Quantifying Uncertainty for the PUNQ-S3 Problem in a Bayesian Setting With RML and EnKF
- Guohua Gao (Chevron Corp.) | Mohammad Zafari (Schlumberger) | Albert C. Reynolds (U. of Tulsa)
- Document ID
- Society of Petroleum Engineers
- SPE Journal
- Publication Date
- December 2006
- Document Type
- Journal Paper
- 506 - 515
- 2006. Society of Petroleum Engineers
- 5.1.2 Faults and Fracture Characterisation, 5.1 Reservoir Characterisation, 5.6.9 Production Forecasting, 5.6.3 Deterministic Methods, 4.3.4 Scale, 5.6.4 Drillstem/Well Testing, 1.2.3 Rock properties, 5.5.8 History Matching, 3.3 Well & Reservoir Surveillance and Monitoring, 5.5 Reservoir Simulation, 5.2.1 Phase Behavior and PVT Measurements, 5.1.5 Geologic Modeling
- 1 in the last 30 days
- 709 since 2007
- Show more detail
- View rights & permissions
|SPE Member Price:||USD 10.00|
|SPE Non-Member Price:||USD 30.00|
The well known PUNQ-S3 reservoir model represents a synthetic problem which was formulated to test the ability of various methods and research groups to quantify the uncertainty in the prediction of cumulative oil production. Previous results reported on this project suggest that the randomized maximum likelihood (RML) method gives a biased characterization of the uncertainty. A major objective of this paper is to show that this is incorrect. With a correct implementation of the RML method within a Bayesian framework, we show that RML does an adequate job of sampling the a posteriori distribution for the PUNQ problem. In particular, the true predicted oil production lies within the band of predictions generated with the RML method and is not biased. We also apply the ensemble Kalman Filter (EnKF) method to the PUNQ data set, and show that this method also gives a reasonable quantification of the uncertainty in performance predictions with an uncertainty range similar to the one obtained with RML.
We consider conditioning models to production data in a Bayesian framework and wish to generate a suite (ensemble) of models which represent a correct sampling of the conditional probability density function (pdf). By predicting future reservoir performance with each realization, we obtain a characterization of the uncertainty in predicted performance. Both the rejection algorithm and Markov chain Monte Carlo (MCMC) are theoretically sound sampling procedures, but they are too computationally inefficient for practical applications (Liu and Oliver 2003). Oliver et al. (1996) and Kitanidis (1986) independently proposed the randomized maximum likelihood (RML) method to generate an approximate sampling of the a posteriori pdf. Two different proofs (Oliver 1996; Reynolds et al. 1999) have been presented which show that the RML method samples the posterior probability density function (pdf) correctly if data are linearly related to the model; however, no rigorous theoretical foundation exists for the method when the relation between data and model is nonlinear, which is the case when the data represent production data. Computational results indicate that the RML method generates reasonable characterization of uncertainty for single-phase flow (Oliver et al. 1996; Reynolds et al. 1999; Liu and Oliver 2003). Our first objective is to show that, contrary to a previous claim (Floris 2001), RML gives a reasonable characterization of the uncertainty in predicted performance for the PUNQ-S3 problem; our second objective is to compare the quantification of uncertainty obtained with RML with the one obtained with the ensemble Kalman filter (EnKF).
The PUNQ-S3 reservoir represents a synthetic model based on an actual reservoir (Floris et al. 2001; Barker et al. 2001). The problem was set up as a test case to allow various research groups to test their own methodology for the characterization of the uncertainty in reservoir performance predictions given some geologic information on the reservoir, hard data at well gridblocks and some scattered production data from the first 8 years of production. Then participants were asked to predict cumulative oil production for 16.5 years of total production and characterize the uncertainty in this prediction.
|File Size||4 MB||Number of Pages||10|
Barker, J.W., Cuypers, M. and Holden, L. 2001. Quantifying Uncertainty in ProductionForecasts: Another Look at the PUNQS3 Problem. SPEJ 6 (4):433-441. SPE-74707-PA.DOI:http://www.spe.org/elibrary/servlet/spepreview?id=74707-PA.
Chavent, G., Dupuy, M. and Lemonnier, P. 1975. History Matching by Use of OptimalTheory. SPEJ 15 (1): 74-86. Trans., AIME, 259.SPE-4627-PA. DOI:http://www.spe.org/elibrary/servlet/spepreview?id=4627-PA.
Chen, W.H., Gavalas, G.R., Seinfeld, J.H. and Wasserman, M.L. 1974.A New Algorithm for AutomaticHistory Matching. SPEJ 14 (6): 593-608. Trans., AIME, 257.SPE-4545-PA.DOI:http://www.spe.org/elibrary/servlet/spepreview?id=4545-PA.
Evensen, G. 1994. Sequential Data Assimilation with a NonlinearQuasi-Geostrophic Model using Monte Carlo Methods to Forecast Error Statistics.J. of Geophysical Research 99: 10143-10162.
Evensen, G. 2003. The Ensemble Kalman Filter: Theoretical Automatic HistoryMatching of Production Data. Paper presented at the European Conference on theMathematics of Oil Recovery, Frieberg, Germany, 3-6 September.
Evensen, G. 2003. The Ensemble Kalman Filter: Theoretical Formulation andPractical Implementation. Ocean Dynamics 53: 343-367. DOI:http://dx.doi.org/10.1007/s10236-003-0036-9.
Evensen, G. 2005. The Combined Parameter and State Estimation Problem.Submitted to Computational Geosciences, 39 p.
Floris, F.J.T., Bush, M.D., Cuypers, M., Roggero, F. and Syversveen, A.R.2001. Methods for Quantifying the Uncertainty of Production Forecasts: AComparative Study. Petroleum Geoscience 7 (Supp.): 87-96.
Gao, G. 2005. Data Integration and Uncertainty Evaluation for Large ScaleAutomatic History Matching Problems. PhD dissertation, U. of Tulsa, Tulsa,Oklahoma.
Gao, G. and Reynolds, A.C. 2006. An Improved Implementation of theLBFGS Algorithm for Automatic History Matching. SPEJ 11 (1):5-17. SPE-90058-MS. DOI:http://www.spe.org/elibrary/servlet/spepreview?id=90058-MS.
Gao, G., Zafari, M. and Reynolds, A.C. 2005. Quantifying Uncertainty for thePUNQ-S3 Problem in a Bayesian Setting With RML and EnKF. Paper SPE 93324presented at the SPE Reservoir Simulation Symposium, Houston, Texas, 31January-2 February. DOI:http://www.spe.org/elibrary/servlet/spepreview?id=93324-MS.
Gu, Y. and Oliver, D.S. 2004a. The Ensemble Kalman Filter for ContinuousUpdating of Reservoir Simulation Models. TUPREP Research Report 21, 150.
Gu, Y. and Oliver, D.S. 2004b. History Matching of the PUNQ-S3Reservoir Model Using the Ensemble Kalman Filter. Paper SPE 89942 presentedat the SPE Annual Technical Conference and Exhibition, Houston, Texas, 26-29September. DOI:http://www.spe.org/elibrary/servlet/spepreview?id=89942-MS.
Kidanidis, P.K. 1986. Parameter Uncertainty in Estimation of Spatial Functions:Bayesian Estimation. Water Resour. Res. 22 (4): 499-507.
Li, R., Reynolds, A.C., and Oliver, D.S. 2003. History Matching of Three-Phase FlowProduction Data. SPEJ 8 (4): 328-340.SPE-87336-PA.DOI:http://www.spe.org/elibrary/servlet/spepreview?id=87336-PA.
Liu, N. and Oliver, D.S. 2003. Evaluation of Monte Carlo Methods forAssessing Uncertainty. SPEJ 8 (2): 188-195.SPE-84936-PA.DOI:http://www.spe.org/elibrary/servlet/spepreview?id=84936-PA.
Naevdal, G., Johnsen, L.M., Aanonsen, S.I. and Vefring, E.H. 2005. Reservoir Monitoring and ContinuousModel Updating Using Ensemble Kalman Filter.SPEJ10 (1): 66-74. SPE-84372-PA.DOI:http://www.spe.org/elibrary/servlet/spepreview?id=84372-PA.
Naevdal, G., Mannseth, T. and Vefring, E.H. 2002. Near-Well Reservoir MonitoringThrough Ensemble Kalman Filter. Paper SPE 75235 presented at the 2002SPE/DOE Improved Oil Recovery Symposium, Tulsa, Oklahoma, 13-17 April.DOI: http://www.spe.org/elibrary/servlet/spepreview?id=75235-MS.
Nocedal, J. 1980. Updating Quasi-Newton Matrices with Limited Storage. Math.Comp. 35 (151): 773-782. DOI: http://dx.doi.org/10.2307/2006193.
Oliver, D.S. 1996. On Conditional Simulation to Inaccurate Data. Math.Geology 28 (6): 811-817. DOI: http://dx.doi.org/10.1007/BF02066348.
Oliver, D.S., He, N. and Reynolds, A.C. 1996. Conditioning Permeability Fieldsto Pressure Data. Paper presented at the 5th European Conference for theMathematics of Oil Recovery, Leoben, Austria, 3-6 September.
Reynolds, A.C., He, N. and Oliver, D.S. 1999. Reducing Uncertainty inGeostatistical Description with Well Testing Pressure Data. ReservoirCharacterization: Recent Advances, American Association of Petroleum Geologists149-162.
Wen, X.H. and Chen, W.H. 2005. Real-Time Reservoir Model UpdatingUsing Ensemble Kalman Filter. Paper SPE 92991 presented at the 2005SPE Reservoir Simulation Symposium, Houston, 31 January-2 February.DOI:http://www.spe.org/elibrary/servlet/spepreview?id=92991-MS.
Zafari, M. and Reynolds, A.C. 2005a. EnKF Versus RML, Theoretical Comments andNumerical Experiments. TUPREP Research Report 22, 195.
Zafari, M. and Reynolds, A.C. 2005b. Assessing the Uncertainty inReservoir Description and Performance Predictions With the Ensemble KalmanFilter. Paper SPE 95750 presented at the SPE Annual Technical Conferenceand Exhibition, Dallas, 9-12 October. DOI:http://www.spe.org/elibrary/servlet/spepreview?id=95750-MS.
Zhang, F. and Reynolds, A.C. 2002. Optimization Algorithms for AutomaticHistory Matching of Production Data. Paper presented at the European Conferenceon the Mathematics of Oil Recovery, Frieberg, Germany, 3-6 September.
Zhang, F., Reynolds, A.C. and Oliver, D.S. 2002. Evaluation of the Reduction inUncertainty Obtained by Conditioning a 3D Stochastic Channel to MultiwellPressure Data. Mathematical Geology 34 (6): 713-740. DOI:http://dx.doi.org/10.1023/A:1019805310025.