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The dynamics of miscible viscous fingering from onset to shutdown

Published online by Cambridge University Press:  05 January 2018

Japinder S. Nijjer Open the ORCID record for Japinder S. Nijjer [Opens in a new window] ,
Duncan R. Hewitt Open the ORCID record for Duncan R. Hewitt [Opens in a new window]  and
Jerome A. Neufeld Open the ORCID record for Jerome A. Neufeld [Opens in a new window]
Japinder S. Nijjer*
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
Duncan R. Hewitt
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK
Jerome A. Neufeld
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, UK Department of Earth Sciences, University of Cambridge, Bullard Laboratories, Madingley Road, Cambridge CB3 0EZ, UK BP Institute, University of Cambridge, Bullard Laboratories, Madingley Road, Cambridge CB3 0EZ, UK
*
†Email address for correspondence: [email protected]
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Abstract

We examine the full ‘life cycle’ of miscible viscous fingering from onset to shutdown with the aid of high-resolution numerical simulations. We study the injection of one fluid into a planar two-dimensional porous medium containing another, more viscous fluid. We find that the dynamics are distinguished by three regimes: an early-time linearly unstable regime, an intermediate-time nonlinear regime and a late-time single-finger exchange-flow regime. In the first regime, the flow can be linearly unstable to perturbations that grow exponentially. We identify, using linear stability theory and numerical simulations, a critical Péclet number below which the flow remains stable for all times. In the second regime, the flow is dominated by the nonlinear coalescence of fingers which form a mixing zone in which we observe that the convective mixing rate, characterized by a convective Nusselt number, exhibits power-law growth. In this second regime we derive a model for the transversely averaged concentration which shows good agreement with our numerical experiments and extends previous empirical models. Finally, we identify a new final exchange-flow regime in which a pair of counter-propagating diffusive fingers slow exponentially. We derive an analytic solution for this single-finger state which agrees well with numerical simulations. We demonstrate that the flow always evolves to this regime, irrespective of the viscosity ratio and Péclet number, in contrast to previous suggestions.

JFM classification

Low-Reynolds-number flows: Low-Reynolds-number flows Low-Reynolds-number flows: Porous media
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 837 , 25 February 2018 , pp. 520 - 545
Copyright
© 2018 Cambridge University Press 

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References

Adams, J. C.1999 Mudpack: Multigrid software for elliptic partial differential equations. Computational information systems laboratory. https://www2.cisl.ucar.edu/resources/legacy/mudpack.Google Scholar
Almarcha, C., Trevelyan, P. M. J., Grosfils, P. & De Wit, A. 2010 Chemically driven hydrodynamic instabilities. Phys. Rev. Lett. 104, 144501.Google Scholar
Blackwell, R. J., Rayne, J. R. & Terry, W. M. 1959 Factors influencing the efficiency of miscible displacements. Petrol. Trans. AIME 216, 1–8.Google Scholar
Boait, F. C., White, N. J., Bickle, M. J., Chadwick, R. A., Neufeld, J. A. & Huppert, H. E. 2012 Spatial and temporal evolution of injected CO2 at the Sleipner field, North Sea. J. Geophys. Res. 117, B03309.Google Scholar
Booth, R. J. S. 2010 On the growth of the mixing zone in miscible viscous fingering. J. Fluid Mech. 655, 527–539.Google Scholar
Chui, J. Y. Y., De Anna, P. & Juanes, R. 2015 Interface evolution during radial miscible viscous fingering. Phys. Rev. E 92, 041003(R).Google Scholar
De Wit, A. & Homsy, G. M. 1997a Viscous fingering in periodically heterogeneous porous media. Part I. Formulation and linear instability. J. Chem. Phys. 107 (22), 9609–9618.Google Scholar
De Wit, A. & Homsy, G. M. 1997b Viscous fingering in periodically heterogeneous porous media. Part II. Numerical simulations. J. Chem. Phys. 107 (22), 9619–9628.10.1063/1.475259Google Scholar
Hewitt, D. R., Neufeld, J. A. & Lister, J. R. 2013 Convective shutdown in a porous medium at high Rayleigh number. J. Fluid Mech. 719, 551–586.Google Scholar
Hill, S. 1952 Channelling in packed columns. Chem. Engng Sci. 1, 247–253.10.1016/0009-2509(52)87017-4Google Scholar
Huppert, H. E. & Neufeld, J. A. 2014 The fluid mechanics of carbon dioxide sequestration. Annu. Rev. Fluid Mech. 46, 255–272.Google Scholar
Islam, M. N. & Azaiez, J. 2005 Fully implicit finite difference pseudo-spectral method for simulating high mobility-ratio miscible displacements. Intl J. Numer. Meth. Fluids 1, 161–183.Google Scholar
Jha, B., Cueto-Felgueroso, L. & Juanes, R. 2011a Fluid mixing from viscous fingering. Phys. Rev. Lett. 106 (19), 194502.Google Scholar
Jha, B., Cueto-Felgueroso, L. & Juanes, R. 2011b Quantifying mixing in viscously unstable porous media flows. Phys. Rev. E 84, 066312.Google Scholar
van Keken, P. E., Hauri, E. H. & Ballentine, C. J. 2002 Mantle mixing: the generation, preservation, and destruction of chemical heterogeneity. Annu. Rev. Earth Planet. Sci. 30, 493–525.Google Scholar
Koval, E. J. 1963 A method for predicting the performance of unstable miscible displacement in heterogeneous media. Soc. Petrol. Engng J. 3 (2), 145–154.Google Scholar
Lajeunesse, E., Martin, J., Rakotomalala, N., Salin, D. & Yortsos, Y. C. 1999 Miscible displacement in a Hele-Shaw cell at high rates. J. Fluid Mech. 398, 299–319.Google Scholar
Lake, L. W. 1989 Enhanced Oil Recovery. Prentice Hall.Google Scholar
Lele, S. K. 1992 Compact finite difference schemes with spectral-like resolution. J. Comput. Phys. 103 (1), 16–42.Google Scholar
Loggia, D., Rakotomalala, N., Salin, D. & Yortsos, Y. C. 1998 The effect of mobility gradients on viscous instabilities in miscible flows in porous media. Phys. Fluids 11 (3), 740–742.Google Scholar
Malhotra, S., Sharma, M. M. & Lehman, E. R. 2015 Experimental study of the growth of mixing zone in miscible viscous fingering. Phys. Fluids 27, 014105.Google Scholar
McCloud, K. V. & Maher, J. V. 1995 Experimental perturbations to Saffman–Taylor flow. Phys. Rep. 260, 139–185.Google Scholar
Neufeld, J. A., Hesse, M. A., Riaz, A., Hallworth, M. A., Tchelepi, H. A. & Huppert, H. E. 2010 Convective dissolution of carbon dioxide in saline aquifers. Geophys. Res. Lett. 37 (22), L22404.Google Scholar
Nicolaides, C., Jha, B., Cueto-Felgueroso, L. & Juanes, R. 2015 Impact of viscous fingering and permeability heterogeneity on fluid mixing in porous media. Water Resour. Res. 51 (4), 2634–2647.Google Scholar
Paterson, L. 1985 Fingering with miscible fluids in a Hele-Shaw cell. Phys. Fluids 28 (1), 26–30.Google Scholar
Pramanik, S. & Mishra, M. 2015a Effect of Péclet number on miscible rectilinear displacement in a Hele-Shaw cell. Phys. Rev. E 91, 033006.Google Scholar
Pramanik, S. & Mishra, M. 2015b Nonlinear simulations of miscible viscous fingering with gradient stresses in porous media. Chem. Engng Sci. 122, 523–532.Google Scholar
Ruith, M. & Meiburg, E. 2000 Miscible rectilinear displacements with gravity override. Part 1. Homogeneous porous medium. J. Fluid Mech. 420, 225–257.Google Scholar
Saffman, P. G. & Taylor, G. 1958 The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid. Proc. R. Soc. Lond. A 245 (1242), 312–329.Google Scholar
Stone, H. A., Stroock, A. D. & Ajdari, A. 2004 Engineering flows in small devices: microfluidics toward a lab-on-a-chip. Annu. Rev. Fluid Mech. 36, 381–411.Google Scholar
Tan, C. T. & Homsy, G. M. 1986 Stability of miscible displacements in porous media: rectilinear flow. Phys. Fluids 29 (11), 3549–3556.Google Scholar
Tan, C. T. & Homsy, G. M. 1987 Stability of miscible displacements in porous media: radial source flow. Phy. Fluids 30 (5), 1239–1245.Google Scholar
Tan, C. T. & Homsy, G. M. 1988 Simulation of nonlinear viscous fingering in miscible displacement. Phys. Fluids 31 (6), 1330–1338.Google Scholar
Tan, C. T. & Homsy, G. M. 1992 Viscous fingering with permeability heterogeneity. Phys. Fluids 4 (6), 1099–1101.Google Scholar
Yang, Z. & Yortsos, Y. C. 1997 Asymptotic solutions of miscible displacements in geometries of large aspect ratio. Phys. Fluids 9, 286–298.Google Scholar
Yortsos, Y. C. & Salin, D. 2006 On the selection principle for viscous fingering in porous media. J. Fluid Mech. 557, 225–236.Google Scholar
Zhou, Q. 2013 Temporal evolution and scaling of mixing in two-dimensional Rayleigh–Taylor turbulence. Phys. Fluids 25, 085107.Google Scholar
Zimmerman, W. B. & Homsy, G. M. 1991 Nonlinear viscous fingering in miscible displacement with anisotropic dispersion. Phys. Fluids 3 (8), 1859–1872.Google Scholar
Zimmerman, W. B. & Homsy, G. M. 1992 Viscous fingering in miscible displacements: unification of effects of viscosity contrast, anisotropic dispersion, and velocity dependence of dispersion on nonlinear finger propagation. Phys. Fluids 4 (11), 2348–2359.Google Scholar