On the selection of viscosity to suppress the Saffman–Taylor instability in a radially spreading annulus

Abstract

We examine the stability of a system with two radially spreading fronts in a Hele-Shaw cell in which the viscosity increases monotonically from the innermost to the outermost fluid. The critical parameters are identified as the viscosity ratio of the inner and outer fluids and the viscosity difference between the intermediate and outer fluids as a fraction of the viscosity difference between the inner and outer fluids. There is a minimum viscosity ratio of the inner and outer fluids above which, for each azimuthal mode, the system is stable to perturbations of that mode at any flow rate. This condition is directly analogous to the result for a single interface. Below this minimum ratio, the system may be stable at any flow rate early in the flow. However, once the inner radius reaches a critical fraction of the outer radius, this absolute stability ceases to apply owing to the coupling of the inner and outer interfaces. We determine the maximum flow rate, as a function of time, in order that all modes remain stable due to the effects of interfacial tension. These criteria for stability are then used to select the viscosity of the intermediate fluid so that a fixed volume of the intermediate and then inner fluid can be added to the system in the minimum time with the system remaining stable throughout. The optimal viscosity for this intermediate fluid depends on the relative volume of the inner and intermediate fluid and also on the overall viscosity ratio of the innermost fluid and the original fluid in the cell, with the balance being to suppress the early time instability of the outer interface and the late time instability of the inner interface. We discuss application of this approach to a problem of injection of treatment fluid in an oil well.

Viscous fingering of miscible annular ring

“…The ever increasing area available for VF and the spatially varying velocity in radial displacement affects the instability in many ways. However, the limited amount of the sample in the annulus results in features different from classical radial VF, as also observed for immiscible fluids by Beeson-Jones & Woods (2015).…”

“…Recently, in context with the treatment of fluids in an oil well, Beeson-Jones & Woods (2015) considered the overall stability of the radially spreading annulus considering three fluids each having different viscosity. They reported that the results for an annulus are similar to the single interface radial flow, but depend on the viscosity difference between the inner and outer fluid.…”

Spreading or contraction of viscous drops between plates: single, multiple or annular drops

“…For the case of contraction (F < 0), we shall find that the fingering instability persists and is driven primarily from the outer boundary. The annular problem is related to the 'three-layer problem' involving the radial spread of one fluid followed by another of different viscosity in a Hele-Shaw cell (Cardoso & Woods 1995;Beeson-Jones & Woods 2015). Work in this area has been much motivated by the problem of enhanced oil recovery through a porous medium.…”

Stability Results on Radial Porous Media and Hele-Shaw Flows with Variable Viscosity Between Two Moving Interfaces

“…Cardoso & Woods (1995) studied the stability of three-layer radial flows in the limiting case in which the inner interface is completely stable and looked at the break up of the middle layer into drops. Beeson-Jones & Woods (2015) analyzed general three-layer flows, and Gin & Daripa (2015) performed a linear stability analysis for an arbitrary number of fluid layers in radial geometry.…”