Pore-Level Network Modeling of Three-Phase Capillary Pressure and Relative Permeability Curves
Authors: Mani V. , Mohanty K.K. 
Published: December 2007

Abstract
A three-dimensional pore-level network model is developed to calculate three-phase capillary pressure and relative permeability curves. The model combines a description of pore-space morphological features and three-phase displacement physics to model capillarity-controlled gas invasion into a water-wet medium containing oil and water. The gas-water capillary pressure curves in three-phase systems depend on the fluid saturations, the spreading coefficient, and the oil-water capillary pressure. Water relative permeability for the strongly water-wet system modeled, is a function of its saturation alone. Gas and oil relative permeability curves depend on the gas and oil saturation, respectively, the saturation history and the spreading coefficient. The displacement of oil as a response to changing the capillary pressure in a three-phase system cannot be described by existing models for multiphase flow. A modification of Darcy's law is proposed to describe capillary pressure induced oil mobilization in three-phase systems.

Introduction
Simultaneous flow of gas, oil and water in porous media is prevalent in many applications such as oil recovery and subsurface contaminant transport. Numerous experiments show that higher oil recovery is possible following gravity drainage or gas-injection of reservoirs than by water-flooding alone. However, little success has been achieved in the mechanistic modeling of oil mobilization and the rate of oil recovery in three-phase systems at the pore scale or the continuum scale.
Unlike three-phase flow models, continuum-scale models (usually based on Darcy's law) have had much success in describing two-phase flow in porous media. Darcy's law defines the superficial velocity of each phase ui, as a function of the phase pressure gradient Pi, viscosity i, relative permeability Kri, and the medium absolute permeability K as follows:
(1)
The pressures in the two phases are related by the capillary pressure which is the difference between the pressure in the non-wetting phase and the wetting phase. Therefore to model multiphase flow, in addition to system parameters such as the medium permeability and the phase viscosities, the capillary pressure and relative permeability values are required.
To develop better mathematical models of three-phase systems, identification of the pore-level mechanisms involved has been the focus of many studies. The amount of oil that can be recovered following gravity drainage or gas injection is shown to be closely related to the presence of continuous oil layers in the system, which in turn depends on the capillary pressure and the spreading coefficient. The spreading coefficient, S, is defined as
(2)
where, , and are the gas-water, gas-oil and oil-water interfacial tensions, respectively.
Three pore-level mechanisms have been identified in three-phase gas invasion processes by etched-glass micromodel experiments in water-wet media:direct water drainage,direct oil drainage, anddouble drainage.
Direct water drainage results when gas displaces water and the displaced water is recovered at the outlet. Direct oil drainage occurs in a similar fashion when oil is directly displaced by gas out of the medium. In the double drainage mechanism, gas displaces oil, which in turn displaces water. It is possible to relate the gas-water capillary pressure required for the occurrence of each of the three displacement mechanisms to the fluid-fluid interfacial tensions, local pore geometry and the oil-water capillary pressure.
A number of pore-level models have been proposed based on the displacement mechanisms of three-phase flow. Capillarity-controlled gas invasion is typically modeled by increasing the gas-water capillary pressure in small steps. P. 405^