# Probability density function (PDF) models for particle transport in porous media

## TL;DR

This paper develops a PDF-based mathematical framework using a joint position-velocity Fokker-Planck equation to model particle transport in heterogeneous porous media, aiming to improve understanding and modeling of dispersion phenomena.

## Contribution

It derives a novel joint position-velocity Fokker-Planck equation for particle transport, providing a rigorous basis for stochastic velocity models in porous media.

## Key findings

- Derivation of a joint position-velocity Fokker-Planck equation
- Potential to replace Lagrangian simulations with PDF models
- Insights into closure problems and anomalous dispersion

## Abstract

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.