Statistical Network Theory of Three-Phase Relative Permeabilities

Abstract

ABSTRACT The statistical network model and percolation theory of capillary pressure and relative permeabilities in low capillary number displacements account for much of the behavior of two fluid phases under the gamut of wetting conditions encountered in reservoir rocks (SPE 11015, SPE 12172). The model and theory are here extended to three fluid phases, viz. gas, oil, and water. For convenience in illustrating the theory the gas is taken as everywhere nonwetting against the oil and water, and the oil as everywhere nonwetting against the water. The starting point is the representation of the chaotic connectivity of porespace by a bond-disordered Bethe tree; also as before, the bonds are characterized by a single statistically distributed parameter, porethroat radius, with which other pore dimensions are regarded as completely correlated. The distribution of the three phases among the poresegments, or bonds, is taken as controlled by capillary pressure, wettability, and prior occupancy criteria of allowability to a locally invading phase: the complication is the competition among three phases. These criteria and accessibilities evaluated with percolation theory in turn define the pore filling sequence in terms of phase-occupancy-conditioned poresize distributions. From this follow occupancy-conditioned distributions of segment volumes, and thus saturations; and occupancy-conditioned distributions of hydraulic conductance, from which relative permeabilities are again calculated using Stinchcombe's method from percolation theory. Examples include simultaneous displacement of oil by gas and water, and waterflooding in the presence of an evolved gas saturation. Effects of oil wettability and of phase swelling by mass transfer are touched on. The predictions of relative permeabilities and saturation states accessible by a given process from a given initial state highlight the sensitivity of displacements to the history of the process, or to the details of a laboratory experiment, when three phases are present. Limited comparisons are made with the few published data.