Use Of An Exact Mathematical Formulation To Plan Three Dimensional Directional Wells
Authors: Planeix Michele Y. , Fox Richard C. 
Published: December 2007

Abstract
The paper briefly reviews common planning practices for three-dimensional directional wells, and practices for three-dimensional directional wells, and shows the need for a better method. Exact mathematical expressions are derived for departures in (X, Y, Z) at any point of a constant build rate - turn rate well. An exact expression for instantaneous dogleg is included. A resolution algorithm is derived for finding the final angle and direction of a well which turns and builds angle to get from a known surface location to a known target. Emphasis is placed on the fact that the method eliminates unnecessary interpolation completely and trial-and-error to a degree. A computer program (on CMS) using the method has been prepared.

Introduction
Although a review of the literature on directional calculation methods yields a wealth of papers on "survey computations" (See references), few authors to date have addressed "directional planning", particularly for "three-dimensional wells". particularly for "three-dimensional wells". G. J. Wilson, however, suggests a mathematical method for vertical plane wells (two-dimensional) in the appendix to his "Improved Method for Computing Directional Surveys" paper. This type of logical, minimum error approach is most desirable for three-dimensions as well.
From discussions with some of the leading directional service companies, it has become apparent that in the best of cases a three-dimensional well plan will be handled by a trial-and-error method. A starting point for build and turn and a build rate and turn rate are chosen (or specified by the operator). From these an artificial survey will be generated, at say 100' intervals of the measured depth, incrementing hole angle and direction accordingly. Departures in TVD, east-west and northsouth are then calculated using one of the widely accepted interpolation methods of survey computation. By trial and error a final angle and a final direction are found when it is possible to make the well curve fit the target coordinates.
This method can even be programmed for the computer, and has been. Unfortunately, it does not handle the case where the build ends before the turn does. In this case, below the end-of-build point the well course is still curved. A projection downhole cannot be made along a straight line above this point to verify if the hole angle is "lined out". The projection should follow the actual curve. An projection should follow the actual curve. An interpolation method is not well suited to this type of problem. problem. The following work was pursued with the conviction that there must be a "better way" to plan three-dimensional directional wells. The basic mathematical equations can be found in most of the literature. It is the use of their integrated form to plan wells which are not in a vertical plane which constitutes what we believe to be a "better way".
By three-dimensional at the planning stage we mean a well course which turns more than 20 degrees over a substantial distance making it impossible to contain the well in one vertical plane over that distance.

EXACT DEPARTURES FORMULATION
The position of any point along a borehole/well curve is defined by its curvilinear distance, S (measured depth) and a three component vector r(S), having the top of the hole for origin. If we define the three components X(S), Y(S), and Z(S) of r(S) as:
X: Departure along N-S axis (north positive) Y: Departure along E-W axis (east positive) Z: True vertical depth (positive down)