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Flow regimes for fluid injection into a confined porous medium

Published online by Cambridge University Press:  24 February 2015

Zhong Zheng ,
Bo Guo ,
Ivan C. Christov Open the ORCID record for Ivan C. Christov [Opens in a new window] ,
Michael A. Celia  and
Howard A. Stone
Zhong Zheng
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA
Bo Guo
Affiliation:
Department of Civil and Environmental Engineering, Princeton University, Princeton, NJ 08544, USA
Ivan C. Christov
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
Michael A. Celia
Affiliation:
Department of Civil and Environmental Engineering, Princeton University, Princeton, NJ 08544, USA
Howard A. Stone*
Affiliation:
Department of Mechanical and Aerospace Engineering, Princeton University, Princeton, NJ 08544, USA
*
†Email address for correspondence: hastone@princeton.edu
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Abstract

We report theoretical and numerical studies of the flow behaviour when a fluid is injected into a confined porous medium saturated with another fluid of different density and viscosity. For a two-dimensional configuration with point source injection, a nonlinear convection–diffusion equation is derived to describe the time evolution of the fluid–fluid interface. In the early time period, the fluid motion is mainly driven by the buoyancy force and the governing equation is reduced to a nonlinear diffusion equation with a well-known self-similar solution. In the late time period, the fluid flow is mainly driven by the injection, and the governing equation is approximated by a nonlinear hyperbolic equation that determines the global spreading rate; a shock solution is obtained when the injected fluid is more viscous than the displaced fluid, whereas a rarefaction wave solution is found when the injected fluid is less viscous. In the late time period, we also obtain analytical solutions including the diffusive term associated with the buoyancy effects (for an injected fluid with a viscosity higher than or equal to that of the displaced fluid), which provide the structure of the moving front. Numerical simulations of the convection–diffusion equation are performed; the various analytical solutions are verified as appropriate asymptotic limits, and the transition processes between the individual limits are demonstrated. The flow behaviour is summarized in a diagram with five distinct dynamical regimes: a nonlinear diffusion regime, a transition regime, a travelling wave regime, an equal-viscosity regime, and a rarefaction regime.

JFM classification

Geophysical and Geological Flows: Geophysical and Geological Flows Geophysical and Geological Flows: Gravity currents Low-Reynolds-number flows: Porous media
Type
Papers
Information
Journal of Fluid Mechanics , Volume 767 , 25 March 2015 , pp. 881 - 909
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Copyright
© 2015 Cambridge University Press 

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References

Acton, J. M., Huppert, H. E. & Worster, M. G. 2001 Two-dimensional viscous gravity currents flowing over a deep porous medium. J. Fluid Mech. 440, 359–380.Google Scholar
Barenblatt, G. I. 1979 Similarity, Self-Similarity, and Intermediate Asymptotics. Consultants Bureau.CrossRefGoogle Scholar
Bear, J. 1972 Dynamics of Fluids in Porous Media. Elsevier.Google Scholar
Boussinesq, J. V. 1904 Recherches theoretique sur l’ecoulement des nappes d’eau infiltrees dans le sol et sur le debit des sources. J. Math. Pures Appl. 10, 5–78.Google Scholar
Gasda, S. E., Bachu, S. & Celia, M. A. 2004 Spatial characterization of the location of potentially leaky wells penetrating a deep saline aquifer in a mature sedimentary basin. Environ. Geol. 46, 707–720.CrossRefGoogle Scholar
Gunn, I. & Woods, A. W. 2011 On the flow of buoyant fluid injected into a confined, inclined aquifer. J. Fluid Mech. 672, 109–129.CrossRefGoogle Scholar
Hallez, Y. & Magnaudet, J. 2009 A numerical investigation of horizontal viscous gravity currents. J. Fluid Mech. 630, 71–91.CrossRefGoogle Scholar
Hesse, M. A., Orr, F. M. Jr & Tchelepi, H. A. 2008 Gravity currents with residual trapping. J. Fluid Mech. 611, 35–60.Google Scholar
Hesse, M. A., Tchelepi, H. A., Cantwell, B. J. & Orr, F. M. Jr 2007 Gravity currents in horizontal porous layers: transition from early to late self-similarity. J. Fluid Mech. 577, 363–383.Google Scholar
Hinch, E. J. 1991 Perturbation Methods. Cambridge University Press.Google Scholar
Homsy, G. M. 1987 Viscous fingering in porous media. Annu. Rev. Fluid Mech. 19, 271–311.CrossRefGoogle Scholar
Huppert, H. E. 1982a Flow and instability of a viscous current down a slope. Nature 300, 427–429.Google Scholar
Huppert, H. E. 1982b The propagation of two-dimensional and axisymmetric viscous gravity currents over a rigid horizontal surface. J. Fluid Mech. 121, 43–58.CrossRefGoogle Scholar
Huppert, H. E. & Woods, A. W. 1995 Gravity driven flows in porous layers. J. Fluid Mech. 292, 55–69.Google Scholar
Kurganov, A. & Tadmor, E. 2000 New high-resolution central schemes for nonlinear conservation laws and convection–diffusion equations. J. Comput. Phys. 160, 241–282.Google Scholar
Lake, L. W. 1989 Enhanced Oil Recovery. Prentice Hall.Google Scholar
Lenormand, R., Touboul, E. & Zarcone, C. 1988 Numerical models and experiments on immiscible displacements in porous media. J. Fluid Mech. 189, 165–187.Google Scholar
LeVeque, R. J. 2002 Finite Volume Methods for Hyperbolic Problems. Cambridge University Press.Google Scholar
Lister, J. R. 1992 Viscous flows down an inclined plane from point and line sources. J. Fluid Mech. 242, 631–653.CrossRefGoogle Scholar
Lyle, S., Huppert, H. E., Hallworth, M., Bickle, M. & Chadwick, A. 2005 Axisymmetric gravity currents in a porous medium. J. Fluid Mech. 543, 293–302.CrossRefGoogle Scholar
MacMinn, C. W., Szulczewski, M. L. & Juanes, R. 2010 $\text{CO}_{2}$ migration in saline aquifers. Part 1. Capillary trapping under slope and groundwater flow. J. Fluid Mech. 662, 329–351.Google Scholar
MacMinn, C. W., Szulczewski, M. L. & Juanes, R. 2011 $\text{CO}_{2}$ migration in saline aquifers. Part 2. Combined capillary and solubility trapping. J. Fluid Mech. 688, 321–351.CrossRefGoogle Scholar
Metz, B., Davidson, O., De Connick, H., Loos, M. & Meyer, L. 2005 IPCC Special Report on Carbon Dioxide Capture and Storage. Cambridge University Press.Google Scholar
Neufeld, J. A., Vella, D., Huppert, H. E. & Lister, J. R. 2011 Leakage from gravity currents in a porous medium. Part 1. A localized sink. J. Fluid Mech. 666, 391–413.CrossRefGoogle Scholar
Nordbotten, J. M. & Celia, M. A. 2006 Similarity solutions for fluid injection into confined aquifers. J. Fluid Mech. 561, 307–327.Google Scholar
Pegler, S. S., Huppert, H. E. & Neufeld, J. A. 2014 Fluid injection into a confined porous layer. J. Fluid Mech. 745, 592–620.Google Scholar
Pritchard, D. 2007 Gravity currents over fractured substrates in a porous medium. J. Fluid Mech. 584, 415–431.CrossRefGoogle Scholar
Rottman, J. W. & Simpson, J. E. 1983 Gravity currents produced by instantaneous releases of a heavy fluid in a rectangular channel. J. Fluid Mech. 135, 95–110.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 liquid. Proc. R. Soc. Lond. A 245, 312–329.Google Scholar
Saha, S., Salin, D. & Talon, L. 2013 Low Reynolds number suspension gravity currents. Eur. Phys. J. E 36, 10385.Google Scholar
Shin, J. O., Dalziel, S. B. & Linden, P. F. 2004 Gravity currents produced by lock exchange. J. Fluid Mech. 521, 1–34.Google Scholar
Taghavi, S. M., Alba, K., Seon, T., Wielage-Burchard, K., Martinez, D. M. & Frigaard, I. A. 2012 Miscible displacement flows in near-horizontal ducts at low Atwood number. J. Fluid Mech. 696, 175–214.Google Scholar
Taghavi, S. M., Seon, T., Martinez, D. M. & Frigaard, I. A. 2009 Buoyancy-dominated displacement flows in near-horizontal channels: the viscous limit. J. Fluid Mech. 639, 1–35.Google Scholar
Vella, D. & Huppert, H. E. 2006 Gravity currents in a porous medium at an inclined plane. J. Fluid Mech. 555, 353–362.Google Scholar
Verdon, J. & Woods, A. W. 2007 Gravity-driven reacting flows in a confined porous aquifer. J. Fluid Mech. 588, 29–41.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
Zheng, Z., Christov, I. C. & Stone, H. A. 2014 Influence of heterogeneity on second-kind self-similar solutions for viscous gravity currents. J. Fluid Mech. 747, 218–246.Google Scholar
Zheng, Z., Soh, B., Huppert, H. E. & Stone, H. A. 2013 Fluid drainage from the edge of a porous reservoir. J. Fluid Mech. 718, 558–568.CrossRefGoogle Scholar