Mixing Equations in Short Laboratory Cores
Authors: Brigham W.E. 
Published: December 2007

Abstract
This paper investigates mixing in short cores where the mixed zone is large compared with the core length. In such systems the boundary conditions used affect the resulting solutions. Two models, the diffusion equation and the dead-end pare equation, are commonly used to match such displacements. Although the solutions using the diffusion equations differ considerably in form, this paper shows that, if the boundary conditions are interpreted correctly, and proper allowance is made for the difference between in-situ concentration and flowing concentration, the predicted results from these equations are nearly identical. The simplest way to analyze effluent data is to use the volume modifying junction, U, and to plot the data on probability paper. probability paper. The dead-end pore model has been used incorrectly in the past. It must be adjusted at the effluent boundary to account for the difference between in-situ concentration and flowing concentration. In addition, one must check the experimental data to make sure a material balance is preserved. The data and calculations in the paper show the disastrous predictions that can result if these corrections are not made.

Introduction
When miscible displacement occurs in short cores with a large dispersion coefficient, the effluent mixed zone is large compared with the length of the core. The effluent concentration curve may extend over more than 2 PV. Under this experimental condition it is prudent to worry about be validity of the various equations used to describe mixing in the core. For instance, if the displacement follows the diffusion equation, there are several solutions that differ, depending on the boundary conditions one imposes. The question arises, "How do these boundary conditions differ, and which is the correct solution to use?" A more complex mixing equation, the dead-end pore model, is also available to describe the pore model, is also available to describe the displacement process. This model also will produce differing results when differing boundary conditions are applied. Thus these same questions should arise. The practical significance of this problem is that native-state core displacements must nearly always be run in short cores. If a displacement model is used incorrectly in such a system, it will seriously affect the mixed-zone volume predicted for long distances. In turn, the predicted size of the solvent slug for vertical miscible floods will be wrong.

DIFFUSION EQUATIONS
When one fluid is miscibly displacing another in a linear porous medium and when the displacement is stable so that viscous fingers do not form, the diffusion equation is often used to describe the displacement. This equation is(1)
Generally, for convenience, the concentration limits are shown from 0 to 1. I will use 0 for the concentration of the fluid originally in place, and 1 for the concentration of the displacing fluid. Several solutions to Eq. 1 can be found in the literature. In general, they are various combinations of the error function, differing according to the boundary conditions imposed. Ordinarily, when the porous medium is long compared with the length of porous medium is long compared with the length of the mixed zone, it makes little difference which solution one uses, for they all give virtually identical results. On the other hand, when the mixed zone is about the same length as the porous medium, the boundary conditions can have a noticeable effect on the results, and we need to worry about which of the several solutions to use. The dimensionless grouping that characterizes this worry is the dimensionless dispersion, uL/K, called gamma by Coats and Smith. Smaller gamma's show greater dispersion, and greater boundary condition effects. In this paper we will look at various solutions in some detail for a particular case in which the dimensionless group, gamma, is equal to 14.0. This is a particularly small value of gamma, where the boundary conditions will be quite important. Breakthrough occurs at about 0.4 PV injection. PV injection.
SPEJ
P. 91