  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.

Fluid Dynamics