A New Design of Steady-State Type Experiments for Simultaneous Estimation of Two-Phase Flow Functions
- H. Urkedal (RF-Rogaland Research) | E. Ebeltoft (RF-Rogaland Research) | J.E. Nordtvedt (RF-Rogaland Research) | A.T. Watson (Texas A&M U.)
- Document ID
- Society of Petroleum Engineers
- SPE Reservoir Evaluation & Engineering
- Publication Date
- June 2000
- Document Type
- Journal Paper
- 230 - 238
- 2000. Society of Petroleum Engineers
- 4.1.5 Processing Equipment, 5.6.9 Production Forecasting, 4.1.2 Separation and Treating, 5.3.2 Multiphase Flow, 5.2 Reservoir Fluid Dynamics, 5.5.8 History Matching, 5.3.1 Flow in Porous Media, 6.1.5 Human Resources, Competence and Training, 5.6.2 Core Analysis, 5.5.2 Core Analysis, 1.6.9 Coring, Fishing
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We have developed a new design of steady-state type experiments for simultaneous estimation of relative permeability and capillary pressure functions(collectively called multiphase flow functions) from measured pressure-drop and production data. The multiphase flow functions are represented by B-splines to ensure a flexible representation, and the coefficients in the representation are determined using a regression-based approach. The methodology is demonstrated using synthetic data. Further, we show that both relative permeability and capillary pressure functions can be estimated, and we were able to reconcile all pressure-drop and production data from the steady-state type experiment conducted. An analysis of the accuracy of the estimated functions showed that they were accurately determined over a large saturation interval.
Relative permeability and capillary pressure functions are important properties of porous media and essential for understanding multiphase flow behavior. Accurate estimates of these properties are important input for reservoir production forecasting. Since the reservoir itself is inaccessible for determination of relative permeability and capillary pressure functions, these properties are commonly determined through laboratory experiments on small core samples. Relative permeability and capillary pressure functions are then inferred from analysis of various experimental data.
The multiphase flow functions are defined through the system of equations that describes flow in porous media.1 The relative permeabilities enter into Darcy's law (or the flow equation) which relates the superficial velocities of each individual phase to the corresponding pressure gradient and viscosity [ui=-(kkri/µi)×(?Pi/?x)], i.e., the relative permeabilities are empirical properties defined by these equations. Darcy's law is generally assumed to be adequate for describing capillary dominated flow through porous media, i.e., flow for which the capillary number (ratio of viscous-to capillary forces) is relatively low.2 Because the reservoir flow generally will be capillary dominated(with the possible exception of the near-well flow), the relative permeabilities should be determined in the corresponding capillary number region. Another common assumption when interpreting core analysis data is to assume homogeneity in the core sample. Several authors3,4 have recently discussed the effects of rock heterogeneity on relative permeability and capillary pressure estimates. However, in this paper a single rock sample has been used in the synthetic cases and computerized tomography (CT)-scan images of the rock sample in the experimental part show no heterogeneities. Hence, the impact of heterogeneity on experimental data is not discussed in this paper.
Conventional methods (unsteady-state and steady-state) are generally incapable of determining relative permeabilities under capillary dominated conditions. In the most commonly used method for analyzing unsteady-state data, the Johnson, Bossler, Naumann5 method, capillary pressure is neglected altogether. This has computational advantages, since the system of equations describing flow through porous media can be solved analytically and relative permeability values explicitly calculated. However, elimination of capillary effects would require high injection rates with correspondingly large capillary numbers, possibly outside the range of interest. In this method, relative permeabilities are computed as points after breakthrough, i.e., on a very limited saturation interval. High flow rates are also required in steady-state experiments2,6to overcome capillary effects and produce uniform saturation profiles. These effects can also be overcome by placing the core sample between porous disks(semipermeable membranes). The relative permeability points are computed directly from Darcy's law once a constant average saturation and pressure drop across the core are measured. This method gives relative permeability points distributed over the entire saturation interval. Newly developed steady-state methods compensate for the capillary end effects by retaining the capillary term in the equations.7 However, this method7 requires measurement of each phase pressure and it is as time consuming as the traditional steady-state method. For both unsteady-state and steady-state methods, the capillary pressure must be found by independent experiments.8-10 This requires multiple experiments on the same core sample, which necessitates reestablishing the same initial states and wetting conditions. This procedure is both difficult and time consuming. Therefore, neighboring core samples are often used to determine the flow functions. However, this may lead to errors of unknown magnitude since the properties of core samples may differ even on small scales. It is much more desirable to determine the relative permeability and capillary pressure functions simultaneously and from a single experiment.
By designing experiments so the measured data contain information of both relative permeability and capillary pressure effects, simultaneous estimates of these functions can be found through the solution of the appropriate inverse problem. In such an approach, we estimate the multiphase flow functions so that the solution of the mathematical model for the process"matches"the measured data. This methodology has been demonstrated by analyzing both unsteady-state (pressure drop and production data) as well as centrifuge displacement(production data) experiments.11,12 Although the relative permeability and capillary pressure curves can, in principle, be identified in those conventional experiments,11,12the accuracy may not be adequate due to insufficient "information content"of the measured data. Conventional methods can be improved by measuring additional data, such as in situ saturation or pressure.13-16However, this approach requires substantial investments in equipment and training, and it is desirable to be able to arrive at accurate estimates of the relative permeability and capillary pressure functions utilizing conventional data alone.
In this work, we present a method that provides accurate estimates of relative permeability and capillary pressure functions using steady-state equipment and pressure-drop and production measurements. We first design a low-rate steady-state type experiment, for which the measured data contain information about relative permeability and capillary pressure effects. During the experiment, we need not wait until a steady-state saturation distribution is obtained since we use all the measured data (both transient and near steady-state data) when estimating the relative permeability and capillary pressure functions. Hence, both the time and cost of the experiment are reduced compared to the conventional steady-state experiment. The proposed method reconciles the experimental data and we show it is superior to the steady-state method.
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