Lattice Boltzmann simulation of pre-Darcy flow in porous media

Fluid flow in porous media, the theoretical basis of which is a linear law proposed by Darcy, plays a vital role in the energy industry. Much evidence suggests that the correlation of velocity and pressure gradient deviates from Darcy's law in the low-flux region due to the existence of viscous (sticky) boundary layers (also termed pre-Darcy flow). Using Darcy's law to characterize the fluid flow process will impede the efficient exploitation of unconventional resources and energy utilization. However, the pore-scale simulation method of pre-Darcy flow was less reported. Based on microtube experiments, we first built a mathematical model of boundary layer thickness, accounting for the effect of pressure gradient and the viscosity difference of distinct regions. Then we incorporated this model into a lattice Boltzmann framework to simulate the pre-Darcy flow and analyzed the influences of different factors. Our results indicate that the boundary layer thickness and throat aperture dominate the pre-Darcy flow behavior, but the boundary layer viscosity shows less impact. As the boundary layer thickness increases, the apparent liquid permeability of porous media decreases, and the pseudo-threshold pressure gradient alters distinctly. In comparison to classical Darcy flow, the streamlines in the pre-Darcy flow are concentrated in the central region of the throat and may redistribute in some throats under the boundary layer effect. This study advances our understanding of pre-Darcy flow and provides a useful methodology to simulate the process in porous media, which is favorable for understanding the transport physics in shale and tight matrices, and vital for accurate dynamic performance prediction and production optimization in unconventional reservoirs.