Mathematical Basis of Two-Phase, Incompressible, Vertical Flow Through Porous Media and Its Implications in the Study of Gravity-Drainage-Type Petroleum Reservoirs
Authors: Hiatt William N. 
Published: December 2007

Abstract
The mathematical theory of the flow of two-phase, incompressible fluid through porous media is clarified, and the development of a general fluid displacement equation for two-phase, incompressible vertical flow through porous media is outlined. The relationship between the three components of the equation - the "fluid drive effect", the "gravity effect", and the "capillary effect" - is discussed, as well as implications of the analysis in connection with the study of certain types of petroleum reservoirs.

Introduction
The purpose of this paper is to clarify and summarize the mathematical basis of the theory of two-phase incompressible fluid flow through porous media, particularly as it pertains to vertical flow, and to present a qualitative exposition of the controlling factors involved in circumstances where this type of flow may exist.
Although the generally accepted theory of heterogeneous fluid flow through porous media was developed a number of years ago by Muskat, and Buckley and Leverett using the basic premise of Darcy, (and since then expounded upon by many others) there is a need for an over-all summary and clarification of this theory, especially regarding certain types of applications in connection with so-called "gravity-drainage-type reservoirs". The source and derivation is outlined for what is referred to here as the "General Fluid Displacement Equation for Incompressible Vertical Two-Phase Flow through Porous Media". This equation would apply under the Porous Media". This equation would apply under the condition where the pressure drop along the path of flow is negligible in comparison with the pressure level on the system, or where compressibility and solubility effects are negligible. These conditions exist ordinarily in gravity-drainage-type reservoirs.
Specifically, it will be shown that this general equation may be developed directly from the basic equations, which, according to present theory, govern the general case of viscous heterogeneous flow through porous media. This general equation has appeared in the literature previously, but in most cases its development from basic theory is not indicated clearly, or the equation is presented in terms of groups of constants and variables involving reduced or dimensionless quantities. This presentation is expedient in cases of experimental or computer applications; but it may make it difficult for the practicing petroleum engineer to identify all the factors involved or to understand the relationships between them. Therefore, in the following development, only ordinary mathematical parlance will be used, and no terms involving groups of other equations or reduced or dimensionless quantities will be used.

DERIVATION
The following equations for two-phase oil and gas flow result from analogy with Muskat's general equations, if solubility and compressibility effects are neglected, and if all volumes are considered in terms of reservoir conditions, and the water phase, if present, is considered to be immobile (or effectively a part of the porous medium.) In this case, the sum of the oil and gas saturations may be taken as unity, and the saturation of each phase is in terms of the total hydrocarbon saturation.
SPEJ
P. 225