Velocity fluctuations and dispersion in a simple porous medium

We model a fluid-filled disordered porous medium by a lattice-Boltzmann system with randomly broken links. The broken links exert a friction on the fluid without excluding volume. Such a model closely mimics the idealized picture of a porous medium, which is often used in the theoretical analysis of hydrodynamic dispersion. We find that the Brinkman equation describes both the mean flow characteristics and the spatial decay of velocity fluctuations in the system. However, the temporal decay of the velocity correlations (that a particle experiences as it moves with the fluid), cannot be simply related to the spatial decay. It is this temporal decay that determines the dispersivity. Thus, hydrodynamic dispersion is generally greater than theories based on spatial correlations would imply. This is particularly true at high densities, where such theories considerably underestimate both the magnitude and transient time scale for dispersion. Nonetheless, temporal velocity correlations are still ultimately screened and the hydrodynamic dispersion coefficient converges exponentially. The long-lived transients reported for more realistic systems must therefore be due explicitly to the presence of excluded volume.