Computational Geosciences 5: 257–278, 2001.  2001 Kluwer Academic Publishers. Printed in the Netherlands. Delineation of microscale regimes of fully-developed drainage and implications for continuum models Yanis C. Yortsos a , Baomin Xu a and Dominique Salin b a Petroleum Engineering Program, Department of Chemical Engineering, University of Southern California, Los Angeles, CA 90089-1211, USA E-mail: yortsos@euclid.usc.edu b Laboratoire Fluides, Automatique et Systemes Thermiques, Université Paris XI, 91405 Orsay Cedex, France Received 4 December 2000; accepted in June 2001 Despite significant progress made in recent years, a fundamental understanding of immis- cible displacements at the macroscale is lacking. In this paper we use a version of percolation theory, based on invasion percolation in a gradient, to connect drainage processes at the pore- network scale with the displacement at the macroscale. When the mobility ratio M is suffi- ciently small, the displacement is stabilized and can be described by invasion percolation in a stabilizing gradient. In the opposite case, it has common features with invasion percolation in a destabilizing gradient. A diagram delineating the regimes of fully developed drainage is developed. The transition between stabilized displacement and fingering is controlled by the change of the sign of the gradient of the percolation probability, and the transition boundary is described by a scaling law involving the capillary number and the viscosity ratio. We review recent work for random networks and extend the method to correlated pore networks. As the regimes of stabilized displacement are also those for which conventional theories (such as the Buckley–Leverett equation) are expected to apply, the phase diagram helps to delineate their validity. Keywords: drainage, immiscible displacement, invasion percolation, porous media, upscaling 1. Introduction The mathematical description of immiscible displacements in porous media is based on a classical methodology developed several decades ago (for example, see [9]). This is founded on two postulates: that capillary equilibrium between the fluids exists to relate their pressure difference to saturation-dependent capillary pressure functions, and that saturation-dependent relative permeabilities can be defined. These allow extending Darcy’s law to multi-phase flow. The resulting non-linear differential equations have been the subject of extensive analysis. For one-dimensional flows, in particular, this for- mulation reduces to the celebrated Buckley–Leverett equation in petroleum engineering, or to the Richards equation in hydrology, respectively. These two equations have been