Impact of velocity correlations on longitudinal dispersion in space-Lagrangian advective transport models

Space-Lagrangian random walk models conceptualize advective transport in terms of collections of particles undergoing fixed-length steps along flow streamlines. The statistics and correlation structure of the underlying flow velocity statistics determine the tran-sit times of particles undergoing advective transport. Broad velocity distributions lead to broadly distributed step transit times, reproducing commonly observed anomalous transport features such as superdiffusive plume growth, which are not captured by clas-sical Fickian theories. Early space-Lagrangian models considered uncorrelated velocities across steps. These approaches were later extended to account for correlations through a spatial-Markov process. Here, we compare longitudinal dispersion dynamics in an uncorrelated continuous-time random walk with fixed-space steps to a Bernoulli relaxation spatial-Markov model exhibiting exponential decay of spatial velocity correlations along streamlines. We provide rigorous theoretical derivations, validated against numerical sim-ulations. We find that, although the scaling forms of asymptotic dispersion agree between the two models, exact asymptotic equivalence requires employing different correlation lengths, which depend on the underlying Eulerian velocity statistics. The two models become equivalent in the limit of very broad velocity distributions, corresponding to a new quasiballistic regime recently identified in unsaturated porous media, which we rigorously characterize here.