Rapid Updating of Stochastic Models by Use of an Ensemble-Filter Approach
- Reza Tavakoli (University of Texas at Austin) | Sanjay Srinivasan (University of Texas at Austin) | Mary F. Wheeler (University of Texas at Austin)
- Document ID
- Society of Petroleum Engineers
- SPE Journal
- Publication Date
- June 2014
- Document Type
- Journal Paper
- 500 - 513
- 2013.Society of Petroleum Engineers
- 5.5.8 History Matching, 5.4.1 Waterflooding, 5.1.8 Seismic Modelling
- multidimensional scaling, ensemble kalman filter, multipoint statistics
- 2 in the last 30 days
- 349 since 2007
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Applying an ensemble Kalman filter (EnKF) is an effective method for reservoir history matching. The underlying principle is that an initial ensemble of stochastic models can be progressively updated to reflect measured values as they become available. The EnKF performance is only optimal, however, if the prior-state vector is linearly related to the predicted data and if the joint distribution of the prior-state vector is multivariate Gaussian. Therefore, it is challenging to implement the filtering scheme for non-Gaussian random fields, such as channelized reservoirs, in which the continuity of permeability extremes is well-preserved. In this paper, we develop a methodology by combining model classification with multidimensional scaling (MDS) and the EnKF to create rapidly updating models of a channelized reservoir. A dissimilarity matrix is computed by use of the dynamic responses of ensemble members. This dissimilarity matrix is transformed into a lower-dimensional space by use of MDS. Responses mapped in the lower-dimension space are clustered, and on the basis of the distances between the models in a cluster and the actual observed response, the closest models to the observed response are retrieved. Model updates within the closest cluster are performed using EnKF equations. The results of an update are used to resample new models for the next step. Two-dimensional, waterflooding examples of channelized reservoirs are provided to demonstrate the applicability of the proposed method. The obtained results demonstrate that the proposed algorithm is viable both for sequentially updating reservoir models and for preserving channel features after the data-assimilation process.
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Aanonsen, S.I., Nævdal, G., Oliver, D.S. et al. 2009. The Ensemble Kalman Filter in Reservoir Engineering—A Review. SPE J. 14 (3):393-412. http://dx.doi.org/10.2118/117274-PA.
Agbalaka, C. and Oliver, D.S. 2008. Application of the EnKF and Localization to Automatic History Matching of Facies Distribution and Production Data. Math. Geosci. 40.
Anderson, J.L. 2007. Exploring the Need for Localization in Ensemble Data Assimilation Using a Hierarchical Ensemble Filter. Physica D: Nonlinear Phenomena 230 (1-2): 99-111.
Borg, I. and Groenen, P. 1997. Modern Multidimensional Scaling: Theory and Applications. New York: Springer.
Chang, H., Zhang, D., and Lu, Z. 2010. History Matching of Facies Distribution With the EnkF and Level Set Parameterization. J. Computational Physics 229 (20): 8011-8030.
Dovera, L. and Rossa, E.D. 2011. Multimodal Ensemble Kalman Filtering Using Gaussian Mixture Models. Computational Geosci. 15 (2):307-323.
Emerick, A. and Reynolds, A. 2010. Combining Sensitivities and Prior Information for Covariance Localization in the Ensemble Kalman Filter for Petroleum Reservoir Applications. Computational Geosci. 15(2): 251-269.
Evensen, G. 1994. Sequential Data Assimilation With a Nonlinear Quasi-Geostrophic Model Using Monte Carlo Methods to Forecast Error Statistics.J. Geophysical Research 99 (C5): 10143-10162.
Evensen, G. 2005. The Combined Parameter and State Problem. In TechnicalReport. Norsk Hydro Research Center.
Evensen, G. 2007. Data Assimilation: The Ensemble Kalman Filter. NewYork: Springer.
Evensen, G. 2009. The Ensemble Kalman Filter for Combined State and Parameter Estimation. IEEE Control Systems Magazine 29 (3):83-104.
Evensen, G., Hove, J., Meisingset, H.C. et al. 2007. Using the EnKF for Assisted History Matching of a North Sea Reservoir Model (SPE 106184). In Proceedings of the 2007 SPE Reservoir Simulation Symposium.
Gao, G., Zafari, M., and Reynolds, A.C. 2006. Quantifying Uncertainty for the PUNQ-S3 Problem in a Bayesian Setting With RML and Enkf. SPE J. 11 (4): 506-515.
Gu, Y. and Oliver, D.S. 2005. History Matching of the PUNQ-S3 Reservoir Model Using the Ensemble Kalman Filter. SPE J. 10 (2):51-65.
Honarkhah, M. and Caers, J. 2010. Stochastic Simulation of Patterns Using Distance-Based Pattern Modeling. Math. Geosci. 42 (5):487-517.
Houtekamer, P.L. and Mitchell, H.L. 2001. A Sequential Ensemble Kalman Filter for Atmospheric Data Assimilation. Monthly Weather Review 129 (1): 123-137.
Jafarpour, B. and Khodabakhshi, M. 2011. A Probability Conditioning Method(PCM) for Nonlinear Flow Data Integration Into Multiple-Point Statistical Facies Simulation. Math. Geosci. 43 (2): 133-164. http://dx.doi.org/10.1007/s11004-011-9316-y.
Jafarpour, B. and McLaughlin, D.B. 2009. Estimating Channelized Reservoir Permeabilities With the Ensemble Kalman Filter: The Importance of Ensemble Design. SPE J. 14 (2): 374-388.
Kalman, R.E. 1960. A New Approach to Linear Filtering and Prediction Problems. Trans. of the ASME, J. Basic Eng. 82:35-45.
Kohonen, T. 1984. Self-Organization and Associative Memory (2ndedition). Berlin: Springer-Verlag.
Kruskal, J. and Wish, M. 1978. Multidimensional Scaling. In Sage University Paper Series on Quantitative Applications in the Social Sciences (07-011).
Liu, N. and Oliver, D.S. 2005. Ensemble Kalman Filter for Automatic History Matching of Geologic Facies. J. Petrol. Sci. Eng. 47 (3-4):147-161.
MacQueen, J.B. 1967. Some Methods for Classification and Analysis of Multivariate Observations. In Proceedings of 5th Berkeley Symposiumon Mathematical Statistics and Probability, 281-297.
Mariethoz, G., Renard, O., and Straubhaar, J. 2010. The Direct Sampling Method to Perform Multiple-Point Geostatistical Simulations. Water Resources Research 46.
Moreno, D. and Aanonsen, S.I. 2007. Stochastic Facies Modelling Using the Level Set Method. In Extended Abstracts Book of Petroleum Geostatistics2007. Cascais, Portugal. A18.
Remy, N., Boucher, A., and Wu, J. 2009. Applied Geostatistics With SGeMS: A User's Guide. Cambridge University Press.
Reynolds, A.C., Zafari, M., and Li, G. 2006. Iterative Forms of the EnsembleKalman Filter. In 10th European Conference on the Mathematics of Oil Recovery. Amsterdam, The Netherlands. A030.
Romney, A.K., Shepard, R.N., and Nerlove, S.B. 1972. Multidimensional Scaling, Theory and Applications in the Behavioral Sciences. New York:Seminar Press. 2 Vols.
Sammon, J.W. Jr. 1969. A Nonlinear Mapping for Data Structure Analysis.IEEE Trans. on Computers c-18 (5): 401-409.
Sarma, P. and Chen, W.H., 2009. Generalization of the Ensemble Kalman Filter Using Kernels for Non-Gaussian Random Fields. Paper SPE 119177 presentedat the SPE Reservoir Simulation Symposium, The Woodlands, Texas, 2-4 February.http://dx.doi.org/10.2118/119177-MS.
Skjervheim, J.-A., Evensen, G., Aanonsen, S.I. et al. 2007. Incorporating 4D Seismic Data in Reservoir Simulation Models Using Ensemble Kalman Filter.SPE J. 12 (3): 282-292. http://dx.doi.org/10.2118/95789-PA.
Strebelle, S. 2002a. Conditional Simulation of Complex Geological Structures Using Multiple-Point Statistics. Math. Geol. 34 (1):1-22.
Strebelle, S. 2002b. Conditional Simulation of Complex Geological Structures Using Multiple-Point Statistics. Math. Geol. 34 (1).
Suzuki, S. and Caers, J. 2008. A Distance-Based Prior Model Parameterization for Constraining Solutions of Spatial Inverse Problems. Math. Geosci. 40 (4): 445-469.
Zhang, T., Switzer, P., and Journel, A. 2006. Filter-Based Classification of Training Image Patterns for Spatial Simulation. Math. Geol. 38 (1): 63-80.
Zhao, Y., Reynolds, A.C., and Li, G. 2008. Generating Facies Maps by Assimilating Production Data and Seismic Data With the Ensemble Kalman Filter.Paper SPE 113990 presented at the SPE/DOE Symposium on Improved Oil Recovery,Tulsa, Oklahoma, 20-23 April. http://dx.doi.org/10.2118/113990-MS.
Zhou, H., Gomez-Hernandez, J., Franssen, H.-J. H. et al. 2011. An Approach to Handling Non-Gaussianity of Parameters and State Variables in Ensemble Kalman Filtering. Advances in Water Resources 34:844-864.