Graphical Techniques for Determining Relative Permeability From Displacement Experiments
Authors: Jones S.C. , Roszelle W.O. 
Published: December 2007

This paper presents graphical constructions that simplify the calculation of relative permeability from displacement data. These constructions convert raw data to relative permeability in a less tedious, more accurate manner than the usual computations. Fractional-flow saturation curves derived from waterflood displacements are always concave downward and never yield multiple-value saturations.

Introduction
To find oil and water relative permeabilities by the displacement or unsteady-state method, a small linear core usually is saturated with water, then oilflooded to irreducible water saturation. Subsequently, the core is waterflooded, and during the process, pressure drop (either constant or variable) across the entire core and water injection rate (constant or variable) are determined. Effluent fractions are collected and the amount of water and oil in each is measured. Augmented by the absolute permeability and pore volume of the core and by oil and permeability and pore volume of the core and by oil and water viscosities, these data are sufficient to develop relative permeability curves.
The average saturation in the core at any time in the flood can be found from an over-all material balance. However, to calculate relative permeability, the saturation history at some point in the core must be determined, not the average saturation history. The Welge equation yields saturations at the effluent end of the core when the average saturation history is known.
Similarly, to compute relative permeability, the point pressure gradient per unit injection rate is needed, not the pressure gradient per unit injection rate is needed, not the average. The equation developed by Johnson et al. converts average relative injectivity to a point value, accomplishing the required task.
While the equations of Welge and Johnson et al. have been used successfully for years, they require tedious computation and are subject to error because of the evaluation of derivatives. The graphical techniques presented in this study are equivalent to these equations, but are easier to use and can give a more accurate evaluation of relative permeability.
Lefebvre du Prey has presented graphical constructions based on curves of volume of oil produced vs time and pressure drop vs time to develop the required point functions. These constructions are limited to constant rate displacements. The constructions presented here are general and apply to constant rate, constant pressure, or variable rate-pressure displacements. Constant-rate and constant-pressure examples are given to help clarify the methods.
The graphical techniques make it easy to see that double or triple saturation values, so extensively discussed in the past simply do not result from the fractional flow curve generated by a single displacement, such as a waterflood or an oilflood.

Theory
Ignoring gravity effects and capillary pressure, water and oil relative permeabilities (expressed as functions of saturation) are
(1)
(2)
To use these equations, the fractional flow of water or oil and effective viscosity, lambda-1, must be determined as functions of saturation.
JPT
P. 807