A Mathematical Theory for the Displacement of Oil and Water by Alcohol
Authors: Wachmann C. 
Published: December 2007

Abstract
The theory presented in this paper makes possible examination of the phenomenological aspect of secondary oil recovery by alcohol flooding. Limitations of the theory are contingent on three primary assumptions:
chemical equilibrium, no fingering, and no dispersion.
The second assumption implies one-dimensional flow. In a geometric sense the theory extends the Buckley-Leverett construction to systems of two slightly miscible fluids and to systems of three components. This is made possible by considering single-phase fluids to be made up of two phases, namely of an excess of one phase and a negative amount of the other phase. The procedure is discussed in the paper. The theory also permits direct determination of the relationship between the aqueous phase saturation and the concentration of water in the aqueous phase in a zone of variable composition. The velocity of planes of constant water concentration and of constant saturation can be computed likewise and the existence of saturation discontinuities inferred. Consequently the behavior of such a zone, which may exist at some time between a displaced and a displacing fluid, may be determined. Mathematical details are given in Appendices. Problems of experimental verification are pointed out on the basis of a supporting experiment. Implications of the theory and effects of violations of primary assumptions are discussed briefly. The major conclusion is that, if the primary assumptions are met, with pure alcohol and alcohol. rich slugs piston-like displacement of oil and water is possible.
Introduction
The recovery of oil from porous media by alcohol displacement has been studied for several years; Extensive experimental investigations into alcohol flooding have been carried out in the laboratory although little has been done in the field. A large amount of practical information is available, the extent being indicated by the first three papers cited and their respective bibliographies. This has permitted the formulation of inferential theories. A more analytical approach in which theories are developed primarily by reasoning from known physical principles and relationships has, however, lagged behind. In fact, an analytically derived theory of alcohol displacement has not been available. The theory presented in this paper on oil and water displacement by alcohol from a porous medium is a result of the analytical approach. The theory is idealized, as certain assumptions had to be made to make the mathematics tractable and reduce complexity of argument. In particular, major assumptions concerning conditions of equilibrium, interfacial stability, longitudinal mixing, etc., led to the exclusion of fingering, for example, from all considerations. In many laboratory experiments reported in the literature fingering could not be avoided. There is, therefore, disagreement between theories based on experimental evidence available and the analytically derived theory presented here. This theory presented is based on known physical data and makes use of the Buckley-Leverett construction and its suggested extension. The mathematical derivation of the theory is given in Appendices. Three examples relevant to alcohol displacement are discussed to explain the use of this theory:
system of two immiscible fluids (the Buckley-Leverett case); system of two slightly miscible fluids involving an extension of the Buckley-Leverett construction; and system of three components, including injection of pure alcohol or alcohol-rich fluid as a special case, which involves a more general extension of the Buckley-Leverett construction.
In these discussions strict assumptions were made to impose equilibrium conditions and exclude mixing and fingering. Effects of relaxing these assumptions and consequent extensions of the theory are indicated. The complexity of the alcohol displacement process makes experimental verification of the theory essential. Some relevant evidence is offered in this paper.
SPEJ
P. 250ˆ