Properties of exact and approximate traveling wave solutions for transport with nonlinear and nonequilibrium sorption

Abstract

Nonlinear sorption leads to the existence of moving concentration fronts of substances transported in porous media that do not change shape. The cause for the existence of such traveling wave fronts is a balance between the self-sharpening effect of nonlinear sorption and the spreading effects of dispersion and sorption kinetics. While analytical solutions for the transport of nonlinearly sorbing substances are usually not available, the asymptotic front speed and the shape of traveling waves can in some situations be determined analytically. This is done by deriving a traveling wave equation in a moving coordinate system. In the case of simultaneous linear equilibrium and nonlinear nonequilibrium sorption, this traveling wave equation is a second-order differential equation. Because this equation cannot be solved analytically for typical nonlinear sorption isotherms, the second-order term is usually neglected. In this paper the significance of the second-order term is discussed for a simple piecewise linear sorption isotherm that allows the analytical solution of both the full and the simplified differential equations. These analytical solutions make it possible to calculate the approximation error as a function of flow and isotherm parameters and thus to localize domains where the error may be significant. In addition, the analytical solutions are used to discuss the identifiability of model parameters from asymptotic front shape data.