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 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 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 flowThe 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-Taylor instability and layer instability follow profiles. The Saffman-Taylor instability and layer instability 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., 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. Introduction Several important processes involve displacement of a fluid in a porous medium by continuous injection of another immiscible fluid. Examples of current interest are processes for the recovery of oil from reservoirs by the processes for the recovery of oil from reservoirs by the injection of fluids of suitable chemical composition or thermal energy content. Because of the inherent heterogeneity of porous-medium properties, flow stability is an essential requirement for the development of an efficient displacement. The frequently observed significant reduction in process efficiency that results from unstable displacement (viscous fingering ) has spurred intensive research to obtain a fundamental understanding of the instability mechanisms. The majority of past investigations on the stability of immiscible flow in porous media are based on the description of flow in terms of two macroscopically distinct, single-phase flow regions separated by an abrupt macroscopic interface. Guided by the premise that, as in a Hele-Shaw cell, the potential of incompressible single-phase flow in porous media satisfies the Laplace equation, Chuoke et al. extended the Saffman-Taylor stability analysis for Hele-Shaw flows. On the basis of the abrupt interface approximation, their results show that in the case of low IFT, the displacement is stable when the mobility of the displaced phase is higher than that of the displacing phase and unstable otherwise. To assess the contribution of capillary effects, Chuoke et al. proceeded further by assigning to the macroscopic interface an effective IFT, gamma e, that allows for the pressures in the two regions across the interface to be related pressures in the two regions across the interface to be related to the curvature of the interface. The resulting stability condition showed that capillarity exerts a stabilizing influence at small wavelengths with the onset of instability occurring at wavelengths larger than a critical value X,. With the abrupt interface approximation, several previous investigators examined related aspects of immiscible previous investigators examined related aspects of immiscible displacement, such as the growth and shape of unstable macroscopic fingers, the linear and weakly nonlinear stability of immiscible layers, and the stability of bounded flows. In a study that considers the two-phase-flow transition zone, Hagoort used an energy method to assess the contribution to stability of viscous and capillary effects. The objective of this and the companion paper is to initiate an investigation of the linear stability of immiscible, two-phase flows in porous media that includes a representation of the effects of simultaneous flow and capillarity. With commonly accepted principles for multiphase flow in porous media, a linear-stability analysis in terms of normal modes is implemented. The resulting eigenvalue problems are similar to the equations describing the flow stability of unbounded shear layers. For instance, noncapillary flow stability is the analog to the porous-medium Rayleigh problem for an unbounded porous-medium Rayleigh problem for an unbounded non-viscous shear layer, while capillary flow stability is the analog of the Orr-Sommerfeld problem for an unbounded viscous shear layer. Such an analogy allows the implementation in the present context of several of the methods developed for the study of the hydrodynamic stability of shear layers. SPERE P. 378