The Onset Of Instability During Immiscible Displacements In Cylindrical Models
Authors: Chakrabarty Chayan , Bentsen R.G. 
Published: March 2010

Abstract
The immiscible displacement of one viscous fluid by another within a porous medium forms the basis of most secondary-recovery schemes. The stability of such displacement processes is one of the factors which controls displacement efficiency. Consequently, it is of interest to be able to predict the onset ofil1Slability during such displacement processes.
Using a force-potential approach and defining a macroscopic capillary pressure in the mobile-oil zone, a stability theory has been developed for a cylindrical flow system. The instability number has been defined along with its critical value beyond which the displacement is unstable. The stability boundary has been validated by means of published experimental data.

Introduction
During water injection in an oil-bearing reservoir, the displacement velocity is maximal at the point of injection and decreases monotonically as the radial distance from the wellbore increases. Also from potentiometric studies it can be seen that, near the wellbore, the displacement process is essentially radial. Therefore, the cylindrical displacement model can be considered to be more appropriate to practical au-recovery situations than the linear-displacernent model, especially during the initial stage of the displacement process. Hence, it is important to be able to develop a stability theory for immiscible displacement in cylindrical systems.
Unlike linear systems, previous investigation of instability in radial displacement systems is meagre. Bataille(1) made a theoretical and experimenta1 examination of the problem of instability of an initially circular interface between two immiscible fluids in radial flow in a Hele-Shaw cell. He showed that the condition for stability of the interface was Equation (1) (Available in full paper)
where Ø is the interfacial tension, k is the absolute permeability n is the number of fingers, µ1 and µ2 are the viscosities of the displacing and the displaced fluids, respectively and rF is the radius of the hypothetical undisturbed interface when there is no fingering.
Wilson(2) studied the mechanism of dynamic instability of an initially circular interface between two immiscible fluids in a Hele- Shaw coli. Adopting an analysis similar to the one described by Saffman and Taylor(3), he arrived at a stability criterion identical to that derived by Bataille(1).
Paterson(4) used the analysis of Bataille and Wilson to examine the initial stages of immiscible interfacial fingering for bath advancing and contracting displacement fronts. He derived the defining equation for the critical wavelength of the perturbation of the front. Also he argued that, when the circumference of the front is less than the critical wavelength, the displacement is stable and the front remains circular. Ni et al. (3) extended Paterson's theory for immiscible radial fingering in a Hele-Shaw cell to that in a porous medium confined between two horizontal parallel plates.
Nasr-El-Din et (6) studied the effect of flow rate on radial displacement patterns in water-wet porous media. They observed, as did Ni el al.(5), that at intermediate flow rates, recovery at breakthrough was independent of flaw rate; and at higher flow rates, recovery decreased with increasing flow rate to a second stabilized value independent of flow rate,