discussion of different calculational procedures and is, therefore, pertinent to the effect of errors of the first type. A statistical analysis is given in Ref. 2 that deals with consequences of errors of the second type.
JPT
P. 935