Stochastic Dynamics of Two‐Dimensional Particle Motion in Darcy‐Scale Heterogeneous Porous Media

We study the upscaling and prediction of ensemble dispersion in two-dimensional heterogeneous porous media with focus on transverse dispersion. To this end, we study the stochastic dynamics of the motion of advective particles that move along the streamlines of the heterogeneous flow field. While longitudinal dispersion may evolve super-linearly with time, transverse dispersion is characterized by ultraslow diffusion, that is, the transverse displacement variance grows asymptotically with the logarithm of time. This remarkable behavior is linked to the solenoidal character of the flow field, which needs to be accounted for in stochastic models for the two-dimensional particle motion. Here, we derive an upscaled model based on the statistical characterization of the motion of solute particles. To this end, we analyze particle velocities and orientations through equidistant sampling along the particle trajectories obtained from direct numerical simulations. This sampling strategy respects the flow structure, which is organized on a characteristic length scale. Perturbation theory shows that the longitudinal particle motion is determined by the variability of travel times, while the transverse motion is governed by the fluctuations of the space increments. The latter turns out to be strongly anticorrelated with a correlation structure that leads to ultraslow diffusion. Based on this analysis, we derive a stochastic model that combines a correlated Gaussian noise for the transverse motion with a spatial Markov model for the particle speeds. The model results are contrasted with detailed numerical simulations in two-dimensional heterogeneous porous media of different heterogeneity variances.