The stability of immiscible viscous fingering in Hele-Shaw cells with spatially varying permeability

Jackson, S.J.; Power, H.; Giddings, D.; Stevens, D.

Abstract

In this paper, we investigate the stability of immiscible viscous fingering in Hele-Shaw cells with spatially varying permeability, across a range of capillary numbers. We utilise a coupled boundary element - radial basis function (BE –RBF) numerical method that adapts and moves with the growing interface, providing an efficient, high accuracy scheme to track the interfacial displacement of immiscible fluids. By comparing the interfacial evolution and growth rate in varying permeability cells to that in uniform cells, we can assess the relative stability of the perturbations as a consequence of the variable permeability.

Numerical experiments in Hele-Shaw cells with gradually varying permeability highlight 3 aperture effects that control the interfacial stability: (1) Gradients in the capillary pressure (2) Local changes in fluid mobility (3) Variation in the viscous pressure gradient. In low capillary number regimes, we find that aperture effect 1 and 2 dominate, which (relatively) stabilise interfacial perturbations in converging geometries and destabilise perturbations in diverging geometries. In high capillary number regimes, aperture effect 3 dominates meaning the relative stability transitions; the interface is destabilised in converging cells and stabilised in diverging cells. We find an upper bound critical capillary number Cagt at which the relative stability transitions in our gradually varying cell as 1000<Cagt<1250, which is independent of both α and ϵ0. This result is much lower than the value of Cagt=9139 predicted by linear stability theory, due to significant non-linear perturbation growth. This transition links the results found in previous works performed at low and high capillary numbers, providing new insight into the viscous fingering instability in variable permeability cells.

To conclude, we present simulations in Hele-Shaw cells with large geometric heterogeneities and anisotropy, in order to demonstrate the significant fluid re-distribution that can occur due to localised variations in cell permeability. Using periodic permeability distributions, we show the significant re-distribution of fluid that can occur due to large capillary pressure gradients in the capillary limit, and the channelling of flow that can occur in the viscous limit along anisotropic features.

Citation

Jackson, S., Power, H., Giddings, D., & Stevens, D. (2017). The stability of immiscible viscous fingering in Hele-Shaw cells with spatially varying permeability. Computer Methods in Applied Mechanics and Engineering, 320, 606-632. https://doi.org/10.1016/j.cma.2017.03.030

• Journal Article Type: Article
• Acceptance Date: Mar 21, 2017
• Online Publication Date: Mar 31, 2017
• Publication Date: Jun 15, 2017
• Deposit Date: Apr 4, 2017
• Publicly Available Date: Apr 4, 2017
• Journal: Computer Methods in Applied Mechanics and Engineering
• Print ISSN: 0045-7825
• Electronic ISSN: 1879-2138
• Publisher: Elsevier
• Peer Reviewed: Peer Reviewed
• Volume: 320
• Pages: 606-632
• DOI: https://doi.org/10.1016/j.cma.2017.03.030
• Keywords: Spatially varying permeability, Capillary number, Viscous fingering instability, BE -rbf
• Public URL: https://nottingham-repository.worktribe.com/output/865899
• Publisher URL: http://www.sciencedirect.com/science/article/pii/S0045782516312373
• Additional Information: This article is maintained by: Elsevier; Article Title: The stability of immiscible viscous fingering in Hele-Shaw cells with spatially varying permeability; Journal Title: Computer Methods in Applied Mechanics and Engineering; CrossRef DOI link to publisher maintained version: https://doi.org/10.1016/j.cma.2017.03.030; Content Type: article; Copyright: © 2017 Elsevier B.V. All rights reserved.
• Contract Date: Apr 4, 2017