Reservoir model history matching is a multiobjective optimization problem. It involves the adjustment of relevant reservoir model input parameters, to minimize the mismatch between simulated and observed reservoir responses and to obtain a diverse set of geologically plausible reservoir simulation models. Typically, single objective optimization methods are adopted during history matching. This requires weighted sum scalarization. However, scalarization biases the optimization search, limiting the diversity of the recovered solutions. In this work, a computer assisted history matching procedure based on transform parameterization and a multiobjective evolutionary algorithm with dominance and decomposition (MOEA/DD) is proposed. In the procedure, history matching is treated as a multiobjective optimization problem, parameterized in terms of a small number of kernel principal component analysis (KPCA) variables. KPCA provides efficient parameterization of the reservoir model input property fields. Concurrently, MOEA/DD provides robust and unbiased optimization over multiple objectives. The effectiveness of the proposed procedure is demonstrated with the UNISIM-I-H history matching benchmark problem.
Acknowledgements
The Author thanks Nexen Petroleum UK Limited for supporting this work. The Author also thanks Sultan Djabbarov for reviewing the draft manuscript.
| [1] |
C.S. Kabir, M.C.H. Chien, J.L. Landa, Experiences with Automated History Matching, Society of Petroleum Engineers, 2003, https://doi.org/10.2118/79670-MS.
|
| [2] |
R.W. Rwechungura, M. Dadashpour, J. Kleppe, Advanced History Matching Techniques Reviewed, Society of Petroleum Engineers, 2011, https://doi.org/10.2118/142497-MS.
|
| [3] |
O. Ilamah, M. Ebere, Fast Tracking Field Development Optimisation with Nature Inspired Heuristics, Society of Petroleum Engineers, 2017, https://doi.org/10.2118/189173-MS.
|
| [4] |
P. Jacquard, C. Jain, Permeability distribution from field pressure data, SPE J. 5 (1965) 281-294 https://doi.org/10.2118/1307-PA.
|
| [5] |
M.L. Wasserman, A.S. Emanuel, J.H. Seinfeld, Practical applications of optimalcontrol theory to history-matching multiphase simulator models, SPE J. 15 (1975) 347-355 https://doi.org/10.2118/5020-PA.
|
| [6] |
P. Sarma, L.J. Durlofsky, K. a. Aziz, Efficient real-time reservoir management using adjoint-based optimal control and model updating, Comput. Geosci. 10 (2006) 3-36 https://doi.org/10.1007/s10596-005-9009-z.
|
| [7] |
C.E. Romero, J.N. Carter, R.W. Zimmerman, A. Gringarten, Improved Reservoir Characterization through Evolutionary Computation, Society of Petroleum Engineers, 2000, https://doi.org/10.2118/62942-MS.
|
| [8] |
R.W. Schulze-Riegert, J.K. Axmann, O. Haase, D.T. Rian, Y.-L. You, Evolutionary algorithms applied to history matching of complex reservoirs, SPE Reserv. Eval. Eng. 5 (2002) 163-173 https://doi.org/10.2118/77301-PA.
|
| [9] |
Y. Hajizadeh, M.A. Christie, V. Demyanov, Application of Differential Evolution as a New Method for Automatic History Matching, Society of Petroleum Engineers, 2009, https://doi.org/10.2118/127251-MS.
|
| [10] |
F. Olalotiti-Lawal, T. Onishi, H. a. Kim, Post-combustion CO2 EOR Development in a Mature Oil Field: Model Calibration Using a Hierarchical Approach, Society of Petroleum Engineers, 2017, https://doi.org/10.2118/187116-MS.
|
| [11] |
R.W. Schulze-Riegert, M. Krosche, A. Fahimuddin, S.G. Ghedan, Multi-Objective Optimization with Application to Model Validation and Uncertainty Quantification, Society of Petroleum Engineers, 2007, https://doi.org/10.2118/105313-MS.
|
| [12] |
Y. Hajizadeh, M.A. Christie, V. Demyanov, Towards Multiobjective History Matching: Faster Convergence and Uncertainty Quantification, Society of Petroleum Engineers, 2011, https://doi.org/10.2118/141111-MS.
|
| [13] |
H.Y. Park, A. Datta-Gupta, M.J. King, Handling conflicting multiple objectives using pareto-based evolutionary algorithm during history matching of reservoir performance, J. Pet. Sci. Eng. 125 (2015) 48-66 http://doi.org/10.1016/j.petrol.2014.11.006.
|
| [14] |
B. Min, J.M. Kang, S. Chung, C. Park, I. Jang, Pareto-based multi-objective history matching with respect to individual production performance in a heterogeneous reservoir, J. Pet. Sci. Eng. 122 (2014) 551-566 https://doi.org/10.1016/j.petrol.2014.08.023.
|
| [15] |
F. Olalotiti-Lawal, A. Datta-Gupta, A Multi-Objective Markov Chain Monte Carlo Approach for History Matching and Uncertainty Quantification, Society of Petroleum Engineers, 2015, https://doi.org/10.2118/175144-MS.
|
| [16] |
A. Cominelli, F. Ferdinandi, P.C. de Montleau, R. Rossi, Using gradients to refine parameterization in field-case history-matching projects, SPE Reserv. Eval. Eng. 10 (2007) 233-240 https://doi.org/10.2118/93599-PA.
|
| [17] |
R. Tavakoli, A.C. Reynolds, History matching with parametrization based on the SVD of a dimensionless sensitivity matrix, SPE J. 15 (2010) 495-508 https://doi. org/10.2118/118952-PA.
|
| [18] |
D.S. Oliver, A.C. Reynolds, Z. Bi, Y. Abacioglu, Integration of production data into reservoir models, Pet. Geosci. 7 (2001) S65-S73 https://doi.org/10.1144/petgeo.7. S.S65.
|
| [19] |
E.W. Bhark, B. Jafarpour, A. Datta-Gupta, A generalized grid connectivity-based parameterization for subsurface flow model calibration, Water Resour. Res. 47 (2011), https://doi.org/10.1029/2010WR009982.
|
| [20] |
D.S. Oliver, Albert C. Reynolds, N. Liu, Inverse Theory for Petroleum Reservoir Characterization and History Matching, Cambridge University Press, 2008.
|
| [21] |
M.K. Sen, P.L. Stoffa, Global Optimization Methods in Geophysical Inversion, Cambridge University Press, 2013, https://doi.org/10.1017/CBO9780511997570.
|
| [22] |
K.H. Coats, J.R. Dempsey, J.H. Henderson, A new technique for determining reservoir description from field performance data, SPE J. 10 (1970) 66-74 https://doi.org/10.2118/2344-PA.
|
| [23] |
R.C. Bissell, O. Dubrule, P. Lamy, P. Swaby, O. Lepine, Combining Geostatistical Modelling with Gradient Information for History Matching: the Pilot Point Method, Society of Petroleum Engineers, 1997, https://doi.org/10.2118/38730-MS.
|
| [24] |
M.R.M. Khaninezhad, B. Jafarpour, Hybrid parameterization for robust history matching, SPE J. 19 (2014), https://doi.org/10.2118/146934-PA.
|
| [25] |
E.W. Bhark, A. Datta-Gupta, B. Jafarpour, History Matching with a Multiscale Parameterization Based on Grid Connectivity and Adaptive to Prior Information, Society of Petroleum Engineers, 2011, https://doi.org/10.2118/147372-MS.
|
| [26] |
B. Jafarpour, D. McLaughlin, Reservoir characterization with the discrete cosine transform, SPE J. 14 (2009) 181-202 https://doi.org/10.2118/106453-PA.
|
| [27] |
K. Li, K. Deb, Q. Zhang, S. Kwong, An evolutionary many-objective optimization algorithm based on dominance and decomposition, IEEE Trans. Evol. Comput. 19 (2015) 694-716 https://doi.org/10.1109/TEVC.2014.2373386.
|
| [28] |
Y. Yuan, H. Xu, B. Wang, B. Zhang, X. Yao, Balancing convergence and diversity in decomposition-based many-objective optimizers, IEEE Trans. Evol. Comput. 20 (2016) 180-198 https://doi.org/10.1109/TEVC.2015.2443001.
|
| [29] |
G.G. Yen, Z. He, Performance metric ensemble for multiobjective evolutionary algorithms, IEEE Trans. Evol. Comput. 18 (2014) 131-144 https://doi.org/10.1109/TEVC.2013.2240687.
|
| [30] |
Q. Zhang, A. Zhou, S. Zhao, P.N. Suganthan, W. Liu, S. Tiwari,Multiobjective Optimization Test Instances for the CEC 2009 Special Session and Competition, The School of Computer Science and Engineering, University of Essex Technical, 2009 Report CES-487.
|
| [31] |
O.R. Castro, R. Santana, J.A. Lozano, A. Pozo, Combining CMA-ES and MOEA/DD for many-objective optimization, 2017 IEEE Congress on Evolutionary Computation (CEC), 2017, pp. 1451-1458 https://doi.org/10.1109/CEC.2017.7969474.
|
| [32] |
Q. Zhang, H. Li, Moea/d: a multiobjective evolutionary algorithm based on decomposition, IEEE Trans. Evol. Comput. 11 (2007) 712-731 https://doi.org/10.1109/TEVC.2007.892759.
|
| [33] |
K. Deb, H. Jain, An evolutionary many-objective optimization algorithm using reference-point-based nondominated sorting approach, part i: solving problems with box constraints, IEEE Trans. Evol. Comput. 18 (2014) 577-601 https://doi.org/10.1109/TEVC.2013.2281535.
|
| [34] |
K. Deb, L. Thiele, M. Laumanns, E. Zitzler,Scalable multi-objective optimization test problems, Proceedings of the 2002 Congress on Evolutionary Computation. CEC’02, vol. 1, 2002, pp. 825-830 https://doi.org/10.1109/CEC.2002.1007032.
|
| [35] |
C.M. Bishop, Pattern Recognition and Machine Learning, Springer, New York, 2006.
|
| [36] |
Q. Wang,Kernel principal component analysis and its applications in face recognition and active shape models, arXiv preprint arXiv:1207.3538, 2012. https://arxiv.org/abs/1207.3538.
|
| [37] |
P. Sarma, L.J. Durlofsky, K. Aziz, Kernel principal component analysis for efficient, differentiable parameterization of multipoint geostatistics, Math. Geosci. 40 (2008) 3-32 https://doi.org/10.1007/s11004-007-9131-7.
|
| [38] |
X. Ma, N. Zabaras, Kernel principal component analysis for stochastic input model generation, J. Comput. Phys. 230 (2011) 7311-7331 https://doi.org/10.1016/j.jcp.2011.05.037.
|
| [39] |
B. Schölkopf, A. Smola, K.R. Müller, Kernel principal component analysis, Proceedings of the 7th International Conference on Artificial Neural Networks, Springer, 1997, pp. 583-588 http://dl.acm.org/citation.cfm?id=646257.685385.
|
| [40] |
H.X. Vo, L.J. Durlofsky, Regularized kernel PCA for the efficient parameterization of complex geological models, J. Comput. Phys. 322 (2016) 859-881 https://doi.org/10.1016/j.jcp.2016.07.011.
|
| [41] |
B. Schölkopf, The kernel trick for distances, Advances in Neural Information Processing Systems, 2001, pp. 301-307.
|
| [42] |
C.K. Williams, On a connection between kernel PCA and metric multidimensional scaling, Mach. Learn. 46 (2002) 11-19 https://doi.org/10.1023/A:1012485807823.
|
| [43] |
S. Mika, B. Schölkopf, A.J. Smola, K.-R. Müller, M. Scholz, G. Rätsch, Kernel PCA and de-noising in feature spaces, Advances in Neural Information Processing Systems, 1999, pp. 536-542 http://dl.acm.org/citation.cfm?id=340534.340729.
|
| [44] |
J.T. Kwok, I.W. Tsang,The pre-image problem in kernel methods, Proceedings of the 20th International Conference on Machine Learning (ICML-03), 2003, pp. 408-415.
|
| [45] |
Y. Rathi, S. Dambreville, A. Tannenbaum, Statistical shape analysis using Kernel PCA, Image Processing: Algorithms and Systems, Neural Networks, and Machine Learning, vol. 6064, International Society for Optics and Photonics, 2006, p. 60641B.
|
| [46] |
M. Laumanns, L. Thiele, K. Deb, E. Zitzler, Combining convergence and diversity in evolutionary multiobjective optimization, Evol. Comput. 10 (2002) 263-282.
|
| [47] |
K. Deb, A. Pratap, S. Agarwal, T. Meyarivan, A fast and elitist multiobjective genetic algorithm: NSGA-II, IEEE Trans. Evol. Comput. 6 (2002) 182-197 https://doi.org/10.1109/4235.996017.
|
| [48] |
G.D. Avansi, D.J. Schiozer, UNISIM-I: synthetic model for reservoir development and management applications, International Journal of Modeling and Simulation for the Petroleum Industry 9 (2015) 21-30.
|
| [49] |
C. Maschio, G.D. Avansi, A.A. Santos, D.J. Schiozer, UNISIM-I-H: Case Study for History Matching, Benchmark Dataset, 2013, https://www.unisim.cepetro.unicamp.br/benchmarks/br/unisim-i/unisim-i-h.
|