Analytical Solution of the Mobile-Immobile Model for Reactive Solute Transport with Scale-Dependent Dispersion

This study presents a novel mathematical model to describe reactive solute transport with scale-dependent dispersion in heterogeneous porous media within a fraction of immobile water. The model is based on the mobile-immobile model (MIM) but the dispersivity increases with the solute travel distance from its input source, considering linear equilibrium sorption and first-order degradation for continuous and instantaneous contaminant sources. Two kinds of scale-dependent dispersivity relationship are considered. One is that dispersivity increases linearly with distance without limit. Another is that dispersivity increases exponentially up to some constant limiting value. The Laplace transform technique and its numerical inversion method are applied to solve the proposed MIM with spatial variable coefficients. The breakthrough curves (BTCs) and solute concentration profiles obtained from the MIM with scale-dependent dispersion are compared to both those from the MIM with constant dispersivity and those from the convection-dispersion equation (CDE) with scale-dependent dispersion. The differences between the MIM with linear and exponential scale-dependent dispersivity are also characterized. The results indicate that scale-dependent dispersion has substantial effect on solute transport behavior which can't be described by appropriately chosen constant dispersivity. The proposed model could provide reasonable description of solute transport process in heterogeneous media where scale-dependent dispersion is of concern and be used as a suitable model for the inversion problem to obtain the true dispersion coefficients.