Abstract
Instability theory is applied to diffusion-convection phenomenon in porous media, where the area in direction of transfer is large and viscosity of the oil varies due to gas dissolution. An important application of this theory arises where diffusion-convection is employed as an EOR technique in oil reservoirs.
As a bed of gas is formed below a column of oil, gas starts to diffuse into the oil. Therefore, the oil becomes lighter and an inverse gradient of density is developed as more gas diffuses in. Although this inverse density gradient is potentially unstable, convection will not initiate until the gradient extends to a certain value. The condition at which convection begins is known as “the onset of convection” and is well specified by the dimensionless Rayleigh number.
In this study, an instability analysis is made for convection-diffusion in large porous media. Unlike other studies where viscosity is assumed constant, in this work viscosity is postulated to be a function of gas concentration. It is shown that the mathematical model developed reduces to previous models if the viscosity variations are ignored.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Arroyo, M.P., Quintanilla, M. and Saviron, J. M., “Three-dimensional Study of Rayleigh-Benard Convection by Particle Image Velocimetry,” Experimental Heat Transfer, 5, No. 2, pp.216–224, (1992).
Bahrami, A., Rashidi, F., “Convection Onset in Horizontal Porous Layer Saturated with Two Diffusing Fluids with the Lighter One Lying Underside,” Submitted to International Journal of Heat and Mass Transfer
Bau, H.H., Torrance, K.E., “Low Rayleigh Number Thermal Convection in a Vertical Cylinder Filled with Porous Materials and Heated from Below,” Journal of Heat Transfer, 104, pp.166–172, (1982).
Bénard, H., “Les Tourbillons cellulaires dans une nappe liquide,” Revue générale des Sciences pures et appliquées, 11, pp.1261–71 and 1309–28, (1900).
Bisshopp, F.E., “On Two-dimensional Cell Patterns,” J. Math. Analysis and Applications, 1, pp.373–85, (1960).
Chandrasekhar, S., Hydrodynamic and Hydromagnetic Stability, Oxford University Press, (1961).
Hales, A.L., Mon. Not. R. Astr. Soc. Geophys. Suppl., 4, p.122, (1937).
Hashim, L. Wilson, S.K., “The Onset of Benard-Marangoni Convection in a Horizontal Layer of Fluid,” International Journal of Engineering Science, 37, No.5, pp.829–840, (1999).
Kassoy, D.R; Zebib, A., “Variable viscosity effects on the onset of convection in porous media,” The Physics of Fluids, 18, No.12, pp. 649–1651, (1975).
Katto, Y., Masuoka, T., “Criterion for Onset of Convection Flow in a Fluid in a Porous Medium,” International Journal of Heat and Mass Transfer, 10, pp.297–309, (1967).
Kaviany, M., “Onset of Thermal Convection in a Saturated Porous Medium: Experiment and Analysis,” Int. J. Heat Mass Transfer, 27, No.11, pp.2101–2110, (1984).
Lapwood, E.R., “Convection of a Fluid in a Porous Medium,” Proc. Cambridge Phil. Soc., 44, pp. 508–521, (1948).
Lord Rayleigh, “On Convection Currents in a Horizontal Layer of Fluid When the Higher Temperature is on the Under Side,” Phil. Mag., Cambridge, England, 32, pp.529–46, (1916); also Scientific Papers, 6, pp.432–46, (1920).
Nadolin, K.A., “Boussinesq Approximation in the Rayleigh-Benard Problem,” Fluid Dynamics, 30, No.5, pp.641–651, (1995).
Pellew, A., Southwell, R.V., “On Maintained Convective Motion in a Fluid Heated from Below,” Proc. Roy. Soc. (London) A, 176, pp.312–43, (1940).
Rashidi, F., Soroush, H. and Bahrami, A., “Prediction of Time Required for Onset of Convection in a Porous Medium Saturated with Oil and a Layer of Gas Underlying the Oil,” Submitted to Journal of Petroleum Science & Engineering
Saidi, M.A., Fractured Reservoir Engineering, Total Press, Paris, (1980).
Wooding, R.A, “The Stability of a Viscous Liquid in a Vertical Tube Containing Porous Material,” Proc. Roy. Soc. A, 252, p.120, (1959).
Zebib, A., Kassoy, D.R., “Onset of Natural Convection in a Box of Water-saturated Porous Media with Large Temperature Variation,” The Physics of Fluids, 20, No.1, pp.4–9, (1977).
Zebib, A., “Onset of Natural Convection in a Cylinder of Water Saturated Porous Media,” Phys. Fluids, 21, No.4, pp.699–700, (1978).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Rashidi, F., Bahrami, A. Mathematical modeling of onset of convection in a porous layer with viscosity variation. J. of Therm. Sci. 9, 141–145 (2000). https://doi.org/10.1007/s11630-000-0008-z
Issue Date:
DOI: https://doi.org/10.1007/s11630-000-0008-z