Skip to main content

Flow regimes for fluid injection into a confined porous medium

  • Zhong Zheng (a1), Bo Guo (a2), Ivan C. Christov (a1) (a3), Michael A. Celia (a2) and Howard A. Stone (a1)...

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.

Corresponding author
Email address for correspondence:
Hide All
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, 359380.
Barenblatt G. I. 1979 Similarity, Self-Similarity, and Intermediate Asymptotics. Consultants Bureau.
Bear J. 1972 Dynamics of Fluids in Porous Media. Elsevier.
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, 578.
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, 707720.
Gunn I. & Woods A. W. 2011 On the flow of buoyant fluid injected into a confined, inclined aquifer. J. Fluid Mech. 672, 109129.
Hallez Y. & Magnaudet J. 2009 A numerical investigation of horizontal viscous gravity currents. J. Fluid Mech. 630, 7191.
Hesse M. A., Orr F. M. Jr & Tchelepi H. A. 2008 Gravity currents with residual trapping. J. Fluid Mech. 611, 3560.
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, 363383.
Hinch E. J. 1991 Perturbation Methods. Cambridge University Press.
Homsy G. M. 1987 Viscous fingering in porous media. Annu. Rev. Fluid Mech. 19, 271311.
Huppert H. E. 1982a Flow and instability of a viscous current down a slope. Nature 300, 427429.
Huppert H. E. 1982b The propagation of two-dimensional and axisymmetric viscous gravity currents over a rigid horizontal surface. J. Fluid Mech. 121, 4358.
Huppert H. E. & Woods A. W. 1995 Gravity driven flows in porous layers. J. Fluid Mech. 292, 5569.
Kurganov A. & Tadmor E. 2000 New high-resolution central schemes for nonlinear conservation laws and convection–diffusion equations. J. Comput. Phys. 160, 241282.
Lake L. W. 1989 Enhanced Oil Recovery. Prentice Hall.
Lenormand R., Touboul E. & Zarcone C. 1988 Numerical models and experiments on immiscible displacements in porous media. J. Fluid Mech. 189, 165187.
LeVeque R. J. 2002 Finite Volume Methods for Hyperbolic Problems. Cambridge University Press.
Lister J. R. 1992 Viscous flows down an inclined plane from point and line sources. J. Fluid Mech. 242, 631653.
Lyle S., Huppert H. E., Hallworth M., Bickle M. & Chadwick A. 2005 Axisymmetric gravity currents in a porous medium. J. Fluid Mech. 543, 293302.
MacMinn C. W., Szulczewski M. L. & Juanes R. 2010 inline-graphic $\text{CO}_{2}$ migration in saline aquifers. Part 1. Capillary trapping under slope and groundwater flow. J. Fluid Mech. 662, 329351.
MacMinn C. W., Szulczewski M. L. & Juanes R. 2011 inline-graphic $\text{CO}_{2}$ migration in saline aquifers. Part 2. Combined capillary and solubility trapping. J. Fluid Mech. 688, 321351.
Metz B., Davidson O., De Connick H., Loos M. & Meyer L. 2005 IPCC Special Report on Carbon Dioxide Capture and Storage. Cambridge University Press.
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, 391413.
Nordbotten J. M. & Celia M. A. 2006 Similarity solutions for fluid injection into confined aquifers. J. Fluid Mech. 561, 307327.
Pegler S. S., Huppert H. E. & Neufeld J. A. 2014 Fluid injection into a confined porous layer. J. Fluid Mech. 745, 592620.
Pritchard D. 2007 Gravity currents over fractured substrates in a porous medium. J. Fluid Mech. 584, 415431.
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, 95110.
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, 312329.
Saha S., Salin D. & Talon L. 2013 Low Reynolds number suspension gravity currents. Eur. Phys. J. E 36, 10385.
Shin J. O., Dalziel S. B. & Linden P. F. 2004 Gravity currents produced by lock exchange. J. Fluid Mech. 521, 134.
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, 175214.
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, 135.
Vella D. & Huppert H. E. 2006 Gravity currents in a porous medium at an inclined plane. J. Fluid Mech. 555, 353362.
Verdon J. & Woods A. W. 2007 Gravity-driven reacting flows in a confined porous aquifer. J. Fluid Mech. 588, 2941.
Yortsos Y. C. & Salin D. 2006 On the selection principle for viscous fingering in porous media. J. Fluid Mech. 557, 225236.
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, 218246.
Zheng Z., Soh B., Huppert H. E. & Stone H. A. 2013 Fluid drainage from the edge of a porous reservoir. J. Fluid Mech. 718, 558568.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *



Full text views

Total number of HTML views: 8
Total number of PDF views: 129 *
Loading metrics...

Abstract views

Total abstract views: 356 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 17th January 2018. This data will be updated every 24 hours.