Stability results for multi-layer radial Hele-Shaw and porous media flows

Abstract

Motivated by stability problems arising in the context of chemical enhanced oil recovery, we perform linear stability analysis of Hele-Shaw and porous media flows in radial geometry involving an arbitrary number of immiscible fluids. Key stability results obtained and their relevance to the stabilization of fingering instability are discussed. Some of the key results, among many others, are (i) absolute upper bounds on the growth rate in terms of the problem data; (ii) validation of these upper bound results against exact computation for the case of three-layer flows; (iii) stability enhancing injection policies; (iv) asymptotic limits that reduce these radial flow results to similar results for rectilinear flows; and (v) the stabilizing effect of curvature of the interfaces. Multi-layer radial flows have been found to have the following additional distinguishing features in comparison to rectilinear flows: (i) very long waves, some of which can be physically meaningful, are stable; and (ii) eigenvalues can be complex for some waves depending on the problem data, implying that the dispersion curves for one or more waves can contact each other. Similar to the rectilinear case, these results can be useful in providing insight into the interfacial instability transfer mechanism as the problem data are varied. Moreover, these can be useful in devising smart injection policies as well as controlling the complexity of the long-term dynamics when drops of various immiscible fluids intersperse among each other. As an application of the upper bound results, we provide stabilization criteria and design an almost stable multi-layer system by adding many layers of fluid with small positive jumps in viscosity in the direction of the basic flow.

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

“…In the limit of a stable inner interface (µ 1 µ 2 , µ 3 ) the result matches that given by Cardoso & Woods (1995). These asymptotic limits were also derived by Gin & Daripa (2015).…”

“…However, in the present problem of well treatment, both interfaces are likely to be unstable. Recently, Gin & Daripa (2015) examined the growth rates of instabilities in a dual-interface system as a function of the viscosity of the three fluids in the system. In that analysis less attention was placed on the conditions for overall stability; however, this is relevant for the injection of treatment fluid followed by a volume of post-treatment clean-up fluid, and forms the focus of the present work.…”

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

“…Work in this area has been much motivated by the problem of enhanced oil recovery through a porous medium. The linear stability problem for the radial spread resulting from successive injection of two fluids at a constant rate into the cell has been investigated in a series of papers by Daripa (2008a, b), Daripa & Ding (2012) and Gin & Daripa (2015). A weakly nonlinear treatment of this three-layer situation has been recently presented by and further developed numerically in the nonlinear regime by Zhao et al (2020).…”

Simultaneous Flow of n-Immiscible Fractional Maxwell Fluids with Generalized Thermal Flux and Robin Boundary Conditions

“…It has been seen that the Kelvin-Helmholtz instability can occur due to viscosity and density stratification. Several scientists subsequently researched the stability/instability of the immiscible fluid flow in two or multifaceted layers [6][7][8]. The existence and uniqueness of the simultaneous multilayered Couette/Poiseuille fluid motions in channel/pipes were investigated by Le Meur [9], and the approximated Oldroyd differential component and the viscosity proportions were found important to a unique result.…”