Direct numerical simulation and macroscopic modeling of heat and fluid flow in porous media

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Two- and three-dimensional numerical calculations have been conducted to simulate the viscous and porous inertia effects on the pressure drop in Newtonian and non-Newtonian fluid flows through porous media. Various periodic arrangements of square and circular rods are proposed as two-dimensional models of microscopic porous structure, whereas a collection of cubes placed in a region of infinite extent is proposed as its three-dimensional model. A full set of two- and three-dimensional momentum equations are 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. It has been found that the modified permeability determined by reading the intercept in the dimensionless pressure gradient versus Reynolds number plot 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. Thermal dispersion in porous media is also investigated using a two-dimensional periodic array. The resulting correlation for high Peclet number range agrees well with available experimental data.