Explicit Calculation of Expansion Factors for Collision Avoidance Between Two Coplanar Survey-Error Ellipses
- Steven J. Sawaryn (BP Exploration Operating Company Limited) | Angus L Jamieson (Tech21) | Andrew E. McGregor (Tech21)
- Document ID
- Society of Petroleum Engineers
- SPE Drilling & Completion
- Publication Date
- March 2013
- Document Type
- Journal Paper
- 75 - 85
- 2013. Society of Petroleum Engineers
- 4.1.2 Separation and Treating, 4.3.4 Scale, 6.1 HSSE & Social Responsibility Management, 1.9 Wellbore positioning, 4.1.5 Processing Equipment
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- 511 since 2007
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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. With the use of 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 standards organizations to define minimal acceptable separation distances between wellbores (e.g., the Norwegian NORSOK D-10 Standard and Oil and Gas UK Well Integrity Guidelines). 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. This paper presents explicit equations for determining the exact condition in which the ellipses touch, expressing the result as an expansion scale factor. Methods are presented for the expansion of either ellipse or both, 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 while 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. Because 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 3 November 2011.
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Bromwich, T. 1906. Quadratic Forms and Their Classification by Means ofInvariant-Factors (Cambridge Tracts in Mathematics and MathematicalPhysics, Vol. 3). Cambridge: Cambridge University Press.
Choi, Y.K. 2008. Collision Detection for Ellipsoids and Other Quadrics. PhDThesis, University of Hong Kong, Pokfulam, Hong Kong (March 2008).
Herbison-Evans, D. 2011. Solving Quartics and Cubics for Graphics, TechnicalReport TR94-487 (updated 31 March 2011), University of Sydney, Australia.
Snapper, E. and Troyer, R.J. 1971. Metric Affine Geometry. 1, 36-55.London: Academic Press.
Thorogood, J.L. and Sawaryn, S.J. 1991. The Traveling-Cylinder Diagram: APractical Tool for Collision Avoidance. SPE Drill Eng 6(1): 31-36. http://dx.doi.org/10.2118/19989-PA.
Wang, W., Wang, J., and Kim, M.-S. 2001. An Algebraic Condition for theSeparation of Two Ellipsoids. Computer-Aided Geometric Design 18 (6): 531-539. http://dx.doi.org/10.1016/S0167-8396(01)00049-8.
Weisstein, E.W. 2012. Ellipse, http://mathworld.wolfram.com/Ellipse.html(last updated 12 December 2012).
Williamson, H.S. 1998. Towards Risk-Based Well Separation Rules. SPEDrill & Compl 13 (1): 47-51. http://dx.doi.org/10.2118/36484-PA.
Williamson, H.S. 2000. Accuracy Prediction for Directional Measurement WhileDrilling. SPE Drill & Compl 15 (4): 221-233. http://dx.doi.org/10.2118/67616-PA.
Zheng, X., Iglesias, W., and Palffy-Muhoray, P. 2009. Distance of ClosestApproach of Two Arbitrary Hard Ellipsoids. Phys. Rev. EStat. Nonlin.Soft Matter Phys. 79: 057702. http://dx.doi.org/10.1103/PhysRevE.79.057702.
Zheng, X. and Palffy-Muhoray, P. 2007. Distance of Closest Approach of TwoArbitrary Hard Ellipses in Two Dimensions. Phys. Rev. E Stat. Nonlin. SoftMatter Phys. 75: 061709. http://dx.doi.org/10.1103/PhysRevE.75.061709.