Heat and mass transport in random velocity fields with application to dispersion in porous media

The effective equations describing the transport of a Brownian passive tracer in a random velocity field are derived, assuming that the lengthscales and timescales on which the transport process takes place are much larger than the scales of variations in the velocity field. The effective equations are obtained by applying the method of homogenization, that is a multiple-scale perturbative analysis in terms of the small ratio between the characteristic micro- and macro-lengthscales. After expanding the dependent variable and both space and time gradients in terms of , equating coefficients of like powers of yields expressions to determine the dependent variable up to any order of approximation. Finally, a Fickian constitutive relation is determined, where the effective transport coefficients are expressed in terms of the ensemble properties of the velocity field. Our results are applied to the transport of passive tracers in the stationary flow field generated in dilute fixed beds of randomly distributed spheroids, finding the effective diffusivity as a function of the spheroid eccentricity. Our result generalizes the expression of Koch and Brady (1985), who considered spherical inclusions, and is readily applied to the cases of random beds of slender fibers and flat disks.