Mathematical Geology

, Volume 36, Issue 6, pp 743–758 | Cite as

Transient Modeling of Hyperfiltration Effects

  • Peter Oduor
  • T. M. Whitworth


Transient models are needed to analyze time-dependent problems like hyperfiltration associated with head differences across clay barriers. Hyperfiltration (solute-sieving) effects create an increased concentration of natural groundwater solutes outside the clay barrier due to the inward head gradient. The purpose of our model is to predict solute buildup and distribution during hyperfiltration providing a basis for time analysis of solute migration. Required input parameters for the model include membrane properties like reflection coefficient, hydraulic conductivity, and solute concentration on the high-pressure side of the membrane before the onset of steady state, solution flux, and the effluent concentration. Model verification is based on published experimental results. The transient hyperfiltration model presented herein may prove useful in elucidating clay membrane (hyperfiltration) effects in the subsurface, however the sole purpose of this paper is to develop a transient model of hyperfiltration effects and test it by using published experimental data.

concentration polarization layer membrane influent effluent 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Carslaw, H. S., and Jaeger, J. C., 1959, Conduction of heat in solids, 2nd edn.: Clarendon Press, Oxford, UK, 482 p.Google Scholar
  2. Charbeneau, R. J., 2000, Groundwater hydraulics and pollutant transport: Prentice-Hall, Upper Saddle River, NJ, 370 p.Google Scholar
  3. Dankwerts, P. V., 1953, Continuous flow system distribution of residence times: Che. Eng. Sci., v. 2, p. 1–13.CrossRefGoogle Scholar
  4. Day, P. R., 1956, Dispersion of a moving salt-water boundary advancing through saturated sand: Am. Geophys. Union Trans., v. 37, p. 595–601.Google Scholar
  5. DeGroot, S. R., and Mazur, P., 1963, Non-equilibrium thermodynamics: Dover, North-Holland, Amsterdam, 105 p.Google Scholar
  6. Fritz, S. J., 1986, Ideality of clay membranes in osmotic processes; a review: Clays Clay Miner., v. 34, no. 2, p. 214–232.Google Scholar
  7. Fritz, S. J., and Marine, I. W., 1983, Experimental support for a predictive osmotic model of clay membranes: Geochimica et Cosmochimica Acta, v. 47, p. 1515–1522.CrossRefGoogle Scholar
  8. Fritz, S. J., and Whitworth, T. M., 1994, Hyperfiltration-induced fractionation of lithium isotopes: Ramifications relating to representativeness of aquifer sampling: Water Resour. Res., v. 30, p. 225–235.CrossRefGoogle Scholar
  9. Irving, J., and Mulinneux, N., 1959, Mathematics in physics and engineering: Academic Press, New York, 883 p.Google Scholar
  10. Katchalsky, A., and Curran, P. F., 1965, Biophysics: Harvard University Press, Cambridge,MA, 248 p.Google Scholar
  11. Kedem, O., and Katchalsky, A., 1962, Thermodynamics of flow processes in biological systems: Biophys. J., v. 2, p. 53–54.Google Scholar
  12. Lakshminarayanaiah, N., 1984, Equations of membrane biophysics: Academic Press, Orlando, FL, 426 p.Google Scholar
  13. Liangxiong, L., Whitworth, T. M., and Lee, R., 2003, Separation of inorganic solutes from oil-field produced water using a compacted bentonite membrane: J. Membr. Sci., v. 217, p. 215–225.CrossRefGoogle Scholar
  14. Marine, J. W., and Fritz, S. J., 1981, Osmotic model to explain anomalous hydraulic heads: Water Resources Research, v. 17, no. 1, p. 73–82.Google Scholar
  15. Ogata, A., and Banks, R. B., 1961, A solution of the differential equation of longitudinal dispersion in porous media: U.S. Geol. Survey, Prof. Paper, no. 411-A, p. A1–A7.Google Scholar
  16. Porter, M. C., 1979, Membrane filtration, inP. A. Schweitzer, ed., Handbook of separation techniques for chemical engineers: McGraw-Hill, New York, p. 225–235.Google Scholar
  17. Sourirajan, S., 1970, Reverse osmosis: Academic Press, Orlando, FL, 547 p.Google Scholar
  18. Srivastava, R. C., and Jain, A. K., 1975, Non-equilibrium thermodynamics of electro-osmosis of water through composite clay membranes, 1. The electro-kinetic energy conversion: J. Hydrol. v. 25, p. 307–324.CrossRefGoogle Scholar
  19. Staverman, A. J., 1952, Non-equilibrium thermodynamics of membrane processes: Trans. Faraday Soc., v. 48, p. 176–185.CrossRefGoogle Scholar
  20. Whitworth, T. M., 1998, Steady-state mathematical modeling of geologic membrane processes in aquifer systems, it in WERC/WRHSRC/NMHWMS Joint Conference on the Environment, Proceedings, Albuquerque, NM, p. 37–41.Google Scholar
  21. Whitworth, T. M., and DeRosa, G., 1997, Geologic membrane controls on saturated zone heavy metal transport: Las Cruces, New Mexico WRRI: Technical Completion Report, no. 303, 51 p.Google Scholar

Copyright information

© International Association for Mathematical Geology 2004

Authors and Affiliations

  • Peter Oduor
    • 1
  • T. M. Whitworth
    • 1
  1. 1.Department of Geological and Petroleum Engineering, School of Mines and MetallurgyUniversity of Missouri – RollaRollaUSA

Personalised recommendations