Three-dimensional numerical experiments have been conducted to investigate the viscous and porous inertia effects on the pressure drop in a non-Newtonian fluid flow through a porous medium. A collection of cubes placed in a region of infinite extent has been proposed as a three-dimensional model of microscopic porous structure. A full set of three-dimensional momentum equations is treated along with the continuity equation at a pore scale, so as to simulate a flow through an infinite number of obstacles arranged in a regular pattern. The microscopic numerical results, thus obtained, are processed to extract the macroscopic relationship between the pressure gradient-mass flow rate. The modified permeability determined by reading the intercept value in the plot showing the dimensionless pressure gradient versus Reynolds number closely follows Christopher and Middleman’s formula based on a hydraulic radius concept. Upon comparing the results based on the two- and three-dimensional models, it has been found that only the three-dimensional model can capture the porous inertia effects on the pressure drop, correctly. The resulting expression for the porous inertia possesses the same functional form as Ergun’s, but its level is found to be only one third of Ergun’s.
Generalized Newtonian fluid flow through a porous medium
