Reply to comments on “Two-dimensional concentration distribution for mixing-controlled bioreactive transport in steady state” by H. Shao et al.

Shao et al. [Shao H, Centler F, Biase CD, Thullner M, Kolditz O, Comments on two-dimensional concentration distribution for mixing-controlled bioreactive transport in steady state by O.A. Cirpka and A.J. Valocchi, Adv Water Res, submitted for publication.] test our analytical approach of computing steady state concentration distributions in bioreactive transport [Cirpka OA, Valocchi AJ, Two-dimensional concentration distribution for mixing-controlled bioreactive transport in steady state. Adv Water Res 2007;30(6–7):1668–79.] by comparison to a numerical model. They observe that the analytical solution is met by the numerical model in the region of high biomass concentrations, but not at the plume fringe. From this, they conclude that our criterion View the MathML sourceω=cAtotKA+cAtotcBtotKB+cBtotμmaxkdec>1, with variables defined in the article, is not a sufficient condition to guarantee the existence of biomass at steady state. Instead, they define a new criterion, View the MathML sourceω∗=cAKA+cAcBKB+cBμmaxkdec, which is unity where biomass exists at steady state, and smaller than unity elsewhere. In this response, we show that the critique by Shao et al. (submitted for publication) is justified in that ω>1ω>1 is not sufficient to guarantee non-zero steady state biomass, and that the new criterion ω∗ω∗ does not help deriving an analytical expression. We present a correction of our analytical solution, in which the region of non-zero steady state biomass is reduced and a smooth transition of reactive-species concentrations from the region with to those without biomass is achieved. The new solution provides a unique mapping between the mixing ratio and the reactive-species concentrations throughout the entire range of the mixing ratio.