Linear Stability of Immiscible Displacement in Porous Media

The linear stability of immiscible displacement in consolidated porous media is examined. The flow of the two immiscible phases is represented by a continuum model in terms of capillary pressure and effective permeability functions. Upper bounds on the rate of growth and asymptotic results at low values of the wavenumber are derived. The analysis is carried out for several different process conditions. The predictions obtained are qualitatively different than those for Hele–Shaw flows. Specifically, most dangerous and cutoff modes are shown to scale with the capillary number ( NCa ) as opposed to the NCa1/ 2 scaling in Hele–Shaw flows. The asymptotic expansions indicate that long-wave instability is driven by an unfavorable contrast in the total mobility, thus generalizing the Hill-Saffman–Taylor result. A combination of capillary and graded mobility effects act to stabilize the displacement at higher wavenumbers. The relationship of the problem to that of miscible displacement in the presence of equilibrium adsorption is also discussed.