Scaling Analysis for the Tracer Flow Problem in Self-Similar Permeability Fields

The spatial variations in porous media (aquifers and petroleum reservoirs) occur at all length scales (from the pore to the reservoir scale) and are incorporated into the governing equations for multiphase flow problems on the basis of random fields (geostatistical models). As a consequence, the velocity field is a random function of space. The randomness of the velocity field gives rise to a mixing region between fluids, which can be characterized by a mixing length $\ell=\ell(t)$. Here we focus on the scale-up problem for tracer flows. Under very general conditions, in the limit of small heterogeneity strengths it has been derived by perturbation theories that the scaling behavior of the mixing region is related to the scaling properties of the self-similar (or fractal) geological heterogeneity through the scaling law $\ell(t)\sim t^{\gamma}$, where $\gamma=\max\{1/2,$ $1-\beta/2\}$; $\beta$ is the scaling exponent that controls the relative importance of short vs. large scales in the geology. The goals...