Macroscopic Dispersion
Authors: Warren Joseph E. , Skiba Francis F. 
Published: December 2007

Abstract
An idealized miscible displacement process in a three-dimensional, heterogeneous porous medium has been studied via experimental computation based on a "Monte Carlo" model. The macroscopic dispersion which results solely from variations in the permeability of the porous system is related to the scale of the heterogeneity as well as the distribution function for the permeabilities. The interpretation of laboratory data is often subjective and may be misleading, since ambiguities are not always recognized. Furthermore, an experiment performed on a conventional oil-field core does not yield a valid measure of macroscopic dispersion for reservoir engineering purposes since the scale of heterogeneity that is significant in the laboratory is not significant in the reservoir. The analysis of field-performance data is discussed and it is shown that a unique characterization of the reservoir can not be obtained from this in formation alone. A possible method, based on a minimum of data, for estimating the spatial distribution of permeabilities in a reservoir is postulated.
Introduction
Many studies concerned with diffusion and dispersion in porous media have been undertaken because of the potential importance of miscible displacement processes for increasing the recovery of oil. The results from a large number of these studies have been published; in a recent paper Perkins and Johnston presented a comprehensive survey of the pertinent literature and excellent appraisal of the "state of the art".In this investigation, dispersion resulting from macroscopic variations in the properties of the porous medium is of primary interest. Dispersion, in this sense, is caused by fluctuations in the velocities of the individual fluid elements as they move through the porous system. This phenomenon is the result of inhomogeneities in the permeability and/or porosity, but it can be amplified by differences in the characteristics of the fluids involved. The objectives of this study are the following:
- to determine the qualitative manner in which macroscopic rock properties affect the observed dispersion coefficient;
- to evaluate the possible influence of macroscopic dispersion on laboratory experiments; and
- to appraise the significance of macroscopic dispersion with regard to reservoir performance.
Investigations of this type are essential since many miscible displacement processes, while seemingly understood for a mechanistic point of view, do not behave in a predictable manner under field conditions. Although a plausible explanation is that the process is dominated by the environment, little serious effort has been devoted either to describing the reservoir itself or to determining the probable effect of the reservoir properties on the process. This study is intended to be exploratory in nature; however, it is hoped that it will provide additional insight into the physical problem and that it will suggest areas suitable for further investigation.
EXPERIMENTAL COMPUTATION
MODEL
To limit the observed dispersion to that which results solely from variations in the properties of the porous medium, an idealized process in which one incompressible fluid displaces another identical fluid is simulated. This simulation is achieved by means of a modified Monte Carlo method - by tracking a number of mathematical particles (tracer) through the system according to the steady-state, single-phase velocity distribution. Then the system is characterized by the distribution of the residence times of the particles. Since any number of particles can be injected simultaneously (a true delta-function distribution) and only pressure conditions are imposed on the boundaries, the model is completely determinate. Furthermore molecular diffusion, microscopic dispersion, adsorption and/or reaction are automatically precluded from the model.
SPEJ
P. 215^