Numerical Solution Of The Perturbation Equations For Miscible Displacement
Authors: Perrine Richard L. , Lammers Donald L. 
Published: December 2007

This paper is to be presented at the 39th Annual Fall Meeting of the Society of Petroleum Engineers on Oct. 11–14, 1964, in Houston, Tex., and is considered the property of the Society of Petroleum Engineers. Permission to publish is hereby restricted to an abstract of not more than 300 words, with no illustrations, unless the paper is specifically released to the press by the Editor of JOURNAL OF PETROLEUM TECHNOLOGY or the Executive Secretary. Such abstract should contain conspicuous acknowledgment of where and by whom the paper is presented. Publication elsewhere after publication in JOURNAL OF PETROLEUM TECHNOLOGY or SOCIETY OF PETROLEUM ENGINEERS JOURNAL is granted on request, providing proper credit is given that publication and the original presentation of the paper.
Discussion of this paper is invited. Three copies of any discussion should be sent to the Society of Petroleum Engineers office. Such discussion may be presented at the above meeting and considered for publication in one of the two SPE magazines with the paper.

Abstract
A modified analysis of the perturbation equations for miscible displacement has been completed, avoiding previous difficulties. The results obtained are stability coefficients. These define initial growth of a disturbance at a particular time and set of conditions. The initial change with time of these coefficients was also obtained. Results agree well with those from the drastically, simpler earlier method, and agree reasonably well with experiments. Errors in the present calculations are several percent or less.

Introduction
A number of experimental and theoretical studies, and field experience as well, has shown that unstable viscous fingering is inherent in most miscible displacement oil recovery operations. Recovery efficiency can be drastically affected by the change from diffusion-like to fingering displacement. Thus complete knowledge of both stable and unstable flow mechanisms, and the transition from one regime to the other, is an ultimate need as a basis for reservoir engineering predictions. Two approaches can be used to develop a mathematical description of the transition from stable to unstable displacement. The simplest approach is useful when the thickness of the transition zone from oil to solvent is small compared with the scale of inhomogeneities which trigger the fingering process. The first approach considers transition to be discontinuous, and thus the process to be like a non-capillary immiscible displacement process. The second approach is useful throughout most of a true miscible solvent flood, for which transition zone thickness is large compared with the scale of individual inhomogeneities. Following this second approach a stability theory for miscible liquid-liquid displacement has been developed by application of perturbation methods. Validity of the results has been questioned, however, because of the drastic mathematical simplifications used to obtain an answer of engineering utility.
A modified analysis of the perturbation equations for miscible displacement which avoids previous difficulties has been completed.