A Simple Model for Deep Bed Filtration

TL;DR

This paper introduces a simple, mean-field model for deep bed filtration that captures the transition between different steady states and describes the evolution of trapped particles using directed percolation theory.

Contribution

The paper presents a novel simple model for deep bed filtration, linking steady states to directed percolation and analyzing particle trapping dynamics.

Findings

Two distinct steady states depending on trap fraction

Trapped particle distribution is robust to model details

Numerical solutions align with theoretical predictions

Abstract

We present a simple model for deep bed filtration, where particles suspended in a fluid are trapped while passing through a porous filter. A steady state of the model is reached when filter can not trap additional particles. We find the model has two qualitatively different steady states depending on the fraction of traps, and the steady states can be described by directed percolation. We study in detail the evolution of the distribution of trapped particles, as the number of trapped particles increases. To understand the evolution, we formulate a mean field equation for the model, whose numerical solution is consistent with the behavior of the model. We find the trapped particle distribution is insensitive to details of the formulation of the model.