# Effective rheology of two-phase flow in a capillary fiber bundle model

## TL;DR

This paper analyzes how two immiscible fluids flow through a bundle of capillary tubes with varying diameters, revealing a transition from linear to non-linear flow regimes and deriving scaling laws for flow rate near critical thresholds.

## Contribution

The study provides an analytical and numerical investigation of the transition from linear to non-linear flow in a capillary bundle model with random threshold distributions, including new scaling laws.

## Key findings

- Flow rate transitions from linear to non-linear with an exponent of 2.
- Introduction of a lower cutoff in threshold distribution alters the non-linear exponent.
- Flow rate scales as (|ΔP|-Pm)^{3/2} near the threshold Pm.

## Abstract

We investigate the effective rheology of two-phase flow in a bundle of parallel capillary tubes carrying two immiscible fluids under an external pressure drop. The diameter of each tube varies along its length and the corresponding capillary threshold pressures are considered to be distributed randomly according to a uniform probability distribution. We demonstrate through analytical calculations that a transition from a linear Darcy regime to a non-linear behavior occurs while decreasing the pressure drop $\Delta P$, where the total flow rate $\langle Q \rangle$ varies with $\Delta P$ with an exponent $2$. This exponent for the non-linear regime changes when a lower cut-off $P_m$ is introduced in the threshold distribution. We demonstrate analytically that, in the limit where $\Delta P$ approaches $P_m$, the flow rate scales as $\langle Q \rangle \sim (|\Delta P|-P_m)^{3/2}$. We have also provided some numerical results in support to our analytical findings.