Reservoir Capillary Equilibrium: A Derived Equation
Authors: Eddy D. 
Published: March 2010

A capillary equilibrium equation has been derived for porous rock. The basis for the derivation is thermodynamic equilibrium, accounting for hydrostatic equilibrium, phase equilibrium and minimization of Gibbs free energy. The equation provides a relationship between oil saturation, pore size and height in the reservoir, relative to the oil-water contact. In the lateral direction the saturation in one pore can be related to another of a different size. Given the pore size distribution, the average saturation in a rock sample can be found at an elevation above the oil water contact. Furthermore, the equation provides insights into the laboratory imbibition and drainage pressure curves.

The equation is developed for a water wet, oil-water reservoir. A similar derivation could be carried out for different systems.

Introduction

Virgin oil saturations within a reservoir increase rapidly with height from the base of the oil column. This rapid change occurs in the transition zone, which in most reservoirs is less than 6 m thick. The transition zone is followed by constant water (and therefore constant oil) saturation with height, called the irreducible water saturation (Swirr). Capillary pressure defines the water (and oil) saturation versus height, also referred to as the initial equilibrium state. Capillary pressure is also an important element of flow, causing crossflow between beds, crossflow between fractures and the matrix, mitigating the formation of fingers and other factors affecting the nature of the flow.

Capillary pressure is not a fundamental potential of nature and as such must be the sum of other fundamental potentials. If we consider these other potentials to be pressure potentials, it would be possible to carry out a force (pressure) balance to determine the initial equilibrium state. However, the pressure balance becomes very complex, when trying to balance pressure in pores of different sizes, shapes, oil saturations and at different heights. On the other hand if we work in the realm of energy and entropy the problem becomes tractable. The total energy-entropy balance about a pore is the methodology used here to derive capillary pressure for a water-wet reservoir.

Capillary Pressure Equation

A derived capillary equilibrium equation, based on an energyentropy balance about a pore is provided in detail in Appendix A. This equation in the form of the water pressure is:

Equation 1 (available in full paper)

To understand the various terms in the equation, five elements need to be defined:Adsorption Potential, H, represents the interaction between the water and the rock. The adsorption potential arises from the unbalanced electrostatic charge density at the surface of the solid. Water's polar nature allows the dissipation of this electrostatic energy by aligning with the pore wall. The dissipating of the surface energy releases heat, called the heat of adsorption. The effect of the heat of adsorption on the water phase is to align the water molecules into a more ordered state (a decrease in entropy). The measured heat of adsorption of pure water onto a silica gel or a glass surface is approximately 0.070 N/m at room temperature(1).