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The effect of temperature-dependent solubility on the onset of thermosolutal convection in a horizontal porous layer

Published online by Cambridge University Press:  04 January 2007

DAVID PRITCHARD  and
CHRIS N. RICHARDSON
DAVID PRITCHARD
Affiliation:
Department of Mathematics, University of Strathclyde, 26 Richmond St, Glasgow G1 1XH, UKdtp@maths.strath.ac.uk.
CHRIS N. RICHARDSON
Affiliation:
B.P. Institute for Multiphase Flow, Department of Earth Sciences, University of Cambridge, Madingley Rise, Madingley Road, Cambridge CB3 0EZ, UK
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Abstract

We consider the onset of thermosolutal (double-diffusive) convection of a binary fluid in a horizontal porous layer subject to fixed temperatures and chemical equilibrium on the bounding surfaces, in the case when the solubility of the dissolved component depends on temperature. We use a linear stability analysis to investigate how the dissolution or precipitation of this component affects the onset of convection and the selection of an unstable wavenumber; we extend this analysis using a Galerkin method to predict the structure of the initial bifurcation and compare our analytical results with numerical integration of the full nonlinear equations. We find that the reactive term may be stabilizing or destabilizing, with subtle effects particularly when the thermal gradient is destabilizing but the solutal gradient is stabilizing. The preferred spatial wavelength of convective cells at onset may also be substantially increased or reduced, and strongly reactive systems tend to prefer direct to subcritical bifurcation. These results have implications for geothermal-reservoir management and ore prospecting.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 571 , 25 January 2007 , pp. 59 - 95
Copyright
Copyright © Cambridge University Press 2007

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References

REFERENCES

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
Bear, J. 1972 Dynamics of Fluids in Porous Media. Dover.Google Scholar
Bodenschatz, E., Pesch, W. & Ahlers, G. 2000 Recent developments in Rayleigh–Bénard convection. Annu. Rev. Fluid Mech. 32, 709–778.CrossRefGoogle Scholar
Drazin, P. G. 2002 Introduction to Hydrodynamic Stability. Cambridge University Press.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. Transport Porous Media 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
Horton, C. W. & Rogers, F. T. 1945 Convection currents in a porous medium. J. Appl. Phys. 16, 367–370.CrossRefGoogle Scholar
Jupp, T. E. & Woods, A. W. 2003 Thermally-driven reaction fronts in porous media. J. Fluid Mech. 484, 329–346.CrossRefGoogle Scholar
Kaufman, J. 1994 Numerical models of fluid flow in carbonate platforms: implications for dolomitization. J. Sed. 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
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
Murray, B. T. & Chen, C. F. 1989 Double-diffusive convection in a porous medium. J. Fluid Mech. 201, 147–166.CrossRefGoogle Scholar
Nield, D. A. 1968 Onset of thermohaline convection in a porous medium. Water Resour. Res. 4 (3), 553–560.CrossRefGoogle Scholar
Oldenburg, C. M. & Pruess, K. 1998 Layered thermohaline convection in hypersaline geothermal systems. Transport Porous Media 33, 29–63.CrossRefGoogle Scholar
Palm, E., Weber, J. E. & Kvernvold, O. 1972 On steady convection in a porous medium. J. Fluid Mech. 54 (1), 153–161.CrossRefGoogle Scholar
Phillips, O. M. 1991 Flow and Reactions in Permeable Rocks. Cambridge University Press.Google Scholar
Raffensperger, J. P. & Garven, G. 1995a 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. 1995b The formation of unconformity-type uranium ore deposits. 2. Coupled hydrochemical modelling. Am. J. Sci. 295 (6), 639–696.CrossRefGoogle Scholar
Rudraiah, N., Siddheshwar, P. G. & Masuoka, T. 2003 Nonlinear convection in porous media: a review. J. Porous Med. 6 (1), 1–32.Google Scholar
Rudraiah, N., Srimani, P. K. & 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
Schoofs, S. 1999 Thermochemical convection in porous media: an application to hydrothermal systems and magmatic intrusions. PhD thesis, Universiteit Utrecht.Google Scholar
Schoofs, S. & Spera, F. J. 2003 Transition to chaos and flow dynamics of thermochemical porous medium convection. Transport Porous Media 50, 179–195.CrossRefGoogle Scholar
Spiegelman, M. & Katz, R. F. 2006 A semi-Lagrangian Crank-Nicolson algorithm for the numerical solution of advection-diffusion problems. Geochem. Geophys. Geosystems 7, Q04014, doi: 10.1029/2005GC001073.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. R. 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
Turner, J. S. 1979 Buoyancy Effects in Fluids. Cambridge University Press.Google Scholar
Turner, J. S. 1985 Multicomponent convection. Annu. Rev. Fluid Mech. 17, 11–44.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. 1971a Chemical instabilities: I. A heated horizontal layer of dissociating fluid. Phys. Fluids 14 (1), 13–18.CrossRefGoogle Scholar
Wollkind, D. J. & Frisch, H. L. 1971b Chemical instabilities. III. Nonlinear stability analysis of a heated horizontal layer of dissociating fluid. Phys. Fluids 14 (3), 482–487.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