An improved lattice Boltzmann model for variable-order time-fractional generalized Navier-Stokes equations with applications to permeability prediction

The classic generalized Navier-Stokes (GNS) equations with integer-order calculus are not capable of capturing anomalous transport phenomena within porous media. Fractional calculus is able to character transport processes related to long-term memory and is commonly incorporated into various model equations for describing anomalous transport. The fractional order typically demonstrates spatial variation due to the spatial variability of complex microstructures within porous media. In this study, variable-order time-fractional GNS equations are presented to describe anomalous dynamics in porous flows by introducing the time-fractional derivative with a space-dependent fractional order. A fresh lattice Boltzmann (LB) model is developed to solve the variable-order time-fractional GNS equations. The key point is to propose a new equilibrium distribution function and a modified discrete force term so that the LB model can recover the correct macroscopic equations. Numerical simulations are carried out to verify the present model, and a strong consistency is found between the LB and analytical solutions. The present LB model is employed to simulate fluid flow through porous media and predict the permeability at the representative elementary volume (REV) scale. In contrast to previous research that focused solely on the REV-scale permeability under stable-state conditions, this study provides a comprehensive analysis of the REV-scale permeability under both unstable and stable states. Furthermore, the impact of the fraction order on the REV-scale permeability is thoroughly investigated. An increase in the fractional order is observed to result in a shorter time for the REV-scale permeability to reach a stable state, while having little impact on the REV-scale permeability in the stable state. The spatial distribution of the fraction order affects the spatial distribution of the velocity field, and then influences the REV-scale permeability in the stable state.