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Onset of natural convection in layered aquifers

Published online by Cambridge University Press:  23 February 2015

Don Daniel Open the ORCID record for Don Daniel [Opens in a new window] ,
Amir Riaz  and
Hamdi A. Tchelepi
Don Daniel
Affiliation:
Department of Mechanical Engineering, University of Maryland, College Park, MD 20742, USA
Amir Riaz*
Affiliation:
Department of Mechanical Engineering, University of Maryland, College Park, MD 20742, USA
Hamdi A. Tchelepi
Affiliation:
Energy Resources Engineering, Stanford University, Stanford, CA 94305, USA
*
†Email address for correspondence: ariaz@umd.edu
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Abstract

The stability of gravitationally unstable, transient boundary layers in heterogeneous saline aquifers is examined with respect to the onset of natural convection. Permeability is assumed to vary periodically across the thickness of the aquifer. We study the interaction between permeability variation and concentration perturbations within the boundary layer. We observe that the instability decreases with an increase in the permeability variance if the boundary layer thickness is large compared with the permeability wavelength. On the other hand, when the boundary layer thickness is smaller than the permeability wavelength, the behaviour of instability as a function of variance depends on the phase of permeability variation. Such behaviours are shown to result from the interaction of two modes of vorticity production related to the coupling of concentration and velocity perturbations with the magnitude and gradient of permeability variation, respectively. We show that these two modes of vorticity production, when coupled with the transient nature of the boundary layer, determine the evolutionary paths followed by the most amplified perturbations that trigger the onset of convection. When the permeability variance is large, we find that small changes in the permeability field can lead to large changes in the onset times for convection.

JFM classification

Convection: Buoyancy-driven instability Convection: Convection Interfacial Flows (free surface): Fingering instability
Type
Papers
Information
Journal of Fluid Mechanics , Volume 767 , 25 March 2015 , pp. 763 - 781
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Copyright
© 2015 Cambridge University Press 

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References

Bando, S., Takemura, F., Nishio, M., Hihara, E. & Akai, M. 2004 Viscosities of aqueous NaCl solutions with dissolved $\text{CO}_{2}$ at (30–60) °C and (10 to 20) MPa. J. Chem. Engng Data 49 (5), 1328–1332.Google Scholar
Ben, Y., Demekhin, E. A. & Chang, H. C. 2002 A spectral theory for small-amplitude miscible fingering. Phys. Fluids 14 (3), 999–1010.CrossRefGoogle Scholar
Bickle, M., Chadwick, A., Huppert, H. E., Hallworth, M. & Lyle, S. 2007 Modelling carbon dioxide accumulation at Sleipner: implications for underground carbon storage. Earth Planet. Sci. Lett. 255 (1–2), 164–176.CrossRefGoogle Scholar
Camhi, E., Meiburg, E. & Ruith, M. 2000 Miscible rectilinear displacements with gravity override. Part 2. Heterogeneous porous media. J. Fluid Mech. 420, 259–276.CrossRefGoogle Scholar
Chadwick, R. A., Arts, R. & Eiken, O. 2005 4D seismic quantification of a growing $\text{CO}_{2}$ plume at Sleipner, North Sea. In 6th Petroleum Geology Conference, Geological Society London, vol. 6, pp. 1385–1399. Geological Society London.Google Scholar
Chen, C. Y. & Meiburg, E. 1998 Miscible porous media displacements in the quarter five-spot configuration. Part 2. Effect of heterogeneities. J. Fluid Mech. 371, 269–299.Google Scholar
Christie, M. A., Muggeridge, A. H. & Barley, J. J. 1993 3D simulation of viscous fingering and wag schemes. SPE Res. Engng 8 (1), 19–26.Google Scholar
Clifton, P. M. & Neuman, S. P. 1982 Effects of kriging and inverse modeling on conditional simulation of the Avra valley aquifer in Southern Arizona. Water Resour. Res. 18 (4), 1215–1234.CrossRefGoogle Scholar
Dagan, G. 1984 Solute transport in heterogeneous porous formations. J. Fluid Mech. 145, 151–177.CrossRefGoogle Scholar
Daniel, D. & Riaz, A. 2014 Effect of viscosity contrast on gravitationally unstable diffusive layers in porous media. Phys. Fluids 26, 116601.Google Scholar
Daniel, D., Tilton, N. & Riaz, A. 2013 Optimal perturbations of gravitationally unstable, transient boundary layers in porous media. J. Fluid Mech. 727, 456–487.Google Scholar
Daripa, P. 2012 On stabilization of multi-layer Hele–Shaw and porous media flows in the presence of gravity. Transp. Porous Med. 95 (2), 349–371.Google Scholar
Delhomme, J. P. 1979 Spatial variability and uncertainty in groundwater-flow parameters – geostatistical approach. Water Resour. Res. 15 (2), 269–280.Google Scholar
DeWit, A. & Homsy, G. M. 1997 Viscous fingering in periodically heterogeneous porous media. 1. Formulation and linear instability. J. Chem. Phys. 107 (22), 9609–9618.Google Scholar
Fleury, M. & Deschamps, H. 2008 Electrical conductivity and viscosity of aqueous NaCl solutions with dissolved $\text{CO}_{2}$ . J. Chem. Engng Data 53 (11), 2505–2509.CrossRefGoogle Scholar
Freeze, R. A. 1975 Stochastic-conceptual analysis of one-dimensional groundwater flow in nonuniform homogeneous media. Water Resour. Res. 11 (5), 725–741.CrossRefGoogle Scholar
Gjerde, K. M. & Tyvand, P. A. 1984 Thermal convection in a porous medium with continuous periodic stratification. Intl J. Heat Mass Transfer 27 (12), 2289–2295.Google Scholar
Green, C. P. & Ennis-King, J. 2010 Effect of vertical heterogeneity on long-term migration of $\text{CO}_{2}$ in saline formations. Transp. Porous Med. 82, 31–47.Google Scholar
Hewitt, D. R., Neufeld, J. A. & Lister, J. R. 2014 High Rayleigh number convection in a porous medium containing a thin low-permeability layer. J. Fluid Mech. 756, 844–869.Google Scholar
Hoeksema, R. J. & Kitanidis, P. K. 1985 Analysis of the spatial structure of properties of selected aquifers. Water Resour. Res. 21 (4), 563–572.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
Neufeld, J. A. & Huppert, H. E. 2009 Modelling carbon dioxide sequestration in layered strata. J. Fluid Mech. 625, 353–370.CrossRefGoogle Scholar
Nomeli, M. A., Tilton, N. & Riaz, A. 2014 A new model for the density of saturated solutions of $\text{CO}_{2}{-}\text{H}_{2}\text{O}{-}\text{NaCl}$ in saline aquifers. Intl J. Greenh. Gas Control 31, 192–204.Google Scholar
Orr, F. M. 2009 Onshore geologic storage of $\text{CO}_{2}$ . Science 325, 1656–1658.Google 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.Google Scholar
Ranganathan, P., Farajzadeh, R., Bruining, H. & Zitha, P. L. J. 2012 Numerical simulation of natural convection in heterogeneous porous media for $\text{CO}_{2}$ geological storage. Transp. Porous Med. 95 (1), 25–54.CrossRefGoogle Scholar
Rapaka, S., Pawar, R. J., Stauffer, P. H., Zhang, D. & Chen, S. 2009 Onset of convection over a transient base-state in anisotropic and layered porous media. J. Fluid Mech. 641, 227–244.Google Scholar
Riaz, A. & Cinar, Y. 2014 Carbon dioxide sequestration in saline formations: part I – review of the modeling of solubility trapping. J. Petrol. Sci. Engng 124, 367–380.Google 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, 87–111.Google Scholar
Riaz, A. & Meiburg, E. 2004 Vorticity interaction mechanisms in variable-viscosity heterogeneous miscible displacements with and without density contrast. J. Fluid Mech. 517, 1–25.Google Scholar
Schincariol, R. A. 1998 Dispersive mixing dynamics of dense miscible plumes: natural perturbation initiation by local-scale heterogeneities. J. Contam. Hydrol. 34 (3), 247–271.CrossRefGoogle Scholar
Schincariol, R. A., Schwartz, F. W. & Mendoza, C. A. 1997 Instabilities in variable density flows: stability and sensitivity analyses for homogeneous and heterogeneous media. Water Resour. Res. 33 (1), 31–41.Google 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 (1–4), 245–275.Google Scholar
Tan, C. T. & Homsy, G. M. 1992 Viscous fingering with permeability heterogeneity. Phys. Fluids 4 (6), 1099–1101.CrossRefGoogle Scholar
Tchelepi, H. A. & Orr, F. M. 1994 Interaction of viscous fingering, permeability heterogeneity, and gravity segregation in 3-dimensions. SPE Res. Engng 9 (4), 266–271.CrossRefGoogle Scholar
Tchelepi, H. A., Orr, F. M., Rakotomalala, N., Salin, D. & Woumeni, R. 1993 Dispersion, permeability heterogeneity, and viscous fingering – acoustic experimental-observations and particle-tracking simulations. Phys. Fluids A 5 (7), 1558–1574.Google Scholar
Tilton, N., Daniel, D. & Riaz, A. 2013 The initial transient period of gravitationally unstable diffusive boundary layers developing in porous media. Phys. Fluids 25, 092107.Google Scholar
Tilton, N. & Riaz, A. 2014 Nonlinear stability of gravitationally unstable, transient, diffusive boundary layers in porous media. J. Fluid Mech. 745, 251–278.Google Scholar
Trefethen, L. N. & Embree, M. 2005 Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators. Princeton University Press.CrossRefGoogle Scholar
Wessel-Berg, D. 2009 On a linear stability problem related to underground $\text{CO}_{2}$ storage. SIAM J. Appl. Maths 70 (4), 1219–1238.Google Scholar
Yang, C. D. & Gu, Y. G. 2006 Accelerated mass transfer of $\text{CO}_{2}$ in reservoir brine due to density-driven natural convection at high pressures and elevated temperatures. Ind. Engng Chem. Res. 45 (8), 2430–2436.CrossRefGoogle Scholar
Zhang, H. R., Sorbie, K. S. & Tsibuklis, N. B. 1997 Viscous fingering in five-spot experimental porous media: new experimental results and numerical simulations. Chem. Engng Sci. 52, 37–54.CrossRefGoogle Scholar