Macroscopic modeling of two-phase flow in porous media is based on the use of appropriate constitutive equations, the most important of which correlates the capillary pressure with the saturation. Several semi-empirical relationships of this type have been proposed in the literature, which include adjustable parameters. In the present work the dependence of capillary pressure on saturation is calculated as a function of the pertinent parameters (capillary number, viscosity ratio, dynamic contact angles, pore structure) using an advanced pore network simulator. The advantage of this approach is the correlation of the mesoscopic properties of the process, such as capillary pressure and saturation, with the main flow mechanisms at pore scale. The calculations involve the actual pore mechanisms and do not use adjustable parameters. The pore medium is modeled as a three-dimensional network of randomly sized unit cells of the constricted-tube type. Solving the problem of forced imbibition and drainage, the time evolution of the capillary pressure is calculated as a function of the saturation and the system parameters.
The capillary number, the viscosity ratio and the wettability affect the capillary pressure-saturation relation strongly and their effects should not be neglected. We find that the shape of the capillary pressure-saturation curve depends strongly on the step increase of the capillary number. The capillary pressure is an increasing function of the viscosity ratio at the initial stages of displacement and a decreasing function of the viscosity ratio at the final stages. The effects of wettability are quite complex and are explained in the full text. The results are correlated with the flow mechanisms at the pore scale (mechanisms of oil disconnection, action of wetting films flowing through pore wall microroughness, distribution of oil in the porous medium, etc.).
Multiphase flows in porous media are complex processes encountered in many applications of practical interest, such as oil recovery from underground reservoirs, soil and aquifer remediation from organic contaminants and risk assessment, imbibition and drainage processes, hydrology, etc. Macroscopic phenomenological mathematical modeling and simulation of multiphase flows in porous media is typically carried out using constitutive and conservation relations based on the macroscopic representation of porous media. In the macroscopic representation all properties correspond to local volume averages and are usually continuous functions of position and time. The use of the macroscopic representation makes it necessary to neglect the detailed microscopic geometry of the porous medium (information which is usually unavailable or difficult to obtained) and flow or transport phenomena taking place at the pore scale. As a result, these equations employ phenomenological parameters that can only be obtained by data fitting methods.