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

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 medium, 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 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.

The diffusing-velocity random walk: a spatial-Markov formulation of heterogeneous advection and diffusion

“…2016; Puyguiraud et al. 2019 a ; Icardi & Dentz 2020). Under purely advective transport, particles experience this variability as they move along streamlines.…”

“…As a scalar tracer is transported through a heterogeneous medium, the statistics of velocity sampled by the tracer plume evolve in space and time as the underlying heterogeneity is sampled by moving tracer particles (Dentz et al 2016;Puyguiraud et al 2019a;Icardi & Dentz 2020). Under purely advective transport, particles experience this variability as they move along streamlines.…”

Population Balance Models for Particulate Flows in Porous Media: Breakage and Shear-Induced Events

“…In the presence of complex geometries and porous materials, the detailed resolution of the flow structures, while in principle possible (Marcato et al 2021), it is often impractical and upscaled porous media models need to be introduced. While the derivation and calibration of single-and multi-phase flow and transport models have been widely studied in the porous media and fluid dynamics community Boccardo et al (2017); Icardi and Dentz (2020), there is currently no general population balance theory at the Darcy/continuum scale. Krehel et al (2015) studied the upscaling of a discrete number of particle sizes and derived upscaled equations in the assumption of slow aggregation kinetics.…”

Dilution of Reactive Plumes: Evolution of Concentration Statistics Under Diffusion and Nonlinear Reaction

“…These methods were originally developed in the context of turbulent flows, where they have been extensively employed to model both conservative transport [12][13][14][15][16][17] and reactive transport with an emphasis on combustion [18][19][20][21][22][23]. More recently, these methods have gained popularity in the context of porous media to study conservative [2,[24][25][26][27][28][29][30][31][32][33] and reactive [34][35][36][37][38][39][40] mixing. The statistics of concentrations have been studied in relation to different types of fluctuations.…”