Multiphase flows in porous media are encountered in several contexts-e.g., hydrocarbon recovery operations, battery electrodes, microfluidic devices, etc. Capillary-dominated flows are interesting due to the complex interplay of interfacial properties and pore geometries. Conventional hydrodynamic flow solvers are computationally inefficient in the capillary-dominated regime, particularly in complex pore structures. The algorithm developed here specifically targets this regime to reduce simulation times. We minimise the fluid-fluid and fluid-solid interaction energies through an approach inspired by the ferromagnetic Ising model. We validate the algorithm on (1) model pore geometries with analytical solutions for capillary action, and (2) rocks with available mercury porosimetry data. We validate its predictions for model geometries and sandstones using (1) curvatures calculated from theories developed by Mayer-Stowe-Princen, Ma and Morrow, and Mason and Morrow; (2) predictions from GeoDict, a commercial software package, which also includes a state-of-the-art drainage simulator; (3) mercury porosimetry data. Drainage capillary pressure curves predicted for Bentheimer and Fontainebleau rocks reasonably match porosimetry data.