Relative Permeability from Unsteady-State Displacements: An Analytical Interpretation

Abstract

ABSTRACT An analytic method for calculation of individual relative permeabilities from unsteady state displacement data was developed which includes the capillary pressure changes as a function of saturation. The method is applicable to linear displacement of two-phase fluids in laboratory cores. The displacement experiments may be conducted at low flow rates because capillary pressure is included, thus removing one of the restrictions of previous methods. The model is an integro-differential equation that relates the two-phase relative permeabilities to the pressure difference and the cumulative productions of the two phases at the outlet, which are readily measurable quantities. The solution is accomplished by replacing the integrals and derivatives with quadrature approximations. First, the method was compared to recent calculations of relative permeability in the literature which are partially accomplished by graphical methods. When capillary pressure is ignored in our model, it reproduces the literature values almost exactly (since graphical methods are involved in the literature methods, their solutions from experimental data are subject to random error). Second, we show that our model gives accurate relative permeability data when low flow rates are used in the unsteady state displacement experiments, which is not possible with other methods because the high flow rates are necessary to overcome capillary end-effects. Our method incorporates capillary pressure behavior and thus corrects for capillary end effects. This analytic method for calculating two phase relative permeabilities from unsteady state displacement data is not restricted to high flow rate experimental conditions which are used to overcome capillary end effects. Removal of this restriction allows the analysis of low permeability cores using water and oil where the flow rates are necessarily low and capillary end effects are not severe. Our method allows direct numerical solution from the experimental data without intermediate interpretations of graphs.