Abstract
The effect of strong heterogeneity on the onset of convection induced by a vertical density gradient in a saturated porous medium governed by Darcy’s law is investigated. The general case, where there is heterogeneity in both the vertical and horizontal directions, and where there is heterogeneity in permeability, thermal conductivity, and applied temperature gradient, is considered. A computer package has been developed to implement an algorithm giving a criterion for instability, and this is now employed to investigate the case where there is two-dimensional variation in a horizontal plane and the case where the variation is generated by a log-normal distribution. In the latter case, spatially correlated fields with known stochastic properties are generated, and the results are analyzed in a statistical framework. We now test cases that are representative of natural, field-scale, geologic conditions—both in terms of the correlated structure and the much larger standard deviation of the permeability distribution.
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References
Adams D.: The Hitchhiker’s Guide to the Galaxy. Pan Macmillan, London (1979)
Barlebo H.C., Hill M.C., Rosbjer D.: Investigating the macrodispersion experiment (MADE) site in Columbus, Mississippi, using a three-dimensional inverse flow and transport model. Water Resour. Res. 40, W04211 (2004). doi:10.1029/2002WR001935
Beck J.L.: Convection in a box of porous material saturated with fluid. Phys. Fluids 15, 1377–1383 (1977)
Dagan G.: Flow and Transport in Porous Formations. Springer, New York (1989)
Dagan G.: Transport in heterogeneous porous formations: spatial moments, ergodicity and effective dispersion. Water Resour. Res. 26, 1281–1290 (1990)
Dagan G.: The significance of heterogeneity of evolving scales to transport in porous formations. Water Resour. Res. 30, 3327–3336 (1994)
Deutsch C.V., Journel A.G.: GSLIB Geostatistical Software Library and Users Guide. Oxford University Press, New York (1992)
Freeze R.A.: A stochastic-conceptual analysis of one-dimensional groundwater flow in nonuniform homogeneous media. Water Resour. Res. 11, 725–741 (1975)
Gelhar L.W.: Stochastic subsurface hydrology from theory to applications. Water Resour. Res. 22, 135S–145S (1986)
Gelhar L.W.: Stochastic Subsurface Hydrology. Prentice-Hall, New Jersey (1993)
Kreyszig E.: Advanced Engineering Mathematics. Wiley, New York (1988)
Kuznetsov A.V., Nield D.A.: The effects of combined horizontal and vertical heterogeneity on the onset of convection in a porous medium: double diffusive case. Transp. Porous Media 72, 157–170 (2008)
Kuznetsov, A.V., Nield, D.A., Simmons, C.T.: The effect of strong heterogeneity on the onset of convection in a porous medium: periodic and localized variation. Transp. Porous Media (2009). doi:10.1007/s11242-009-9390-5
LeBlanc D.R., Garabedian S.P., Hess K.M., Gelhar L.W., Quadri R.D., Stollenwerk K.G., Wood W.W.: Large-scale natural gradient tracer test in sand and gravel, Cape Cod, Massachusetts. 1. Experimental design and observed tracer movement. Water Resour. Res. 27, 895–910 (1991)
Nield D.A., Kuznetsov A.V.: The effects of combined horizontal and vertical heterogeneity on the onset of convection in a porous medium. Int. J. Heat Mass Transf. 50, 1909–1915 (2007a) Erratum, 4512–4512
Nield D.A., Kuznetsov A.V.: The onset of convection in a shallow box occupied by a heterogeneous porous medium with constant flux boundaries. Transp. Porous Media 67, 441–451 (2007b)
Nield D.A., Kuznetsov A.V.: The effects of combined horizontal and vertical heterogeneity and anisotropy on the onset of convection in a porous medium. Int. J. Therm. Sci. 46, 1211–1218 (2007c)
Nield D.A., Kuznetsov A.V.: The effects of combined horizontal and vertical heterogeneity on the onset of convection in a bidisperse porous medium. Int. J. Heat Mass Transf. 50, 3329–3339 (2007d)
Nield D.A., Kuznetsov A.V.: The onset of convection in a porous medium occupying an enclosure of variable width. ASME J. Heat Transf. 129, 1714–1718 (2007e)
Nield D.A., Kuznetsov A.V.: The effects of combined horizontal and vertical heterogeneity on the onset of transient convection in a porous medium. J. Porous Media 11, 377–387 (2008a)
Nield D.A., Kuznetsov A.V.: The effects of combined horizontal and vertical heterogeneity on the onset of convection in a porous medium: moderate heterogeneity. Int. J. Heat Mass Transf. 50, 1909–1915 (2008b)
Nield D.A., Simmons C.T.: A discussion on the effect of heterogeneity on the onset of convection in a porous medium. Transp. Porous Media 68, 413–421 (2007)
Nield D.A., Kuznetsov A.V., Simmons C.T.: The effect of strong heterogeneity on the onset of convection in a porous medium; non-periodic global variation. Transp. Porous Media 77, 169–186 (2009)
Prasad A., Simmons C.T.: Unstable density-driven flow in heterogeneous porous media: a stochastic study of the Elder [1967b] “short heater” problem. Water Resour. Res. 39(1), #1007 (2003)
Rees, D.A.S.: Private communication (2008)
Simmons C.T., Fenstemaker T.R., Sharp J.M.: Variable-density flow and solute transport in heterogeneous porous media: approaches, resolutions and future challenges. J. Contam. Hydrol. 52, 245–275 (2001)
Smith L., Freeze R.A.: Stochastic analysis of steady state groundwater flow in a bounded domain. 2. Two-dimensional simulations. Water Resour. Res. 15, 1543–1559 (1979)
Smith L., Schwartz F.W.: Mass transport. 1. A stochastic analysis of macroscopic dispersion. Water Resour. Res. 16, 303–313 (1980)
Smith L., Schwartz F.W.: Mass transport. 2. Analysis of uncertainty in prediction. Water Resour. Res. 17, 351–369 (1981)
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Nield, D.A., Kuznetsov, A.V. & Simmons, C.T. The Effect of Strong Heterogeneity on the Onset of Convection in a Porous Medium: 2D/3D Localization and Spatially Correlated Random Permeability Fields. Transp Porous Med 83, 465–477 (2010). https://doi.org/10.1007/s11242-009-9455-5
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DOI: https://doi.org/10.1007/s11242-009-9455-5