Journal of Fluid Mechanics

Cambridge Journals Online - CUP Full-Text Page

Journal of Fluid Mechanics (2007), 577:363-383 Cambridge University Press

Copyright © Cambridge University Press 2007

doi:10.1017/S0022112007004685


Papers

Gravity currents in horizontal porous layers: transition from early to late self-similarity


M. A. HESSEa1, H. A. TCHELEPIa1 c1, B. J. CANTWELa2 and F. M. ORR Jra1



a1 Department of Energy Resources Engineering, Stanford University, Stanford CA 94305, USA

a2 Department of Aeronautics and Astronautics, Stanford University, Stanford CA 94305, USA

Article author query

HESSE MA [Google Scholar]
TCHELEPI HA [Google Scholar]
CANTWEL BJ [Google Scholar]
ORR FM [Google Scholar]

Abstract

We investigate the evolution of a finite release of fluid into an infinite, two-dimensional, horizontal, porous slab saturated with a fluid of different density and viscosity. The vertical boundaries of the slab are impermeable and the released fluid spreads as a gravity current along a horizontal boundary. At early times the released fluid fills the entire height of the layer, and the governing equation admits a self-similar solution that is a function of the viscosity ratio between the two fluids. This early similarity solution describes a tilting interface with tips propagating as x xs221D t1/2. At late times the released fluid has spread along the boundary and the height of the current is much smaller than the thickness of the layer. The governing equation simplifies and admits a different similarity solution that is independent of the viscosity ratio. This late similarity solution describes a point release of fluid in a semi-infinite porous half-space, where the tip of the interface propagates as x xs221D t1/3. The same simplification of the governing equation occurs if the viscosity of the released fluid is much higher than the viscosity of the ambient fluid. We have obtained an expression for the time when the solution transitions from the early to the late self-similar regime. The transition time increases monotonically with increasing viscosity ratio. The transition period during which the solution is not self-similar also increases monotonically with increasing viscosity ratio, for mobility ratios larger than unity. Numerical computations describing the full evolution of the governing equation show good agreement with the theoretical results. Estimates of the spreading of injected fluids over long times are important for geological storage of CO2, and for the migration of pollutants in aquifers. In all cases it is important to be able to anticipate when the spreading regime transitions from x xs221D t1/2 to x xs221D t1/3.

(Received March 30 2006)

(Revised October 26 2006)

Correspondence:

c1 Author to whom correspondence should be addressed.



REFERENCES

Bachu, S. 2003 Screening and ranking of sedimentary basins for sequestration of CO2 in geological media in response to climate change. Environmental Geology 44, 277–289. [OpenURL Query Data]  [CrossRef]  [Google Scholar]

Barenblatt, G. I. 1952 On some unsteady motions of fluids and gases in a porous medium. Appl. Math. and Mech. (PMM) 16, 67–78. [OpenURL Query Data]  [Google Scholar]

Barenblatt, G. I. 1996 Scaling, Self-Similarity, and Intermediate Asymptotics. Cambridge University Press.[Google Scholar]

Barenblatt, G. I. & Vishik, M. I. 1956 On the finite speed of propagation in the problems of unsteady filtration of fluid and gas in a porous medium. Appl. Math. Mech. (PMM) 20, 411–417. [OpenURL Query Data]  [Google Scholar]

Barenblatt, G. I. & Zeldovich, Ya. B. 1972 Self-similar solutions as intermediate asymptotics. Ann. Rev. Fluid Mech. 4, 285–312. [OpenURL Query Data]  [CrossRef]  [Google Scholar]

Batchelor, G. K. 1973 An Introduction to Fluid Mechanics. Cambridge University Press.[Google Scholar]

Bear, J. 1972 Dynamics of Fluids in Porous Media. American Elsevier.[Google Scholar]

Bear, J. & Ryzhik, V. 1998 On displacement of NAPL lenses and plumes in a phreatic aquifer. Transp. Porous Media 33, 227–255. [OpenURL Query Data]  [CrossRef]  [Google Scholar]

Dietz, D. N. 1953 A theoretical approach to the problem of encroaching and by-passing edge water. Akad. van Wetenschappen, Proc. V. 56 B, 83–92. [OpenURL Query Data]  [Google Scholar]

Hunt, J. R., Sitar, N. & Udell, K. S. 1995 Nonaqueous phase liquid transport and cleanup. Part 1. Analysis of mechanisms. Water Resour. Res. 24, 1247–1258. [OpenURL Query Data]  [CrossRef]  [Google Scholar]

Huppert, H. E. 1982 Propagation of two-dimensional viscous gravity currents over a rigid horizontal surface. J. Fluid Mech. 121, 43–58. [OpenURL Query Data]  [CrossRef]  [Google Scholar]

Huppert, H. E. & Woods, A. W. 1995 Gravity-driven flows in porous media. J. Fluid Mech. 292, 55–69. [OpenURL Query Data]  [CrossRef]  [Google Scholar]

Lake, L. L. 1989 Enhanced Oil Recovery. Prentice-Hall.[Google Scholar]

Leveque, R. J. 2002 Finite Volume Methods for Hyperbolic Problems. Cambridge University Press.[Google 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. [OpenURL Query Data]  [CrossRef]  [Google Scholar]

Metz, B. Davidson, O. de Coninck, H. Loos, M., & Meyer, L., (Eds.) 2006 Special Report on Carbon Dioxide Capture and Storage. Cambridge University Press.[Google Scholar]

Nordbotten, J. M., Celia, M. A. & Bachu, S. 2005 Injection and storage of CO2 in deep saline aquifers: Analytical solution for the CO2 plume evolution during plume injection. Transp. Porous Media 58, 339–360. [OpenURL Query Data]  [CrossRef]  [Google Scholar]

Ozah, R. C., Lakshminarasimhan, S., Pope, G. A., Sepehrnoori, K. & Bryant, S. L. 2005 Numerical simulation of the storage of pure CO2 and CO2-H2S gas mixtures in deep saline aquifers. In SPE Annual Technical Conference and Exhibition (Dallas, TX).

Riaz, A. & Tchelepi, H. A. 2006 Numerical simulation of immiscible two-phase flow in porous media. Phys. Fluids 18 014104. [OpenURL Query Data]  [CrossRef]  [Google Scholar]

Spiteri, E. J., Juanes, R., Blunt, M. J. & Orr, F. M. Jr 2005 Relative permeability hysteresis: Trapping models and application to geological CO2 sequestration. In SPE Annual Technical Conference and Exhibition (Dallas, TX).

Tchelepi, H. A. 1994 Viscous fingering, gravity segregation and permeability heterogeneity in two-dimensional and three-dimensional flows. PhD thesis, Stanford University.

Yortsos, Y. C. 1995 A theoretical analysis of vertical flow equilibrium. Transp. Porous Media 18, 107–129. [OpenURL Query Data]  [CrossRef]  [Google Scholar]