Abstract

Despite the significant progress made in recent years, a fundamental understanding of immiscible 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 sufficiently small, the displacement is stabilized and can be described by Invasion Percolation in a Stabilizing Gradient. In the opposite case, the displacement has common features with Invasion Percolation in a Destabilizing Gradient. A phase diagram of fully developed drainage is then 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. The theory is subsequently extended to correlated pore networks and a phase diagram involving the correlation length A is constructed. As the regimes of stabilized displacement are also those for which conventional theories (such as the Buckley-Leverett equation) apply, the phase diagram helps to delineate their validity.

P. 877

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