Effect of Capillary Number and Its Constituents on Two-Phase Relative Permeability Curves
Authors: Fulcher R.A. , Ertekin Turgay , Stahl C.D. 
Published: December 2007

Summary
One primary goal of any enhanced recovery project is to maximize the ability of the fluids to flow through a porous medium (i.e., the reservoir). This paper discusses the effect of capillary number, a dimensionless group describing the ratio of viscous to capillary forces, on two-phase (oil-water) relative permeability curves. Specifically, a series of steady-state relative permeability measurements were carried out to determine whether the capillary number causes changes in the two-phase permeabilities or whether one of its constituents, such as flow velocity, fluid viscosity, or interfacial tension (IFT), is the controlling variable.
For the core tests, run in fired Berea sandstone, a Soltrol 170™ oil/calcium chloride (CaCl2) brine/isopropyl alcohol (IPA)/glycerin system was used. Alcohol was the IFT reducer and glycerin was the wetting-phase viscosifier.
The nonwetting-phase (oil) relative permeability showed little correlation with the capillary number. As IFT decreased below 5.50 dyne/cm [5.50 N/m], the oil permeability increased dramatically. Conversely, as the water viscosity increased, the oil demonstrated less ability to flow. For the wetting-phase (water) relative permeability, the opposite capillary number effect was shown. For both the tension decrease and the viscosity increase (i.e., a capillary number increase) the water permeability increased. However, the water increase was not as great as the increase in the oil curves with an IFT decrease. No velocity effects were noted within the range studied.
Other properties relating to relative permeabilities were also investigated. Both the residual oil saturation (ROS) and the imbibition-drainage hysteresis were found to decrease with an increase in the capillary number. The irreducible water saturation was a function of IFT tension only.
A relative permeability model was developed from the experimental data, based on fluid saturations, IFT, fluid viscosities, and the residual saturations, by using regression analysis. Both phases were modeled for both the imbibition and the drainage processes. These models demonstrated similar or better fits with experimental data of other water- and oil-wet systems, when compared with existing relative permeability models. The applicability of these regression models was tested with the aid of a two-phase reservoir simulator.

Introduction
As world oil reserves dwindle, the need to develop EOR techniques to maximize recovery is of great importance. Methods such as chemical flooding, miscible flooding, and thermal recovery involve altering the mobility and/or the IFT between the displacing the displaced fluids.
Recovery efficiency was found to be dependent on the capillary number, defined asEquation 1
The viscous forces were defined as the fluid viscosity, flow velocity, and the flow path length. Capillary forces vary with the fluid IFT and the pore geometry of the medium.1
Taber defined the capillary number in terms of the pressure drop between two points, the flow length, and the IFT.2Equation 2
He concluded that as this ratio increased to a value of 5 psi/ft/dyne/cm [0.2 kPa/m/N/m] the ROS was reduced significantly. By decreasing the IFT by using surface-active agents, or by decreasing the path length by altering the field geometry, the capillary number could be increased.
Others have shown similar results. Melrose and Brandner,3 for example, indicated that as the capillary number rose to a value of 10–4, the microscopic displacement efficiency, which accounts for the residual saturations to both oil and water, increased. The effects of the capillary number on the recovery of residual oil are given by Chatzis and Morrow4 and by other authors5 (Fig. 1).
Few studies, however, have shown the effect of the capillary number on the two-phase flow between the residuals. The variables within this group have been researched, but their combined effect on relative permeabilities has been largely ignored.
Several authors have noted that the viscosity ratio of oil and water alters the oil relative permeability but has little effect on that of water.6–8 Few or no changes by fluid flow velocity were observed, provided that no boundary effects were present during the core tests.9–11