Analytical Miscible Relative Permeability Curves And Their Usage With Compositional And Pseudo-Miscible Simulators
Authors: Jankovic M.S. 
Published: March 2010

Abstract
One of the unsolved problems in miscible displacement studies has been the analytical representation of solvent/oil relative permeability curves here referred to as miscible relative permeability curves (MRPC). A few analytical expressions for MRPC have been suggested and none has been shown to match experimental data.
The analytical MRPC proposed here are simple functions of oil (solvent) saturation and two non-dimensional real parameters ??? and β which describe mixing effects and the pore structure. Moreover, the logarithm of the solvent/oil miscible relative permeability ratio (MRPR) is shown to be a straight line. Excellent matches with experimental solvent/oil miscible relative permeability ratios are then obtained without trial and error.
These curves are tested in compositional simulators, and the results again show excellent agreement with various experimental data.
A two-parameter mathematical model based on application of the proposed analytical MRPC in a pseudo-miscible simulator is also presented. All the matches of experimental data obtained with this new model are shown to be significantly better than those previously obtained with the mixing parameter model(1).

Introduction
Relative permeability curves are empirical, scalar single valued functions of oil (solvent) saturation, and are used to described mathematically multi-phase flow in porous media using Darcy's law. Similarly, MRPC permit further application of Darcy's law, although approximately, to two-phase flow of solvent and oil when solubility exists between the phases under certain conditions. They describe the simultaneous flow of solvent and oil on a microscopic level, taking into account a multitude of microscopic factors that affect the miscible displacement. Such factors include pore structure (heterogeneity, homogeneity), viscosity ratio, interfacial tension, displacement velocity, and the effect of first contact or multiple contact mixing. MRPC are among the most influential functions governing the miscible displacement, being modeled either with compositional or pseudo-miscible simulators.
Since the miscible relative permeability ratio always enters the equations of multi-phase flow, finding an appropriate one is very important for the success of any simulation study of miscible displacement. It is evident that its unjudicious selection will lead to erroneous results. The analytical expressions that have been suggested(1,2) have not been shown to match experimental data. Another alternative, often used, is to obtain non-analytical MRPC by trial and error, a method which is costly and which has an uncertain outcome. Regardless of the type, any acceptable MRPC used in simulation study must produce a satisfactory match of some meaningful experimental data.
New analytical MRPC are now introduced. Thereafter, the application of MRPC in compositional and pseudo-miscible simulators is discussed.

Analytical Miscible Relative

Permeability Expressions
The proposed analytical representation for MRPR (Appendix A) is
Equation (1) (Available in full paper)
where Kro ≠ O, and I – S – Sor ≠ O.
Under the assumption that
Equation (2) (Available in full paper)
the solution of equations (1) and (2) is obtained as
Equation (3a) (Available in full paper)
Equation (3b) (Available in full paper)
FIGURE 1 Solvent-oil permeability curves vs solvent saturation. (Available in full paper)
FIGURE 2 Solvent-oil relative permeability ratio vs solvent saturation ratio.(Available in full pap