Mechanics of Viscous Fingering in Miscible Systems
Authors: Perkins Thomas K. , Johnston Orville C. , Hoffman Robert N. 
Published: December 2007

Abstract
A simplified theory of viscous fingering in miscible systems has been developed. It predicts the correct functional relationship between pertinent variables and permits the calculation of the order of magnitude of fingering behavior for simple systems, such as linear or radial.
The theory is based on the following four observations:cross-flow takes place only near the ends of fingers;the relative finger width is about 0.5;fingers can be suppressed by transverse dispersion, the suppression being quantitatively described by a critical ratio of dispersion times; andthe widths of extending fingers are close to the minimum size finger that can grow at any stage of displacement.
Fingering is studied in two-dimensional linear and diverging radial systems, both theoretically and experimentally. For linear systems, the length of the fingered region is proportional to mean displacement; the finger width is proportional to square root of mean displacement; and there is a small initial region void of fingers because of suppression by longitudinal dispersion.
For the radial system, two limiting conditions are recognized. If the mean displacement is small compared with the wellbore radius, the length of the fingered region is proportional to the mean displacement. The width is proportional to the square root of mean displacement. If the mean displacement is large compared with wellbore radius, length of the fingered region is proportional to mean displacement, but the number of fingers approaches a constant value. Also, in the radial case there is a small initial region void of fingers because of longitudinal dispersion.

Introduction
The behavior of viscous fingers in miscible systems has been of interest to the oil industry for many years. Previous studies have clearly shown the existence and nature of fingers in small models,1,2,3,4 Engineering techniques for extrapolating to reservoir situations have been proposed.5 Still, because of the lack of a real understanding of the mechanics of fingering, there remains uncertainty and disagreement as to the best way to scale or calculate fingering behavior in the reservoir.
In this paper we discuss a study of the mechanics of fingering in miscible systems (i.e., why do fingers behave as they do?). A simplified theory is developed which we believe willpredict the correct functional relationship between pertinent variables,permit us to calculate the order of magnitude of fingering behavior for simple systems such as linear and radial, andgive further insight into the problem of scaling or otherwise extending the results to more complicated reservoir conditions.
The paper includes the following sections:brief summary of the mixing behavior of miscible fluids in linear and radial systems;four fundamental observations of fluid flow (under fingering conditions);based on these four observations, a simplified theory of fingering in linear and radial systems is developed; andthe theoretical equations are compared with experimental fingering data measured in laboratory models.

A Review of Diffusion and Dispersion
As will be shown later, the behavior of viscous fingers in miscible systems is controlled in large degree by the mixing between the two fluids. A quantitative description of fingers will first require a quantitative description of mixing. Fortunately, much work has been done to clarify this subject; a fairly comprehensive review has been presented in a previous paper.7 For present purposes let usbriefly summarize the quantitative description of diffusion and dispersion (both longitudinal and transverse) within a differential element;present a simplification of the "width of mixed zone"; anddescribe the effect of geometry on width of mixed zone.