Spatial Markov processes for modeling Lagrangian particle dynamics in heterogeneous porous media

We investigate the representation of Lagrangian velocities in heterogeneous porous media as Markov processes. We use numerical simulations to show that classical descriptions of particle velocities using Markov processes in time fail because low velocities are much more strongly correlated in time than high velocities. We demonstrate that Lagrangian velocities describe a Markov process at fixed distances along the particle trajectories (i.e., a spatial Markov process). This remarkable property has significant implications for modeling effective transport in heterogeneous velocity fields: (i) the spatial Lagrangian velocity transition densities are sufficient to fully characterize these complex velocity field organizations, (ii) classical effective transport descriptions that rely on Markov processes in time for the particle velocities are not suited for describing transport in heterogeneous porous media, and (iii) an alternative effective transport description derives from the Markovian nature of the spatial velocity transitions. It expresses particle movements as a random walk in space time characterized by a correlated random temporal increment and thus generalizes the continuous time random walk model to transport in correlated velocity fields.