Influence of the imposed flow rate boundary condition on the flow of Bingham fluid in porous media

TL;DR

This study investigates how imposing a flow rate boundary condition affects Bingham fluid flow in porous media, revealing boundary layer behavior and flow path statistics through pore network modeling and theoretical mapping.

Contribution

It introduces a detailed analysis of flow under imposed flow rate conditions, highlighting boundary layer formation and connecting flow path roughness to directed polymer models.

Findings

Boundary layer size scales as Q^{-rac{eta}{ u}}

Flow path density increases as Q^{eta}

Identifies a connection between flow roughness and directed polymer models

Abstract

The flow of yield stress fluids in porous media presents interesting complexity due to the interplay between the non-linear rheology and the heterogeneity of the medium. A remarkable consequence is that the number of flow paths increases with the applied pressure difference and is responsible for a non-linear Darcy law. Previous studies have focused on the protocol where the pressure difference is imposed. Here we consider instead the case of imposed flow rate, $Q$. In contrast to Newtonian fluids, the two types of boundary conditions have an important influence on the flow field. Using a two-dimensional pore network model we observe a boundary layer of merging flow paths of size $\ell(Q) \sim Q^{-\mu/\delta}$ where $\mu = 0.42 \pm 0.02$ and $\delta \simeq 0.63 \pm 0.05$. Beyond this layer the density of the flow paths is homogeneous and grows as $Q^\mu$. Using a mapping to the directed polymer model we identify $\delta$ with the roughness exponent of the polymer. We also characterize the statistics of non-flowing surfaces in terms of avalanches pulled at one end.