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

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.