A Stochastic Optimization Algorithm for Automatic History Matching
- Guohua Gao (Chevron Corp.) | Gaoming Li (U. of Tulsa) | Albert Coburn Reynolds (U. of Tulsa)
- Document ID
- Society of Petroleum Engineers
- SPE Journal
- Publication Date
- June 2007
- Document Type
- Journal Paper
- 196 - 208
- 2007. Society of Petroleum Engineers
- 5.5.1 Simulator Development, 3.3 Well & Reservoir Surveillance and Monitoring, 4.1.5 Processing Equipment, 5.2.1 Phase Behavior and PVT Measurements, 5.6.3 Deterministic Methods, 5.6.4 Drillstem/Well Testing, 4.1.2 Separation and Treating, 5.5.8 History Matching, 5.6.9 Production Forecasting, 5.5 Reservoir Simulation, 5.3.2 Multiphase Flow, 1.2.3 Rock properties, 5.1.5 Geologic Modeling, 4.6 Natural Gas, 5.1 Reservoir Characterisation, 4.3.4 Scale
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For large- scale history- matching problems, optimization algorithms which require only the gradient of the objective function and avoid explicit computation of the Hessian appear to be the best approach. Unfortunately, such algorithms have not been extensively used in practice because computation of the gradient of the objective function by the adjoint method requires explicit knowledge of the simulator numerics and expertise in simulation development. Here we apply the simultaneous perturbation stochastic approximation (SPSA) method to history match multiphase flow production data. SPSA, which has recently attracted considerable international attention in a variety of disciplines, can be easily combined with any reservoir simulator to do automatic history matching. The SPSA method uses stochastic simultaneous perturbation of all parameters to generate a down hill search direction at each iteration. The theoretical basis for this probabilistic perturbation is that the expectation of the search direction generated is the steepest descent direction.
We present modifications for improvement in the convergence behavior of the SPSA algorithm for history matching and compare its performance to the steepest descent, gradual deformation and LBFGS algorithm. Although the convergence properties of the SPSA algorithm are not nearly as good as our most recent implementation of a quasi-Newton method (LBFGS), the SPSA algorithm is not simulator specific and it requires only a few hours of work to combine SPSA with any commercial reservoir simulator to do automatic history matching.
To the best of our knowledge, this is the first introduction of SPSA into the history matching literature. Thus, we make considerable effort to put it in a proper context.
|File Size||5 MB||Number of Pages||13|
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: 10.2118/74707-PA.
Beck, J.V. and Arnold, K.J. 1977. Parameter Estimation in Engineering andScience. Chichester, England, UK: John Wiley & Sons.
Chin, D.C. 1997. Comparative study of stochastic algorithms for systemoptimization based on gradient approximations. IEEE Transactions on Systems,Man and Cybernetics-Part B: Cybernetics 27 (2): 244-249.
Evensen, G. 1994. Sequential data assimilation with a nonlinearquasigeostrophic model using Monte Carlo methods to forecast error statistics.Journal of Geophysical Research 99 (10): 143-162.
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. Tulsa: University ofTulsa.
Gao, G. and Reynolds, A.C. 2004. An Improved Implementation of theLBFGS Algorithm for Automatic History Matching. Paper SPE 90058 presentedat the SPE Annual Technical Conference and Exhibition, Houston, 26-29September. DOI: 10.2118/90058-MS.
Gao, G. and Reynolds, A.C. 2006. An Improved Implementation of theLBFGS Algorithm for Automatic History Matching. SPEJ 11 (1):5-17. SPE-90058-PA. DOI: 10.2118/90058-PA.
Gao, G., Zafari, M., and Reynolds, A.C. 2006. Quantifying Uncertainty for thePUNQ-S3 Problem in a Bayesian Setting With RML and EnKF. SPEJ11 (4): 506-515. SPE-93324-PA. DOI: 10.2118/93324-PA.
Gerencser, L., Hill, S.D., and Vagoo, Z. 2001. Discrete optimization viaSPSA. Proc., American Control Conference.
Hu, L.Y. 2000. Gradual deformation and iterative calibration of gaussian-related stochastic models. Math. Geol. 32 (1): 87-108.
Hu, L.Y., Blanc, G., and Noetinger, B. 2001. Gradual deformation anditerative calibration of sequential stochastic simulations. Math. Geol.33 (4): 475-489.
Hutchison, D.W. and Hill, S.D. 1997. Simulation optimization of airlinedelay with constraints. Proc., IEEE Conf. on Decision and Control.
Kiefer, J. and Wolfowitz, J. 1952. Stochastic estimation of a regressionfunction. Ann. Math. Statist. 23: 462-466.
Kushner, H.J. and Yang, J. 1995. Stochastic approximation with averaging andfeedback: Rapidly convergence online algorithms. IEEE Trans. Autom.Control 40: 24-34.
Le Ravalec, M., Hu, L.Y., and Noetinger, B. 1999. Stochastic Reservoir ModelingConstrained to Dynamic Data: Local Calibration and Inference of the StructuralParameters. Paper SPE 56556 presented at the SPE Annual TechnicalConference and Exhibition, Houston, 3-6 October. DOI: 10.2118/56556-MS.
Le Ravalec, M., Hu, L.Y., and Noetinger, B. 2000. Sampling the conditionalrealization space using the gradual deformation method. Geostatistics 2000Cape Town 1: 176-186.
Le Ravalec, M. and Noeinger, B. 2002. Optimization with the gradualdeformation method. Math. Geol. 34 (2): 125-142.
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:10.2118/87336-PA.
Liu, N. and Oliver, D.S. 2003. Evaluation of Monte Carlo Methods forAssessing Uncertainty. SPEJ 8 (2): 188-195. SPE-84936-PA.DOI: 10.2118/84936-PA.
Naevdal, G., Johnsen, L.M., Aanonsen, S.I., and Vefring, E.H. 2003. Reservoir Monitoring and ContinuousModel Updating Using Ensemble Kalman Filter. SPEJ 10 (1):66-74. SPE-84372-PA. DOI: 10.2118/84372-PA.
Naevdal, G., Mannseth, T., and Vefring, E.H. 2002. Near-Well Reservoir MonitoringThrough Ensemble Kalman Filter. Paper SPE 75235 presented at the SPE/DOEImproved Oil Recovery Symposium, Tulsa, 13-17 April. DOI:10.2118/75235-MS.
Nocedal, J. and Wright, S.J. 1999. Numerical Optimization. New York:Springer.
Oliver, D.S. 1994. Incorporation of transient pressure data into reservoircharacterization. In Situ 18 (3): 243-275.
Omre, H., Hegstad, B.K., and Tjelmeland, H. 1996. Alternative historymatching approaches, technical report. Trondheim, Norway Dept. of MathematicalSciences, Norwegian University of Science and Technology
Reynolds, A.C., He, N., and Oliver, D.S. 1999. Reducing uncertaintainty ingeostatistical description with well testing pressure data. In ReservoirCharacterization-Recent Advances, ed. R.A. Schatzinger and J.F. Jordan.American Association of Petroleum Geologists, 149-162.
Robbins, H. and Monro, S. 1951. A stochastic approximation method. Ann.Math. Statist. 22: 400-407.
Skjervheim, J.A. 2002. Automatic history matching. MS thesis. Trondheim:University of Trondheim.
Spall, J.C. 1992. Multivariate stochastic approximation using a simultaneousperturbation gradient approximation. IEEE Transactions Automat. Control.37 (3): 332-341.
Spall, J.C. 1998. Implementation of the simultaneous perturbation algorithmfor stochastic optimization. IEEE Transactions on Aerospace and ElectronicSystems 34 (3): 817-823.
Spall, J.C. 2000. Adaptive stochastic approximation by the simultaneousperturbation method. IEEE Transactions on Automatic Control 45(10): 1839-1853.
Spall, J.C. 2001. Accelerated second-order stochastic optimization usingonly function measurements. Proc., Winter Simulation Conference.
Tarantola, A. 1987. Inverse Problem Theory: Methods for Data Fitting andModel Parameter Estimation. Amsterdam: Elsevier.
Wu, Z., Reynolds, A.C., and Oliver, D.S. 1999. Conditioning Geostatistical Models toTwo-Phase Production Data. SPEJ 4 (3): 142-155. SPE-56855-PA.DOI: 10.2118/56855-PA.
Zhang, F. and Reynolds, A.C. 2002. Optimization algorithms for automatichistory matching of production data. Proc., European Conference on theMathematics of Oil Recovery.
Zhang, F., Reynolds, A.C., and Oliver, D.S. 2002. Evaluation of thereduction in uncertainty obtained by conditioning a 3D stochastic channel tomultiwell pressure data. Math. Geol. 34 (6): 713-740.