Simulation is a powerful tool to better understand physical systems, but generally requires computationally expensive numerical methods. Downstream applications of such simulations can become computationally infeasible if they require many forward solves, for example in the case of inverse design with many degrees of freedom. In this work, we investigate and extend neural PDE solvers as a tool to aid in scaling simulations for two-phase flow problems, and simulations of oil expulsion from a pore specifically. We extend existing numerical methods for this problem to a more complex setting involving varying geometries of the domain to generate a challenging dataset. Further, we investigate three prominent neural PDE solver methods, namely the UNet, DRN and U-FNO, and extend them for characteristics of the oil-expulsion problem: (1) spatial conditioning on the geometry; (2) periodicity in the boundary; (3) approximate mass conservation. We scale all methods and benchmark their speed-accuracy trade-off, evaluate qualitative properties, and perform an ablation study. We find that the investigated methods can accurately model the droplet dynamics with up to three orders of magnitude speed-up, that our extensions improve performance over the baselines, and that the introduced varying geometries constitute a significantly more challenging setting over the previously considered oil expulsion problem.