An accurate description of the mechanics of pore level displacement of immiscible fluids could significantly improve the predictions from pore network models of capillary pressure-saturation curves, interfacial areas and relative permeability in real porous media. If we assume quasi-static displacement, at constant pressure and surface tension, pore scale interfaces are modeled as constant mean curvature surfaces, which are not easy to calculate. Moreover, the extremely irregular geometry of natural porous media makes it difficult to evaluate surface curvature values and corresponding geometric configurations of two fluids. Finally, accounting for the topological changes of the interface, such as splitting or merging, is nontrivial. We apply the level set method for tracking and propagating interfaces in order to robustly handle topological changes and to obtain geometrically correct interfaces. We describe a simple but robust model for determining critical curvatures for throat drainage and pore imbibition. The model is set up for quasi-static displacements but it nevertheless captures both reversible and irreversible behavior (Haines jump, pore body imbibition). The pore scale grain boundary conditions are extracted from model porous media and from imaged geometries in real rocks. The method gives quantitative agreement with measurements and with other theories and computational approaches. (c) 2006 Elsevier Inc. All rights reserved.