Continuous time random walks for the evolution of Lagrangian particle velocities in heterogeneous porous and fractured media flows
Abstract
We develop a continuous time random walk (CTRW) approach for the evolution of Lagrangian velocities in steady
heterogeneous 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 temporal
dynamics of Lagrangian velocities. Unlike classical CTRW formulations, the proposed approach quantifies both stationary
and non-stationary Lagrangian velocity statistics, and their evolution from arbitrary
initial velocity distributions. We provide explicit expressions for the Lagrangian velocity distributions,
and determine the behaviors of the mean particle velocity, velocity covariance and
particle dispersion. We find strong correlation and anomalous dispersion for velocity distributions which are tailed
toward low velocities. The developed CTRW approach and thus the Lagrangian particle dynamics are fully determined by the Eulerian velocity
distribution and the characteristic correlation scale. The developed framework is applied to particle transport in two-dimensional
random fracture networks.