Estimation of Capillary Pressure and Relative Permeability Functions From Centrifuge Experiments

Abstract

Summary A method to estimate capillary pressure and relative permeability curves simultaneously from transient, multirate permeability curves simultaneously from transient, multirate centrifuge experiments performed on homogeneous rock is presented. The unknown functions are represented with B-splines, and the number of knots and coefficient values are selected through a regression-based procedure. The method is demonstrated by use of experimental data for dolomite and sandstone samples with air/water and oil/water fluid systems. An analysis of the accuracy of the estimates is presented. The capillary pressure curve and regions of both relative permeability curves can be estimated accurately. Introduction In recent years, methods have been proposed to estimate relative permeability curves from centrifuge experiments. This type of permeability curves from centrifuge experiments. This type of experiment offers advantages over the more frequently used unsteady-state displacement experiments. Displacement experiments in the centrifuge are stable, so this likely would be the preferred method if unstable displacements were encountered in an unsteady state experiment. Information about properties can be obtained at very low values of the wetting-phase saturation. In addition, centrifuge experiments are believed to provide the most realistic estimates of relative permeability functions for a recovery process dominated by gravity forces. An increased interest in the centrifuge experimental setup has resulted from new measurement techniques, such as collection of nonequilibrium production data, local saturation determination during production data, local saturation determination during centrifuging, and new methods for estimating relative permeability and capillary pressure functions from measured data. Traditionally, centrifuge data consist of the mean wetting-phase saturation at equilibrium as a function of successively higher constant angular velocities. These data are used to estimate the capillary pressure curve. Hassler and Brunner reported the use of the centrifuge to calculate the capillary pressure curve. They used an explicit method in which derivative estimates were used to compute point wise capillary pressure values. Other solutions of the explicit problem have been proposed. Several authors use implicit methods for estimating the capillary pressure curve. In this approach, a functional representation is pressure curve. In this approach, a functional representation is selected for S (P), or its inverse, and coefficients in the representation are estimated by minimizing the sum of squared differences between measured and simulated data. Hagoort used nonequilibrium data (i.e., data from the transient part of the production profile) to estimate the wetting-phase relative permeability curve. He measured the average wetting-phase saturation as a function of time in a single-rate centrifuge experiment. To calculate the wetting-phase relative permeability, the mathematical model for the centrifuge permeability, the mathematical model for the centrifuge displacement process was simplified by neglecting capillary pressure and assuming that the mobility of the nonwetting phase was pressure and assuming that the mobility of the nonwetting phase was infinite compared with that of the wetting phase. An explicit procedure that requires differentiation of measured data was used procedure that requires differentiation of measured data was used to calculate relative permeability values. It is well-known that measurement errors are amplified in the differentiation process; the magnitude of (he resulting errors in the relative permeability estimates have been quantified for unsteady-state displacement experiments. Guo and Nordtvedt provide more detailed discussions of the centrifuge experimental setup and data reduction methods. The errors resulting from simplification of the mathematical model and differentiation of data are avoided with an implicit formulation. Several such centrifuge studies are reported. In these studies, however, only simple, Corey-type functional relationships are used to represent the relative permeability curves. In unsteady-state displacement experiments, for which the estimation process has been studied extensively, such representations have been found to be unsuitable and result in severe (bias) errors in estimating unknown functions. Such biaserrors are eliminated through the use of a regression-based method so that, in principle, maximum likelihood estimates are obtained. B-spline representations for the unknown relative permeability and capillary pressure functions are used, and the numbers and locations of knots are chosen to reduce the weighted sum of squared differences between measured and calculated quantities. In this work, we use the regression-based method to estimate capillary pressure and relative permeability functions simultaneously from transient data gathered from centrifuge experiments. This method provides essentially bias-free estimates of the curves from centrifuge experiments. Measures of the accuracy of estimates of these functions from centrifuge experiments also are presented. Such measures can be used to evaluate the effectiveness of centrifuge experiments in determining both the capillary pressure and the relative permeability curves, as well as for the purpose of experimental design (e.g., selection of desirable rotational speeds), Background This section briefly reviews the procedures for conducting the centrifug eexperiment and the regression-based estimation approach. Centrifuge Experiment. Fig. 1 is a schematic of the centrifuge experimental setup for a drainage experiment with a denser wetting than nonwetting phase. The core sample is initially filled with the wetting phase, then spun at one or several consecutively higher constant angular velocities. The effluent as a function of time is measured as detailed in Ref. 2. The surrounding nonwetting phase enters the core at the inner face and displaces the wetting phase. which exits at the outer end face. The side walls are sealed to justify the ID assumption made in the modeling of the centrifuge displacement process. In this work, primary-drainage experiments are performed on water-wet porous samples with both air/water and oil/water fluid systems as shown in Fig. 1. The data were collected by Shell Development Co., with an automated centrifuges Fluid properties, porosity, and absolute permeability were obtained from independent porosity, and absolute permeability were obtained from independent measurements.

Impact of Capillary and Bond Numbers on Relative Permeability

“…[3][4][5][6][7][8] O'Meara & Crump 4 developed a semi-implicit method for estimating wetting phase relative permeability and capillary pressure simultaneously from a single experiment in a gas-oil system. Estimation of the oil (wetting phase) relative permeability and capillary pressure was achieved by history matching the production data.…”

“…Nordtvedt, J.E. et al 7 used production data to estimate wetting and non-wetting phase relative permeabilities in addition to capillary pressure.…”

“…[5][6][7][8][9][10][11][12] This is an inverse problem where the objective is to minimize the sum of the least square difference between measured data and simulated data from a mathematical model.…”

Two‐phase flow in porous media: Property identification and model validation

“…They developed a generalized, regression-based method for estimating the two-phase flow functions (relative permeabilities and capillary pressure) that is directed to obtaining the most accurate estimates of the unknown functions from the available data, and is not limited by the form of the mathematical model for the experiments, the experimental design, or by which unknown functions are to be determined. Its use has been explored with a wide variety of experimental scenarios (Richmond and Watson, 1990;Nordtvedt et al, 1993Nordtvedt et al, ,1994Mejia et al, 1995) and it has been extended to the estimation of three-phase flow functions (Mejia et al, 1996;Nordtvedt et al, 1996). This methodology is summarized in the following subsections.…”

Evaluation of methods for measuring relative permeability of anhydride from the Salado Formation: Sensitivity analysis and data reduction

“…Firoozabadi and AziZ (1991) matched production behavior using nonlinear regression on the parameters of relationships similar to those of Equations 16, 17, and 18 that were incorporated into a numerical model of the centrifuge process. Nordtvedt et al (1993) used the Marquardt-Levenberg algorithm to adjust B-spline representations of relative permeability and capillary pressure.…”