Effective Rheology of Two-Phase Flow in a Capillary Fiber Bundle Model - Roy, Subhadeep; Hansen, Alex; Sinha, Santanu (Frontiers in Physics, 2019)
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 Δ P, where the total flow rate Q varies with Δ P with an exponent 2 as Q Δ 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 Δ P approaches P_m, the flow rate scales as Q (|Δ P|-P_m)^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 P_m=0 and γ=α+1/2 for P_m>0. We provide numerical results in support of our analytical findings.