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Mixing at the interface between two fluids in porous media: a boundary-layer solution

Published online by Cambridge University Press:  25 July 2007

AMIR PASTER  and
GEDEON DAGAN
AMIR PASTER
Affiliation:
School of Mechanical Engineering, Tel Aviv University, Ramat Aviv 69978, Israel
GEDEON DAGAN
Affiliation:
School of Mechanical Engineering, Tel Aviv University, Ramat Aviv 69978, Israel
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Abstract

A lighter fluid (fresh water) flows steadily above a body of a standing heavier one (sea water) in a porous medium. If mixing by transverse pore-scale dispersion is neglected, a sharp interface separates the two fluids. Solutions for interface problems have been derived in the past, particularly for the case of interest here: sea-water intrusion in coastal aquifers. The Péclet number characterizing mixing, Pe = b′/αT where b′ is the aquifer thickness and αT is transverse dispersivity, is generally much larger than unity. Mixing is nevertheless important in a few applications, particularly in the development of a transition layer near the interface and in entrainment of sea water within this layer. The equations of flow and transport in the mixing zone comprise the unknown flux, pressure and concentration fields, which cannot be separated owing to the presence of density in the gravity term. They are nonlinear because of the advective term and the dependence of the dispersion coefficients on flux, the latter making the problem different from that of mixing between streams in laminar viscous flow.

The aim of the study is to solve the mixing-layer problem for sea-water intrusion by using a boundary-layer approximation, which was used in the past for the case of uniform flow of the upper fluid, whereas here the two-dimensional flux field is non-uniform. The boundary-layer solution is obtained in a few steps: (i) analytical potential flow solution of the upper fluid above a sharp interface is adopted; (ii) the equations are reformulated with the potential and streamfunction of this flow serving as independent variables; (iii) boundary-layer approximate equations are formulated in terms of these variables; and (iv) simple analytical solutions are obtained by the von Káarmán integral method. The agreement with an existing boundary-layer solution for uniform flow is excellent, and similarly for a solution of a particular case of sea-water intrusion with a variable-density code. The present solution may serve for estimating the thickness of the mixing layer and the rate of sea-water entrainment in applications, as well as a benchmark for more complex problems.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 584 , 10 August 2007 , pp. 455 - 472
Copyright
Copyright © Cambridge University Press 2007

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References

REFERENCES

Bakker, M. 2003 A Dupuit formulation for modelling seawater intrusion in regional aquifer systems. Water Resour. Res. 39, 1131. doi:10.1029/2002WR001710.CrossRefGoogle Scholar
Batchelor, G. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.Google Scholar
Bear, J. 1979 Hydraulics of Groundwater. McGraw-Hill.Google Scholar
Bear, J. & Dagan, G. 1964 Some exact solutions of interface problems by means of the hodograph method. J. Geophys. Res. 69, 1563–1572.CrossRefGoogle Scholar
Cole, J. D. 1968 Perturbation Methods in Applied Mathematics. Blaisdell.Google Scholar
Dagan, G. 1971 Perturbation solutions of the dispersion equation in porous mediums. Water Resour. Res. 7, 135–142.CrossRefGoogle Scholar
Dagan, G. 1989 Flow and Transport in Porous Formations. Springer.Google Scholar
Dentz, M., Tartakovsky, D. M., Abarca, E., Guadagnini, A., Sanchez-Vila, X. & Carrera, J. 2006 Variable density flow in porous media. J. Fluid Mech. 561, 209–235.Google Scholar
Fiori, A. & Dagan, G. 1999 Concentration fluctuations in transport by groundwater: comparison between theory and field experiments. Water Resour. Res. 35, 102–112.Google Scholar
Glover, R. E. 1959 The pattern of freshwater flow in a coastal aquifer. J. Geophys. Res. 64, 457–459.CrossRefGoogle Scholar
Henry, H. R. 1964 Effects of dispersion on salt encroachment in coastal aquifers. Water Supply Paper 1613-C. US Geol. Surv.Google Scholar
Holzbecher, E. 1998 Modeling Density-Driven Flow in Porous Media. Springer.CrossRefGoogle Scholar
List, E. J. & Brooks, N. H. 1967 Lateral dispersion in porous media. J. Geophys. Res. 72, 2531–2541.CrossRefGoogle Scholar
Lock, R. C. 1951 The velocity distribution in the laminar boundary layer between parallel streams, Q. J. Mech. Appl. Maths 4, 42–63.Google Scholar
Paster, A., Dagan, G. & Guttman, J. 2006 The salt-water body in the Northern part of Yarkon–Taninim aquifer: field data analysis, conceptual model and prediction. J. Hydrol. 323, 154–167.CrossRefGoogle Scholar
Rubin, H. 1983 On the application of the boundary layer approximation for the simulation of density stratified flows in aquifers. Adv. Water Resour. 6, 96–105.CrossRefGoogle Scholar
Rugner, H., Holder, T., Maier, U., Bayer-Raich, M., Grathwohl, P. & Teutsch, G. 2004 Natural attenuation investigation at the former landfill ‘Osterhofen’. Grundwasser 2, 98–108 (in German).Google Scholar
Smith, A. J. 2004 Mixed convection and density-dependent seawater circulation in coastal aquifers. Water Resour. Res. 40, W08309. doi:10.1029/2003WR002977.CrossRefGoogle Scholar
VanDuijn, C. J. Duijn, C. J. & Peletier, L. A. 1992 A boundary-layer problem in fresh–salt groundwater flow. Q. J. Mech. Appl. Maths 45, 1–24.Google Scholar