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

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.