A mobile–immobile model with an asymptotic scale-dependent dispersion function

This study combined the mobile–immobile model (MIM) with an asymptotic dispersivity function of travel distance to embrace the concept of scale-dependent dispersion during solute transport in finite heterogeneous porous media. The proposed MIM with an asymptotic scale-dependent dispersivity (MIMA) was analytically solved by the Laplace transform technique and the extended power series method. The developed semi-analytical solution of MIMA was compared with the Laplace transformed finite-difference numerical solution, and they agreed well with each other. The applicability of MIMA was tested with concentration data from a 1250-cm long and highly heterogeneous soil column. The transport process was also simulated by the MIM with a constant dispersivity (MIMC) for comparison. The simulation results of MIMC were found to depart significantly from the measured breakthrough curves (BTCs) in the column, whereas the fitting of MIMA with measured BTCs improved substantially as it considered that the increase of dispersivity with distance was limited. Our results indicated that MIMA was efficient to capture the evolution of scale-dependent dispersion behavior, and it was more useful for describing anomalous solute transport in heterogeneous porous media. We also generally evaluated the advantages and limitations of MIMA with respect to other up-scaling models for describing solute transport in heterogeneous media, and indicated several issues deserving further investigation for MIMA.