  Mathematical models based on probability density functions (PDF) have been
extensively used in hydrology and subsurface flow problems, to describe the
uncertainty in porous media properties (e.g., permeability modelled as random
field). Recently, closer to the spirit of PDF models for turbulent flows, some
approaches have used this statistical viewpoint also in pore-scale transport
processes (fully resolved porous media models). When a concentration field is
transported, by advection and diffusion, in a heterogeneous media, in fact,
spatial PDFs can be defined to characterise local fluctuations and improve or
better understand the closures performed by classical upscaling methods. In the
study of hydrodynamical dispersion, for example, PDE-based PDF approach can
replace expensive and noisy Lagrangian simulations (e.g. trajectories of
drift-diffusion stochastic processes). In this work we derive a joint
position-velocity Fokker-Planck equation to model the motion of particles
undergoing advection and diffusion in in deterministic or stochastic
heterogeneous velocity fields. After appropriate closure assumptions, this
description can help deriving rigorously stochastic models for the statistics
of Lagrangian velocities. This is very important to be able to characterise the
dispersion properties and can, for example, inform velocity evolution processes
in Continuous Time Random Walk (CTRW) dispersion models. The closure problem
that arises when averaging the Fokker Planck equation shows also interesting
similarities with the mixing problem and can be used to propose alternative
closures for anomalous dispersion.
