Relative Permeability From Unsteady-State Displacements With Capillary Pressure Included

Abstract

Summary A semianalytic method for calculation of relative permeabilities from unsteady-state displacement data is developed. The method is based on the simultaneous solution of the fractional-flow equation and an integro-differential equation, derived from the material-balance and Darcy equations for immiscible displacement of two incompressible phases, including the capillary pressure by use of various volumetric relationships. This method for calculating two-phase relative permeabilities from unsteady-state displacement data is not restricted to the high-flow-rate experimental conditions used to overcome capillary end effects. Removal of this restriction allows the analysis of low-permeability cores with water and oil where the flow rates are low and capillary end effects are important. Our method allows direct calculation from the experimental data without intermediate interpretations of graphs. Introduction Relative permeability data are necessary for reservoir simulation involving multiphase flow of fluids in porous media. The relative-permeability-vs.-saturation data are usually obtained from displacement experiments with cores in the laboratory. A comprehensive summary of various experimental methods is presented by Honarpour et al. Two methods have emerged for general interpretation of laboratory-derived displacement data:the steady- state method in which both of the immiscible fluids are simultaneously injected into the core andthe unsteady-state method where one of the fluids displaces the other. Interpretation of the data from steady-state experiments is simple, but it is difficult to obtain a constant average saturation of the fluids, and a long time (many hours) is required to establish the saturation after each change. On the other hand, the unsteady-state method can be carried out in a relatively short time, but the interpretation of the data is more complex. In both cases, the analysis is complicated by adverse capillary end effects at the entrance and exit of the core unless the fluid displacements are conducted at high flow rates to diminish the end effects. High flow rates are frequently difficult and impractical to attain. For example, the requirement of high flow rates confines the measurements to oils with relatively low viscosity. High flow rates may also cause undesirable alterations in porous media, such as fines migration and other complications. Hence, low flow rates are preferred for gathering displacement data. Consequently, if the capillary-pressure-vs. saturation relationship is not included in the calculation of relative permeabilities, the capillary end effects remain as a source of error. For high-flow-rate displacements, where the capillary pressure end effects are negligible, the Johnson-Bossier-Naumann (JBN) method, or some modification of it, is available to interpret the data. Judging from the number of papers that have appeared in the literature, however, development of interpretation methods for low- rate displacement data is still under investigation. One rapid method for inclusion of the capillary end effects has been presented by Odeh and Dotson. In their method, relative permeability data are first obtained by the unsteady-state method according to Jones and Roszelle, neglecting the influence of end effects. Then a plot of the ratio of oil flow rate to relative permeability vs. the average water saturation is prepared. Odeh and Dotson postulate that in the absence of capillary end effects, the relationship will be a straight line. The deviations of data from the straight line are used to correct for the end effects. Thus, the applicability of the method depends on the existence of a straight-line region in the measured data, which is not always guaranteed. Several other investigators, such as Richmond and Watsons and Qadeer et al., have resorted to numerical solutions of partial-differential equations for unsteady-state displacement of phases to obtain relative permeability data with capillary pressure effects. Such approaches require cumbersome methods to search for the best correlations for relative permeability. In this paper, the limitations and difficulties of the above-mentioned methods, which include the capillary pressure effects, are alleviated by extension of the analytic method presented by Marle for calculation of two-phase relative permeabilities from unsteady-state displacement data to include capillary pressure. Formulation Consider one-dimensional displacement of two incompressible and immiscible phases in a homogeneous cylindrical core, Phase 1 being wetting and Phase 2 nonwetting. Material- and Momentum-Balance Equations. The Phase I and Phase 2 volumetric continuity equations are ..........................................(1) ..........................................(2) ..........................................(3) ..........................................(4) ..........................................(5) The pressures of the two phases, as a function of Phase I saturation, are related by the capillary pressure as follows: ..........................................(6) Eq. 6 is substituted into Eq. 5 to introduce the capillary pressure term, deriving Eq. 7: ..........................................(7) Volumetric Relationships. The relationship of the volumetric flux to the cumulative volumetric flow is ..........................................(8) The relationship of volumetric flow rate to volumetric flux is ..........................................(9) The fractional flow of the phases is defined by ..........................................(10) Substituting Eq. 8 into Eq. 10 gives ..........................................(11) The total volumetric flux and total cumulative flow are defined, respectively, as follows: ..........................................(12) ..........................................(13)