Skip to main content
SpringerLink
Log in

Contaminant transport with adsorption and their inverse problems

  • Regular Article
  • Published:
Computing and Visualization in Science

Abstract

In Constales et al. (water Resources Res. 39(30), 1303, 2003) dual-well tests are used to reconstruct the flow and dispersion parameters in contaminant transport. A tracer is introduced by the injection well, which is considered to be in steady-state regime with the extraction well. Then, from measurements of the time evolution of the extracted tracer (breakthrough curve) the required model data has been recovered. In Constales et al. (water Resources Res. 39(30), 1303, 2003), a very precise numerical method has been developed for the solution of the direct problem. In Kačur et al. (Comput. Meth. Appl. Mech. Engo. 194(2–5), 479–489, 2005); Remešiková (J. Comp. Appl. Math. 169(1), 101–116, 2004) an extension has been discussed which adds adsorption terms to the model. The inverse problem of determination of sorption isotherms in nonequilibrium mode was solved by a Levenberg–Marquardt iteration method. In the present paper we develop the adjoint system to evaluate the sensitivity of the solution (via the breakthrough curve) on the sorption parameters in equilibrium and nonequilibrium modes. Possible use of the adjoint system in determining the several parameters occuring in the model is a crucial point for iteration methods. The obtained model parameters then can be used in a 3D flow and transport model with adsorption. The numerical experiments we present, justify the used method.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Barrett J.W. and Knabner P. (1997). Finite element approximation of transport of reactive solutes in porous media. Part 2: error estimates for equilibrium adsorption processes. SIAM J. Numer. Anal. 34: 49–72

    Google Scholar 

  2. Bear J. (1972). Dynamics of Fluid in Porous Media. Elsevier, New York

    Google Scholar 

  3. Constales D., Kačur J. and Malengier B. (2003). A precise numerical scheme for contaminant transport in dual-well flow. Water Resources Res. 39(30): 1303

    Article  Google Scholar 

  4. Constales D., Kačur J. and Van Keer F. (2002). Parameter identification by a single injection-extraction well. Inv. probl. 18: 1605–1620

    Article  MATH  Google Scholar 

  5. Crandall M.G. and Majda A. (1980). The method of fractional steps for conservation laws. Numer. Math. 34: 285–314

    Article  MATH  MathSciNet  Google Scholar 

  6. Gelhar L.W. and Collins M.A. (1971). General analysis of longitudinal dispersion in nonuniform flow. Water Resource Res. 7(6): 1511–1521

    Google Scholar 

  7. Hajdjema H.M. (1995). Analytic modelling of groundwater flow. Academic, New York

    Google Scholar 

  8. Kačur, J., Frolkovič, P.: Semi-analytical solutions for contaminant transport with nonlinear sorption in 1D. Preprint 24, Interdis. Zentrum fur Wissens. Rechnen, Heidelberg

  9. Kačur J., Malengier B. and Remešíková M. (2005). Solution of contaminant transport with equilibrium and non-equilibrium adsorption. Comp. Meth. Appl. Mech. Eng. 194(2–5): 479–489

    MATH  Google Scholar 

  10. Karlsen K.H. and Lie K.-A. (1999). An unconditionally stable splitting for a class of nonlinear parabolic equations. IMA J. Numer. Anal. 19(4): 609–635

    Article  MATH  MathSciNet  Google Scholar 

  11. Lee T.-C. (1998). Applied Mathematics in Hydrogeology. Lewis Publishers, Boca Raton

    Google Scholar 

  12. Lockwood, E.H.: Bipolar Coordinates, chap. 25. In: A Book of Curves, pp. 186–190. Cambridge University Press, Cambridge (1967)

  13. Mercado A. (1967). The spreading problem of injected water in a permeability stratified aquifer. Int. Assoc. Sci. Hydrol. Publ. 72: 23–36

    Google Scholar 

  14. Patankar S.V. (1980). Numerical Heat Transfer and Fluid Flow. Taylor & Francis, London

    MATH  Google Scholar 

  15. Pickens J.F. and Grisak G.E. (1981). Scale-dependent dispersion in a stratified granular aquifer. Water Resources Res 17(4): 1191–1211

    Article  Google Scholar 

  16. Polak E. (1997). Optimization: algorithms and consistent approximations. Springer, Berlin Heidelberg Newyork

    MATH  Google Scholar 

  17. Remešíková M. (2004). Solution of convection-difffusion problems with non-equilibrium adsorption. J. Comput. Appl. Math. 169(1): 101–116

    Article  MathSciNet  MATH  Google Scholar 

  18. Schroth M.H., Istok J.D. and Haggerty R. (2001). In situ evaluation of solute retardation using single-well push-pull tests. Adv. Water Resources 24: 105–117

    Google Scholar 

  19. Spiegel M.R. (1968). Mathematical Handbook of Formulas and Tables. McGraw Hill, Newyork

    Google Scholar 

  20. Stewart, D.E., Leyk, Z.: Meschach: Matrix computations in C, http://www.math.uiowa.edu/~dstewart/ meschach/

  21. Sun N.-Z. (1996). Mathematical model of groundwater pollution. Springer, Berlin Heidelberg Newyork

    Google Scholar 

  22. Welly C. and Gelhar L.W. (1971). Evaluation of longitudinal dispersivity from nonuniform flow tracer tests. J. Hydrol. 153: 71–102

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Mathematics, Physics and Informatics, Comenius University, Bratislava, Slovakia

    J. Kačur & M. Remešíková

  2. Department of Mathematical Analysis, Research Group NfaM2, Ghent University, Galglaan 2, 9000, Gent, Belgium

    B. Malengier

Authors
  1. J. Kačur
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Remešíková
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. B. Malengier
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J. Kačur.

Additional information

Communicated by P. Frolkovic.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kačur, J., Remešíková, M. & Malengier, B. Contaminant transport with adsorption and their inverse problems. Comput. Visual Sci. 10, 29–42 (2007). https://doi.org/10.1007/s00791-006-0049-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00791-006-0049-2

Keywords

Search

Navigation