Continuous time random walks for the evolution of Lagrangian particle velocities in heterogeneous porous and fractured media flows

International audienceWe develop a continuous time random walk (CTRW) approach for the evolution of Lagrangian velocities in steadyheterogeneous porous and fractured media flows based on a stochastic relaxation process.This approach describes persistence of velocities over a characteristic spatial scale, unlike classical random walk methods,which model persistence over a characteristic time scale.We first establish the relations between Eulerian and Lagrangian velocities for both equidistant and isochronal sampling along streamlines,under transient and stationary conditions. Based on this, we develop the CTRW approach for the spatial and temporaldynamics of Lagrangian velocities. Unlike classical CTRW formulations, the proposed approach quantifies both stationaryand non-stationary Lagrangian velocity statistics, and their evolution from arbitraryinitial velocity distributions. We provide explicit expressions for the Lagrangian velocity distributions,and determine the behaviors of the mean particle velocity, velocity covariance andparticle dispersion. We find strong correlation and anomalous dispersion for velocity distributions which are tailedtoward low velocities. The developed CTRW approach and thus the Lagrangian particle dynamics are fully determined by the Eulerian velocitydistribution and the characteristic correlation scale. The developed framework is applied to particle transport in two-dimensionalrandom fracture networks