Estimation of Three-Phase Relative Permeability And Residual Oil Data
Authors: Stone H.L. 
Published: March 2010

Abstract
The probability model presented in an earlier paper has been revised to yield improved estimates of three phase relative permeabilities. Both the old and the new models use two sets of two-phase data ? water-oil and gas-oil – as a basis for estimating the more difficult-to measure three-phase data. The relative permeabilities predicted by the new model compare favorably with the experimental data available in the literature, including data on the dependence of waterflood residual oil saturations on trapped gas saturations.

Introduction
A PROBABILITY MODEL for the estimation of threephase relative permeability data, which was described in an earlier paper,(2) has been revised and improved. The revised model uses two sets of two-phase data water displacing oil and gas displacing oil – as the basis for making such estimates. Hysteresis is taken into consideration by employing the appropriate two-phase data, as discussed in the section titled "Data Required."
Relative permeabilities predic.ted by this model are in better agreement with experimental data than those resulting from the earlier model, especially in the region of low oil saturations. In fact, the new approach is capable of providing estimates of residual oil data in the three-phase region which are needed, for example, to determine the effect on oil recovery of an initial free gas saturation during a water-flood. In theprevious model, these residual oil data had to be provided as input to supplement the two sets of two-phase data.
Both the old and the new model have a desirable property in that they yield the correct two-phase data when only two phases are flowing and yet provide interpolated data for three-phase flow that are consistent and continuous functions of the phase saturations. As will be shown later, these interpolated values agree with the available three-phase data within experimental uncertainty.
The method can be used to estimate the relative permeability to oil in a preferentially water-wet system in which water and gas relative permeabilities depend on only the water and gas saturations, respectively. Its applicability to preferentially oil-wet and mixed-wettability systems is a matter of conjecture, because the data used to evaluate it were measured on cores or sand packs which the experimenters attempted to keep fully waterwet. However, the probabilistic basis of the model suggests that it should be useful for estimating the relative permeability of one phase if the relative permeability of the other twophases is a function only of their respective phase saturations. Thus, in a fully oil-wet system where oil relative permeability is a function of oil saturation and gas relative permeability a function of gas saturation, one would expect the probability model to be effective in predicting the water relative permeability. For systems in which mixed wettability is caused by a fraction of the pore spaces ?being strongly oil wet while the remainder are strongly water wet, a reasonable, but not proven, hypothesis is that in one saturation range equations applicable to a water-wet system can be used, whereas elsewhere the oil-wet equations will apply.