We develop a continuous time random walk (CTRW) approach for the evolution of Lagrangian velocities in steady heterogeneous flows based on a stochastic relaxation process for the streamwise particle velocities. 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 relation between Eulerian and Lagrangian velocities for both equidistant and isochrone sampling along streamlines, under transient and stationary conditions. Based on this, we develop a space continuous CTRW approach for the spatial and temporal dynamics of Lagrangian velocities. While classical CTRW formulations have non-stationary Lagrangian velocity statistics, the proposed approach quantifies the evolution of the Lagrangian velocity statistics under both stationary and non-stationary conditions. We provide explicit expressions for the Lagrangian velocity statistics, and determine the behaviors of the mean particle velocity, velocity covariance and particle dispersion. We find strong Lagrangian correlation and anomalous dispersion for velocity distributions which are tailed toward low velocities as well as marked differences depending on the initial conditions. The developed CTRW approach predicts the Lagrangian particle dynamics from an arbitrary initial condition based on the Eulerian velocity distribution and a characteristic correlation scale.

The understanding of the dynamics of Lagrangian velocities is key for the understanding and upscaling of solute transport in heterogeneous porous media. The prediction of large‐scale particle motion in a stochastic framework implies identifying the relation between the Lagrangian velocity statistics and the statistical characteristics of the Eulerian flow field and the hydraulic medium properties. In this paper, we approach both challenges from numerical and theoretical points of view. Direct numerical simulations of Darcy‐scale flow and particle motion give detailed information on the evolution of the statistics of particle velocities both as a function of travel time and distance along streamlines. Both statistics evolve from a given initial distribution to different steady‐state distributions, which are related to the Eulerian velocity probability density function. Furthermore, we find that Lagrangian velocities measured isochronally as a function of travel time show intermittency dominated by low velocities, which is removed when measured equidistantly as a function of travel distance. This observation gives insight into the stochastic dynamics of the particle velocity series. As the equidistant particle velocities show a regular random pattern that fluctuates on a characteristic length scale, it is represented by two stationary Markov processes, which are parametrized by the distribution of flow velocities and a correlation distance. The velocity Markov models capture the evolution of the Lagrangian velocity statistics in terms of the Eulerian flow properties and a characteristics length scale and shed light on the role of the initial conditions and flow statistics on large‐scale particle motion.