Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-26T16:37:45.243Z Has data issue: false hasContentIssue false
  • English
  • Français

Natural convection and the evolution of a reactive porous medium

Published online by Cambridge University Press:  17 February 2011

LINDSEY T. RITCHIE  and
DAVID PRITCHARD
LINDSEY T. RITCHIE*
Affiliation:
Department of Mathematics and Statistics, University of Strathclyde, 26 Richmond Street, Glasgow G1 1XH, Scotland, UK
DAVID PRITCHARD
Affiliation:
Department of Mathematics and Statistics, University of Strathclyde, 26 Richmond Street, Glasgow G1 1XH, Scotland, UK
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We describe a mathematical model of buoyancy-driven flow and solute transport in a saturated porous medium, the porosity and permeability of which evolve through precipitation and dissolution as a mineral is lost or gained from the pore fluid. Imposing a vertically varying equilibrium solubility creates a density gradient which can drive convective circulation. We characterise the onset of convection using linear stability analysis, and explore the further development of the coupled reaction–convection system numerically. At low Rayleigh numbers, the effect of the reaction–permeability feedback is shown to be destabilising through a novel reaction–diffusion mechanism; at higher Rayleigh numbers, the precipitation and dissolution have a stabilising effect. Over longer time scales, reaction–permeability feedback triggers secondary instabilities in quasi-steady convective circulation, leading to rapid reversals in the direction of circulation. Over very long time scales, characteristic patterns of porosity emerge, including horizontal layering as well as the development of vertical chimneys of enhanced porosity. We discuss the implications of these findings for more comprehensive models of reactive convection in porous media.

JFM classification

Convection: Convection in porous media Geophysical and Geological Flows: Geophysical and Geological Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 673 , 25 April 2011 , pp. 286 - 317
Copyright
Copyright © Cambridge University Press 2011

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

REFERENCES

Anderson, D. M. & Worster, M. G. 1996 A new oscillatory instability in a mushy layer during the solidification of binary alloys. J. Fluid Mech. 307, 245–267.CrossRefGoogle Scholar
Bdzil, J. B. & Frisch, H. L. 1971 Chemical instabilities. II. Chemical surface reactions and hydrodynamic instability. Phys. Fluids 14 (3), 475–482.CrossRefGoogle Scholar
Bdzil, J. B. & Frisch, H. L. 1980 Chemically driven convection. J. Chem. Phys. 72 (3), 1875–1886.CrossRefGoogle Scholar
Bolton, E. W., Lasaga, A. C. & Rye, D. M. 1996 A model for the kinetic control of quartz dissolution and precipitation in porous media flow with spatially variable permeability: formulation and examples of thermal convection. J. Geophys. Res. 101 (B10), 22157–22187.CrossRefGoogle Scholar
Bolton, E. W., Lasaga, A. C. & Rye, D. M. 1997 Dissolution and precipitation via forced-flux injection in a porous medium with spatially variable permeability: kinetic control in two dimensions. J. Geophys. Res. 102 (B6), 12159–12171.CrossRefGoogle Scholar
Bolton, E. W., Lasaga, A. C. & Rye, D. M. 1999 Long-term flow/chemistry feedback in a porous medium with heterogenous permeability: kinetic control of dissolution and precipitation. Am. J. Sci. 299, 1–68.CrossRefGoogle Scholar
Chadam, J., Hoff, D., Merino, E., Ortoleva, P. & Sen, A. 1986 Reactive infiltration instabilities. IMA J. Appl. Math. 36, 207–221.CrossRefGoogle Scholar
Chadam, J., Ortoleva, P., Qin, Y. & Stamicar, R. 2001 The effect of hydrodynamic dispersion on reactive flows in porous media. Eur. J. Appl. Maths 12, 557–569.CrossRefGoogle Scholar
Chandrasekhar, S. 1961 Hydrodynamic and Hydromagnetic Stability. Oxford University Press.Google Scholar
Cochepin, B., Trotignon, L., Bildstein, O., Steefel, C. I., Lagneau, V. & Van der lee, J. 2008 Approaches to modelling coupled flow and reaction in a 2D cementation experiment. Adv. Water Resour. 31, 1540–1551.CrossRefGoogle Scholar
Daccord, G. 1987 Chemical dissolution of a porous medium by a reactive fluid. Phys. Rev. Lett. 58, 479–482.CrossRefGoogle ScholarPubMed
Ennis-King, J. & Paterson, L. 2007 Coupling of geochemical reactions and convective mixing in the long-term geological storage of carbon dioxide. Intl J. Greenh. Gas Control 1, 86–93.CrossRefGoogle Scholar
Gatica, J. E., Viljoen, H. J. & Hlavacek, V. 1989 Interaction between chemical reaction and natural convection in porous media. Chem. Engng Sci. 44 (9), 1853–1870.CrossRefGoogle Scholar
Gilman, A. & Bear, J. 1994 The influence of free convection on soil salinization in arid regions. Transp. Porous Med. 23, 275–301.Google Scholar
Gutkowicz-Krusin, D. & Ross, J. 1980 Rayleigh–Bénard instability in reactive binary fluids. J. Chem. Phys. 72 (6), 3577–3587.CrossRefGoogle Scholar
Hallworth, M. A., Huppert, H. E. & Woods, A. W. 2005 Dissolution-driven convection in a reactive porous medium. J. Fluid Mech. 535, 255–285.CrossRefGoogle Scholar
Hinch, E. J. & Bhatt, B. S. 1990 Stability of an acid front moving through porous media. J. Fluid Mech. 212, 279–288.CrossRefGoogle Scholar
Hoefner, M. L. & Fogler, H. S. 1988 Pore evolution and channel formation during flow and reaction in porous media. AIChE J. 34 (1), 45–54.CrossRefGoogle Scholar
Horton, C. W. & Rogers, F. T. 1945 Convection currents in a porous medium. J. Appl. Phys. 16, 367–370.CrossRefGoogle Scholar
Kaufman, J. 1994 Numerical models of fluid flow in carbonate platforms: implications for dolomitization. J. Sedim. Res. A 64, 128–139.Google Scholar
Lapwood, E. R. 1948 Convection of a fluid in a porous medium. Proc. Camb. Phil. Soc. 44, 508–521.CrossRefGoogle Scholar
Lowell, R. P., Cappellen, P. V. & Germanovitch, L. N. 1993 Silica precipitation in fractures and the evolution of permeability in hydrothermal upflow zones. Science 260 (5105), 192–194.CrossRefGoogle ScholarPubMed
Mamou, M. & Vasseur, P. 1999 Thermosolutal bifurcation phenomena in porous enclosures subject to vertical temperature and concentration gradients. J. Fluid Mech. 395, 61–87.CrossRefGoogle Scholar
Mamou, M., Vasseur, P. & Hasnaoui, M. 2001 On numerical stability analysis of double-diffusive convection in confined enclosures. J. Fluid Mech. 433, 209–250.CrossRefGoogle Scholar
Nield, D. A. & Bejan, A. 2006 Convection in Porous Media, 3rd edn. Springer.Google Scholar
Oldenburg, C. M. & Pruess, K. 1998 Layered thermohaline convection in hypersaline geothermal systems. Transp. Porous Med. 33, 29–63.CrossRefGoogle Scholar
Phillips, O. M. 1991 Flow and Reactions in Permeable Rocks. Cambridge University Press.Google Scholar
Phillips, O. M. 2009 Geological Fluid Dynamics: Sub-Surface Flow and Reactions. Cambridge University Press.CrossRefGoogle Scholar
Pritchard, D. & Richardson, C. N. 2007 The effect of temperature-dependent solubility on the onset of thermosolutal convection in a horizontal porous layer. J. Fluid Mech. 571, 59–95.CrossRefGoogle Scholar
Raffensperger, J. P. & Garven, G. 1995 a The formation of unconformity-type uranium ore deposits. 1. Coupled groundwater flow and heat transport modelling. Am. J. Sci. 295 (5), 581–636.CrossRefGoogle Scholar
Raffensperger, J. P. & Garven, G. 1995 b The formation of unconformity-type uranium ore deposits. 2. Coupled hydrochemical modelling. Am. J. Sci. 295 (6), 639–696.CrossRefGoogle Scholar
Raw, A. W. V. & Woods, A. W. 2003 On gravity-driven flow through a reacting porous rock. J. Fluid Mech. 474, 227–243.CrossRefGoogle Scholar
Rudraiah, N., Srimani, P. & Friedrich, R. 1982 Finite amplitude convection in a two-component fluid saturated porous layer. Intl J. Heat Mass Transfer 25 (5), 715–722.CrossRefGoogle Scholar
Sharp, J. M. & Shi, M. 2009 Heterogeneity effects on possible salinity-driven free convection in low-permeability strata. Geofluids 9, 263–274.CrossRefGoogle Scholar
Steinberg, V. & Brand, H. 1983 Convective instabilities of binary mixtures with fast chemical reaction in a porous medium. J. Chem. Phys. 78 (5), 2655–2660.CrossRefGoogle Scholar
Steinberg, V. & Brand, H. 1984 Amplitude equations for the onset of convection in a reactive mixture in a porous medium. J. Chem. Phys. 80 (1), 431–435.CrossRefGoogle Scholar
Stevens, J. D., Sharp, J. M., Simmons, C. T. & Fenstemaker, T. R. 2009 Evidence of free convection in groundwater: field-based measurements beneath wind-tidal flats. J. Hydrology 375, 394–409.CrossRefGoogle Scholar
Verdon, J. & Woods, A. W. 2007 Gravity-driven reacting flows in a confined porous aquifer. J. Fluid Mech. 588, 29–41.CrossRefGoogle Scholar
Viljoen, H. J., Gatica, J. E. & Hlavacek, V. 1990 Bifurcation analysis of chemically driven convection. Chem. Engng Sci. 45 (2), 503–517.CrossRefGoogle Scholar
Wollkind, D. J. & Frisch, H. L. 1971 Chemical instabilities. I. A heated horizontal layer of dissociating fluid. Phys. Fluids 14 (1), 13–18.CrossRefGoogle Scholar
Wooding, R. A., Tyler, S. W. & White, I. 1997 Convection in groundwater below an evaporating salt lake. 1. Onset of instability. Water Resour. Res. 33 (6), 1199–1217.CrossRefGoogle Scholar
Worster, M. G. 1997 Convection in mushy layers. Annu. Rev. Fluid Mech. 29, 91–122.CrossRefGoogle Scholar
Zhao, C., Hobbs, B. E., Ord, A., Hornby, P. & Peng, S. 2008 Morphological evolution of three-dimensional chemical dissolution front in fluid-saturated porous media: a numerical simulation approach. Geofluids 8, 113–127.CrossRefGoogle Scholar