Previous theoretical researchers successfully generated viscous finger patterns by assuming a randomly distributed boundary condition in their numerical models. Our objective is to identify a natural source of the randomness that underlies their success. A source of fluid flow instability is discerned by viewing fingering as a chaotic (nonlinear dynamical) phenomenon.
We begin by showing that miscible displacement models can be expressed as nonlinear generalizations of the linear convection-dispersion equation. A nonlinear dynamical analysis technique is used to study the stability of the nonlinear system. Detailed study of several 1D and 2D cases illustrates the applicability of the stability analysis.
Viscous fingers play an important role in determining the success or failure of a fluid displacement process. Much effort has been expended in the study of viscous fingering, e.g., Refs. 1 and 2, yet several important problems remain to be solved. Among the problems awaiting explanation is the question of what causes the disturbances that propagate as viscous fingers. A widely held notion in the industry is that reservoir heterogeneities are the source of the disturbances. Previous work3 suggests that nonlinear dynamics provide an alternative perspective on the source of disturbances in porous media or analogs such as Hele-Shaw cells.
Christie and Bond4 have recently applied chaotic concepts to the description of fingering. Their idea was to impose a boundary condition with a random distribution about a user-specified average value. They were able to numerically generate finger patterns using a nonlinear miscible displacement model. This work may be viewed as a numerical experiment that provides justification for applying chaos concepts to the viscous fingering problem.
The introduction of a randomly distributed boundary condition is an assumption. Why does the assumption appear to work? What mechanism is responsible for the randomness? One obvious explanation for floods performed in porous media is the randomness associated with the flow paths of a porous medium.5–7 This answer is incomplete, however, because the presence of a porous medium is not a prerequisite for the appearance of viscous fingers. Indeed, pioneering studies of viscous fingering8–9 did not even use a porous medium. These studies used a seemingly simple system called a Hele-Shaw cell.10 Fluid flow in a Hele-Shaw cell is restricted to the space between two parallel plates. A well-constructed radial Hele-Shaw cell should exhibit radial symmetry. It would seem that the uniformity of a Hele-Shaw cell would preclude any cell related source of instability. That this is so is evidenced by the early time radial behavior observed at the inlet of 3 radial Hele-Shaw cell.2 Yet explanations of fingering in Hele-Shaw cells usually presuppose the presence of a perturbation in the system. Paterson, for example, introduced a sinusoidal perturbation in his analysis of a radial Hele-Shaw cell.11 What is the source of the perturbation?
Kelkar and Gupta 12 reported that they were unable to initiate a viscous finger without introducing some type of perturbation such as a permeability variation or a concentration perturbation.