Variable-order anomalous heat transport mathematical models in disordered and heterogeneous porous media

Anomalous transport models based on constant-order fractional derivatives equations have been employed to capture the complexities and irregularities encountered in modelling real-world applications with varying levels of success. However, recent findings show that there exist some diffusion phenomena where the constant-order approach with variable coefficients may fail to predict the reality. Take, for instance, in describing transport processes in fractured rocks or in unconventional reservoirs. In this paper, three classes of variable-order anomalous diffusion models (time fractional Fokker-Planck equation) are proposed to predict the temperature evolution in a fractured porous medium. The three classes of variable-order fractional Fokker-Planck equations presented differ in terms of the underlying physics controlling the diffusive behavior of the system. Furthermore, existing numerical discretization method is utilized to handle the resulting mathematical model(s). The results of the numerical simulations are presented to illustrate the effect of a time-dependent, space-dependent, and a temperature-dependent diffusive behavior. The variable order fractional approach presented in this study contains the constant-order fractional approach and the classic continuum approach as special cases. The variable-order fractional approach employed herein exhibits several interesting features some of which cannot be described by existing continuum based mathematical models. The numerical results reveal that prior knowledge or information of the nature of the anomalous heat transport behavior through the porous media is essential for accurate heat transport prediction or modelling. This research exhibits the application of fractional calculus as a sound mathematical tool for describing the anomalous effects in heat transport in porous media.