Upscaling and Prediction of Lagrangian Velocity Dynamics in Heterogeneous Porous Media

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.SCOPUS: ar.j