Spatial fractional permeability and fractional thermal conductivity models of fractal porous medium

Abstract

In order to describe the seepage and heat transfer problems of non-Newtonian fluids in porous media, a spatial fractional permeability model and a fractional thermal conductivity model for a fractal porous medium are developed based on the fractional non-Newtonian constitutive equation and the fractional generalized Fourier law. It is an innovative attempt to link fractional operators to the microstructure of pore porous media. The predictive capability of the proposed permeability and thermal conductivity model is verified by comparing with experimental data and the conventional capillary model, and the effects of fractal dimension, fractional parameters, and microstructural parameters on permeability and thermal conductivity are discussed. The results are as follows: (a) These two new models have higher accuracy than the conventional capillary model and reveal the relationship between the nonlocal memory and microstructural properties of complex fluids. (b) The permeability and thermal conductivity increase with increase in the fractional parameter α and radius ratio β and decrease with the increase in the fractal dimension (Dτ and Df) and microstructural parameters (length ratio γ, branching angle θ, and branching level m) of the porous medium. (c) When the radius ratio is larger than a certain value, the growth rate of permeability (β > 0.46) and thermal conductivity (β > 0.3) increases significantly, while the branch angle has the smallest influence on permeability and thermal conductivity, which can be ignored.