Skip to main content Accessibility help

Analysis and computation of gravity-induced migration in porous media

  • R. DE LOUBENS (a1) and T. S. RAMAKRISHNAN (a1)


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.


Corresponding author

Email address for correspondence:


Hide All

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



Hide All
Barenblatt, G. I. 1996 Scaling, Self-Similarity, and Intermediate Asymptotics. Cambridge University Press.
Bear, J. 1972 Dynamics of Fluids in Porous Media. Elsevier.
Doll, H. G. 1955 Filtrate invasion in highly permeable sands. The Petroleum Engineer, January.
Dussan, E. B. & Auzerais, F. M. 1993 Buoyancy-induced flow in porous media generated near a drilled oil well. Part 1. The accumulation of filtrate at a horizontal impermeable boundary. J. Fluid Mech. 254, 283311.
Farcas, A. & Woods, A. W. 2009 The effect of drainage on the capillary retention of CO2 in a layered permeable rock. J. Fluid Mech. 618, 349359.
Fayers, F. J. & Muggeridge, A. H. 1990 Extensions to Dietz theory and behavior of gravity tongues in slightly tilted reservoirs. SPE Reservoir Engineering 5, 487494.
Gilding, B. H. 1977 Properties of solutions of an equation in the theory of infiltration. Arch. Rat. Mech. Anal. 65, 203225.
Gilding, B. H. & Peletier, L. A. 1976 The Cauchy problem for an equation in the theory of infiltration. Arch. Rat. Mech. Anal. 61, 127140.
Grundy, R. E. 1983 Asymptotic solution of a model non-linear convective diffusion equation. IMA J. Appl. Math. 31, 121137.
Hesse, M. A., Orr, F. M. Jr & Tchelepi, H. A. 2008 Gravity currents with residual trapping. J. Fluid Mech. 611, 3560.
Hesse, M. A., Tchelepi, H. A., Cantwell, B. J. & Orr, F. M. Jr. 2007 Gravity currents in horizontal porous layers: transition from early to late self-similarity. J. Fluid Mech. 577, 363383.
Huppert, H. E. & Woods, A. W. 1995 Gravity-driven flows in porous layers. J. Fluid Mech. 292, 5569.
Juanes, R., MacMinn, C. W. & Szulczewski, M. L. 2010 The footprint of the CO2 plume during carbon dioxide storage in saline aquifers: storage efficiency for capillary trapping at the basin scale. Trans. Porous Med. 82, 1930.
Kevorkian, J. & Cole, J. D. 1981 Perturbation Methods in Applied Mathematics. Springer.
Kochina, I. N., Mikhailov, N. N. & Filinov, M. V. 1983 Groundwater mound damping. Intl J. Engng Sci. 21, 413421.
Laurençot, P. & Simondon, F. 1998 Long-time behaviour for porous medium equations with convection. Proc. R. Soc. Edin. A 128, 315336.
Lenoach, B., Ramakrishnan, T. S. & Thambynayagam, R. K. M. 2004 Transient flow of a compressible fluid in a connected layered permeable medium. Trans. Porous Med. 57, 153169.
de Loubens, R. & Ramakrishnan, T. S. 2011 Asymptotic solution of a nonlinear advection–diffusion equation. Q. Appl. Math. (in press).
Ramakrishnan, T. S. 2007 On reservoir fluid-flow control with smart completions. SPE Prod. Oper. 22 (1), 412.
Sheldon, J. W. & Fayers, F. J. 1962 The motion of an interface between two fluids in a slightly dipping porous medium. Soc. Petrol. Engng J. 2, 275282.
Vella, D. & Huppert, H. E. 2006 Gravity currents in a porous medium at an inclined plane. J. Fluid Mech. 555, 353362.
Woods, A. W. & Farcas, A. 2009 Capillary entry pressure and the leakage of gravity currents through a sloping layered permeable rock. J. Fluid Mech. 618, 361379.
Yortsos, Y. C. 1995 A theoretical analysis of vertical flow equilibrium. Trans. Porous Med. 18 (2), 107129.
MathJax is a JavaScript display engine for mathematics. For more information see

JFM classification


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed