Dynamics of Moving Interfaces in Porous Media: III. Extension of Buckley-Leverett Theory

Abstract

ABSTRACT In this paper Buckley-Leverett frontal advance theory is extended to two dimensional flows. Two dimensional analog of the front is a closed curve c across which the water saturation changes abruptly. Behind the front water saturation increases monotonically towards the injection well. Equisaturation contours behind the front are also closed curves. The continuous change in water saturation is approximated by several small steps. Then we get several equisaturation zones behind the front separated by contours across which water saturation changes abruptly in small steps. We shall refer to these contours as fronts. This approximation corresponds to linearizing the fractional flow curve beyond the frontal saturation by several linear segments. The progress of the fronts with time depends upon the well configuration and the fractional flow curve. The total mobility changes across each front giving rise to a vortex sheet. A complex velocity potential for the instantaneous velocity field is constructed by the method developed in the earlier papers and the efflux vector is found out everywhere in the flow region. The velocity with which the fronts advance is in the same direction as the efflux vector, but the magnitude of the former differs from the latter by a factor Fw' = dFw/dSw evaluated at the corresponding water saturation. Knowing the velocity of the front it can be traced in time. In other words, after finding the efflux vector in the flow region, in an infinitesimal portion of the flow region we can apply the theory of frontal advance by assuming the flow to be linear there. The procedure of calculations is demonstrated for a typical fractional flow curve with some simple well patterns.