Covariance EOU: A Joint-Covariance Framework for Wellbore Positioning and Multisensor Data Fusion

The modern industry method for wellbore-position uncertainty calculation (Williamson, 2000; ISCWSA) describes each error source by a magnitude, a propagation mode, and a weighting function, and sums their contributions into the position covariance at every survey station. The correlation of errors between stations, survey runs, and wells is accounted for through the propagation modes, but it remains inside the calculation: the result is a set of independent ellipsoids of uncertainty (EOU) rather than a joint covariance matrix. The EOUs of a trajectory are therefore not connected to each other as a single object, and in tasks that require the joint treatment of several uncertain objects – MWD–gyro fusion, ranging, and the fusion of the trajectory with the geological model and LWD data – this connection has to be either ignored or reproduced artificially. In addition, the errors of corrections (MSA, BHA sag, the Stockhausen effect) are modeled by assigning toolcodes and adding new error terms, which does not reflect the correction actually performed and leaves the error magnitudes unclear. This paper proposes a covariance EOU approach in which the error of a trajectory, or of a system of trajectories, is treated as a single joint random object. The approach is a matrix expansion of the Williamson method: the weighting function of each error source is replaced by a Jacobian matrix, its magnitude and propagation mode by a covariance matrix of that source, and the summation of errors by an integration operator for the system of wells. The result of the calculation is the full positional covariance matrix of all stations of all wells, whose diagonal blocks coincide with the conventional EOUs and whose off-diagonal blocks retain the connection between stations, runs, and wells; with the off-diagonal blocks set to zero the approach reduces to the standard one. The survey errors can then be modeled explicitly for the correction actually performed – the partial correlation of declination from the correlation table of geomagnetic models, MSA by a full covariance analysis of the linearized system of equations, BHA sag and the Stockhausen effect by the Monte-Carlo method – or, as before, be specified by the error terms of the toolcodes. The resulting covariance is used directly in the maximum-likelihood estimate for data fusion, so that the correction propagates to all correlated stations and wells, and the fusion yields not only a refined trajectory but also its updated covariance. MWD–gyro fusion on this basis has already been implemented by us: a gyro survey in one well refines the position of all wells of the pad correlated with it, including those without gyro surveys. For ranging, a proof of concept has been performed: a relative-position measurement corrects both trajectories simultaneously and leads to a significant reduction of the absolute positional uncertainty of these wells. Beyond that, the approach opens the way to probabilistic synthesis of the geological model from LWD geomarkers, to statistical validation of LWD data against the geological model, to optimal fitting of the trajectory to the logs, and to the use of shared geological constraints in anti-collision analysis – in all of these tasks the covariance EOU serves as the single object that links wellbore positioning and geology.