Transport through the array of vortices: from microscopic model to macroscopic transport with immobilization

Abstract

The paper is devoted to the study of longitudinal solute transport through an array of vortices, bounded by rigid walls. Since the fluid velocity in such system is very heterogeneous, the transfer of solute particles has non-trivial properties. Some particles can flow into the vortex driving by diffusion. These particles do not move in a longitudinal direction. As a result, the observed transport process is similar to the transport with immobilisation. It allows dividing the full solute concentration to mobile and immobile by analogy to the MIM approach. Mobile solute transports with the mean flow in a longitudinal direction. The immobile solute is immobilized by the vortices. The solute transition between these two phases is provided by the diffusion and determines by the concentrations in both phases. The MIM approach is used very often for modelling the transport in porous media. Usually, the particle immobilisation in porous media is explained by the interaction of solute particles with a solid matrix of porous media. However, the flow through porous media is complex and vortices are formed by the interaction of flow with the solid matrix of media. We model the transport of initially heterogeneous distributed solute through the channel by the flow with vortices. The modelling is performed into the terms of special flow by the microscopic methods. The distribution of passage time is compared for the same distribution obtained by standard linear macroscopic MIM model. The comparison is performed by the solution of the inverse problem with minimization the difference between these two distributions. As a result, the parameters of the linear MIM model is defined and its dependence on the vortices structure and the molecular diffusivity is obtained.