Advances in Water Resources 16 (1993) 97-105 Travelling waves during the transport of reactive solute in porous media: combination of Langmuir and Freundlich isotherms C.J. van Duijn Technical University Delft, Faculty of Technical Mathematics and Informatics, PO Box 356, 2600 AJ Delft, The Netherlands P. Knabner Institut ffir Angewandte Analysis und Stochastik, Hausvogteiplatz 5-7, D 0-1086 Berlin, Germany & S.E.A.T.M. van der Zee Agricultural University Wageningen, Department of Soil Science and Plant Nutrition, PO Box 8005, 6700 EC Wageningen, The Netherlands Recently, it has been shown that in the case of nonlinear solute adsorption the displacement may be in the form of a travelling wave. In this paper, we investigate whether a travelling wave type of behaviour can be expected when two different types of sorption sites can be distinguished with different isotherms and kinetics. Illustrations are given for cases where the overall isotherm comprises two contributions that follow the Langmuir and the Freundlich equations, respectively. Boundary conditions are chosen that ensure a decrease in concentration in the direction of flow. Depending on the value of the Freundlich power (p) the travelling wave may exist. For p _< 1, the travelling wave always exists, whereas for 1 < p < 2 it depends on the values of the other adsorption parameters and whether a lower bound of the upstream concentration (at x = -o~) is exceeded. For p > 2, the existence of the travelling wave requires that the upstream concentration does not exceed an (specified) upper bound. Besides illustrating some waves we show that two different rate functions that have the Freundlich isotherm as their limit for an infinite rate parameter result in qualitatively different travelling waves. Key words: Transport, travelling wave, porous media, Langrnuir isotherm, Freundlich isotherm, non-linear adsorption. 1 INTRODUCTION Reactive solute transport is of profound interest from the scope of soil and ground water quality as well as chemical engineering. The incentive for theoretical and practical research may be, for instance, optimizing flow reactor dimensions for industrial purposes or soil and water sanitation. Evaluating the concentration distri- bution in natural soil or ground water is of interest in environmental risk assessment. Advances in Water Resources 0309-1708/93/$06.00 © 1993 Elsevier Science Publishers Ltd. 97 To describe the one-dimensional transport of non- reacting solutes through a homogeneous porous medium, use is made of the convection-dispersion equation (CDE) with constant coefficients. For reacting solutes, an accumulation rate term is added to the nonreactive CDE. Assuming local equilibrium, this term is based on a functional relationship between the accumulated solute on the solid phase (adsorbed amount) and the concentration in solution. Such a description of the sorption process is, for example, valid for a solute present at trace concentrations in a background electrolyte that does not vary as a function of position and time. One often assumes a linear relation