Capillary Pressure During Immiscible Displacement
Authors: Chen T. , Chen O. , Chen P. 
Published: March 2010

Abstract
Experiments performed on the immiscible displacement of heptane and mineral oil by water in capillary tubing showed that capillary pressure during drainage and imbibition were constant over a range of velocities from 10−4 to 10−2 m/s. The results obtained were quite different from the equilibrium capillary pressure. The drainage capillary pressure was lower than its equilibrium value by 25﹪ for heptane, and higher by 10﹪ for mineral oil. In the case of imbibition, the capillary pressure for both oils was lower than their respective equilibrium value. Due to the limitation of the apparatus, velocities lower than 10−4 m/s could not be measured. It is, therefore, not clear if capillary pressure during flow would, in fact, reduce to its equilibrium value at low enough velocities.
While it is quite adequate to describe the capillary behaviour of two fluids inside a tubing by a single number, one needs to consider capillary pressure as a function of the wetting fluid saturation in the porous medium. The problem of considering the effect of flow on this function is going to be more complicated than the single capillary tubing case. The present result indicates that it is inconsistent to use capillary pressure determined from equilibrium conditions and apply it to flowing conditions.

Introduction
In the study of fluid flow in porous media, one starts with the material balance equation for each of the fluids. If they are immiscible, then one needs to consider the interaction between them across the interfaces. A common approach(1–4) is to define a capillary pressure to be equal to the difference in pressure of the two fluids in the same representative elemental volume. This pressure is also assumed to be dependent on the saturation of one of thefluids there. In terms of mathematics, the use of these assumptions completes the system of partial differential equations governing the flow of two fluids in porous media. In actual practice, one needs to provide a physically realistic capillary pressure saturation relation in order to make meaningful predictive calculations.
Very often, capillary pressure is taken to be zero, as in the Buckley-Leverett solution for immiscible displacement(5), or Muskat's model of water coning(6). However, in some reservoir engineering applications, this may not be feasible. For example, when one needs to build a numerical model with more than one layer in the vertical direction, it is necessary to input a non-zero capillary pressure curve to initialize the fluids in the reservoir. A realistic initialization would be to use the drainage portion of the capillary pressure saturation relation. In most commercially available simulators, the same curve is being used for the subsequent displacement calculations. As the drainage curve is determined while the fluids are at equilibrium, using it for flow calculation implies that capillary pressure is independent of the dynamics. We are not aware of any theoretical justification given in the literature for using this assumption. In fact, there was no discussion given on the physical requirements of the capillary pressure saturation relation given in some of the standard texts on numerical reservoir simulation(7, 8).