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On the growth of the mixing zone in miscible viscous fingering

Published online by Cambridge University Press:  16 June 2010

R. J. S. BOOTH
R. J. S. BOOTH*
Affiliation:
Mathematical Institute, Oxford University, 24–29 Saint Giles, Oxford OX1 3LB, UK
*
†Email address for correspondence: rjsbooth@gmail.com
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Abstract

We investigate the growth of a mixing zone in the displacement of oil by a solvent. Such a zone usually takes the form of long thin fingers of solvent which protrude into the oil. However, despite the existence of reasonably good empirical models for the evolution of mixing zones, there is limited theoretical understanding of the observed growth. Of particular interest is the rate at which the leading edge of the mixing zone grows. In this paper we establish the structure of the mixing zone, and reveal a critical mechanism that plays a role in the growth of the leading edge of the mixing zone. It turns out that there is a close link between the growth rate of the mixing zone and a shape selection problem for Saffman–Taylor fingers.

JFM classification

Convection: Convection in porous media Interfacial Flows (free surface): Fingering instability
Type
Papers
Information
Journal of Fluid Mechanics , Volume 655 , 25 July 2010 , pp. 527 - 539
Copyright
Copyright © Cambridge University Press 2010

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References

REFERENCES

Blackwell, R. J., Rayne, J. R. & Terry, W. M. 1959 Factors influencing the efficiency of miscible displacement. Trans. AIME 216, 1–8.CrossRefGoogle Scholar
Booth, R. J. S. 2008 Miscible flow through porous media. PhD thesis, Oxford University, Oxford, UK.Google Scholar
Chapman, S. J. 1999 On the rôle of Stokes lines in the selection of Saffman–Taylor fingers with small surface tension. Eur. J. Appl. Math. 10, 513–534.CrossRefGoogle Scholar
Chapman, S. J. & King, J. R. 2003 The selection of Saffman–Taylor fingers by kinetic undercooling. J. Engng Math. 46, 1–32.CrossRefGoogle Scholar
Chen, C.-Y. & Meiburg, E. 2000 High-accuracy implicit finite-difference simulations of homogeneous and heterogeneous miscible-porous-medium flows. SPE J. 5, 129–137.Google Scholar
Christie, M. A., Muggeridge, A. H. & Barley, J. J. 1993 3D simulation of viscous fingering and WAG schemes. SPE J. 8, 19–26.Google Scholar
Combescot, R., Hakim, V., Dombre, T., Pomeau, Y. & Pumir, A. 1986 Shape selection of Saffman–Taylor fingers. Phys. Rev. Lett. 56, 2036–2039.CrossRefGoogle ScholarPubMed
Fayers, F. J. 1988 An approximate model with physically interpretable parameters for representing viscous fingering. SPE Reservoir Engng 285, 551–558.CrossRefGoogle Scholar
Fayers, F. J., Blunt, M. J. & Christie, M. A. 1992 Comparisons of empirical viscous-fingering models and their calibration for heterogeneous problems. SPE Reservoir Engng, pp. 195–203.Google Scholar
Hickernell, F. J. & Yortsos, Y. C. 1986 Linear stability of miscible displacement processes in porous media in the absence of dispersion. Stud. Appl. Math. 74, 93–115.CrossRefGoogle Scholar
Homsy, G. M. 1987 Viscous fingering in porous media. Annu. Rev. Fluid Mech. 19, 271–311.CrossRefGoogle Scholar
Koval, E. J. 1963 A method for predicting the performance of unstable miscible displacement in heterogenous media. SPE J. 450, 145–154.Google Scholar
Manickam, O. & Homsy, G. M. 1995 Fingering instabilities in vertical miscible displacement flows in porous media. J. Fluid Mech. 288, 75–102.CrossRefGoogle Scholar
Menon, G. & Otto, F. 2005 Dynamic scaling in miscible viscous fingering. Commun. Math. Phys. 257, 303–317.CrossRefGoogle Scholar
Mikelić, A. 1993 Regularity and uniqueness results for two-phase miscible flows in porous media. Intl. Ser. Numer. Math. 114, 139–154.Google Scholar
Peaceman, D. W. & Rachford, H. H. 1962 Numerical calculation of multidimensional miscible displacement. SPE J. 2, 327–339.Google Scholar
Riaz, A. & Meiburg, E. 2003 Three-dimensional miscible displacement simulations in homogeneous porous media with gravity override. J. Fluid Mech. 494, 95–117.CrossRefGoogle Scholar
Riaz, A., Pankiewitz, C. & Meiburg, E. 2004 Linear stability of radial displacements in porous media: influence of velocity-induced dispersion and concentration-dependent diffusion. Phys. Fluids 16, 3592–3598.CrossRefGoogle Scholar
Rogerson, A. & Meiburg, E. 1993 Numerical simulation of miscible displacement processes in porous medium flows under gravity. Phys. Fluids A 5, 2644–2660.CrossRefGoogle Scholar
Ruith, M. & Meiburg, E. 2000 Miscible rectilinear displacements with gravity override. Part 1. Homogeneous porous medium. J. Fluid Mech. 420, 225–257.CrossRefGoogle Scholar
Saffman, P. G. & Taylor, G. I. 1958 The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous fluid. Proc. R. Soc. A 245, 312–329.Google Scholar
Siegel, M., Caflisch, R. E. & Howison, S. D. 2004 Global existence, singular solutions, and ill-posedness for the Muskat problem. Commun. Pure Appl. Math. LVII, 1374–1411.CrossRefGoogle Scholar
Tan, C. T. & Homsy, G. M. 1986 Stability of miscible displacements in porous media – rectilinear flow. Phys. Fluids 29, 3549–3556.CrossRefGoogle Scholar
Tan, C. T. & Homsy, G. M. 1988 Simulation of nonlinear viscous fingering in miscible displacement. Phys. Fluids 31, 1330–1338.CrossRefGoogle Scholar
Tchelepi, H. A. & Orr, F. M. Jr., 1994 Interaction of viscous fingering, permeability heterogeneity, and gravity segregation in three dimensions. SPE Reservoir Engng 9, 266–271.CrossRefGoogle Scholar
Tchelepi, H. A., Orr, F. M. Jr., Rakotomalala, N., Salin, D. & Wouméni, R. 1993 Dispersion, permeability heterogeneity, and viscous fingering: acoustic experimental observations and particle-tracking simulations. Phys. Fluids A 5, 1558–1574.CrossRefGoogle Scholar
Todd, M. R. & Longstaff, W. J. 1972 The development, testing and application of a numerical simulator for predicting miscible flood performance. J. Pet. Tech. 253, 874–882.CrossRefGoogle Scholar
Wooding, R. A. 1969 Growth of fingers at an unstable diffusing interface in a porous medium or Hele-Shaw cell. J. Fluid Mech. 39, 477–495.CrossRefGoogle Scholar
Yang, Z. M., Yortsos, Y. C. & Salin, D. 2002 Asymptotic regimes in unstable miscible displacements in random porous media. Adv. Water Resour. 25, 885–898.CrossRefGoogle Scholar
Yortsos, Y. C. & Salin, D. 2006 On the selection principle for viscous fingering in porous media. J. Fluid Mech. 557, 225–236.CrossRefGoogle Scholar
Zimmerman, W. B. & Homsy, G. M. 1991 Nonlinear viscous fingering in miscible displacement with anisotropic dispersion. Phys. Fluids A 3, 1859–1872.CrossRefGoogle 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 A 4, 2348–2359.CrossRefGoogle Scholar