Automatic History Matching in a Bayesian Framework, Example Applications
- Fengjun Zhang (Chevron Corp.) | Jan-Arild Skjervheim (University of Bergen) | Albert C. Reynolds (U. of Tulsa) | Dean S. Oliver (U. of Oklahoma)
- Document ID
- Society of Petroleum Engineers
- SPE Reservoir Evaluation & Engineering
- Publication Date
- June 2005
- Document Type
- Journal Paper
- 214 - 223
- 2005. Society of Petroleum Engineers
- 3.3.1 Production Logging, 5.8.7 Carbonate Reservoir, 4.3.4 Scale, 5.3.2 Multiphase Flow, 5.5 Reservoir Simulation, 3.3 Well & Reservoir Surveillance and Monitoring, 5.5.8 History Matching, 5.6.4 Drillstem/Well Testing, 3.3.6 Integrated Modeling, 5.1 Reservoir Characterisation, 5.2.1 Phase Behavior and PVT Measurements, 5.4.2 Gas Injection Methods, 5.1.1 Exploration, Development, Structural Geology, 5.5.3 Scaling Methods, 5.6.3 Pressure Transient Testing, 5.6.1 Open hole/cased hole log analysis, 7.6.2 Data Integration, 4.1.2 Separation and Treating, 4.1.5 Processing Equipment, 5.1.5 Geologic Modeling
- 1 in the last 30 days
- 924 since 2007
- Show more detail
- View rights & permissions
|SPE Member Price:||USD 12.00|
|SPE Non-Member Price:||USD 35.00|
The Bayesian framework allows one to integrate production and static datainto an a posteriori probability density function (pdf) for reservoir variables(model parameters). The problem of generating realizations of the reservoirvariables for the assessment of uncertainty in reservoir description orpredicted reservoir performance then becomes a problem of sampling this aposteriori pdf to obtain a suite of realizations. Generation of a realizationby the randomized-maximum-likelihood method requires the minimization of anobjective function that includes production-data misfit terms and a modelmisfit term that arises from a prior model constructed from static data.Minimization of this objective function with an optimization algorithm isequivalent to the automatic history matching of production data, with a priormodel constructed from static data providing regularization. Because of thecomputational cost of computing sensitivity coefficients and the need to solvematrix problems involving the covariance matrix for the prior model, thisapproach has not been applied to problems in which the number of data and thenumber of reservoir-model parameters are both large and the forward problem issolved by a conventional finite-difference simulator.
In this work, we illustrate that computational efficiency problems can beovercome by using a scaled limited-memory Broyden-Fletcher-Goldfarb-Shanno(LBFGS) algorithm to minimize the objective function and by using approximatecomputational stencils to approximate the multiplication of a vector by theprior covariance matrix or its inverse. Implementation of the LBFGS methodrequires only the gradient of the objective function, which can be obtainedfrom a single solution of the adjoint problem; individual sensitivitycoefficients are not needed. We apply the overall process to two examples. Thefirst is a true field example in which a realization of log permeabilities at26,019 gridblocks is generated by the automatic history matching of pressuredata, and the second is a pseudofield example that provides a very roughapproximation to a North Sea reservoir in which a realization of logpermeabilities at 9,750 gridblocks is computed by the automatic historymatching of gas/oil ratio (GOR) and pressure data.
The Bayes theorem provides a general framework for updating a pdf as newdata or information on the model becomes available. The Bayesian setting offersa distinct advantage. If one can generate a suite of realizations thatrepresent a correct sampling of the a posteriori pdf, then the suite of samplesprovides an assessment of the uncertainty in reservoir variables. Moreover, bypredicting future reservoir performance under proposed operating conditions foreach realization, one can characterize the uncertainty in future performancepredictions by constructing statistics for the set of outcomes. Liu and Oliverhave recently presented a comparison of methods for sampling the a posterioripdf. Their results indicate that the randomized-maximum-likelihood method isadequate for evaluating uncertainty with a relatively limited number ofsamples. In this work, we consider the case in which a prior geostatisticalmodel constructed from static data is available and is represented by amultivariate Gaussian pdf.Then, the a posteriori pdf conditional toproduction data is such that calculation of the maximum a posteriori estimateor generation of a realization by the randomized-maximum-likelihood method isequivalent to the minimization of an appropriate objective function.
History-matching problems of interest to us involve a few thousand to tensof thousands of reservoir variables and a few hundred to a few thousandproduction data. Thus, an optimization algorithm suitable for large-scaleproblems is needed. Our belief is that nongradient-based algorithms such assimulated annealing and the genetic algorithm are not competitive withgradient-based algorithms in terms of computational efficiency. Classicalgradient-based algorithms such as the Gauss-Newton and Levenberg-Marquardttypically converge fairly quickly and have been applied successfully toautomatic history matching for both single-phase- and multiphase-flow problems.No multiphase-flow example considered in these papers involved more than 1,500reservoir variables. For single-phase-flow problems, He et al. and Reynolds etal. have generated realizations of models involving up to 12,500 reservoirvariables by automatic history matching of pressure data. However, they used aprocedure based on their generalization of the method of Carter et al. tocalculate sensitivity coefficients; this method assumes that thepartial-differential equation solved by reservoir simulation is linear and doesnot apply for multiphase-flow problems.
|File Size||3 MB||Number of Pages||10|
1. Hegstad, B.K. and Omre, H.: "Uncertainty Assessment in History Matchingand Forecasting," Geostatistics Wollogong 96, E.Y. Baafi and N.A. Schofield(eds.), Kluwer Academic Publishers, Dordrecht, The Netherlands (1997).
2. Reynolds, A.C., He, N., and Oliver, D.S.: "Reducing Uncertainty inGeostatistical Description with Well Testing Pressure Data," ReservoirCharacterization: Recent Advances, R.A. Schatzinger and J.F. Jordan (eds.),American Assn. of Petroleum Geologists, Tulsa (1999) 149-162.
3. Liu, N. and Oliver, D.S.: "Evaluation of Monte Carlo Methods forAssessing Uncertainty," SPEJ (June 2003) 188.
4. Tarantola, A.: Inverse Problem Theory: Methods for Data Fitting and ModelParameter Estimation, Elsevier, Amsterdam (1987).
5. Oliver, D.S.: "Incorporation of Transient Pressure Data into ReservoirCharacterization," In Situ (1994) 18, No. 3, 243.
6. Oliver, D.S., He, N., and Reynolds, A.C.: "Conditioning PermeabilityFields to Pressure Data," Proc., 5th European Conference for the Mathematics ofOil Recovery V, Leoben, Austria (1996).
7. Tan, T.B.: "AComputationally Efficient Gauss-Newton Method for Automatic HistoryMatching," paper SPE 29100 presented at the 1995 SPE Symposium on ReservoirSimulation, San Antonio, Texas, 12-15 February.
8. He, N., Reynolds, A.C., and Oliver, D.S.: "Three-Dimensional ReservoirDescription From Multiwell Pressure Data and Prior Information," SPEJ(September 1997) 312.
9. Wu, Z., Reynolds, A.C., and Oliver, D.S.: "Conditioning Geostatistical Models toTwo-Phase Production Data," SPEJ (June 1999) 142.
10. Li, R., Reynolds, A.C., and Oliver, D.S.: "History Matching of Three-Phase FlowProduction Data," paper SPE 66351 presented at the 2001 SPE ReservoirSimulation Symposium, Houston, 11-14 February.
11. Carter, R.D. et al.: "Performance Matching WithConstraints," SPEJ (April 1974) 187; Trans., AIME, 257.
12. Chavent, G.M., Dupuy, M., and Lemonnier, P.: "History Matching by Use of OptimalTheory," SPEJ (February 1975) 74; Trans., AIME, 259.
13. Chen, W.H. et al.: "A NewAlgorithm for Automatic History Matching," SPEJ (December 1974) 593;Trans., AIME, 257.
14. Anterion, F., Eymard, R., and Karcher, B.: "Use of Parameter Gradients forReservoir History Matching," paper SPE 18433 presented at the 1989 SPESymposium on Reservoir Simulation, Houston, 6-8 February.
15. Jacquard, P. and Jain, C.: "Permeability Distribution From FieldPressure Data," SPEJ (December 1965) 281; Trans., AIME, 234.
16. Jahns, H.O.: "A Rapid Methodfor Obtaining a Two-Dimensional Reservoir Description From Well PressureResponse Data," SPEJ (December 1966) 315; Trans., AIME, 237.
17. Bissell, R.C., Sharma, Y., and Killough, J.E.: "History Matching Using the Method ofGradients: Two Case Studies," paper SPE 28590 presented at the 1994 SPEAnnual Technical Conference and Exhibition, New Orleans, 25-28 September.
18. Bissell, R.: "Calculating Optimal Parameters for History Matching,"Proc., 4th European Conference on the Mathematics of Oil Recovery, Roros,Norway (1994).
19. de Marsily, G. et al.: "Interpretation of Interference Tests in a WellField Using Geostatistical Techniques to Fit the Permeability Distribution in aReservoir Model," Geostatistics for Natural Resources Characterization, Part 2,G. Verly, M. David, A.G. Journel, and A. Marechal (eds.), D. Reidell,Dordrecht, The Netherlands (1984) 831-849.
20. RamaRao, B.S. et al.: "Pilot Point Methodology for Automated Calibrationof an Ensemble of Conditionally Simulated Transmissivity Fields, 1. Theory andComputational Experiments," Water Resour. Res. (March 1995) 31, No. 3, 475.
21. Kennett, B.L.N. and Williamson, P.R.: "Subspace methods for large-scalenonlinear inversion," Mathematical Geophysics, D. Reidell, Dordrecht, TheNetherlands (1988) 139-154.
22. Oldenburg, D.W., McGillivray, P.R., and Ellis, R.G.: "Generalizedsubspace methods for large-scale inverse problems," Geophys. J. Int. (July1993) 114, No. 1, 12.
23. Reynolds, A.C. et al.: "Reparameterization Techniques forGenerating Reservoir Descriptions Conditioned to Variograms and Well-TestPressure Data," SPEJ (December 1996) 413.
24. Abacioglu, Y., Oliver, D.S., and Reynolds, A.C.: "Efficient Reservoir HistoryMatching Using Subspace Vectors," Computational Geosciences (2001) 5, No.2, 151.
25. Chu, L., Komara, M., and Schatzinger, R.A.: "Efficient Technique for Inversion ofReservoir Properties Using Iteration Method," SPEJ (March 2000) 71.
26. Mackie, R.L. and Madden, T.R.: "Three-dimensional magnetotelluricinversion using conjugate gradients," Geophys. J. Int. (October 1993) 15, No.1, 215.
27. Axelsson, O.: Iterative Solution Methods, Cambridge U. Press, New YorkCity (1994).
28. Makhlouf, E.M. et al.: "AGeneral History Matching Algorithm for Three-Phase, Three-Dimensional PetroleumReservoirs," SPE Advanced Technology Series (July 1993) 83.
29. Zhang, F. and Reynolds, A.C.: "Optimization Algorithms for AutomaticHistory Matching of Production Data," Proc., 8th European Conference on theMathematics of Oil Recovery, Freiburg, Germany (2002).
30. Yang, P.H. and Watson, A.T.: "Automatic History Matching WithVariable-Metric Methods," SPERE (August 1988) 995.
31. Masumoto, K.: "PressureDerivative Matching Method for Two Phase Fluid Flow in HeterogeneousReservoir," paper SPE 59462 presented at the 2000 SPE Asia PacificConference on Integrated Modelling for Asset Management, Yokohama, Japan, 25-26April.
32. Savioli, G.B., Grattoni, C.A., and Bidner, M.S.: "On the Inverse Problem Application toReservoir Characterization," paper SPE 25522 available from SPE,Richardson, Texas (1992).
33. Deschamps, T. et al.: "The results of testing six different gradientoptimisers on two history matching problems," Proc., 6th European Conference onthe Mathematics of Oil Recovery, Peebles, Scotland (1998).
34. Kitanidis, P.K.: "Quasi-linear geostatistical theory for inversing,"Water Resour. Res. (October 1995) 31, No. 10, 2411.
35. Nocedal, J.: "Updating Quasi-Newton Matrices with Limited Storage,"Math. Comp. (July 1980) 35, No. 151, 773.
36. Nocedal, J. and Wright, S.J.: Numerical Optimization, Springer, New YorkCity (1999).
37. Meijerink, J.A. and van der Vorst, H.A.: "An iterative solution methodfor linear systems of which the coefficient matrix is a symmetric M-matrix,"Mathematics of Computation (January 1977) 31, No. 137, 148.
38. Oliver, D.S.: "Calculation of the Inverse ofthe Covariance," Mathematical Geology (October 1998) 30, No. 7, 911.
39. Skjerheim, J.A. and Oliver, D.S.: "Approximation of the InverseCovariance for History Matching," TUPREP Research Report 19, Tulsa (9 May2002).
40. Aannonsen, S.I. et al.: "Integration of 4D Data in the History MatchLoop by Investigating Scale Dependent Correlations in the Acoustic ImpedanceCube," Proc., 8th European Conference on the Mathematics of Oil Recovery,Freiburg, Germany (2002).
41. Chambers, K.T. et al.: "Characterization of a CarbonateReservoir With Pressure-Transient Tests and Production Logs: Tengiz Field,Kazakhstan," SPEREE (August 2001) 250.
42. He, N. and Chambers, K.T.: "Calibrate Flow Simulation Models withWell-Test Data To Improve History Matching," paper SPE 56681 presented atthe 1999 SPE Annual Technical Conference and Exhibition, Houston, 3-6October.