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

Effective Rheology of Two-phase Flow in Three-Dimensional Porous Media: Experiment and Simulation