The stability of immiscible viscous fingering in Hele-Shaw cells with spatially varying permeability
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 int...
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| Format: | Article |
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Elsevier
2017
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| Online Access: | https://eprints.nottingham.ac.uk/41734/ |
| _version_ | 1848796341059190784 |
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| author | Jackson, S.J. Power, H. Giddings, Donald Stevens, D. |
| author_facet | Jackson, S.J. Power, H. Giddings, Donald Stevens, D. |
| author_sort | Jackson, S.J. |
| building | Nottingham Research Data Repository |
| collection | Online Access |
| description | 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. |
| first_indexed | 2025-11-14T19:46:26Z |
| format | Article |
| id | nottingham-41734 |
| institution | University of Nottingham Malaysia Campus |
| institution_category | Local University |
| last_indexed | 2025-11-14T19:46:26Z |
| publishDate | 2017 |
| publisher | Elsevier |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | nottingham-417342024-08-15T15:22:47Z https://eprints.nottingham.ac.uk/41734/ The stability of immiscible viscous fingering in Hele-Shaw cells with spatially varying permeability Jackson, S.J. Power, H. Giddings, Donald Stevens, D. 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. Elsevier 2017-06-15 Article PeerReviewed Jackson, S.J., Power, H., Giddings, Donald and 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 . pp. 606-632. ISSN 1879-2138 Spatially varying permeability Capillary number Viscous fingering instability BE -rbf http://www.sciencedirect.com/science/article/pii/S0045782516312373 doi:10.1016/j.cma.2017.03.030 doi:10.1016/j.cma.2017.03.030 |
| spellingShingle | Spatially varying permeability Capillary number Viscous fingering instability BE -rbf Jackson, S.J. Power, H. Giddings, Donald Stevens, D. The stability of immiscible viscous fingering in Hele-Shaw cells with spatially varying permeability |
| title | The stability of immiscible viscous fingering in Hele-Shaw cells with spatially varying permeability |
| title_full | The stability of immiscible viscous fingering in Hele-Shaw cells with spatially varying permeability |
| title_fullStr | The stability of immiscible viscous fingering in Hele-Shaw cells with spatially varying permeability |
| title_full_unstemmed | The stability of immiscible viscous fingering in Hele-Shaw cells with spatially varying permeability |
| title_short | The stability of immiscible viscous fingering in Hele-Shaw cells with spatially varying permeability |
| title_sort | stability of immiscible viscous fingering in hele-shaw cells with spatially varying permeability |
| topic | Spatially varying permeability Capillary number Viscous fingering instability BE -rbf |
| url | https://eprints.nottingham.ac.uk/41734/ https://eprints.nottingham.ac.uk/41734/ https://eprints.nottingham.ac.uk/41734/ |