A Method for Predicting the Performance of Unstable Miscible Displacement in Heterogeneous Media
Authors: Koval E.J. 
Published: December 2007

KOVAL, E.J., CALIFORNIA RESEARCH CORP., LA HABRA, CALIF.

Abstract
Practical miscible displacement processes will be characterized by fingering of the solvent into the oil. The fingering process is brought on by viscosity differences, and can be accentuated by channeling and longitudinal dispersion. The effects of these factors on the efficiency of unstable completely miscible displacements are combined in what is called the K-factor method. This method, analogous to the Buckley-Leverett method, predicts recovery and solvent cut as a function of pore volumes of solvent injected. Experimental data are included and show excellent agreement with theory for a wide variety of sandstone cores and viscosity ratios.

Introduction
Theoretical considerations, laboratory experiments, and pilot tests lead to the conclusion that miscible displacements in the field will be unstable. In an unstable miscible displacement, the solvent fingers through the oil. This fingering leads to early breakthrough of the solvent and an extended period during which both oil and solvent are produced.For such a system, there appear to be four principal factors which bring about or accentuate the effects of instability: longitudinal dispersion (including geometrical effects), channeling, viscosity differences and gravity differences. Other factors, such as diffusion and flooding rate, can also influence the effects of instability, but at the flooding rates considered in this report, circa 15 ft/D, they are unimportant. Longitudinal dispersion can be thought of as a spreading of the solvent front caused by the presence of microscopic inhomogeneities. Channeling of the solvent occurs when a porous medium has macroscopic inhomogeneities; i.e., gross permeability variations. Viscosity differences lead to fingering of the less viscous solvent. This difference in viscosity accelerates the growth of fingers along paths previously developed because of permeability variations. Gravity differences lead to overriding of the usually less dense solvent. Although gravity effects are generally small at a flooding rate of 15 ft/D, they would, nevertheless, unnecessarily complicate the interpretation of flooding experiments. For this reason, all the experiments reported herein were done with matched density fluids.Fingering and the resultant poor areal sweep were recognized early as the dominant influences on the efficiency and the economics of miscible displacement processes. Much research effort has been spent on ways to minimize fingering and increase areal sweep such as the use of graded viscosity slugs or water slugs. Some researchers attempted to work out ways to prevent fingering completely; i.e., to achieve a stable displacement through gravity or latitudinal dispersion stabilization. Others did not attempt to control fingering but obtained an economic process by merely recycling the solvent and sweeping pattern by pattern.During this period, all aspects of fingering came under close scrutiny. Some researchers reported on how fingering looks and how it is affected by viscosity ratio, geometry, and slug size. Peaceman and Rachford suggested a mathematical approach to the prediction of unstable miscible displacements in relatively homogeneous sand packs, but their work cannot be extended conveniently to heterogeneous systems. Hence for a heterogeneous system, no method is presently available for predicting solvent cut and recovery as functions of pore volumes of solvent injected.The purpose of this investigation was to attempt to fill in the gap in our knowledge concerning the prediction of performance of unstable miscible displacements. Necessarily, the system selected for study was a relatively simple one. The restrictions placed on the system were:The system was linear;The solvent was miscible in all proportions with the oil in place;The solvent was continuously injected into the porous medium;Gravitational effects were eliminated by matching densities andAll the flood rates were high and constant at 15 ft/D to avoid any small rate effect and to minimize any diffusion effects.
To simplify and to indicate that both longitudinal dispersion and channeling arise from permeability variations, the effects which they cause or influence have been termed heterogeneity effects.
SPEJ
P. 145^