We study the upscaling and prediction of 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. 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 anti-correlated 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 variance.