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Analysis and computation of gravity-induced migration in porous media

Published online by Cambridge University Press:  29 March 2011

R. DE LOUBENS  and
T. S. RAMAKRISHNAN
R. DE LOUBENS
Affiliation:
Schlumberger-Doll Research Center, One Hampshire Street, Cambridge, MA 02139, USA
T. S. RAMAKRISHNAN*
Affiliation:
Schlumberger-Doll Research Center, One Hampshire Street, Cambridge, MA 02139, USA
*
‡Email address for correspondence: [email protected]
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Abstract

Motivated by the problem of gravity segregation in an inclined porous layer, we present a theoretical analysis of interface evolution between two immiscible fluids of unequal density and mobility, both in two and three dimensions. Applying perturbation theory to the appropriately scaled problem, we derive the governing equations for the pressure and interface height to leading order, obtained in the limit of a thin gravity tongue and a slightly dipping bed. According to the zeroth-order approximation, the pressure profile perpendicular to the bed is in equilibrium, a widely accepted assumption for this class of problems. We show that for the inclined bed two-dimensional problem, in the reference frame moving with the mean gravity-induced advection velocity, the interface motion is dictated by a degenerate parabolic equation, different from those previously published. In this case, the late-time behaviour of the gravity tongue can be derived analytically through a formal expansion of both the solution and its two moving boundaries. In three dimensions, using a moving coordinate along the dip direction, we obtain an elliptic–parabolic system of partial differential equations where the fluid pressure and interface height are the two dependent variables. Although analytical results are not available for this case, the evolution of the gravity tongue can be investigated by numerical computations in only two spatial dimensions. The solution features are identified for different combinations of dimensionless parameters, showing their respective influence on the shape and motion of the interface.

JFM classification

Convection: Convection in porous media Geophysical and Geological Flows: Geophysical and Geological Flows Geophysical and Geological Flows: Gravity currents
Type
Papers
Information
Journal of Fluid Mechanics , Volume 675 , 25 May 2011 , pp. 60 - 86
Copyright
Copyright © Cambridge University Press 2011

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Footnotes

†

Present address: Total, avenue Larribau, 64018 Pau CEDEX, France.

References

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