Hostname: page-component-7c8c6479df-r7xzm Total loading time: 0 Render date: 2024-03-29T10:50:33.141Z Has data issue: false hasContentIssue false

On the analysis of brine transport in porous media

Published online by Cambridge University Press:  26 September 2008

C. J. van Duijn
Affiliation:
Department of Mathematics, Delft University of Technology, Delft, The Netherlands Mathematical Institute, Leiden University, Leiden, The Netherlands
L. A. Peletier
Affiliation:
Mathematical Institute, Leiden University, Leiden, The Netherlands
R. J. Schotting
Affiliation:
Department of Mathematics, Delft University of Technology, Delft, The Netherlands

Abstract

An analysis is given of brine transport through a porous medium, which incorporates the effect of volume changes due to variations in the salt concentration. Two specific situations are investigated which lead to self-similarity. We develop the existence and uniqueness theory for the corresponding ordinary differential equations, and give a number of qualitative properties of the solutions. In particular, we present an asymptotic expression for the solution in terms of the relative density difference (ρs−ρf)/ρf. Finally, we show some numerical results. It is found that the volume changes have a noticeable effect on the mass transport only when salt concentrations are large.

Type
Research Article
Copyright
Copyright © Cambridge University Press 1993

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bear, J. 1972 Dynamics of Fluids in Porous Media. Elsevier.Google Scholar
Atkinson, F. V. & Peletier, L. A. 1974 Similarity solutions of nonlinear diffusion equations. Arch. Rational Mech. Anal. 54, 373392.CrossRefGoogle Scholar
de Josseling de Jong, G. & van Duun, C. J. 1986 Transverse dispersion from an originally sharp fresh-salt interface caused by shear flow. J. Hydrology 84, 5579.CrossRefGoogle Scholar
van Duijn, C. J., Gomez, S. M. & Zhang, H. 1988 On a class of similarity solutions of the equation ut = (|u|m−1ux)x with m > −1. IMA J. Appl. Math. 41, 147163.CrossRefGoogle Scholar
van Duijn, C. J. & Peletier, L. A. 1977a A class of similarity solutions of the nonlinear diffusion equation. Nonlinear Anal. TMA 1, 223233.CrossRefGoogle Scholar
van Duijn, C. J. & Peletier, L. A. 1977b Asymptotic behaviour of solutions of nonlinear diffusion equations. Arch. Rational Mech. Anal. 65, 363377.CrossRefGoogle Scholar
van Duijn, C. J. & Peletier, L. A. 1992 A boundary layer problem in fresh-salt groundwater flow. Quart. J. Appl. Math. Mech. 45, 124.CrossRefGoogle Scholar
Weast, R. C., ed. 1981 Handbook of Chemistry and Physics 62nd Edn, D-232.Google Scholar
Hassanizadeh, S. M. & Leijnse, T. 1988 On the modelling of brine transport in porous media. Water Resources Res. 24, 321330.CrossRefGoogle Scholar
von Mises, R. & Friedrichs, K. O. 1971 Fluid Dynamics. Vol 171 of Applied Mathematics Series 5, Springer-Verlag.CrossRefGoogle Scholar
Oleinik, O. A. 1963 The Prandtl system of equations. Sov. Math. 4, 383386.Google Scholar
Taliaferro, S. D. 1979 A nonlinear singular boundary value problem. Nonlinear Anal. TMA 3, 897904.CrossRefGoogle Scholar
Taylor, G. I. 1953 Dispersion of soluble matter in solvent flowing slowly through a tube. Proc. R. Soc. Lond. Ser A219, 186203.Google Scholar
Taylor, G. I. 1954 The dispersion of matter in turbulent flow through a pipe. Proc. R. Soc. Lond. Ser. A223, 446468.Google Scholar
Zhang, H. 1992 Large time behaviour of the maximal solution of the equation ut = (u m−1ux)x with −1 < m < 0. (Submitted to SIAM J. Math. Anal)CrossRefGoogle Scholar