Fractional Diffusion Analysis of the Electromagnetic Fields Generated by a Straight Current Source over a Porous Geological Media

Abstract

An interpretation based on the Continuous Time Random Walk theory (CTRW) to the diffusion of electromagnetic fields generated by a transient straight current source over a porous geological media is presented here. The CTRW theory is demonstrated to be a powerful tool to concisely and more accurately model a transport process in a fractal medium with complex structures, comparing to the classical transport theory. In the controlled‐source electromagnetic (EM) induction setting, the time dependent evolution of the EM field of some sources over a rough medium are governed by the fractional diffusion EM equation in a CTRW sense. The master equation can be solved for a uniform conducting half‐space in the Laplace domain semi‐analytically. We use 2D finite difference method to calculate the solution numerically for the assigned space and transform to time domain with Gaver‐Stehfest algorithm. Here we adopt a spatially uniform roughness parameter β in the solution to characterize the complexity of the geoelectrical structure of the geological medium. To introduce the heterogeneity to our model, we set up the space as several 2 D blocks with different conductivities and βs. Then we compare our results with the synthetic data we got from the high resolution numerical simulations. We are able to show that by introducing the heterogeneity to the fractional diffusion perspective, our approach is competent for tracing the diffusion process with less model parameters.