Rapid 2D Occam’s Inversion of Ultra-Deep Borehole Azimuthal Resistivity Measurements

Summary A rapid two-dimensional (2D) Occam’s inversion method is developed to interpret ultra-deep borehole azimuthal resistivity (UDAR) measurements acquired in complex dipping formations and along arbitrary well trajectories. While conventional one-dimensional (1D) inversion approaches are computationally efficient, they are inadequate to capture lateral variations of electrical resistivity and structural complexity, hence limiting their accuracy and reliability in realistic geological settings. The developed method extends the inversion into both 2D longitudinal and 2D transverse directions with respect to the well trajectory, providing greater accuracy of inversion results than possible with 1D inversion while maintaining computational efficiency. A new regularization scheme is incorporated into the 2D Occam’s inversion formulation to enforce consistency of the inversion model with the local geological orientation. In contrast to traditional Tikhonov regularization, which often yields overly smooth models, this scheme utilizes 2D Total Variation (TV) model weighting matrices to estimate sharper layer boundaries and reduce structural artefacts. The Jacobian matrix is efficiently computed and updated using a new Block Gauss–Radau modelling algorithm via the adjoint method, which provides improved accuracy, faster convergence, and greater stability compared to the traditional Spectral Lanczos Decomposition Method (SLDM). To overcome the non-uniqueness of inversion, a logit transformation is applied to constrain the inverted model properties within realistic physical limits. Anderson acceleration is incorporated to further enhance convergence speed and stability of the nonlinear inversion process. Lastly, a new uncertainty quantification approach based on Bayesian extrapolation is integrated into the inversion that captures the spatial sensitivity of the measurements to the spatial distribution of electrical resistivity. The method is validated on synthetic and field data, verifying improved performance over 1D inversion results through reduced artefacts, lower data misfit, and decreased model uncertainty, while producing results within minutes for efficient real-time well-geosteering applications.