Effect Of Length On Unstable Miscible Displacements
Authors: Coskuner Gokhan , Bentsen Ramon G. 
Published: March 2010

Abstract
Many secondary recovery schemes involve the displacement of oil by a miscible fluid. The success of such schemes depends. to a large extent, on whether the displacement is stable or not. Thus, it is important to be able to predict the boundary at which a displacement becomes unstable.
In this paper, a dimensionless scaling group and its critical value for predicting the onset of instability during a miscible displacement in porous medium is derived. The perturbation equations used in the derivation are obtained by employing a small perturbations analysis and are solved using a variational technique. As a consequence, it is possible, for the first time, to infer explicitly the effect that the length of a porous medium has on the instability of a miscible displacement. It is demonstrated that in short laboratory systems, the length has an important effect and it should be taken into account. However, under field conditions, the analyses which consider the porous medium as being infinitely long can be used.
The theory developed is used to predict the onset of instability in miscible Hele-Shaw cell displacements. It is shown that the dimensionless scaling group successfully predicts the instability boundary. Moreover, experimental observations indicate that although a displacement initially can be stable, it may become unstable as the interface travels through the system. This, as well, agrees with the theory.

Introduction
Immiscible secondary recovery methods, even when successful, may leave significant amounts of oil in the reservoir due to the presence of capillary forces which are responsible for the entrapment of the oil. Thus, miscible secondary recovery processes, in theory, are considered to be very efficient because they eliminate capillary forces. However, in practically all miscible displacement tests, the displacing fluid (solvent) is less viscous than the crude oil and, as a result, the displacement front becomes unstable and viscous fingers of the displacing fluid penetrate into the crude oil. Thus, in practice, early breakthrough of the solvent and poor sweep efficiencies are encountered due to the phenomenon of viscous fingering(1–4). Therefore, the success of miscible secondary schemes depends on whether the displacement is stable or unstable.
The need to predict the onset of hydrodynamic instability in miscible displacements had led a number of investigators to undertake a stability analysis(5–14). All of these analyses are based on the small perturbations method. The perturbation equations which are obtained in a perturbation type analysis are nonlinear and the analytical solution of them poses problems. Therefore, it is common to linearize the perturbation equations. The time dependence of the perturbations is obtained by expressing an arbitrary disturbance as a superposition of certain basic modes and examining the stability of the system with respect to each of these modes. Because the equations used in the stability analysis have been linearized, only the onset of instability can be predicted. That is, the behaviour of the system, nce it becomes unstable, cannot be evaluated.