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Transport in Porous Media

, Volume 18, Issue 1, pp 65–85 | Cite as

Approximate analytical solutions for solute transport in two-layer porous media

  • F. J. Leij
  • M. Th. Van Genuchten
Article

Abstract

Mathematical models for transport in layered media are important for investigating how restricting layers affect rates of solute migration in soil profiles; they may also improve the analysis of solute displacement experiments. This study reports an (approximate) analytical solution for solute transport during steady-state flow in a two-layer medium requiring continuity of solute fluxes and resident concentrations at the interface. The solutions were derived with Laplace transformations making use of the binomial theorem. Results based on this solution were found to be in relatively good agreement with those obtained using numerical inversion of the Laplace transform. An expression for the flux-averaged concentration in the second layer was also obtained. Zero- and first-order approximations for the solute distribution in the second layer were derived for a thin first layer representing a water film or crust on top of the medium. These thin-layer approximations did not perform as well as the ‘binomial’ solution, except for the first-order approximation when the Peclet number,P, of the first layer, was low (P<5). Results of this study indicate that the ordering of two layers will affect the predicted breakthrough curves at the outlet of the medium. The two-layer solution was used to illustrate the effects of dispersion in the inlet or outlet reservoirs using previously published data on apparatus-induced dispersion.

Key words

Solute transport composite media boundary conditions 

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References

  1. Al-Niami, A. N. S. and Rushton, K. R., 1979, Dispersion in stratified porous media: analytical solutions,Water Resour. Res. 15, 1044–1048.Google Scholar
  2. Barry, D. A. and Parker, J. C., 1987, Approximations for solute transport through porous media with flow transverse to layering,Transport in Porous Media 2, 65–82.Google Scholar
  3. Brenner, H., 1962, The diffusion model of longitudinal mixing in beds of finite length. Numerical values,Chem. Eng. Sci. 17, 229–243.Google Scholar
  4. Carnahan, B., Luther, H. A., and Wilkes, J. O., 1969,Applied Numerical Models, John Wiley, New York.Google Scholar
  5. Carslaw, H. S. and Jaeger, J. C., 1959,Conduction of Heat in Solids, Clarendon Press, Oxford.Google Scholar
  6. Dagan, G. and Bresler, E., 1985, Comment on ‘Flux-averaged and volume-averaged concentrations in continuum approaches to solute transport’, by J. C. Parker and M. Th. van Genuchten,Water Resour. Res. 21, 1299–1300.Google Scholar
  7. Danckwerts, P. V., 1953, Continuous flow systems,Chem. Eng. Sci. 2, 1–13.Google Scholar
  8. De Hoog, F. R., Knight, J. H., and Stokes, A. N., 1982, An improved method for numerical inversion of Laplace transforms,SIAM J. Sci. Stat. Comput. 3, 357–366.Google Scholar
  9. Jacobsen, O. H., Leij, F. J., and van Genuchten, M. Th., 1992, Lysimeter study of anion transport during steady flow through layered coarse-textured soil profiles,Soil Sci. 154, 196–205.Google Scholar
  10. James, R. V. and Rubin, J., 1972, Accounting for apparatus-induced dispersion in analysis of miscible displacement experiments,Water Resour. Res. 8, 717–721.Google Scholar
  11. Jury, W. A. and Utermann, J., 1992, Solute transport through layered soil profiles: Zero and perfect travel time correlation models,Transport in Porous Media 8, 277–297.Google Scholar
  12. Kreft, A., 1981a, On the residence time distribution in systems with axial dispersed flow,Bull. Ac. Pol. Sci. Tech. 29, 509–519.Google Scholar
  13. Kreft, A., 1981b, On the boundary conditions of flow through porous media and conversion of chemical flow reactors,Bull. Ac. Pol. Sci. Tech. 29, 521–529.Google Scholar
  14. Kreft, A. and Zuber, A., 1978, On the physical meaning of the dispersion equation and its solutions for different initial and boundary conditions,Chem. Eng. Sci. 33, 1471–1480.Google Scholar
  15. Leij, F. J., Dane, J. H., and van Genuchten, M. Th., 1991, Mathematical analysis of one-dimensional solute transport in a layered soil profile,Soil Sci. Soc. Am. J. 55, 944–953.Google Scholar
  16. Lindstrom, F. T., Haque, R., Freed, V. H., and Boersma, L., 1967, Theory on the movement of some herbicides in soils: Linear diffusion and convection of chemicals in soils,J. Envir. Sci. Technol. 1, 561–565.Google Scholar
  17. Luikov, A. V., 1968,Analytical heat diffusion, Academic Press, New York.Google Scholar
  18. Mikhailov, M. D. and öziŞik, M. N., 1984,Unified Analysis and Solutions of Heat and Mass Diffusion, John Wiley, New York.Google Scholar
  19. Nakayama, S., Takagi, I., Nakai, K., and Higashi, K., 1984, Migration of radionuclide through two-layered geologic media,J. Nucl. Sci. Technol. 21, 139–147.Google Scholar
  20. Novakowski, K. S., 1992a, An evaluation of boundary conditions for one-dimensional solute transport. 1. Mathematical development,Water Resour. Res. 28, 2399–2410.Google Scholar
  21. Novakowski, K. S., 1992b, An evaluation of boundary conditions for one-dimensional solute transport. 2. Column experiments,Water Resour. Res. 28, 2411–2423.Google Scholar
  22. öziŞik, M. N., 1980,Heat Conduction, John Wiley, New York.Google Scholar
  23. Parker, J. C. and van Genuchten, M. Th., 1985, Reply to G. Dagan and E. Bresler,Water Resour. Res. 21, 1301–1302.Google Scholar
  24. Selim, H. M., Davidson, J. M., and Rao, P. S. C., 1977, Transport of reactive solutes through multilayered soils,Soil Sci. Soc. Am. J. 41, 3–10.Google Scholar
  25. Shamir, U. Y. and Harleman, D. R. F., 1967, Dispersion in layered porous media,J. Hydraul. Div. Proc. ASCE 93(HY5), 237–260.Google Scholar
  26. Taylor, G. I., 1953, Dispersion of soluble matter in solvent flowing through a tube,Proc. Roy. Soc. London, A 219, 186–203.Google Scholar
  27. van der Laan, E. T., 1958, Notes on the diffusion-type model for longitudinal mixing in flow,Chem. Eng. Sci. 7, 187–191.Google Scholar
  28. van Genuchten, M. Th. and Alves, W. J., 1982, Analytical solutions of the one-dimensional convective-dispersive solute transport equation,U.S.D.A. Technical Bulletin No. 1661.Google Scholar
  29. van Genuchten, M. Th. and Parker, J. C., 1984, Boundary conditions for displacement experiments through short laboratory soil columns,Soil Sci. Soc. Am. J. 48, 703–708.Google Scholar
  30. Wehner, J. F. and Wilhelm, R. H., 1956, Boundary conditions of flow reactor,Chem. Eng. Sci. 6, 89–93.Google Scholar

Copyright information

© Kluwer Academic Publishers 1995

Authors and Affiliations

  • F. J. Leij
    • 1
  • M. Th. Van Genuchten
    • 1
  1. 1.U.S. Salinity LaboratoryUSDA-ARSRiversideUSA

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