Time-dependent injection strategies and interfacial stability in multi-layer Hele-Shaw and porous media flows

Abstract

We study the stability of multi-layer radial flows in porous media within the Hele-Shaw model. We perform a linear stability analysis for radial flows consisting of an arbitrary number of fluid layers with interfaces separating fluids of constant viscosity and with positive viscosity jump at each interface in the direction of flow. Several different time-dependent injection strategies are analyzed including the maximal injection rate that maintains a stable flow. We find numerically that flows with more fluid layers can be stable with faster time-dependent injection rates than comparable flows with fewer fluid layers. In particular, the injection rate for a stable flow increases at a rate that is proportional to the number of interfaces to the two-thirds power for large times. Additionally, we show that in any multi-layer radial Hele-Shaw flow, if all of the interfaces are circular except for one perturbed interface then there exists a time-dependent injection rate such that the circular interfaces remain circular as they propagate and the disturbance on the perturbed interface decays.The motion of the interfaces within linear theory is also investigated numerically for the case of constant injection rates. It is found that: (i) A disturbance of one interface can be transferred to the other interface(s); (ii) The disturbances on the interfaces can develop either in phase or out of phase from any arbitrary initial disturbance; and (iii) The dynamics of the flow can change dramatically with the addition of more interfaces.

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

“…The constant viscosity case is given by the solid (black) line and the viscosity of the middle layer fluid is µ = 6. The stability of three-layer constant viscosity flows has been studied extensively by Gin & Daripa (2015, 2018. Note that there is a maximum growth rate and that short waves are stable due to interfacial tension.…”

“…We now turn to three-layer flows in which the fluid in the middle layer also has constant viscosity, µ 1 . This situation has been investigated by Beeson-Jones & Woods (2015) and Gin & Daripa (2018), and it has been found that the magnitudes of interfacial disturbances A n (t) and B n (t) are governed by the following system of ODE's…”

“…1. It was demonstrated by Gin & Daripa (2018) that M r 1 can have complex eigenvalues. However, it is shown in the next section (section 5) that the problem in the ζ-coordinates has real growth rates.…”

“…Recent studies by Beeson-Jones & Woods (2015) and Gin & Daripa (2018) explore the idea of using a variable injection rate Q(t) to stabilize multi-layer constant viscosity flows. In these works, the maximum injection rate which results in a stable flow is investigated.…”