Relative-permeability and capillary-pressure functions define how much oil can be recovered and at what rate. These functions, in turn, depend critically on the geometry and topology of the pore space, on the physical characteristics of the rock grains and the fluids, and on the conditions imposed by the recovery process. Therefore, imaging and characterizing the rock samples and the fluids can add crucial insight into the mechanisms that control field-scale oil recovery. When the fundamental equations of immiscible flow in the imaged samples are solved, one can elucidate how relative-permeability and capillary-pressure functions depend on wettability, interfacial tension, and the interplay among viscous, capillary and gravitational forces. This paper summarizes the development of a complete quasistatic pore-network simulator of two-phase flow, “ANetSim,” and its validation against Statoil’s state-of-the-art proprietary simulator. Most equations presented in this paper are new; therefore, repetition of the published Statoil results is kept to a minimum. In particular, the hydraulic conductance correlations in two-phase flow and the model of cooperative pore-body filling are new. ANetSim has been implemented in MATLAB® and it can run on any platform. Three-dimensional, disordered networks with complex pore geometry have been used to calculate primary-drainage and secondary-imbibition capillary pressures and relative permeabilities. The results presented here agree well with the Statoil simulations and experiments.