  We present an experimental and numerical study of immiscible two-phase flow
in 3-dimensional (3D) porous media to find the relationship between the
volumetric flow rate ($Q$) and the total pressure difference ($\Delta P$) in
the steady state. We show that in the regime where capillary forces compete
with the viscous forces, the distribution of capillary barriers at the
interfaces effectively creates a yield threshold, making the fluids reminiscent
of a Bingham viscoplastic fluid in the porous medium, introducing a threshold
pressure $P_t$. In this regime, $Q$ depends quadratically on an excess pressure
drop ($\Delta P-P_t$). While increasing the flow-rate, there is a transition,
beyond which the flow is Newtonian and the relationship is linear. In our
experiments, we build a model porous medium using a column of glass beads
transporting two fluids -- de-ionized water and air. For the numerical study,
reconstructed 3D pore-networks from real core samples are considered and the
transport of wetting and non-wetting fluids through the network are modeled by
tracking the fluid interfaces with time. We find agreement between our
numerical and experimental results. Our results match the mean-field results
reported earlier.
