Relative Permeabilitties In Two-Phase Flow Through Porous Media: An Application Of Effective Medium Theory
Authors: Levine S. , Cuthiell D.L. 
Published: March 2010

Abstract
A mathematical model of immiscible two-phase (oil/water) flow through a porous medium is proposed. The model is composed of a simple cubic lattice of cylindrical tubes having a distribution of cross-sectional radii. It is assumed that the wetting fluid (water) flows through tubes of radii less than a and the nonwetting fluid (oil) flows through tubes of radii greater than a, where a depends on saturation. Using effective-medium theory (E.M.T.), the relative permeabilities of both fluid phases are determined as functions of saturation. The permeabilities become zero at saturations which correspond to the residual saturation of the non-wetting phase and the irreducible saturation of the wetting phase. The model is extended in an appendix to include an intermediate range of tube radii through which both oil and water flow. This extended model predicts that at fixed saturation the relative permeability of the oil increases with viscosity ratio whereas that of the water is independent of this ratio. The dependence on temperature of the cutoffs in the relative permeabilities (at residual oil and irreducible water saturations) is interpreted as a variation with temperature of the range of tube radii carrying both oil and water.

Introduction
In constructing a mathematical model of the transport properties of disordered materials, such as porous media, a random conductance (e.g. electrical or hydraulic) network is often used to simulate these properties. Perhaps the simplest approximate theory of the transport phenomena is the so-called effective medium theory (E.M.T.), applied to such a model. In E.M.T., a uniform effective network, that is one having the same conductance for each element of the network, is assumed. One then requires the same average local field to be obtained from the average transport parameter as from the original random transport parameter. The E.M.T. has been applied with considerable success to various electrical and optical phenomena in random mixtures.
We base our theory on the work or Kirkpatrick(1) who used E.M.T. to derive the electrical conductance of a mixture or materials, interspersed at random on a regular three dimensional lattice. The model or the mixture is a connected regular network of resistors. A typical resistor has a conductance g, with a normalized distribution f(g), i.e., f(g)dg is the probability that a resistor located anywhere in the network has a conductance in the range g, g + dg. This model or random resistors is replaced by an effective medium consisting or the same network but of equal conductances, gm given by the relation
Equation (1) (Available in full paper)
where z is the coordination number of the lattice network. Consider the special case where a fraction, p, of the resistors have a given conductance which we take as unity and the remaining (1-p) zero conductance. This means
Equation (2) (Available in full paper)
where δ denotes the Dirac delta function. Equation (1) now reduces to
Equation (3) (Available in full paper)
The relation between gm and p is therefore a straight line in which gm goes to zero where p = 2/z.