Percolation Theory of Foam in Porous Media

Abstract

Abstract A statistical network model and percolation theory are presented to describe the behavior of foam in porous media. Central to the present theory are constrained pore throat and pore body size distribution functions which characterize the converging/diverging nature of the pore geometry and the assumption that foam lamellae are generated by snap-off in pores with an aspect ratio greater than some critical value. Foam lamellae are not always stable; they may break-and-remake. Permeability reduction results from blockage of the flow channels. The probability of snap-off and the percolating conductivity of the network are calculated, approximated by a Bethe tree, which relates to two-phase permeabilities. Results show that foam generation critically depends on the pore size distributions and that a strong foam could be generated by the snap-off mechanism, provided that the average pore aspect ratio is sufficiently large. Consistent with experimental observations, the theory predicts that gas relative permeability is greatly decreased by foam and approaches zero at the trapped gas saturation corresponding to the minimum gas saturation below which the gas channels become discontinuous. Because snap-off occurs only in gas-occupied pores, the water relative permeability is not affected by foam. Thus, the theory is able to describe quantitatively gas and water relative permeabilities with or without foam. The predicted relative permeabilities together with Darcy's law of two-phase flow explain much of the anomalous behaviors of foam in porous media, including the near-constancy of steady state saturation in the presence of a strong foam and the seemingly contradictory effects of gas and liquid flow rates on foam. The possibility of foam generation by imbibition and the effects of rock permeability, wettability, and the presence of residual oil on foam displacements are also discussed.