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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 l = l(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 l(t) similar to 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 of this work are the following: (i) The derivation of a new, mathematically rigorous scaling analysis for the tracer flow problem subject to self-similar heterogeneities. This theoretical development relates the large strength to the small strength heterogeneity regime by a simple scaling of solutions. It follows from this analysis that the scaling law derived by perturbation theory is valid for any strength of the underlying geology, thereby extending the current available results. To the best of the knowledge of the authors this is the only rigorous result available in the literature for the large strength heterogeneity regime. (ii) The presentation of a Monte Carlo study of highly resolved simulations, which are in excellent agreement with the predictions of our new theory. This indicates that our Monte Carlo results are accurate and can be applied to other models for stochastic geology.

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Multiscale Modeling & Simulation



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Mathematics Commons