Electromagnetic Induction in Rough Geologic Media: The Classical Approach

Electromagnetic (EM) induction methods are well known to be sensitive indicators of electrically conducting subsurface mineral and fluid phases, responding to both the topological connectedness of the conducting phase and galvanic charge buildup on conductivity boundaries. Key to quantifying the relationship between electromagnetic signatures and the underlying the geohydrology is accurate representation of the infinitely complex, conducting Earth in terms of a finite set of model parameters and their associated physics. However, field observations of the spatial variability of induced EM fields and their inferred rate of diffusive propagation suggest that simple, piecewise smooth or continuous models of electrical conductivity -- as commonly depicted in numerical modeling -- may not fully capture the relevant electrodynamics in all geologic settings, especially those where the subsurface is characterized by multi-scale, hierarchical structures such as fractures. Consistent with such observations is a recasting of the Maxwell Equations in terms of fractional calculus, similar to that done routinely in hydrology for the transport equations to explain anomalous hydrologic diffusion, where the underlying multi-scale complexity is captured efficiently by only a few, simple, model parameters. This study focuses on how geo-complexity ultimately manifests its EM signature in a fractional-calculus sense through three-dimensional modeling of spatially-correlated stochastic realizations of the electrically conducting subsurface. Preliminary results simulating the response of a frequency-domain, loop-loop system suggest that heterogeneity proximal to the transmitting antenna generates a strong, but relatively smooth response in the near-field vertical magnetic induction when compared to that in the far field. This finding suggests an exploration strategy based on multi-offset observations may be relevant to quantifying the length scale over which the fractional calculus model holds.; Induced vertical magnetic induction, quadrature phase, surrounding a vertical magnetic dipole transmitter antenna located on Earth's surface, for two different model realizations: a homogeneous 0.1 S/m Earth (left), and a stochastic Earth with log-conductivity uniformly distributed on (-1.5,-0.5) log_10 S/m.