The relationship between immiscible and miscible displacement in porous media

Abstract

Displacement processes in porous media share several common characteristics. Fundamental among them is the representation of the momentum balance via Darcy’s law and its multiphase extension (Bear, 1972). Such description leads to systems of equations the solution of which parallels the behavior of the solution of processes in multicomponent chromatography. For instance, it has been recognized that noncapillary, multiphase displacement in one-dimensional (rectilinear or radial) porous media can be solved by techniques identical to those used in chromatographic transport (Helfferich, I98 I; Rhee et al., 1986, and references therein). This similarity arises naturally for the case of one-dimensional flow geometries and in the absence of dissipative terms (diffusion, dispersion, or capillarity), where both multiphase and multicomponent chromatography processes are formulated by systems of first-order hyperbolic equations. In this note we pursue further this relationship in mathematical representation between multiphase flow and chromatographic processes in porous media. Specifically, we consider two-phase, immiscible displacement and single-phase, miscible displacement in the presence of equilibrium adsorption. It is demonstrated that the two problems are mathematically equivalent, regardless of the dimensionality of the flow system and the presence of dissipative effects. Such an analogy is of practical importance in the analysis of various process characteristics, for instance in describing the evolution of unstable two-dimensional disturbances during viscous fingering. The latter is a topic of active current investigations (Homsy, 1987, and references therein), and of fundamental importance in process performance. Any similarities between seemingly different processes would be of considerable help in an effort to reduce complexity and to uncover common mechanisms. An extension to multiphase immiscible displacement and multicomponent miscible displacement is also briefly discussed. Mathematical Description