Why Waterflood Works: A Linearized Stability Analysis

Abstract

ABSTRACT The stability of the transient waterflood problem has been examined. The treatment includes both the effects of relative permeability and of capillary pressure. The analysis is formulated for a Buckley-Leverett type base state, although with capillary corrections. This base state is obtained for arbitrary water/oil relative permeabilities and capillary pressure functions, using the method of matched asymptotic expansions. The stability analysis is performed in a coordinate system co-moving with the base state; the transformation is a combined stretching and uniform translation in position. The growth rate of a perturbation is found to depend upon both the spatial position and the time of initiation of the instability. The long and short wavelength asymptotics are obtained analytically, the latter using a WKB-type approach. Finite wavelength stability is studied numerically. The pertubration evolution is found to be a competition between a temporal algebraic instability and an exponential decay, in regimes of practical interest. The success of waterflood can be understood since the algebraic instability remains finite at all wavelengths (short wave growth rates are independent of wavelength), in distinction to the strong exponential instability exhibited by an adverse mobility miscible flood. Essentially, the stretching of the Buckley-Leverett base state is as fast as the growth of the instability, changing the exponential instability to an algebraic one. For highly adverse mobility ratio immiscible floods, the exponential instability remains, although it is spatially localized due to the effects of relative permeability. The method of analysis is also applied to the motion of a steady state solution, with relative permeability and capillary pressure, for comparison to the results of previous researchers. An exponential instability arises, instead of the temporal instability, demonstrating the inadequacy of such base states for the examination of waterflood stability.