Explicit Calculation of Expansion Factors for Collision Avoidance between Two Co-planar Survey Error Ellipses

Abstract

The positional uncertainty about a point on a wellbore is commonly represented as an ellipsoid. The ellipsoid also accounts for the dimensions of the casing or open hole. Using this model, at any time the resulting uncertainty about a wellbore along its trajectory is a curved, continuous cone. To a good approximation, the intersection of the plane normal to a reference well with these cones can be represented as ellipses. This simple geometrical model has been adopted by various standards organisations to define minimum acceptable separation distances between well bores, for example the Norwegian NORSOK D-10 standard. Because of mathematical difficulties, the existing methods for calculating the resulting separation factors are only approximations and may be either too optimistic or too conservative, particularly for ellipses with high eccentricities. The paper presents explicit equations for determining the exact condition where the ellipses touch, expressing the result as an expansion scale factor. Methods are presented for the expansion of either one, or both ellipses, together with implementation notes and other associated tools. The new algorithms are only marginally less efficient than the existing approximation methods and they can be used to increase the allowable proximity of two adjacent wells whilst satisfying the geometrical and probabilistic constraints. The examples included in the paper illustrate this. The proposed calculation method is consistent with existing industry wellbore uncertainty models. Since the determination of the osculating condition is exact, the calculation is neither too optimistic nor too conservative. This paper is a response to discussions held at the SPE Wellbore Positioning Technical Section meeting on 3rd November 2011.