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Freely draining gravity currents in porous media: Dipole self-similar solutions with and without capillary retention

Published online by Cambridge University Press:  01 June 2007

JOCHONIA S. MATHUNJWA  and
ANDREW J. HOGG
JOCHONIA S. MATHUNJWA
Affiliation:
Centre of Environmental and Geophysical Flows School of Mathematics, University of Bristol University Walk, Bristol BS8 1TW, UK email: [email protected]
ANDREW J. HOGG
Affiliation:
Centre of Environmental and Geophysical Flows School of Mathematics, University of Bristol University Walk, Bristol BS8 1TW, UK email: [email protected]
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Abstract

We analyse the two-dimensional, gravitationally-driven spreading of fluid through a porous medium overlying a horizontal impermeable boundary from which fluid can drain freely at one end. Under the assumption that none of the intruding fluid is retained within the pores in the trail of the current, the motion of the current is described by the dipole self-similar solution of the first kind derived by Barenblatt and Zel'dovich (1957). We show that small perturbations of arbitrary shape imposed on this solution decay in time, indicating that the self-similar solution is linearly stable. We use the connection between the perturbation eigenfunctions and symmetry transformations of the self-similar solution to demonstrate that variables can always be specified in terms of which the rate of decay of the perturbations is maximised. Unsaturated flow can be modelled by assuming that a constant fraction of the fluid is retained within the pores by capillary action in the trail of the current. It has been shown (Barenblatt and Zel'dovich, 1998; Ingerman and Shvets, 1999) that in this case, the motion of the current is described by a self-similar solution of the second kind characterised by an anomalous exponent. We derive leading-order analytic expressions for the anomalous exponent and the self-similar quantities valid for small values of the fraction of fluid retained using direct asymptotic analysis and by using a novel application of the method of multiple scales. The latter offers a number of advantages and permits the evolution of the current to be clearly connected with its initial conditions in a way not possible with conventional approaches. We demonstrate that the theoretical predictions provided by these expressions are in excellent agreement with results from the numerical integration of the governing equations.

Type
Papers
Information
European Journal of Applied Mathematics , Volume 18 , Issue 3 , June 2007 , pp. 337 - 362
Copyright
Copyright © Cambridge University Press 2007

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References

Abramowitz, M. & Stegun, I. A. (1965) Handbook of Mathematical Functions, Dover Publications Inc.Google Scholar
Barenblatt, G. I. (1996) Scaling, Self-Similarity, and Intermediate Asymptotics, Cambridge University Press.Google Scholar
Barenblatt, G. I., Ingerman, E. A., Shvets, H. & Vazquez, J. L. (2000) Very intense pulse in the groundwater flow in fissurized-porous stratum. Proc. Nat. Acad. Sci. U.S.A. 97 (4), 1366–1369.Google Scholar
Barenblatt, G. I. & Vazquez, J. L. (1998) A new free boundary problem for unsteady flows in porous media. Eur J. Appl. Math. 9, 37–54.CrossRefGoogle Scholar
Barenblatt, G. I. & Zel'dovich, Ya. B. (1957) On dipole solutions in problems on non-stationary filtration of gas under polytropic regime. Prikladnaia Matematika i Mekhanika 21 (5), 718–720.Google Scholar
Bear, J. (1988) Dynamics of Fluids in Porous Media, Dover.Google Scholar
Bernis, F., Hulshof, J. & King, J. R. (2000) Dipoles and similarity solutions of the thin film equation in the half-line. Nonlinearity 13, 413–439.Google Scholar
Bowen, M. & Witelski, T. P. (2006) The linear limit of the dipole problem for the thin film equation. SIAM J. Appl. Math. 66, 1727–1748.Google Scholar
Fowler, A. C. (1997) Mathematical Models in the Applied Sciences Cambridge University Press.Google Scholar
Goldenfeld, N., Martin, O., Oono, Y. & Liu, F. (1990) Anomalous dimensions and the Renormalization Group in a nonlinear diffusion process. Phys. Rev. Lett. 64, 1361–1364.CrossRefGoogle Scholar
Grundy, R. E. & McLaughlin, R. (1982) Eigenvalues of the Barenblatt–Pattle similarity solution in nonlinear diffusion. Proc. Royal. Soc. Lond. Ser. A 383, 89–100.Google Scholar
Huppert, H. E. & Woods, A. W. (1995) Gravity-driven flows in porous layers. J. of Fluid Mech. 292, 53–69.CrossRefGoogle Scholar
Ingerman, E. A. & Shvets, H. (1999) Numerical Investigation of the Dipole Type Solution for the Unsteady Groundwater Flow With capillary Retention and Forced Drainage, Centre for Pure and Applied Mathematics, University of California, Berkeley, CPAM–775.Google Scholar
Kath, W. L. & Cohen, D. S. (1982) Waiting-time behaviour in a nonlinear diffusion equation. Stud. Appl. Math. 67, 79–105.Google Scholar
King, S. E. & Woods, A. W. (2003) Dipole solutions for viscous gravity currents: Theory and experiments. J. Fluid Mech. 483 91–109.CrossRefGoogle Scholar
Mathunjwa, J. S. & Hogg, A. J. (2006) Self-similar gravity currents in porous media: Linear stability of the Barenblatt-Pattle solution revisited. Eur. J. Mech. B Fluids 25, 360–378.Google Scholar
Wagner, B. (2005) An asymptotic approach to second-kind similarity solutions of modified porous-medium equation. J. Eng. Math. 53, 201–220.CrossRefGoogle Scholar
Woods, A. W. (1998) Vapourizing gravity currents in a superheated porous medium. J. Fluid Mech. 377, 151–168.CrossRefGoogle Scholar