Reply to comment: "Analytical solution for solute transport resulting from instantaneous injection in streams with transient storage."

Florimond De Smedt, W. Brevis, P. Debels

Research output: Contribution to journalArticlepeer-review

2 Citations (Scopus)


We thank J. Huang, M.N. Goltz, and P.V. Roberts to bring to our attention the analytical solution of so-called mobile-immobile zone models, which have been used to simulate solute transport in groundwater, and are mathematically identical to transient storage models. In particular, we agree with the comment that the analytical solution of Goltz and Roberts (1984) is shown to be equivalent to the solution derived in De Smedt et al. (2005). We did not include in our literature review publications of other disciplines that deal with mathematically similar equations, because this field is very wide as it not only covers the so-called mobile-immobile zone transport models in groundwater, but also mass transport in porous media with non-equilibrium sorption, and similar processes in petroleum engineering, chromatography, etc. However, as far as we know the analytical solution presented by us in De Smedt et al. (2005) is unique and has never been presented before. The solution given by Goltz and Roberts (1984) was also presented by Carnahan and Remer, 1984 and van Kooten, 1995. Similar solutions for slightly different boundary conditions were presented e.g. by Lapidus and Amundson, 1952, Coats and Smith, 1964, Ogata, 1964, Villermaux and van Swaaij, 1969, Bennet and Goodridge, 1970, Lindstrom and Boersma, 1973, Lindstrom and Narasimhan, 1973, van Genuchten and Wierenga, 1976, Lindstrom, 1976, De Smedt and Wierenga, 1979 and Veling, 2002 amongst others. The list of references is probably not exhaustive.

In addition, the solution given by Eq. 1 of J. Huang, M.N. Goltz, and P.V. Roberts was also derived independently for tracer transport in rivers with effects of storage in dead zones by Davis et al. (2000), a publication which escaped our attention previously.

We do not agree with the last comment of J. Huang, M.N. Goltz, and P.V. Roberts that the solution presented by Goltz and Roberts (1984) is more straightforward than the solution presented in De Smedt et al. (2005), because the latter solution requires evaluation of a double integral. This argument only holds if the J-function is evaluated by the integral expression given by Goldstein (1953). As pointed out by De Smedt et al. (2005) there are other ways by which this function can be evaluated that do not require integration, as for instance shown by De Smedt and Wierenga (1979).
Original languageEnglish
Pages (from-to)761-762
Number of pages2
JournalJournal of Hydrology
Publication statusPublished - 2006


  • Transient storage
  • Solute transport
  • Analytical solution
  • Tracer experiment
  • Chillán river
  • Chile


Dive into the research topics of 'Reply to comment: "Analytical solution for solute transport resulting from instantaneous injection in streams with transient storage."'. Together they form a unique fingerprint.

Cite this