Liquid-liquid Immiscible Displacement In Unconsolidated Porous Media
Authors: Bentsen Ramon G. , Saeedi Jawaid 
Published: March 2010

Introduction
An improved understanding of how one fluid displaces another in a one-dimensional porous medium can be gained by measuring saturation as a junction of time and distance, a procedure rarely undertaken. To this end, a series of immiscible displacements were conducted, at various viscosity ratios, in which in-situ saturations were measured using a relatively new technique based on microwave attenuation. The experimental saturation profiles obtained in this manner were compared with those obtained by solving numerically a recently derived Lagrangian formulation of the immiscible displacement equation. Finally, the results obtained were analyzed in the light of published scaling groups and a recently proposed stability criterion. The results of this analysis suggest that one-dimensional, immiscible displacement theory reasonably represents displacements carried out at favourable mobility ratios. For adverse mobility ratio displacements∼ in horizontal systems, a stability criterion must be satisfied before the theory can be said to represent the displacement process.

Introduction
Steady-state methods for measuring relative permeability require approximately one day to acquire a complete relative permeability curve. External-drive methods, on the other hand, can acquire the same information in a few hours. However, it is well known that external-drive techniques have severe limitations for determining water-oil flow properties(1.2.3).
These limitations may arise because of a violation of one or more of the assumptions underlying the external-drive methods. As these techniques are based on Buckley-Leverett(4) theory, it is important that displacements used for determining relative permeability curves be both stabilized (steady-state in the sense that fractional flow is invariant· with time) and onedimensional (pressure and saturation uniform in any cross section) in nature. Moreover, a Lagrangian formulation of the fluid displacement problem must be permissible.
During the early part of a displacement experiment, fractional flow is a function not only of saturation, but also of time. As a consequence, it takes a certain amount of time for a saturation profile to develop character. That is, a certain amount of displacing fluid, the amount depending on the capillary number, must be injected before a stabilized, or fully developed, saturation profile can be achieved. If the capillary number is sufficiently small, saturation profiles are essentially stabilized prior to breakthrough(5), and external-drive techniques may be used to obtain relative permeability curves.
Under certain circumstances, the displacement process may be dominated by viscous fingering. If such is the case, pressure and saturation will not be uniform in the transverse direction to bulk flow, and the displacement is said to be unstable. Whether or not a displacement is stable is dictated by the stability number(6,7). If, for the rectangular systems used in this study, the stability number is less than p2, the displacement is stable, and one-dimensional solutions to the displacement problem may be used for determining water-oil flow properties.
Linear immiscible displacement theory, as currently formulated, is based on three fundamental relationships: Darcy"s law written for both the displacing and the displaced phase(8); the defining equation for capillary pressure, which links the two flow relationships(9); and the