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 the tubes vary along the length which introduce capillary threshold pressures. We demonstrate through analytical calculations that a transition from a linear Darcy to a non-linear behavior occurs while decreasing the pressure drop ΔP, where the total flow rate 〈Q〉 varies with ΔP with an exponent 2 as 〈Q〉~ΔP2 for uniform threshold distribution. The exponent changes when a lower cut-off Pm is introduced in the threshold distribution and in the limit where ΔP approaches Pm, the flow rate scales as 〈Q〉 ~ (|ΔP|-Pm)3/2. While considering threshold distribution with a power α, we find that the exponent γ for the non-linear regime vary as γ = α + 1 for Pm = 0 and γ = α + 1/2 for Pm > 0. We provide numerical results in support of our analytical findings.