Hostname: page-component-5d59c44645-mhl4m Total loading time: 0 Render date: 2024-02-27T19:29:53.862Z Has data issue: false hasContentIssue false

Scale analysis of miscible density-driven convection in porous media

Published online by Cambridge University Press:  16 May 2014

Patrick Jenny*
Affiliation:
Institute of Fluid Dynamics, ETH Zurich, Sonneggstrasse 3, 8092 Zurich, Switzerland
Joohwa S. Lee
Affiliation:
Institute of Fluid Dynamics, ETH Zurich, Sonneggstrasse 3, 8092 Zurich, Switzerland
Daniel W. Meyer
Affiliation:
Institute of Fluid Dynamics, ETH Zurich, Sonneggstrasse 3, 8092 Zurich, Switzerland
Hamdi A. Tchelepi
Affiliation:
Department of Energy Resources Engineering, Stanford University, Stanford, CA 94305, USA
*
Email address for correspondence: pajenny@ethz.ch

Abstract

Scale analysis of unstable density-driven miscible convection in porous media is performed. The main conclusions for instabilities in the developed (long time scales) regime are that (i) large-scale structures are responsible for the bulk of the production of concentration variance, (ii) variance dissipation is dominated by the small (diffusive) scales and that (iii) both the production and dissipation rates are independent of the Rayleigh number. These findings provide a strong basis for a new modelling approach, namely, large-mode simulation (LMS), for which closure is achieved by replacing the actual diffusivity with an effective one. For validation, LMS results for vertical flow in a homogeneous rectangular domain are compared with direct numerical simulations (DNS). Some of the analysis is based on the derivation and closure of the concentration mean and variance equations, whereby averaging over the ensemble of all possible initial perturbations is considered. While self-similar solutions are obtained for vertical, statistically one-dimensional fingering, triple correlation of concentration and scalar dissipation rate (rate at which the concentration variance decays due to diffusion) have to be modelled in the general case. For this purpose, an ensemble-averaged Darcy modelling (EADM) approach is proposed.

Type
Papers
Copyright
© 2014 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

Bachu, S., Gunther, W. D. & Perkins, E. H. 1994 Aquifer disposal of ${\mathrm{CO}}_{2}$ : hydrodynamic and mineral trapping. Energy Convers. Manage. 35, 269279.CrossRefGoogle Scholar
Ennis-King, J. & Paterson, L. 2005 Role of convective mixing in the long-term storage of carbon dioxide in deep saline formations. SPE J. 3, 349356.Google Scholar
Ennis-King, J., Preston, I. & Paterson, L. 2005 Onset of convection in anisotropic porous media subject to a rapid change in boundary conditions. Phys. Fluids 17, 084107.Google Scholar
Farajzadeh, R., Salimi, H., Zitha, P. & Bruining, H. 2007 Numerical simulation of density-driven natural convection in porous media with application for ${\mathrm{CO}}_{2}$ injection projects. Intl J. Heat Mass Transfer 50, 50545064.CrossRefGoogle Scholar
Gasda, S. E., Nordbotten, J. M. & Celia, M. A. 2011 Vertically averaged approaches for ${\mathrm{CO}}_{2}$ migration with solubility trapping. Water Resour. Res. 47, W05528.CrossRefGoogle Scholar
Ghesmat, K., Hassanzadeh, H. & Abedi, J. 2011 The effect of anisotropic dispersion on the convective mixing in long-term ${\mathrm{CO}}_{2}$ storage in saline aquifers. AIChE J. 57 (3), 561570.CrossRefGoogle Scholar
Hesse, M. A., Tchelepi, H. A. & Orr, F. M.2006 Natural convection during aquifer ${\mathrm{CO}}_{2}$ storage GHGT-8, 8th International Conference on Greenhouse Gas Control Technologies, Trondheim, Norway.Google Scholar
Hewitt, D. R., Neufeld, J. A. & Lister, J. R. 2013 Convective shutdown in a porous medium at high Rayleigh number. J. Fluid Mech. 719, 551586.Google Scholar
Hidalgo, J. J., Fe, J., Cueto-Felgueroso, L. & Juanes, R. 2012 Scaling of convective mixing in porous media. Phys. Rev. Lett. 109, 264503.CrossRefGoogle ScholarPubMed
Holloway, S. & Savage, D. 1993 The potential for aquifer disposal of carbon-dioxide in the UK. Energy Convers. Manage. 9 (11), 925932.Google Scholar
IPCC, 2005 Special Report on Carbon Dioxide Capture and Storage (ed. Metz, B., Davidson, O., de Coninck, H., Loos, M. & Meyer, L.), Cambridge University Press.Google Scholar
Kneafsey, T. J. & Pruess, K. 2010 Laboratory flow experiments for visualizing carbon dioxide-induced, density-driven brine convection. Trans. Porous Med. 82, 123139.Google Scholar
Kolmogorov, A. N. 1941 The local structure of turbulence in incompressible viscous fluid for very large Reynolds number. C. R. Dokl. Acad. Sci. URSS 30, 301305.Google Scholar
Kolmogorov, A. N. 1962 A refinement of previous hypotheses concerning the local structure of turbulence in a viscous incompressible fluid at high Reynolds number. J. Fluid Mech. 13, 8285.CrossRefGoogle Scholar
Lake, L. 1989 Enhanced Oil Recovery. Prentice Hall.Google Scholar
Lindeberg, E. & Bergmo, P. 2003 The long-term fate of ${\mathrm{CO}}_{2}$ injected into an aquifer. In Greenhouse Gas Control Technologies (ed. Gale, J. & Kaya, Y.), vol. 1, pp. 489495. Elsevier Science Ltd.Google Scholar
MacMinn, C. W., Szulczewski, M. L. & Juanes, R. 2011 ${\mathrm{CO}}_{2}$ migration in saline aquifers. Part 2. Capillary and solubility trapping. J. Fluid Mech. 688, 321351.Google Scholar
Neufeld, J. A., Hesse, M. A., Riaz, A., Hallworth, M. A., Tchelepi, H. A. & Huppert, H. E. 2010 Convective dissolution of carbon dioxide in saline aquifers. Geophys. Res. Lett. 37 (22), L22404.CrossRefGoogle Scholar
Pau, G., Bell, J. B., Pruess, K., Almgren, A. S., Lijewski, M. J. & Zhang, K. 2010 High-resolution simulation and characterization of density-driven flow in ${\mathrm{CO}}_{2}$ storage in saline aquifers. Adv. Water Resour. 33, 443455.CrossRefGoogle Scholar
Pope, S. B. 2000 Turbulent Flows. Cambridge University Press.CrossRefGoogle Scholar
Pruess, K. & Nordbotten, J. 2011 Numerical simulation studies of the long-term evolution of a ${\mathrm{CO}}_{2}$ plume in a saline aquifer with a sloping caprock. Trans. Porous Med. 90, 135151.CrossRefGoogle Scholar
Riaz, A., Hesse, M., Tchelepi, H. A. & Orr, F. M. 2006 Onset of convection in a gravitationally unstable diffusive boundary layer in porous media. J. Fluid Mech. 548, 87111.Google Scholar
Richardson, L. F. 1922 Weather Prediction by Numerical Process. Cambridge University Press, (reprint: Dover, New York 1965).Google Scholar
Slim, A. C., Bandi, M. M., Miller, J. C. & Mahadevan, L. 2013 Dissolution-driven convection in a Hele-Shaw cell. Phys. Fluids 25 (2), 024101.CrossRefGoogle Scholar
Smagorinsky, J. 1963 General circulation experiments with the primitive equations: I. The basic experiment. Mon. Weath. Rev. 91, 99152.Google Scholar
Yang, C. D. & Gu, Y. G. 2006 Accelerated mass transfer of ${\mathrm{CO}}_{2}$ in reservoir brine due to density driven natural convection at high pressures and elevated temperatures. Ind. Engng Chem. Res. 45 (8), 24302436.Google Scholar
Yang, Z. M. & Yortsos, Y. C. 1997 Asymptotic solutions of miscible displacements in geometries of large aspect ratio. Phys. Fluids 9 (2), 286298.CrossRefGoogle Scholar
Yortsos, Y. 1995 A theoretical-analysis of vertical flow equilibrium. Trans. Porous Med. 18 (2), 107129.Google Scholar