Linear-Stability Analysis of Immiscible Displacement: Part 1—Simple Basic Flow Profiles

Abstract

Summary The linear stability of immiscible, two-phase-flow displacement processes in porous media is examined. Multiphase-flow characteristics are included in the stability description through relative-permeability and capillary-pressure functions. A linear-stability analysis of the steady-state saturation and pressure distributions is carried out in terms of normal modes. The resulting linearized eigenvalue problems describing the early evolution of unstable modes show a certain similarity in the respective cases of negligible and non-negligible capillary effects to the Rayleigh and Orr-Sommerfeld equations governing the stability of unbounded shear flows. The stability of noncapillary displacement is first examined. Growth rates of the unstable modes as a function of the wavelength of instability are explicitly obtained for specific classes of initial total-mobility profiles. The Saffman-Taylor1 instability and layer instability2 follow directly as limiting cases of the initial-mobility profiles. The effect of capillarity on flow stability is examined next. Stability curves are obtained for step-saturation initial profiles. Both capillary pressure and a smooth initial-mobility profile exert a stabilizing influence on the flow displacement. The linear-stability-analysis predictions are compared to the results of Chuoke et al.,3 and an estimate for the effective interfacial tension (IFT) is derived. The results find application in prediction of the onset of instability and description of the early stages of unstable growth in immiscible displacement.