Hybrid Sparse Dictionary Construction Using K-SVD and DCT for History Matching by ES-MDA May 30, 2018 Sungil Kim and Baehyun Min Department of Climate and Energy Systems Engineering EwhaWomans University Sungil Kim & Baehyun Min
Contents 1.Introduction 2.Literature review 3.Methodology 4.Results & Discussion 5.Conclusions 2/19 Sungil Kim & Baehyun Min
Introduction (inverse modeling) or π π§ = π π = π π§ Limited information π§ : reservoir parameters with measurement error and expensive cost π : simulation responses Reliable inverse modeling π : a reservoir simulator Production rate Production rate History Unknown History Reliable Updated Model Time Time Past Future Past Future WOPR, WGPR, P ? ? ? P WBHP ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? Given π π©ππ P ? ? ? P Find π§ 3/19 Sungil Kim & Baehyun Min
Introduction (ensemble-based methods) β« Objective function π π§ = π§ β π§ π π π βπ π§ β π§ π + π π©ππ β π π π βπ π π©ππ β π ππ π§ = π J b , Background term J o , Observation term π§ = π§ π + π(π π―π¨π β π(π§ π )) π = π π§π (π ππ + ππ π ) βπ *Assuming Gaussian dist. (Emerick and Reynolds, 2013; Chen and Oliver, 2013) π§ : state vector (model realization) Transformation of parameters of a channel reservoir π§ π : state vector before update β Distribution modification π : covariance matrix of π§ π β’ Normal Score Transformation (Shin et al. 2010) β’ Level Set (Lorentzen et al., 2013) π : simulated response of a state vector β Image process π π©ππ : observation data β’ Discrete Cosine Transform (DCT) π π―π¨π : perturbed observed data (Jafarpour and McLaughlin, 2007) β Learning algorithm π : covariance matrix of observation error β’ K-Singular Value Decomposition (K-SVD) π : inflating coefficient of π π (Kreutz-Delgado et al., 2003; Aharon et al., 2006) 4/19 Sungil Kim & Baehyun Min
DCT and IDCT application 5/19 Sungil Kim & Baehyun Min
K-SVD for a geological dictionary Words selection A sentence βI love cookiesβ Or the book βRomeo & Julietβ Or even every books of library 6/19 Sungil Kim & Baehyun Min
Procedures of K-SVD 7/19 Sungil Kim & Baehyun Min ΰ·
Literature review βͺ Aharon et al. (2006): showed the efficacy of K-SVD in image reconstruction. βͺ Li and Jafarpour (2010): extracted essences of geologic features in DCT domain. βͺ Liu and Jafarpour (2013): investigated coupling effects of DCT and K-SVD for representations of facies connectivity and flow model calibration. βͺ Sana et al. (2016): built geologic dictionaries from thousands of static reservoir models using K-SVD and updated models by EnKF βͺ Proposed method: geologic dictionary update based on DCT and K-SVD in each assimilation of ES-MDA 8/19 Sungil Kim & Baehyun Min
Methodology (Update of a dictionary in ES-MDA) Methodology (Update of a dictionary in ES-MDA) 9/19 Sungil Kim & Baehyun Min
Methodology (Overall procedure) Update dictionary 10/19 Sungil Kim & Baehyun Min
Experimental setting 11/19 Sungil Kim & Baehyun Min
Dictionaries in each assimilation by the proposed method 12/19 Sungil Kim & Baehyun Min
Updated ensemble samples from five methods 13/19 Sungil Kim & Baehyun Min
Gas rate of the updated ensemble 14/19 Sungil Kim & Baehyun Min
Water rate of the updated ensemble 15/19 Sungil Kim & Baehyun Min
Computation time and error of five methods Only for construction of dictionaries Only for 8 wells on sand Initial ensemble 100% error 16/19 Sungil Kim & Baehyun Min
Conclusions 1. This study proposed a framework of ES-MDA coupled with DCT and K-SVD. 2. This study updated geologic dictionaries with qualified reservoir models considering dynamic observed data during each assimilation of ES-MDA. 3. The proposed method remarkably reduced computational cost and complexity. 4. ES-MDA+DCT+i-K-SVD worked properly and gave overall enhanced performance in terms of channel properties and prediction of productions. 17/19 Sungil Kim & Baehyun Min
References β’ Aharon, M., Elad, M., Bruckstein, A., 2006. K-SVD: An algorithm for designing overcomplete dictionaries for sparse representation. IEEE. T. Signal Proces. 54 (11), 4311 β 4322. β’ Chen,Y., Oliver, D.S., 2013. Levenberg-Marquardt forms of the iterative ensemble smoother for efficient history matching and uncertainty quantification. 17 (4), 689 β 703 β’ Emerick, A.A., Reynolds, A.C., 2013. Ensemble smoother with multiple data assimilation. Comput Geosci. 55 (2013), 3 β 15. β’ Kim, S. , Min, B., Lee, K., Jeong, H., 2018. Integration of an iterative update of sparse geologic dictionaries with ES-MDA for history matching of channelized reservoirs. Geofluids (May 2018, Accepted) β’ Li, L., Jafarpour, B., 2010. Effective solution of nonlinear subsurface flow inverse problems in sparse bases. Inverse Probl. 26 (10), 1 β 24. β’ Liu, E., Jafarpour, B., 2013. Learning sparse geologic dictionaries from low-rank representations of facies connectivity for flow model calibration. Water Resour. Res. 49 (10), 7088 β 7101. β’ Sana, F., Katterbauer, K., Al-Naffouri, T.Y., Hoteit, I., 2016. Orthogonal matching pursuit for enhanced recovery of sparse geological structures with the ensemble Kalman filter. IEEE. J. Sel. Top Appl. 9 (4), 1710 β 1724. β’ Shin, Y., Jeong, H., Choe, J., 2010. Reservoir characterization using an EnKF and a non-parametric approach for highly non-Gaussian permeability fields. Energ. Source Part A. 32 (16), 1569 β 1578. 18/19 Sungil Kim & Baehyun Min
Q & A Thank you for your attention Sungil Kim kim@cerfacs.fr kimsnu@ewha.ac.kr Acknowledgements We are thankful for support by KOGAS 19/19 Sungil Kim & Baehyun Min
Recommend
More recommend
