Quantification of Anomalous Transport in Correlated Heterogeneous Media Using Coupled Continuous Time Random Walks

Abstract:

Large scale transport in heterogeneous porous media is in general non-Fickian. Medium heterogeneities lead to transport behaviours that cannot be explained in terms of an equivalent homogeneous medium. This anomalous character manifests itself in a non-linear asymptotic growth of the variance of particle displacements, in the tailed spatial particle density distribution and in the first passage time distribution. Evidence of anomalous transport can be found in several physical processes in which movements of particles in quenched random environment occur, such as transport of solutes in porous and fractured media, optical media, gels or freely diffusing molecules in tissue. The objectives of this work are (i) the identification of the possible causes of anomalous transport in correlated random media and (ii) their quantification in terms of the medium properties. To this end, we consider transport in a porous medium that is characterized by spatially distributed retardation properties. This model accounts for heterogeneous mass transfer between mobile and immobile phases. In particular, our purpose is to distinguish the different characters of anomalous transport arising as a consequence of the point distribution of heterogeneity and the complex geometry of the medium. We consider d = 2 dimensional media in which distribution and correlation disorder affect the transport properties. We derive the scaling laws for the mean value and variance of particle displacements. We make use of stochastic modeling to derive the effective particle motion, which describes a coupled continuous time random walk (CTRW). This CTRW is fully parameterized by the disorder distribution and the medium geometry. Using the coupled CTRW as well as direct Monte-Carlo simulations, we determine the transport behaviours for different heterogeneity scenarios and discuss their impact on non-Fickian large scale transport in terms of the spatial particle distributions and solute breakthrough curves.