Microfluidic devices offer unique opportunities to directly observe multiphase flow in porous media. However, as a representation of flow in geological pore networks, conventional microfluidics face several challenges. One is whether steady simultaneous two-phase flow through a two-dimensional network is possible without fluctuating occupancy of the pore constrictions. Flow without fluctuations can occur only if the flow paths of the two phases can cross on the 2D network; this requires that wetting phase can form a bridge across the gap between grains at a pore constriction while non-wetting phase flows through the constriction. We consider the conditions under which this is possible as a function of the local capillary pressure and the geometry of the constriction. Using the Surface Evolver software, we determine conditions for stable interfaces in constricted geometries, the range of capillary pressures at which bridging can occur, and those where the wetting phase would re-invade the constriction to block the flow of the non-wetting phase (“snap-off”). If a constriction is long and either straight or uniformly curved, snap-off occurs at the same capillary pressure as bridging. For constrictions of concave shape, which we represent as constrictions between cylindrical grains, however, we find a range of capillary pressures at which bridging is stable; the range is greater the narrower the diameter of the cylinders (i.e. the more strongly concave the throat) relative to the width of the constriction. For smaller-diameter pillars, the phenomenon of “Roof” snap-off as non-wetting phase invades a downstream pore body, is predicted not to occur.