History Matching of the PUNQ-S3 Reservoir Model Using the Ensemble Kalman Filter
- Yaqing Gu (U. of Oklahoma) | Dean S. Oliver (U. of Oklahoma)
- Document ID
- Society of Petroleum Engineers
- SPE Journal
- Publication Date
- June 2005
- Document Type
- Journal Paper
- 217 - 224
- 2005. Society of Petroleum Engineers
- 5.1.1 Exploration, Development, Structural Geology, 1.2.3 Rock properties, 4.3.4 Scale, 5.1.5 Geologic Modeling, 3.3 Well & Reservoir Surveillance and Monitoring, 5.6.9 Production Forecasting, 5.5 Reservoir Simulation, 5.1.2 Faults and Fracture Characterisation, 5.1 Reservoir Characterisation, 5.6.4 Drillstem/Well Testing, 7.2.2 Risk Management Systems, 5.5.8 History Matching
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This paper reports the use of the ensemble Kalman filter (EnKF) forautomatic history matching. EnKF is a Monte Carlo method in which an ensembleof reservoir models is used. The correlation between reservoir response (e.g.,water cut and rate) and reservoir variables (e.g., permeability and porosity)can be estimated from the ensemble. An estimate of uncertainty in futurereservoir performance can also be obtained from the ensemble. The PUNQ-S3reservoir model is used to test the method in this paper. It is a small (19 285) reservoir engineering model. One conclusion is that when applied to thePUNQ-S3 synthetic model, the EnKF technique gives satisfactory history-matchingresults while requiring less computation work than traditional methods.
The process of adjusting the variables in a reservoir simulation model tohonor observations of rates, pressures, saturations, and other variables atindividual wells is called history matching. In many cases, general geologicalinformation also needs to be honored, such as the variance-covariance structureof the model parameters. Thus, to do history matching, one typically attemptsto minimize the square of the mismatch between all measurements and computedvalues, and/or the square of the mismatch of the current model parameters andthe prior model parameters. Although the process can now be largely automated,a large computational effort is still required, either in objective functionevaluation (nongradient-based minimization method), or in gradient computation(gradient-based minimization method). If the gradient-based minimizationmethods are employed, the adjoint method may be required to compute thegradient of the objective function. The adjoint system is highly dependent onthe source code of the reservoir simulator, however, and therefore is notflexible; that is, if we want to use a different simulator, development of anadjoint code requires considerable work. On the other hand, the increase indeployment of permanent sensors for monitoring pressure, temperature,resistivity, or flow rate has added impetus to the related problem ofcontinuous model updating. Because the data output frequency in this case canbe very high, to simultaneously use all recorded data to generate a reservoirflow model is impractical. Instead, it has become important to incorporate thedata as soon as they are obtained so that the reservoir model is always up todate. Both the heavy computational burden and the high data-sampling frequencyrequire a new kind of history-matching method.
The Kalman filter has historically been the most widely applied method forassimilating new measurements to continuously update the estimate of statevariables. Although Kalman filters have occasionally been applied to theproblem of estimating values of petroleum model variables, 1,2 they are moresuitable for the cases with small numbers of variables and a linearrelationship between model and observations. Unfortunately, most problems inpetroleum reservoir engineering are highly nonlinear and are characterized bymany variables, often two or more variables per simulator gridblock. Thus, thetraditional Kalman filters are not appropriate.
Application to nonlinear problems was at least partially solved by thedevelopment of the extended Kalman filter. However, it did not solve thecritical problem with nonlinear unstable dynamics, in which it leads to alinear instability in the error covariance evolution. The EnKF was introducedto overcome some of the problems of the extended Kalman filter. 3 Since then,the method has found widespread application in weather forcasting, 3--7oceanography, 8 hydrology, 9 and petroleum engineering. 10,11
The EnKF has two major advantages for large-scale history-matching problems.First, it does not depend on the specific reservoir simulator. It only requiresoutput from the simulator, such as pressure and phase saturation. Second, thecomputational cost is fairly low. A relatively small ensemble might besufficient for most applications of EnKF. Although nongradient-basedminimization methods are also not dependent on the simulator source code, theyusually take thousands of simulation runs (objective function evaluations) toobtain the global minimal point.
Naevdal et al. 11 applied the EnKF to the problem of updating 2D,three-phase reservoir models by continuously adjusting both the permeabilityfield and the saturation and pressure fields at each assimilation step. Intheir application, the porosity field is assumed to be known. One syntheticexample had 1,931 active gridblocks with 14 producers and four gas injectors.Two of the producers obtained measurements of well pressure, oil rate, gas/oilratio, and water cut from the first day. Assimilation occurred at least once amonth as well as when new wells started to produce or when wells were shut in,so in many respects it was quite similar to a traditional history-matchingproblem. They found that the ability to predict future performance got steadilybetter as more data were assimilated.
Ensemble Kalman Filter
The methodology consists of a forecast step (stepping forward in time) andan assimilation step, in which variables describing the state of the system arecorrected to honor the observations.
The evolution of reservoir dynamic variables is dictated by reservoir-flowequations and simulated using a commercial reservoir simulator in thispaper.
The following introduces the building blocks of the methodology.
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