2-D Modeling and Analysis of Time-Domain Electromagnetic Anomalous Diffusion With Space-Fractional Derivative

Recently, the electromagnetic (EM) anomalous diffusion phenomenon has been observed in time-domain EM (TDEM) surveys. Furthermore, the data interpretation accuracy has been reduced by adopting the traditional EM theory and methods. A number of models, such as random medium and roughness electrical conductivity theory, have been adopted to model the EM anomalous diffusion. However, problems such as modeling difficulty and massive discretization exist regarding characterizing the long-range correlation of EM anomalous diffusion. The space-fractional derivative has been proven to preferably describe the long-range correlation characteristic. Only a handful of studies on TDEM anomalous diffusion with space-fractional derivative have been conducted due to the difficulties in computational engineering problems. Therefore, we performed a series of studies about 2-D TDEM anomalous diffusion with space-fractional derivative. The 2-D TDEM space-fractional diffusion equation was constructed based on the space-fractional Ohm’s law model. Furthermore, the discretization and iteration forms of the control equation were derived based on the finite element method (FEM) by introducing the Riemann–Liouville (R–L)-type Riesz fractional derivatives. The 2-D mountain-shaped function and partial integration method (PIM) were combined to convert the fractional derivative into the primitive function form. Hence, the 2-D modeling of the space-fractional EM diffusion was realized. The effectiveness of our method was verified by the function construction method and wavenumber-domain analytical solution. The spatial and temporal characteristics of the space-fractional EM diffusion were analyzed by different geological models. Furthermore, we discuss the differences with the classical EM diffusion. Our method can effectively model the space-fractional EM diffusion in TDEM surveys and provide theoretical bases for improving the TDEM interpretation accuracy with complex geological conditions.