7.3.5 Minimum-Curvature Splines
The minimum-curvature interpolation method, also known as the circular arc method, is used to describe the shape of a wellbore trajectory interpolated between stations (knots) on which the dip and azimuth are specified. The current exposition largely follows that of Taylor and Mason (1972), Zaremba (1973) and the more recent discussions by Sarawyn and Thorogood (2005), suitably generalized beyond the wellbore trajectory domain. There are a number of differences between this mathematical exposition and industry practice:
- The current calculation is performed in the local 3D projected CRS, and ignores curvature effects which may be important, especially for extended reach wells.
- The current calculation does not assume that the parameterization of the spline is with
respect to the measured depth along the trajectory. As a consequence, just as with all of
the other RESQML splines, the tangent vectors (derivative of position with respect to
parameter) must be specified in the local 3D CRS, and need not be unit vectors specified
solely by dip and azimuth.
- In the context of a well trajectory representation, the local 3D CRS is usually not rotated with respect to the projected CRS. As a consequence, the local Y axis is aligned with the projected North direction.
- This generalization allows an interpolant to be defined with mixed units of measure. However, no unit conversions are implied so care must be taken in such cases.
The algorithm is posed as follows. Given an initial station for which we know position, measured depth (arc length), and tangent vector, and a next station for which we know measured depth and tangent vector, what is the position of the next station based upon a minimum-curvature interpolant? Translating from station to spline knot, from measured depth to parameter values, and from dip and azimuth angles to tangent vectors, we have the following expression for the minimum-curvature trajectory:
where, is defined by . The next position is given by :
If the two tangents vectors are close to parallel then will be close to 0, and some care has to be taken when evaluating these expressions. In that limit:
Additional higher order terms in may also be included to improve upon this approximation. Again, in the context of a wellbore trajectory representation, p is interpreted as the measured depth, Δp as the incremental measured depth, and the tangent vectors will be unit vectors.