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Onset of convection over a transient base-state in anisotropic and layered porous media

Published online by Cambridge University Press:  16 November 2009

SAIKIRAN RAPAKA ,
RAJESH J. PAWAR ,
PHILIP H. STAUFFER ,
DONGXIAO ZHANG  and
SHIYI CHEN
SAIKIRAN RAPAKA*
Affiliation:
Department of Mechanical Engineering, Johns Hopkins University, Baltimore, MD 21218, USA
RAJESH J. PAWAR
Affiliation:
EES-6, Los Alamos National Laboratory, Los Alamos, NM 87544, USA
PHILIP H. STAUFFER
Affiliation:
EES-6, Los Alamos National Laboratory, Los Alamos, NM 87544, USA
DONGXIAO ZHANG
Affiliation:
Department of Civil and Environmental Engineering, University of Southern California, Los Angeles, CA 90089, USA
SHIYI CHEN
Affiliation:
Department of Mechanical Engineering, Johns Hopkins University, Baltimore, MD 21218, USA SKLTCS & CAPT, College of Engineering, Peking University, Beijing 100871, P.R. China
*
†Email address for correspondence: [email protected]
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Abstract

The topic of density-driven convection in porous media has been the focus of many recent studies due to its relevance as a long-term trapping mechanism during geological sequestration of carbon dioxide. Most of these studies have addressed the problem in homogeneous and anisotropic permeability fields using linear-stability analysis, and relatively little attention has been paid to the analysis for heterogeneous systems. Previous investigators have reduced the governing equations to an initial-value problem and have analysed it either with a quasi-steady-state approximation model or using numerical integration with arbitrary initial perturbations. Recently, Rapaka et al. (J. Fluid Mech., vol. 609, 2008, pp. 285–303) used the idea of non-modal stability analysis to compute the maximum amplification of perturbations in this system, optimized over the entire space of initial perturbations. This technique is a mathematically rigorous extension of the traditional normal-mode analysis to non-normal and time-dependent problems. In this work, we extend this analysis to the important cases of anisotropic and layered porous media with a permeability variation in the vertical direction. The governing equations are linearized and reduced to a set of coupled ordinary differential equations of the initial-value type using the Galerkin technique. Non-modal stability analysis is used to compute the maximum growth of perturbations along with the optimal wavenumber leading to this growth. We show that unlike the solution of the initial-value problem, results obtained using non-modal analysis are insensitive to the choice of bottom boundary condition. For the anisotropic problem, the dependence of critical time and wavenumber on the anisotropy ratio was found to be in good agreement with theoretical scalings proposed by Ennis-King et al. (Phys. Fluids, vol. 17, 2005, paper no. 084107). For heterogeneous systems, we show that uncertainty in the permeability field at low wavenumbers can influence the growth of perturbations. We use a Monte Carlo approach to compute the mean and standard deviation of the critical time for a sample permeability field. The results from theory are also compared with finite-volume simulations of the governing equations using fully heterogeneous porous media with strong layering. We show that the results from non-modal stability analysis match extremely well with those obtained from the simulations as long as the assumption of strong layering remains valid.

JFM classification

Instability: Instability Low-Reynolds-number flows: Porous media
Type
Papers
Information
Journal of Fluid Mechanics , Volume 641 , 25 December 2009 , pp. 227 - 244
Copyright
Copyright © Cambridge University Press 2009

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References

REFERENCES

Beck, J. L. 1972 Convection in a box of porous material saturated with fluid. Phys. Fluids 15, 1377–1383.CrossRefGoogle Scholar
Braester, C. & Vadasz, P. 1993 The effect of a weak heterogeneity of a porous medium on natural convection. J. Fluid Mech. 254, 345–362.CrossRefGoogle Scholar
Castinel, G. & Combarnous, M. 1977 Natural convection in an anisotropic porous medium. Intl Chem. Engng 17, 605–613.Google Scholar
Cheng, P. 1978 Heat transfer in geothermal systems. Adv. Heat Transfer 14, 1–105.Google Scholar
Currie, I. G. 1967 The effect of heating rate on the stability of stationary fluids. J. Fluid Mech. 29, 337–347.Google Scholar
Elder, J. W. 1967 Transient convection in a porous medium. J. Fluid Mech. 27, 609–623.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 (8), 084107.Google Scholar
Epherre, J. F. 1977 Criterion for the appearance of natural convection in an anisotropic porous layer. Intl Chem. Engng 17, 615–616.Google Scholar
Farrell, B. F. & Ioannou, P. J. 1996 a Generalized stability theory. Part I. Autonomous operators. J. Atmos. Sci. 53 (14), 2025.2.0.CO;2>CrossRefGoogle Scholar
Farrell, B. F. & Ioannou, P. J. 1996 b Generalized stability theory. Part II. Nonautonomous operators. J. Atmos. Sci. 53 (14), 2041.2.0.CO;2>CrossRefGoogle Scholar
Ferziger, J. H. & Peric, M. 2001 Computational Methods for Fluid Dynamics. Springer.Google Scholar
Foster, T. D. 1965 Stability of a homogeneous fluid cooled uniformly from above. Phys. Fluids 8 (7), 1249–1257.CrossRefGoogle Scholar
Garcia, J. E. 2001 Density of aqueous solutions of CO2. Tech Rep. LBNL-49023. Lawrence Berkeley National Laboratory.CrossRefGoogle Scholar
Golub, G. H. & Loan, C. F. Van 1996 Matrix Computations. Johns Hopkins University Press.Google Scholar
Hassanzadeh, H., Pooladi-Darvish, M. & Keith, D. W. 2006 Stability of a fluid in a horizontal saturated porous layer: effect of nonlinear concentration profile, initial, and boundary conditions. Transp. Porous Med. 65 (2), 193–211.Google Scholar
Hitchon, B. 1996 Aquifer Disposal of Carbon Dioxide: Hydrodynamic and Mineral Trapping. Alberta Research Council.Google Scholar
Horton, C. W. & Rogers, F. T. Jr, 1945 Convection currents in a porous media. J. Appl. Phys. 16, 367–370.CrossRefGoogle Scholar
Intergovernmental Panel on Climate Change (IPCC) 2005 IPCC Special Report on Carbon Dioxide Capture and Storage. Prepared by Working Group III of the Intergovernmental Panel on Climate Change (B. Metz, O. Davidson, H. C. de Coninck, M. Loos and L. A. Meyer). Cambridge University Press.Google Scholar
Jhaveri, B. S. & Homsy, G. M. 1982 The onset of convection in fluid layers heated rapidly in a time-dependent manner. J. Fluid Mech. 114, 251–260.Google Scholar
Joseph, D. D. 1976 Stability of Fluid Motions I, Tracts in Natural Philosophy, vol. 27. Springer.Google Scholar
Kim, M. C. & Choi, C. K. 2007 Relaxed energy stability analysis on the onset of buoyancy-driven instability in the horizontal porous layer. Phys. Fluids 19, 088103.CrossRefGoogle Scholar
Kvernvold, O. & Tyvand, P. A. 1979 Nonlinear thermal convection in an anisotropic porous media. J. Fluid Mech. 90, 609–624.CrossRefGoogle Scholar
Lapwood, E. R. 1948 Convection of a fluid in a porous medium. Proc. Camb. Phil. Soc. 44, 508–521.CrossRefGoogle Scholar
Leong, J. C. & Lai, F. C. 2001 Effective permeability of a layered porous cavity. ASME J. Heat Transfer 123, 512–519.CrossRefGoogle Scholar
Lick, W. 1965 The instability of a fluid layer with time-dependent heating profile. J. Fluid Mech. 21, 565.Google Scholar
Mantoglou, A. & Wilson, J. L. 1982 The turning bands method for simulation of random fields using line generation by a spectral method. Water Resour. Res. 18, 1379–1394.CrossRefGoogle Scholar
McKibbin, R. & O'Sullivan, M. J. 1980 Onset of convection in a layered porous medium heated from below. J. Fluid Mech. 96 (2), 375–393.Google Scholar
McKibbin, R. & Tyvand, P. A. 1982 Anisotropic modelling of thermal convection in multilayered porous media. J. Fluid Mech. 118, 315–339.CrossRefGoogle Scholar
Nield, D. A. 1994 Estimation of an effective Rayleigh number for convection in a vertically inhomogeneous porous medium or clear fluid. Intl J. Heat Fluid Flow 15, 337–340.CrossRefGoogle Scholar
Nield, D. A. 1997 Notes on convection in a porous medium: (i) an effective Rayleigh number for an anisotropic layer, (ii) the Malkus hypothesis and wavenumber selection. Transp. Porous Med. 27, 135–142.CrossRefGoogle Scholar
Nield, D. A. & Bejan, A. 2006 Convection in Porous Media, 3rd edn. Springer.Google Scholar
Nield, D. A., Kuznetsov, A. V. & Simmons, C. T. 2009 The effect of strong heterogeneity on the onset of convection in a porous medium: non-periodic global variation. Transp. Porous Med. 77, 169–186.Google Scholar
Nield, D. A. & Simmons, C. T. 2007 A discussion on the effect of heterogeneity on the onset of convection in a porous medium. Transp. Porous Med. 68, 413–421.CrossRefGoogle Scholar
Nilsen, T. & Storesletten, L. 1990 An analytical study on natural convection in isotropic and anisotropic porous channels. J. Heat Transfer 112, 396–401.CrossRefGoogle Scholar
Prasad, A. & Simmons, C. T. 2003 Unstable density-driven flow in heterogeneous porous media: a stochastic study of the Elder (1967b) ‘short heater’ problem. Water Resour. Res. 39 (1), 1007.CrossRefGoogle Scholar
Rapaka, S., Chen, S., Pawar, R. J., Stauffer, P. H. & Zhang, D. 2008 Nonmodal growth of perturbations in density-driven convection in porous media. J. Fluid Mech. 609, 285–303.CrossRefGoogle Scholar
Riaz, A., Hesse, M., Tchelepi, H. A. & Orr, F. M. Jr., 2006 Onset of convection in a gravitationally unstable diffusive boundary layer in porous media. J. Fluid Mech. 548, 87–111.Google Scholar
Schmid, P. J. 2007 Nonmodal stability theory. Annu. Rev. Fluid Mech. 39, 129–162.CrossRefGoogle Scholar
Simmons, C. T., Fenstemaker, T. R. & Sharp, J. M. 2001 Variable-density groundwater flow and solute transport in heterogeneous porous media: approaches, resolutions and future challenges. J. Contam. Hydrol. 52, 245–275.Google Scholar
Stauffer, P. H. 2006 Flux flummoxed: a proposal for consistent usage. Ground Water 44 (2), 125–128.Google Scholar
Straughan, B. 2004 The Energy Method, Stability and Nonlinear Convection. Springer.CrossRefGoogle Scholar
Tan, C. T. & Homsy, G. M. 1986 Stability of miscible discplacements in porous media: rectilinear flow. Phys. Fluids 29 (11), 3549–3556.Google Scholar
Trefethen, L. N. & Embree, M. 2005 Spectra and Pseudospectra: The Behaviour of Nonnormal Matrices and Operators. Princeton University Press.CrossRefGoogle Scholar
Tyvand, P. A. & Storesletten, L. 1991 Onset of convection in an anisotropic porous medium with oblique principal axes. J. Fluid Mech. 226, 371–382.CrossRefGoogle Scholar
Wooding, R. A. 1978 Large-scale geothermal field parameters and convection theory. NZ J. Sci. 27, 219–228.Google Scholar
Xu, X., Chen, S. & Zhang, D. 2006 Convective stability analysis of the long-term storage of carbon dioxide in deep saline aquifers. Adv. Water Resour. 29, 397–407.CrossRefGoogle Scholar