Estimation of Capillary Pressure and Relative Permeability Functions From Centrifuge Experiments
- J.E. Nordtvedt (Texas A and M U.) | Gerardo Mejia (Texas A and M U.) | Pin-Huel Yang (Texas A and M U.) | A.T. Watson (Texas A and M U.)
- Document ID
- Society of Petroleum Engineers
- SPE Reservoir Engineering
- Publication Date
- November 1993
- Document Type
- Journal Paper
- 292 - 298
- 1993. Society of Petroleum Engineers
- 4.1.5 Processing Equipment, 4.1.1 Process Simulation, 4.1.2 Separation and Treating, 5.5.8 History Matching, 1.6.9 Coring, Fishing, 7.2.2 Risk Management Systems, 5.5 Reservoir Simulation
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A method to estimate capillary pressure and relative permeability curvessimultaneously from transient, multirate permeability curves simultaneouslyfrom transient, multirate centrifuge experiments performed on homogeneous rockis presented. The unknown functions are represented with B-splines, and thenumber of knots and coefficient values are selected through a regression-basedprocedure. The method is demonstrated by use of experimental data for dolomiteand sandstone samples with air/water and oil/water fluid systems. An analysisof the accuracy of the estimates is presented. The capillary pressure curve andregions of both relative permeability curves can be estimated accurately.
In recent years, methods have been proposed to estimate relativepermeability curves from centrifuge experiments. This type of permeabilitycurves from centrifuge experiments. This type of experiment offers advantagesover the more frequently used unsteady-state displacement experiments.Displacement experiments in the centrifuge are stable, so this likely would bethe preferred method if unstable displacements were encountered in anunsteadystate experiment. Information about properties can be obtained at verylow values of the wetting-phase saturation. In addition, centrifuge experimentsare believed to provide the most realistic estimates of relative permeabilityfunctions for a recovery process dominated by gravity forces. An increasedinterest in the centrifuge experimental setup has resulted from new measurementtechniques, such as collection of nonequilibrium production data, localsaturation determination during production data, local saturation determinationduring centrifuging, and new methods for estimating relative permeability andcapillary pressure functions from measured data. Traditionally, centrifuge dataconsist of the mean wetting-phase saturation at equilibrium as a function ofsuccessively higher constant angular velocities. These data are used toestimate the capillary pressure curve. Hassler and Brunner reported the use ofthe centrifuge to calculate the capillary pressure curve. They used an explicitmethod in which derivative estimates were used to compute pointwise capillarypressure values. Other solutions of the explicit problem have been proposed.Several authors use implicit methods for estimating the capillary pressurecurve. In this approach, a functional representation is pressure curve. In thisapproach, a functional representation is selected for S (P ), or its inverse,and coefficients in the representation are estimated by minimizing the sum ofsquared differences between measured and simulated data. Hagoort usednonequilibrium data (i.e., data from the transient part of the productionprofile) to estimate the wetting-phase relative permeability curve. He measuredthe average wetting-phase saturation as a function of time in a single-ratecentrifuge experiment. To calculate the wetting-phase relative permeability,the mathematical model for the centrifuge permeability, the mathematical modelfor the centrifuge displacement process was simplified by neglecting capillarypressure and assuming that the mobility of the nonwetting phase was pressureand assuming that the mobility of the nonwetting phase was infinite comparedwith that of the wetting phase. An explicit procedure that requiresdifferentiation of measured data was used procedure that requiresdifferentiation of measured data was used to calculate relative permeabilityvalues. It is well-known that measurement errors are amplified in thedifferentiation process; the magnitude of (he resulting errors in the relativepermeability estimates have been quantified for unsteady-state displacementexperiments. Guo and Nordtvedt provide more detailed discussions of thecentrifuge experimental setup and data reduction methods. The errors resultingfrom simplification of the mathematical model and differentiation of data areavoided with an implicit formulation. Several such centrifuge studies arereported. In these studies, however, only simple, Corey-type functionalrelationships are used to represent the relative permeability curves. Inunsteady-state displacement experiments, for which the estimation process hasbeen studied extensively, such representations have been found to be unsuitableand result in severe (bias) errors in estimating unknown functions. Such biaserrors are eliminated through the use of a regression-based method so that, inprinciple, maximum likelihood estimates are obtained. B-spline representationsfor the unknown relative permeability and capillary pressure functions areused, and the numbers and locations of knots are chosen to reduce the weightedsum of squared differences between measured and calculated quantities. In thiswork, we use the regression-based method to estimate capillary pressure andrelative permeability functions simultaneously from transient data gatheredfrom centrifuge experiments. This method provides essentially bias-freeestimates of the curves from centrifuge experiments. Measures of the accuracyof estimates of these functions from centrifuge experiments also are presented.Such measures can be used to evaluate the effectiveness of centrifugeexperiments in determining both the capillary pressure and the relativepermeability curves, as well as for the purpose of experimental design (e.g.,selection of desirable rotational speeds),
This section briefly reviews the procedures for conducting the centrifugeexperiment and the regression-based estimation approach.
Centrifuge Experiment. Fig. 1 is a schematic of the centrifuge experimentalsetup 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 orseveral consecutively higher constant angular velocities. The effluent as afunction of time is measured as detailed in Ref. 2. The surrounding nonwettingphase enters the core at the inner face and displaces the wetting phase. whichexits at the outer end face. The side walls are sealed to justify the IDassumption made in the modeling of the centrifuge displacement process.
In this work, primary-drainage experiments are performed on water-wet poroussamples with both air/water and oil/water fluid systems as shown in Fig. 1. Thedata were collected by Shell Development Co., with an automated centrifugesFluid properties, porosity, and absolute permeability were obtained fromindependent porosity, and absolute permeability were obtained from independentmeasurements.
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