Oil reservoirs and soil can be homogeneously wet (water-wet, oil-wet, neutralwet) or heterogeneously wet (mixed wet or fractionally wet). The goal of this research is to model the detailed configuration of wetting and non-wetting phases within homogeneously and heterogeneously wet porous media. We use a dense random pack of equal spheres as a model porous medium. The geometry of the sphere pack is complex but it is known. In homogeneously wet porous media we quantify the effect of low saturations of the wetting phase on the non-wetting phase relative permeability by solving analytically the geometry of the wetting phase. At low saturations (at or near the drainage endpoint) the wetting phase exists largely in the form of pendular rings held at grain contacts. Pore throats correspond to the constriction between groups of three grains, each pair of which can be in contact. Thus the existence of these pendular rings decreases the void area available for the flowing non-wetting phase. Consequently, the existence of the pendular rings decreases the permeability of non-wetting phase. Our model explains the significant permeability reduction of the non-wetting phase with a small change in the wetting phase in a low permeability porous medium. To model heterogeneously wet porous medium, we assume that the porous medium is fractionally wet where each grain is either oil-wet or water-wet. These waterwet or oil-wet grains are distributed randomly within the porous medium. We calculate analytically the stable fluid configuration in individual pores and throats of a fractionally wet medium. The calculation is made tractable by idealizing the configurations as locally spherical (menisci) or toroidal (pendular rings.) Because the calculation of the interface position is entirely local and grain-based, it provides a single, generalized, geometric basis for computing pore-filling events during drainage as well as imbibition. This generality is essential for modeling displacements in fractionally wet media. Pore filling occurs when an interface becomes unstable in a pore throat (analogous to the Haines condition for drainage in a uniformly wet throat), when two or more interfaces come into contact and merge to form a single interface (analogous to the Melrose condition for imbibition in uniformly wet medium), or when a meniscus in a throat touches a nearby grain (a new stability criterion). The concept of tracking the fluid/fluid interfaces on each grain means that a traditional pore network is not used in the model. The calculation of phase saturation or other quantities that are conveniently computed in a network can be done with any approach for defining pore bodies and throats. The fluid/fluid interfaces are mapped from the grain-based model to the network as needed. Consequently, the model is robust as there is no difference in the model between drainage and imbibition, as all criteria are accounted for both increasing and decreasing capillary pressure. / text