Diffusion in random velocity fields with applications to contaminant transport in groundwater

•Diffusion in random velocity fields models contaminant transport in groundwater.•Space variable velocity fields induce transitory or persistent memory effects.•In short-range correlated fields the process loses memory and is self-averaging.•Global random walks (GRW) provide efficient approximations of the process.•The evolution of the concentration probability distribution is modeled as a GRW.

The process of diffusion in a random velocity field is the mathematical object underlying currently used stochastic models of transport in groundwater. The essential difference from the normal diffusion is given by the nontrivial correlation of the increments of the process which induces transitory or persistent dependence on initial conditions. Intimately related to these memory effects is the ergodicity issue in subsurface hydrology. These two topics are discussed here from the perspectives of Itô and Fokker–Planck complementary descriptions and of recent Monte Carlo studies. The latter used a global random walk algorithm, stable and free of numerical diffusion. Beyond Monte Carlo simulations, this algorithm and the mathematical frame of the diffusion in random fields allow efficient solutions to evolution equations for the probability density of the random concentration.