Hydrodynamic dispersion in a self-similar geometry

Abstract

The authors investigate the dispersion of a dynamically neutral tracer flowing in a self-similar, hierarchical model of a porous medium. They consider a purely convective limit in which the time for tracer particles to traverse a given bond is strictly proportional to the inverse of the flux in the bond; the effect of molecular diffusion is neglected. Their hierarchical model contains an adjustable 'asymmetry' parameter which controls the width of the distribution of transit times for a tracer to traverse the multiplicity of paths in the network. They derive a functional recursion relation for this distribution, from which exact expressions for the moments can be obtained. They find two regimes of behaviour which are governed by the value of the asymmetry parameter. In one regime, the transit time distribution is characterised by a single timescale, so that a localised pulse of tracer spreads out at the same rate at which the pulse is convected downstream. In the second regime, the small fraction of tracer passing through the slowest bond of the network dominates in the moments of the distribution, leading to highly enhanced dispersion. The consequences of these results for the dispersion coefficient are discussed, both for self-similar and homogeneous systems.