The diffusing-velocity random walk: Capturing the interplay of diffusion and heterogeneous advection within a spatial-Markov framework

The spatial distribution of a solute undergoing advection and diffusion is impacted by the velocity variability sampled by tracer particles. In spatially structured velocity fields, such as porous medium flows, Lagrangian velocities along streamlines are often characterized by a well-defined correlation length and can thus be described by spatial-Markov processes. Diffusion, on the other hand, is generally modeled as a temporal process, making it challenging to capture advective and diffusive dynamics in a single framework. In order to address this limitation, we have developed a description of transport based on a spatial-Markov velocity process along Lagrangian particle trajectories, incorporating the effect of diffusion as a local averaging process in velocity space. The impact 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 with characteristic longitudinal and transverse lengthscales associated with the flow field. For infinite longitudinal correlation length, our framework recovers Taylor dispersion, and in the absence of diffusion it reduces to a standard spatial-Markov velocity model. This novel framework allows us to derive dynamical equations governing the evolution of particle position and velocity, from which we obtain scaling laws for the dependence of longitudinal dispersion on Péclet number. Our results provide new insights into the role of shear and diffusion on dispersion processes in heterogeneous media.

In this presentation, I propose to discuss: (i) Spatial-Markov models and the modeling of diffusion as a spatial rather than temporal process; (ii) The concept of the effective shear rate and its role in the diffusive dynamics of tracer particle velocities; (iii) The role of transverse diffusion and its interplay with velocity heterogeneity on longitudinal solute dispersion.