Modeling anomalous heat transport in geothermal reservoirs via fractional diffusion equations

The aim of this article is to give an overview of the current research towards applications of fractional partial differential equations for the modeling of anomalous heat transfer in porous media. We start with presenting a physical background behind the anomalous processes described by the Continuous Time Random Walk (CTRW) model and arguing its feasibility for modeling of heat transport processes in heterogeneous media. From the CTRW model on the microscopic level, a macroscopic model in form of a generalized fractional diffusion equation is then deduced. Both mathematical analysis of the generalized fractional diffusion equations and some methods for their numerical treatment are presented. Finally, some open questions and directions for further work are suggested.