ANOMALOUS DIFFUSION OF ELECTROMAGNETIC EDDY CURRENTS IN GEOLOGICAL FORMATIONS

Abstract

Controlled-source electromagnetic (EM) induction in some geological formations is shown here to be compactly described by an anomalous subdiffusion process. Such a process, which is not universal, is governed by a fractional diffusion equation or alternatively the convolutional form of Ohm's law. A subdiffusing eddy current vortex, or electromagnetic smoke ring, propagates in such a way that its position of median intensity overruns its position of peak intensity. This behavior is not allowed in classical diffusion but is a simple consequence of diffusion within a stationary fractal medium. A similar analysis has been applied to understand heavy-tailed traveltime distributions that appear in certain hydrological time series. The tell-tale signature of anomalous electromagnetic diffusion is a slope beta of the magnetic zero-crossing moveout curve that is constant with transmitter-receiver (RX) offset and significantly different from unity. Neither lateral heterogeneity nor unixial anisotropy can generate such a constant-slope moveout curve with an economy of model parameters. Controlled-source EM data from two sites in Texas and one in New Mexico are used in this study to test the eddy current subdiffusion hypothesis.