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Natural convection and dispersion in a tilted fracture

Published online by Cambridge University Press:  26 April 2006

Andrew W. Woods  and
Stefan J. Linz
Andrew W. Woods
Affiliation:
Institute of Theoretical Geophysics, Department of Applied Mathematics and Theoretical Physics. University of Cambridge, Silver Street, Cambridge CB3 9EW. UK
Stefan J. Linz
Affiliation:
Department of Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, IL 60208, USA
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Abstract

In many geophysical situations, fluid is contained in long narrow fractures embedded within an impermeable medium of different thermal conductivity: and there may be a uniform vertical temperature gradient imposed upon the system. We show that whenever the slot is tiled to the vertical, convection develops in the fluid, even if the background temperature increases with height. We then investigate the transport of passive material governed by this flow. The dispersion coefficient of a passive contaminant transported by this flow DT = f[R sin2ϕ] k2 cot2 ϕ/D, where k and D are the thermal and compositional diffusivities, ϕ is the angle of tilt and R is a Rayleigh number for the slot.

Using typical values for the physical properties of a water-filled fracture, we show that the Earth's geothermal gradient produces a convective flow in a fracture through the mechanism above; this has an associated dispersion coefficient DT ∼ 102-103D in fractures about a centimetre wide. We show that this shear dispersion could transport radioactive material, of half-life 104 years, tens of metres along the fracture within one half-life; without this dispersion, the material would only diffuse a few metres along the fracture within one half-life.

If there is a background salinity gradient along the slot in addition to the thermal gradient, analogous steady flow solutions exist; if the flow is stable, the salinity may either enhance or reduce the steady flow and associated dispersion of passive tracer.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 241 , August 1992 , pp. 59 - 74
Copyright
© 1992 Cambridge University Press

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