An Exact Continuous Interior Ellipse of Uncertainty for Minimum-Curvature Wellbore Surveys

An open, validated method for the position covariance (ellipse of uncertainty) of a wellbore survey at any measured depth between stations. The ISCWSA/OWSG error model defines the covariance at survey stations; between them, practice either interpolates the assembled covariance or adds a statistical micro-dogleg term, neither being the exact covariance of the resolved arc. This paper propagates each error source onto the minimum-curvature arc through the partial-leg Jacobian, giving the covariance C(f) as a closed form that is exact at the stations and continuous between them. It is validated against an interpolated-position Monte-Carlo oracle: the analytical-vs-Monte-Carlo difference falls as 1/sqrt(N) with no plateau (to N=4e6), the signature of an exact form. The mechanism is standard state-transition covariance propagation (as in aided inertial navigation, Kalman smoothing and geodetic adjustment); the contribution is its exact application to the wellbore interior in an open validated implementation. Linear interpolation mis-states the interior covariance near doglegs by an amount that is survey-interval dependent (negligible at the ~30 m ISCWSA-assumed spacing, growing on coarse or high-dogleg surveys) and of a sign set by the approach geometry; the exact form removes that error at any interior point and, being continuous in measured depth, makes anti-collision (separation factor and probability of collision, evaluated at the interior point of closest approach) and gradient-based well planning exact and smooth. Complements the covariance-EOU framework of K. Bulychenkov, whose between-station term models the unresolved micro-dogleg geometry rather than the resolved arc.