Computational Geosciences 7: 115–135, 2003.  2003 Kluwer Academic Publishers. Printed in the Netherlands. Crossover from nonlinearity controlled to heterogeneity controlled mixing in two-phase porous media flows Frederico Furtado a and Felipe Pereira b a Department of Mathematics, University of Wyoming, Laramie, WY 82071-3036, USA E-mail: furtado@uwyo.edu b Instituto Politécnico, Universidade do Estado do Rio de Janeiro, Rua Alberto Rangel, s/n, Nova Friburgo, RJ, 28601-970, Brazil E-mail: pereira@iprj.uerj.br Received 26 September 2001; accepted 24 July 2002 We use high resolution Monte Carlo simulations to study the dispersive mixing in two- phase, immiscible, porous media flow that results from the interaction of the nonlinearities in the flow equations with geologic heterogeneity. Our numerical experiments show that distinct dispersive regimes occur depending on the relative strength of nonlinearity and heterogeneity. In particular, for a given degree of multiscale heterogeneity, controlled by the Hurst exponent which characterizes the underlying stochastic model for the heterogeneity, linear and nonlin- ear flows are essentially identical in their degree of dispersion, if the heterogeneity is strong enough. As the heterogeneity weakens, the dispersion rates cross over from those of linear heterogeneous flows to those typical of nonlinear homogeneous flows. Keywords: fractals, heterogeneity, mixing, multiphase flow, porous media, random fields, scale up, scaling laws 1. Introduction In this paper we investigate the complex fluid mixing in porous media flows that results from the combined effect of nonlinearities in the flow equations and geologic heterogeneities. This mixing occurs in a number of important scientific and technolog- ical contexts, including environmental remediation and the management of petroleum reservoirs. We focus on two-phase, immiscible, incompressible flow which corresponds physically to waterflooding in a petroleum reservoir. This problem is important in its own right, and also contains some of the complexities of more general reservoir dis- placements. Spatial variations of naturally occurring porous formations (aquifers, petroleum reservoirs) occur at all length scales, but only the variations at the largest length scales are reliably reconstructed from the data available. The heterogeneities occurring on the smaller length scales have to be incorporated stochastically, on the basis of random