Numerical study and parameters estimation of anomalous diffusion process in porous media based on variable-order time fractional dual-phase-lag model

Abstract

The primary objective of this study is to provide an accurate prediction of thermal transport in a porous media, and more significantly develop a model based on variable-order fractional calculus to better describe the anomalous diffusion process in thermal transport. The variable-order time fractional model based on the dual-phase-lag theory (VOFDPL) has been presented, then numerically solved and the results obtained have been compared to experimental data and other theoretical models to highlight the advantage and capability of this model. The comparison reveals that the results of VOFDPL model can well fit the experimental data, which proves the application of this model to conduct the thermal response of porous media, the non-equilibrium state and the microscale interactions between two phases (solid and fluid) with different thermophysical properties. Also, the heat conduction equations based on the Fourier’s law, single/double-phase-lag (SPL/DPL) models and also their fractional forms i.e., fractional single-phase-lag (FSPL) and fractional dual-phase-lag (FDPL) models have been solved and their results have been compared to experimental data, but the deviation between the results obtained and experimental data illustrates that these models cannot accurately predict the temperature response of porous media. We show that the VOFDPL model developed is an improvement over these models, leading to the most accurate prediction of thermal transport in porous media. In fact, the two lagging times and also the time derivative term with variable order as a time-dependent function can describe the local non-equilibrium condition, microscale thermal interaction between two phases and the anomalous diffusion process respectively Moreover, the two lagging times in the VOFDPL model have been estimated by using an inverse analysis based on the Levenberg–Marquardt algorithm, for the thermal response of porous media at different locations of sensors through the computational domain.