Newtonian fluid invasion into a Yield-Stress Fluid-Saturated Medium

We investigate pressure imposed two-phase flows in porous media where a Newtonian fluid displaces an immiscible Bingham yield-stress fluid. Using a pore-network model, we identify invasion patterns governed by three dimensionless parameters: the pressure-based capillary number $Ca_P$, the viscosity ratio $M$ , and a number $R_τ$, the ratio of yield stress to capillary forces. Beyond the classical Lenormand regimes (viscous fingering, capillary fingering, stable displacement), we discover two new regimes: a tree-like pattern at low viscosity ratios and a progressively widening column-like pattern at high viscosity ratios. We characterize all regime transitions and establish a breakthrough condition for the invading fluid.