The effects of capillary force and gravity on the interfacial profile in a reservoir fracture or pore

This paper presents a mathematical model for determining the interfacial profile between two immiscible fluids in a reservoir fracture or pore with a random orientation. This model is derived from the Laplace equation of capillarity by considering the effects of the capillary force, the gravity, and the wettability of fluids on reservoir rocks. The fourth-order Runge–Kutta method is employed to obtain the numerical solution. The mathematical model together with its numerical scheme is verified by the measured interfacial profiles between a heavy oil sample and a water phase in a circular tube. The numerical results show that the equilibrium shape of the interfacial profile depends on the dimensionless Bond number, the orientation of the reservoir fracture or pore and the contact angle. In particular, it is found that the Bond number, which is defined as the ratio of the gravity to the capillary force, has a strong effect on the shape of the interfacial profile. For a sufficiently small Bond number, the interfacial profile is governed by the capillary force and the contact angle formed with reservoir rocks. In this case, the orientation of the reservoir fracture or pore has no influence on the interfacial profile. Based on the numerical results, a universal critical Bond number is obtained and further used in determining the so-called critical permeability of hydrocarbon reservoirs. The interfacial profile in a reservoir fracture or pore is dominated by the capillary force as long as the reservoir permeability is smaller than this critical permeability.