Quasi two-dimensional approximations of interfacial curvature, present in current network models of multiphase flow in porous media, are extended to three dimensions. The new expressions for threshold capillary pressure are validated and calibrated using high-resolution direct numerical simulations on synthetic geometries. The effects of pore-space expansion and sagittal interface curvature on displacement are quantified, and are shown to be a key step in improving the physical accuracy of network models. Finally, the calibrated network model is used to obtain predictions for relative permeability and capillary pressure in a water-wet Bentheimer sandstone. The predictions are compared to experimental measurements, revealing that the inclusion of three-dimensional interfacial curvature leads to more accurate predictions.