A Hele-Shaw Cell Study Of A New Approach To Instability Theory In Porous Media
Authors: Purvis G. , Bentsen R.G. 
Published: March 2010

Abstract
Whether a displacement is stable or unstable has a profound effect on how efficiently one fluid immiscibly displaces another within a porous medium. As a consequence, it is of interest to be able to predict the boundary which separates stable displacements from those which are unstable. Recently, a new approach to stability theory has been developed which is based on the assumption that the immiscible displacement of one fluid by another can be treated as (J moving boundary problem. This paper describes a number of experiments undertaken to validate the newly developed theory. Because the assumption of a sharp front between the two fluids is satisfied exactly in a Hele-Shaw cell, these preliminary experiments were undertaken in such a model.
Sixty-three experiments, using fluids having different viscosity ratios, interfacial tensions and wetting characteristics, have been performed. In general, these experiments have confirmed the validity of the new theory. In particular, the stability boundary and subsequent mode boundaries are predicted correctly. Moreover, the time constant vs wavelength behaviour predicted by the theory has been verified experimentally. Also, the relative widths of oppositely directed viscous fingers are predicted correctly. In addition, the distance travelled by the tip o/the oil and the water fingers, with respect to the initially plane interface separating two fluids, was found to be a linear function of time. Finally, the shape of the interface obtained in the experiments W(2) in good agreement with the postulated functional farm.

Introduction
Early work on the stability analysis of immiscible fluid displacement is normally based on first-order perturbation theory and the concept of a velocity potential(3–10). Also, it is usual to use a sharp interface approximation. While such an approach can deal with an incipient finger, it cannot deal with the subsequent growth of the viscous finger. Moreover, there are problems with the use of first-order perturbation theory and with the concept of a velocity potential.
A velocity potential exists only for flow fields involving a fluid of constant density and viscosity and a porous medium which is homogeneous and isotropic throughout(11). Therefore, it cannot, in principle, deal with a real porous medium. Moreover, it is the force potential and not the velocity potential which governs the flow of fluids through porous media(12). When the velocity potential approach is taken, the postulated velocity potential will satisfy Laplace's equation only if the divergence of the perturbation velocity is zero. But the divergence of the perturbation velocity is zero only when a plane interface separates the displacing fluid from the displaced one. As soon as the interface is perturbed, two fluids will occupy a region where there was previously only one, and the divergence of the perturbation velocity will no longer be zero. Rather, the divergence of the sum of the perturbation velocities, one for each fluid, will be zero(22).
First-order perturbation theory is not rigorous and the conclusions obtained from it are not always correct, as linearization of the equations is valid only if the deviations are small.