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 Delta P, where the total flow rate < Q > varies with Delta P with an exponent 2 as < Q > similar to Delta P-2 for uniform threshold distribution. The exponent changes when a lower cut-off P-m is introduced in the threshold distribution and in the limit where Delta P approaches P-m, the flow rate scales as < Q > similar to (vertical bar Delta P vertical bar - P-m)(3/2). While considering threshold distribution with a power alpha, we find that the exponent gamma for the non-linear regime vary as gamma = alpha + 1 for P-m = 0 and gamma = alpha + 1/2 for P-m > 0. We provide numerical results in support of our analytical findings.