Reservoir Characterization With the Discrete Cosine Transform
- Behnam Jafarpour (Texas A&M University) | Dennis B. McLaughlin (Massachusetts Institute of Technology)
- Document ID
- Society of Petroleum Engineers
- SPE Journal
- Publication Date
- March 2009
- Document Type
- Journal Paper
- 182 - 201
- 2009. Society of Petroleum Engineers
- 4.3.4 Scale, 5.5.8 History Matching, 5.1 Reservoir Characterisation, 5.1.5 Geologic Modeling, 2.4.6 Frac and Pack, 5.2.1 Phase Behavior and PVT Measurements, 4.1.2 Separation and Treating, 5.6.4 Drillstem/Well Testing, 5.3.1 Flow in Porous Media, 7.6.2 Data Integration, 4.1.5 Processing Equipment, 5.5 Reservoir Simulation, 5.3.2 Multiphase Flow, 6.1.5 Human Resources, Competence and Training, 5.4.1 Waterflooding, 5.6.1 Open hole/cased hole log analysis, 5.6.9 Production Forecasting
- 6 in the last 30 days
- 1,128 since 2007
- Show more detail
- View rights & permissions
|SPE Member Price:||USD 12.00|
|SPE Non-Member Price:||USD 35.00|
Inverse estimation (history matching) of permeability fields is commonly performed by replacing the original set of unknown spatially discretized reservoir properties with a smaller (lower-dimensional) group of unknowns that captures the most important features of the field. This makes the inverse problem better posed by reducing redundancy. The Karhunen-Loeve transform (KLT), also known as principal component analysis, is a classical option for deriving low-dimensional parameterizations for history-matching applications. The KLT can provide an accurate characterization of complex-property fields, but it can be computationally demanding. In many respects, this approach provides a benchmark that can be used to evaluate the performance of more-computationally-efficient alternatives. The KLT requires knowledge of the property covariance function and can give poor results when this function does not adequately describe the actual property field. By contrast, the discrete cosine transform (DCT) provides a robust parameterization alternative that does not require specification of covariances or other statistics. It is computationally efficient and, in many cases, is almost as accurate as the KLT. The DCT is also able to accommodate prior information, if desired. Here, we describe the DCT approach and compare its performance to the KLT for a set of geologically relevant examples.
Part 2--History Matching
The DCT provides a flexible and effective method for describing spatially distributed reservoir properties such as permeability. This method represents uncertain properties as weighted sums of predefined spatially variable basis functions. The basis function weights may be estimated with iterative or sequential history-matching methods. The compression power of the DCT and its advantages over alternative parameterization techniques are discussed in Part 1. In Part 2, the history-matching capabilities of the DCT parameterization are illustrated with waterflooding examples for synthetic channelized reservoirs. Two history-matching options are examined: an iterative least-squares method and a sequential ensemble Kalman filter (EnKF). Prior information is incorporated through an ensemble of permeability replicates derived from a specified training image. These replicates are used to compute sample covariances for the EnKF and to select basis functions for the DCT expansion in both the least-squares algorithm and the Kalman filter. Prior information improves estimation performance when it is consistent with the directional trends of the true permeability field but may degrade performance if it is incorrectly specified. The most robust history-matching results are obtained with an iterative least-squares algorithm that uses a DCT basis with no directional preference. The experiments documented in this paper indicate that the DCT makes the history-matching problem better-posed and improves the realism of reservoir property estimates. It is efficient and versatile and can be used with or without prior information.
|File Size||4 MB||Number of Pages||20|
Aanonsen, S.I. 2008. EfficientHistory Matching Using a Multiscale Technique. SPE Res Eval &Eng 11 (1): 154-164. SPE-92758-PA. doi: 10.2118/92758-PA.
Ahmed, N., Natarajan, T., and Rao, K.R. 1974. Discrete Cosine Transform.IEEE Transactions on Computers C-23 (1): 90-93.doi:10.1109/T-C.1974.223784.
Andrews, H.C. 1971. Multidimensional Rotations inFeature Selection. IEEE Transactions on Computers C-20(9): 1045-1051. doi:10.1109/T-C.1971.223400.
Aziz, K. and Settari, A. 1979. Petroleum Reservoir Simulation. Essex,UK: Elsevier Applied Science Publishers.
Bissel, R., Sharma, Y., and Killough, J.E. 1994. History Matching Using the Method ofGradients: Two Case Studies. Paper SPE 28590 presented at the SPE AnnualTechnical Conference and Exhibition, New Orleans, 25-28 September. doi:10.2118/28590-MS.
Brigham, E.O. 1988. The Fast Fourier Transform and Its Applications.Upper Saddle River, New Jersey: Signal Processing Series, Prentice-Hall.
Brouwer, D.R., Nævdal, G., Jansen, J.-D., Vefring, E.H., and van Kruijsdijk,C.P.J.W. 2004. Improved ReservoirManagement Through Optimal Control and Continuous Model Updating. Paper SPE90149 presented at the SPE Annual Technical Conference and Exhibition, Houston,26-29 September. doi: 10.2118/90149-MS.
Caers, J. and Zhang, T. 2002. Multiple-point geostatistics: A quantitativevehicle for integrating geological analogs into multiple reservoir models. InAAPG Memoir 80: Integration of outcrop and modern analog data in reservoirmodels, ed. G.M. Grammer, P.M. Harris, and G.P. Eberli, 383-394.
Chavent, G. and Bissell, R. 1998. Indicator for the refinement ofparameterization. In Inverse Problems in Engineering Mechanics, ed. M.Tanaka and G.S. Dulikravich, 309-314. Oxford, UK: Elsevier Science.
Chilès, J.-P. and Delfiner, P. 1999. Geostatistics: Modeling SpatialUncertainty. New York City: Wiley Series in Probability and Statistics,John Wiley & Sons.
Deutsch, C.V. and Journel, A. 1992. GSLIB Geostatistical Software Libraryand User's Guide. Oxford, UK: Oxford University Press.
ECLIPSE E300 reference manual. 2006. Houston: Schlumberger-GeoQuest.
Ersoy, O.K. 1994. A comparativereview of real and complex Fourier-related transforms. Proc. of theIEEE 82 (3): 429-447. doi:10.1109/5.272147.
Eude, T., Grisel, R., Cherifi, H., and Debrie, R. 1994. On the distributionof the DCT coefficients. Proc., IEEE Int. Conf. Acoustics, Speech,Signal Processing, Adelaide, Australia, Vol. 5, 365-368.
Evensen, G. 1994. Sequential data assimilation with a nonlinearquasi-geostrophic model using Monte Carlo methods to forecast error statistics.J. of Geophysical Research 99 (C5): 10143-10162.
Evensen, G. 2007. Data Assimilation: The Ensemble Kalman Filter.Berlin: Springer Verlag.
Feig, E. and Winograd, S. 1992. Fast algorithms for the discretecosine transform. IEEE Trans. on Signal Processing 40(9): 2174-2193. doi:10.1109/78.157218.
Freeze, R.A. 1975. Astochastic-conceptual analysis of one-dimensional groundwater flow innon-uniform homogeneous media. Water Resources Research11 (5): 725-741. doi:10.1029/WR011i005p00725.
Gavalas, G.R., Shah, P.C., and Seinfeld, J.H. 1976. Reservoir History Matching by BayesianEstimation. SPE J. 16 (6): 337-350. SPE-5740-PA. doi:10.2118/5740-PA.
Gelb, A. ed. 1974. Applied Optimal Estimation. Cambridge,Massachusetts: MIT Press.
Gelfand, I.M. and Fomin, S.V. 2000. Calculus of Variations, trans.R.A. Silverman. Mineola, New York: Dover Publications.
Gonzalez, R.C. and Woods, R.E. 2002. Digital Image Processing, secondedition. Reading, Massachusetts: Addison-Wesley Publishing.
Grimstad, A.A., Mannseth, T., Nævdal, G., and Urkedal, H. 2003. Adaptive multiscalepermeability estimation. Computational Geosciences 7(1): 1-25. doi:10.1023/A:1022417923824.
Gu, Y. and Oliver, D.S. 2006. The ensemble Kalman filter forcontinuous updating of reservoir simulation models. J. of EnergyResources Technology 128 (1): 79-87. doi:10.1115/1.2134735.
Hoeksema, R. and Kitanidis, P. 1985. Analysis of the spatialstructure of properties of selected aquifers. Water ResourcesResearch 21 (4): 563-572. doi:10.1029/WR021i004p00563.
Jacquard, P. and Jain, C. 1965. Permeability Distribution From FieldPressure Data. SPE J. 5 (4): 281-294; Trans.,AIME, 234. SPE-1307-PA. doi: 10.2118/1307-PA.
Jafarpour, B. and McLaughlin, D. In press. Estimating ChannelizedReservoirs Permeabilities With the Ensemble Kalman Filter: The Importance ofEnsemble Design. SPE J. (submitted 2007).
Jahns, O.H. 1966. A Rapid Methodfor Obtaining a Two-Dimensional Reservoir Description From Well PressureResponse Data. SPE J. 6 (4): 315-327; Trans.,AIME, 237. SPE-1473-PA. doi: 10.2118/1473-PA.
Jain, A.K. 1988. Fundamentals of Digital Image Processing. UpperSaddle River, New Jersey: Information and System Sciences Series,Prentice-Hall.
Kalman, R.E. 1960. A new approach to linear filtering and predictionproblems. Transactions of the ASME--Journal of Basic Engineering82 (Series D): 35-45.
Karhunen, K. 1947. Ueber lincare methoden in derwahrscheinlichts-keitsrechnung. Ann. Acad Sci. Fenn., Ser. A137: 3-79.
Li, R., Reynolds, A.C., and Oliver, D.S. 2003. History Matching of Three-Phase FlowProduction Data. SPE J. 8 (4): 328-340. SPE-87336-PA.doi: 10.2118/87336-PA.
Loeve, M.M. 1977. Probability Theory, fourth edition, Vol. 1 and 2.New York City: Springer Verlag.
McLaughlin, D. and Townley, L.R. 1996. A reassessment of the groundwaterinverse problem. Water Resources Research 32 (5):1131-1161. doi:10.1029/96WR00160.
Nævdal, G., Johnsen, L.M., Aanonsen, S.I., and Vefring, E.H. 2005. Reservoir Monitoring and ContinuousModel Updating Using Ensemble Kalman Filter. SPE J. 10(1): 66-74. SPE-84372-PA. doi: 10.2118/84372-PA.
Narasimha, M.J. and Peterson, A.M. 1978. On the computation of thediscrete cosine transform. IEEE Transactions on Communications26 (6): 934-936. doi:10.1109/TCOM.1978.1094144.
Nocedal, J. and Wright, S.J. 2006. Numerical Optimization, secondedition. New York City: Series in Operations Research and FinancialEngineering, Springer.
Oliver, D.S., Reynolds, A.C., and Liu, N. 2008. Inverse Theory forPetroleum Reservoir Characterization and History Matching. Cambridge, UK:Cambridge University Press.
Oliver, D.S., Reynolds, A.C., Bi, Z., and Abacioglu, Y. 2001. Integration ofproduction data into reservoir models. Petroleum Geoscience7 (Supplement, 1 May): 65-73.
Pearl, J., Andrews, H., and Pratt, W. 1972. Performance measures fortransform data coding. IEEE Trans. on Communications 20(3): 411-415. doi:10.1109/TCOM.1972.1091168.
Rao, K.R. and Yip, P. 1990. Discrete Cosine Transform: Algorithms,Advantages, Applications. Boston, Massachusetts: Academic Press.
Remy, N. 2004. S-GeMS: A geostatistical earth modeling library and software.PhD thesis, Stanford University.
Reynolds, A.C., He, N., Chu, L., and Oliver, D.S. 1996. Reparameterization Techniques forGenerating Reservoir Descriptions Conditioned to Variograms and Well-TestPressure Data. SPE J. 1 (4): 413-426. SPE-30588-PA.doi: 10.2118/30588-PA.
Sahni, I. and Horne, R.N. 2005. Multiresolution Wavelet Analysis forImproved Reservoir Description. SPE Res Eval & Eng8 (1): 53-69. SPE-87820-PA. doi: 10.2118/87820-PA.
Sarma, P., Durlofsky, L.J., Khalid, A., and Chen, W.H. 2006. Efficient real-timereservoir management using adjoint-based optimal control and modelupdating. Computational Geosciences 10 (1): 3-36.doi:10.1007/s10596-005-9009-z.
Shah, P.C., Gavalas, G.R., and Seinfeld, J.H. 1978. Error Analysis in History Matching:The Optimal Level of Parameterization. SPE J. 18 (3):219-228. SPE-6508-PA. doi: 10.2118/6508-PA.
Skjervheim, J.-A., Evensen, G., Aanonsen, S.I., and Johansen, T.A. 2007. Incorporating 4D Seismic Data inReservoir Simulation Model Using Ensemble Kalman Filter. SPE J.12 (3): 282-292. SPE-95789-PA. doi: 10.2118/95789-PA.
Strebelle, S.B. and Journel, A.G. 2001. Reservoir Modeling UsingMultiple-Point Statistics. Paper SPE 71324 presented at the SPE AnnualTechnical Conference and Exhibition, New Orleans, 30 September-3 October. doi:10.2118/71324-MS.
Tarantola, A. 2005. Inverse Problem Theory and Methods for ModelParameter Estimation. Philadelphia, Pennsylvania: SIAM.
Wen, X.-H. and Chen, W.H. 2005. Real-Time Reservoir Model UpdatingUsing Ensemble Kalman Filter. Paper SPE 92991 presented at the SPEReservoir Simulation Symposium, The Woodlands, Texas, USA, 31 January-2February. doi: 10.2118/92991-MS.
Zhang, F. and Reynolds, A.C. 2002. Optimization algorithms for automatichistory matching of production data. Proc., 8th European Conference onthe Mathematics of Oil Recovery (ECMOR VIII), Freiburg, Germany, 3-6 September,1-10.