Fractional diffusion analysis of the electromagnetic field in fractured media — Part 2: 3D approach

Abstract

We have proposed a 3D finite-difference (FD) approach to discretize the frequency-domain fractional-derivative Maxwell equation on a staggered grid. The Maxwell equation was reformulated to include a fractional-order time derivative that described multiscale electromagnetic (EM) induction in fractured formations exhibiting a fractal geometry. The roughness [Formula: see text] that appeared in the theory described the falloff of the power spectrum of the heterogeneity of a subsurface region in the wavenumber domain, disclosing the geologic model structure in an explicit way. The fractional-derivative Maxwell equation was transformed into the frequency domain and solved by the FD method. To further probe the controlled-source EM response of a power law length-scale distribution of natural fractures, a stochastic random medium model was generated using the von Kármán correlation function. The usual deterministic EM response to such a fractured block model was fitted by a zero-[Formula: see text] fractional EM response at multiple frequencies, indicating that the von-Kármán-type fractured model response is classical. This confirmed the expectation that a fractional diffusion EM response was not reproduced by piecewise constant models based on the classical Maxwell equation.