The diffusing-velocity random walk: Capturing the interplay ofdiffusion and heterogeneous advection within a spatial-Markovframework
Abstract
The spatial distribution of a solute undergoing advection and diffusion is impacted by the velocityvariability sampled by tracer particles. In spatially structured velocity fields, such as porousmedium flows, Lagrangian velocities along streamlines are often characterized by a well-definedcorrelation length and can thus be described by spatial-Markov processes. Diffusion, on the otherhand, is generally modeled as a temporal process, making it challenging to capture advective anddiffusive dynamics in a single framework. In order to address this limitation, we have developed adescription of transport based on a spatial-Markov velocity process along Lagrangian particletrajectories, incorporating the effect of diffusion as a local averaging process in velocity space. Theimpact of flow structure on this diffusive averaging is quantified through an effective shear rate.The latter is fully determined by the point statistics of velocity magnitudes together withcharacteristic longitudinal and transverse lengthscales associated with the flow field. For infinitelongitudinal correlation length, our framework recovers Taylor dispersion, and in the absence ofdiffusion it reduces to a standard spatial-Markov velocity model. This novel framework allows us toderive dynamical equations governing the evolution of particle position and velocity, from whichwe obtain scaling laws for the dependence of longitudinal dispersion on Péclet number. Ourresults provide new insights into the role of shear and diffusion on dispersion processes inheterogeneous media.In this presentation, I propose to discuss: (i) Spatial-Markov models and the modeling of diffusionas a spatial rather than temporal process; (ii) The concept of the effective shear rate and its role inthe diffusive dynamics of tracer particle velocities; (iii) The role of transverse diffusion and itsinterplay with velocity heterogeneity on longitudinal solute dispersion.
