Hostname: page-component-76fb5796d-skm99 Total loading time: 0 Render date: 2024-04-26T02:54:54.577Z Has data issue: false hasContentIssue false
  • English
  • Français

Two-phase gravity currents resulting from the release of a fixed volume of fluid in a porous medium

Published online by Cambridge University Press:  26 October 2017

Madeleine J. Golding Open the ORCID record for Madeleine J. Golding [Opens in a new window] ,
Herbert E. Huppert  and
Jerome A. Neufeld
Madeleine J. Golding*
Affiliation:
Institute of Theoretical Geophysics, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, CambridgeCB3 0WA, UK
Herbert E. Huppert
Affiliation:
Institute of Theoretical Geophysics, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, CambridgeCB3 0WA, UK School of Mathematics, University of New South Wales, Kensington, NSW 2052, Australia Faculty of Science, University of Bristol, BristolBS8 1UH, UK
Jerome A. Neufeld
Affiliation:
Institute of Theoretical Geophysics, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, CambridgeCB3 0WA, UK BP Institute, University of Cambridge, Madingley Road, CambridgeCB3 0EZ, UK Department of Earth Sciences, University of Cambridge, Madingley Road, CambridgeCB3 0EZ, UK
*
†Email address for correspondence: mjgolding@gmail.com
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider the instantaneous release of a finite volume of fluid in a porous medium saturated with a second, immiscible fluid of different density. The resulting two-phase gravity current exhibits a rich array of behaviours due to both the residual trapping of fluid as the current recedes and the differing effects of surface tension between advancing and receding regions of the current. We develop a framework for the evolution of such a current with particular focus on the large-scale implications of the form of the constitutive relation between residual trapping and initial saturation. Pore-scale hysteresis within the current is represented by families of scanning curves relating capillary pressure and relative permeability to saturation. In the resulting vertically integrated model, all capillary effects are incorporated within specially defined saturation and flux functions specific to each region. In the long-time limit, when the height of the current and the saturations within it are low, the saturation and flux functions can be approximated by mathematically convenient power laws. If the trapping model is approximately linear at low saturations, the equations admit a similarity solution for the propagation rate and height profile of the late-time gravity current. We also solve the governing partial differential equation numerically for the nonlinear Land’s trapping model, which is commonly used in studies of two-phase flows. Our investigation suggests that for trapping relations for which the proportion of trapped to initial fluid saturation increases and tends to unity as the initial saturation decreases, both of which are properties of Land’s model, a gravity current slows and eventually stops. This trapping behaviour has important applications, for example to the ultimate distance contaminants or stored carbon dioxide may travel in the subsurface.

JFM classification

Geophysical and Geological Flows: Gravity currents Multiphase and Particle-laden Flows: Multiphase flow Low-Reynolds-number flows: Porous media
Type
Papers
Information
Journal of Fluid Mechanics , Volume 832 , 10 December 2017 , pp. 550 - 577
Check for updates
Copyright
© 2017 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Barenblatt, G. I. 1996 Scaling, Self-Similarity, and Intermediate Asymptotics. Cambridge University Press.Google Scholar
Bear, J. & Ryzhik, V. 1998 On the displacement of NAPL lenses and plumes in a phreatic aquifer. Trans. Porous Med. 33, 227–255.CrossRefGoogle Scholar
Bear, J., Ryzhik, V., Braester, C. & Entov, V. 1996 On the movement of an LNAPL lens on the water table. Trans. Porous Med. 25, 283–311.CrossRefGoogle Scholar
Bennion, D. B. & Bachu, S. 2006 Supercritical CO2 and H2S – brine drainage and imbibition relative permeability relationships for intergranular sandstone and carbonate formations. SPE Res. Engng 11 (03), 487–496.Google Scholar
Brooks, R. H. & Corey, A. T. 1964 Hydraulic properties of porous media. In Hydrology Papers (Colorado State University), vol. 3, p. 127. Colorado State University.Google Scholar
Cowton, L. R., Neufeld, J. A., White, N. J., Bickle, M. J., White, J. C. & Chadwick, R. A. 2016 An inverse method for estimating thickness and volume with time of a thin CO2 -filled layer at the Sleipner field, North Sea. J. Geophys. Res. 121, 5068–5085.Google Scholar
Doster, F., Nordbotten, J. M. & Celia, M. A. 2013 Impact of capillary hysteresis and trapping on vertically integrated models for CO2 storage. Adv. Water Resour. 62, 465–474.Google Scholar
Gasda, S. E., Nordbotten, J. M. & Celia, M. A. 2009 Vertical equilibrium with sub-scale analytical methods for geological CO2 sequestration. Comput. Geosci. 13, 469–481.Google Scholar
Gerhard, J. I. & Kueper, B. H. 2003 Capillary pressure characteristics necessary for simulating DNAPL infiltration, redistribution, and immobilization in saturated porous media. Water Resour. Res. 39 (8), 1–17.Google Scholar
Golding, M. J., Huppert, H. E. & Neufeld, J. A. 2013 The effects of capillary forces on the propagation of axisymmetric, two-phase gravity currents in porous media. Phys. Fluids 25, 036602.CrossRefGoogle Scholar
Golding, M. J., Neufeld, J. A., Hesse, M. A. & Huppert, H. E. 2011 Two-phase gravity currents in porous media. J. Fluid Mech. 678, 248–270.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, 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
Huppert, H. E. & Woods, A. W. 1995 Gravity-driven flows in porous layers. J. Fluid Mech. 292, 55–69.Google Scholar
Juanes, R., MacMinn, C. W. & Szulczewski, M. L. 2010 The footprint of the CO2 plume during carbon dioxide storage in saline aquifers: storage efficiency for capillary trapping at the basin scale. Trans. Porous Med. 82, 19–30.CrossRefGoogle Scholar
Juanes, R., Spiteri, E. J., Orr, F. M. Jr. & Blunt, M. J. 2006 Impact of relative permeability hysteresis on geological CO2 storage. Water Resour. Res. 42, W12418.Google Scholar
Killough, J. E. 1976 Reservoir simulation with history-dependent saturation functions. Soc. Petrol. Engng J. 16 (1), 37–48.Google Scholar
Kochina, I. N., Mikhailov, N. N. & Filinov, M. V. 1983 Groundwater mound damping. Intl J. Engng Sci. 21 (4), 413–421.CrossRefGoogle Scholar
Land, C. S. 1968 Calculation of imbibition relative permeability for two- and three-phase flow from rock properties. Soc. Petrol. Engng J. 8 (2), 149–156.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.CrossRefGoogle Scholar
MacMinn, C. W. & Juanes, R. 2009 Post-injection spreading and trapping of CO2 in saline aquifers: impact of the plume shape at the end of injection. Comput. Geosci. 13, 483–491.Google Scholar
Neufeld, J. A., Vella, D. & Huppert, H. E. 2009 The effect of a fissure on storage in a porous medium. J. Fluid Mech. 639, 239–259.CrossRefGoogle Scholar
Nordbotten, J. M. & Dahle, H. K. 2011 Impact of the capillary fringe in vertically integrated models for CO2 storage. Water Resour. Res. 47, W02537.CrossRefGoogle Scholar
Parker, J. C. & Lenhard, R. J. 1989 Vertical integration of three-phase flow equations for analysis of light hydrocarbon plume movement. Trans. Porous Med. 5, 187–206.CrossRefGoogle 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
Pentland, C. H., Al-Mansoori, S., Iglauer, S., Bijeljic, B. & Blunt, M. J. 2008 Measurement of non-wetting phase trapping in sand packs. SPE J. 15 (02), 115697.Google Scholar
Van Dijke, M. I. J. & Van der Zee, S. E. A. T. M. 1997 A similarity solution for oil lens redistribution including capillary forces and oil entrapment. Trans. Porous Med. 29, 99–125.CrossRefGoogle Scholar
Zhao, B., Macminn, C. W., Huppert, H. E. & Juanes, R. 2014 Capillary pinning and blunting of immiscible gravity currents in porous media. Water Resour. Res. 50 (9), 7067–7081.Google Scholar
Zhao, B., Macminn, C. W., Szulczewski, M. L., Neufeld, J. A., Huppert, H. E. & Juanes, R. 2013 Interface pinning of immiscible gravity-exchange flows in porous media. Phys. Rev. E 87 (2), 023015.Google ScholarPubMed