ABSTRACT
CONTENTS 3.1 Definition of Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 3.2 Volume Averaging. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 3.3 Volume Averaging of Transport Equation and Closures for
Isotropic Porous Media . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 3.4 Tortuosity and Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
3.4.1 Volume Averaging of Electrical Conduction Equation . . . . . . . . . . 97 3.4.2 Propagation of Small Amplitude Low-Frequency Waves . . . . . . 99 3.4.3 Tortuosity Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
3.5 Volume Averaged Navier-Stokes Equation for an Isotropic Porous Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
3.6 Dispersion or Volume Averaged Advection-Diffusion Equation . . . . .112 3.7 Volume Averaged Heat Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .114 3.8 Microscopic Inertia and Flow Induced Dispersion . . . . . . . . . . . . . . . . . . . . . .118 3.9 Summary and Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
Dispersion is a well-known phenomenon in porous media primarily for heat and mass transfer. Like viscosity in momentum transfer, heat conductivity in heat transfer, and diffusion coefficient in mass transfer, dispersion coefficient is a property valid only under continuum assumptions. Dispersion causes fluid (velocity, molecules, and temperature) to distribute evenly, which is directly analogous to mass diffusion (Fickian diffusion) and viscous stress. Fickiandiffusion causesmolecules todistribute evenly, whereasviscous stress causes flow velocity to be distributed evenly. However, dispersion is caused due to the fluctuations of bulk flow, whereas diffusion is caused due to random molecular motion.
