Rapid Modeling of Borehole Induction Measurements in Layered Media along Vertical Wells using an Adaptive Born Approximation

Summary The Born approximation offers a computationally efficient alternative to full electromagnetic (EM) forward modeling, but suffers from limited accuracy due to its reliance on a fixed background conductivity. In this work, we develop an adaptive Born approximation that treats the background medium as a tunable parameter to enhance accuracy in a goal-oriented manner. The background conductivity is selected locally for each measurement configuration using spatial sensitivity functions, enabling accurate modeling in both isotropic and anisotropic media. In this study, we primarily focus on horizontally layered earth models penetrated by a vertical well to investigate the fundamental behavior of the approximation in a simplified setting. We formulate our approach to be applicable to general anisotropic media by using the Green’s function defined for a homogeneous medium. Furthermore, the approach extends to cases where the background conductivity is isotropic while the actual medium is anisotropic. For a layered medium, the orientation of induced current densities relative to the layering provides physical intuition for background selection, drawing analogies to Voigt- and Reuss-type bounds. While these analogies offer useful guidance, our numerical results do not always conform to the expectations derived from them. Among the averaging schemes evaluated, arithmetic averaging generally yields the most accurate results. Numerical experiments indicate that the adaptive approach significantly outperforms fixed-background models across a range of frequencies, spacings, and conductivity contrasts. Furthermore, an example with a 3D structure illustrates the method’s broader applicability beyond the horizontally-layered earth setting. This framework provides a principled and efficient path toward fast, accurate EM borehole modeling for real-time well geosteering and subsurface electrical imaging.