An ensemble-based kernel le learning framework to handle data assimilation problems wit ith im imperfect forw rward simulators Xiaodong Luo, NORCE
Outline • Background and motivation • An ensemble-based kernel algorithm for supervised learning • From supervised learning to data assimilation with model errors • Synthetical examples and a real field application • Discussion and conclusion
Seismic survey for hydrocarbon reservoir monitoring and management More advanced techniques available e.g., Ocean Bottom Cable (OBC) or even Permanent Reservoir Monitoring (PRM) system Source: https://oilnow.gy/
Seismic history matching (S (SHM) SHM involves using (3D or 4D) seismic data to estimate properties of reservoir formations Forward seismic simulator Input: Output: petrophysical seismic data parameters History matching algorithm
Forw rward seismic simulation and inversion Prone to model errors Impedance Impedance ( 𝑤 𝑞 , 𝑤 𝑡 , 𝜍 ) ( 𝑤 𝑞 , 𝑤 𝑡 , 𝜍 ) Forward simulation Rock physics Inversion model Saturation Pressure Reservoir simulation Petrophysical Petrophysical parameters parameters
Motivation Develop a workflow to account for model errors in rock physics models (RPM)
Outline • Background and motivation • An ensemble-based kernel algorithm for supervised learning • From supervised learning to data assimilation with model errors • Synthetical examples and a real field application • Discussion and conclusion
Supervised learning (1 (1/3) Outputs (y) 𝑂 𝑡 with 𝑂 𝑡 • We have a set of inputs 𝑌 ≡ {𝑦 𝑗 } 𝑗=1 𝑧 1 samples; and a corresponding set of noisy 𝑧 3 𝑧 2 𝑧 𝑂 𝑡 𝑂 𝑡 outputs 𝑍 ≡ {𝑧 𝑗 } 𝑗=1 … • We want to learn a function ℎ so that ℎ 𝑦 𝑗 match 𝑧 𝑗 to a good extent, 𝑦 1 for 𝑗 = 1, 2, … , 𝑂 𝑡 𝑦 3 𝑦 2 𝑦 𝑂 𝑡 Inputs (x)
Supervised learning (2 (2/3) To this end, we solve a functional optimization (known as empirical risk minimization , ERM ) problem to find the optimal ℎ ∗ 1 2 + 𝛿 𝑆(||ℎ||) ℎ ∗ = argmin ∑ 𝑧 𝑗 − ℎ 𝑦 𝑗 𝑂 𝑡 ℎ 𝑗 • 𝛿 : regularization parameter regularization functional to avoid overfitting, e.g., 𝑆 𝑦 = 𝑦 2 • R: • ||ℎ|| : functional norm in a certain function space
Supervised learning (3 (3/3) To solve the ERM problem, in practice, one strategy is to adopt a parametric model that can be used to approximate a functional Then the ERM problem is converted to a parameter estimation problem, i.e., 1 2 + 𝛿 𝑆(||ℎ||) ℎ ∗ = argmin ∑ 𝑧 𝑗 − ℎ 𝑦 𝑗 𝑂 𝑡 ℎ 𝑗 ℎ 𝜄; 𝑦 𝑗 ≈ ℎ 𝑦 𝑗 1 2 + 𝛿 𝑆 𝜄 ; 𝜄 ∗ = argmin ∑ 𝑧 𝑗 − ℎ 𝜄; 𝑦 𝑗 𝑂 𝑡 𝑗 𝜄 ℎ 𝜄; 𝑦 𝑗 ≈ ℎ 𝑦 𝑗 Examples of parametric model for functional approximation: generalized linear models support vector machines (SVM) (shallow or deep) neural networks
Ensemble-based supervised learning (1/2) (1 1 2 + 𝛿 𝑆 𝜄 𝜄 ∗ = argmin ∑ 𝑧 𝑗 − ℎ 𝜄; 𝑦 𝑗 𝑂 𝑡 𝑗 𝜄 Similar to a Variational Data vectorize Assimilation (Var) problem 2 + 𝛿 𝑆 𝜄 𝜄 ∗ = argmin 𝑍 − 𝐼 𝜄; 𝑌 𝜄 Naturally, in light of the developments of ensemble based data assimilation methods, instead of estimating a single set 𝜄 of parameters, we can estimate an ensemble 𝑂 𝑓 Θ ≡ {𝜄 𝑘 } 𝑘=1 of such parameters
Ensemble-based supervised learning (2 (2/2) We will obtain all the benefits in using 2 + 𝛿 𝑆 𝜄 𝜄 ∗ = argmin ensemble based methods: 𝑍 − 𝐼 𝜄; 𝑌 • Adjoint free 𝜄 ensemblize • Uncertainty quantification 2 Θ ∗ = argmin 1 𝑂 𝑓 { ∑ 𝑍 − 𝐼 𝜄 𝑘 ; 𝑌 + 𝛿 𝑆 𝜄 𝑘 } • Fast implementation 𝑂𝑓 𝑘 Θ={𝜄 𝑘 } 𝑘=1 Iterative ensemble smoothers, e.g., Luo et al. 2015*, can be used to solve the ensemble-based (supervised) learning problem *Luo, X., Stordal, A. S., Lorentzen, R. J., & Naevdal, G. (2015). Iterative Ensemble Smoother as an Approximate Solution to a Regularized Minimum-Average-Cost Problem: Theory and Applications. SPE Journal , 20 , 962-982.
Kernel method for fu functional approximation 𝑑𝑞 ℎ 𝑦; 𝜄 = 𝑑 𝑙 𝐿( 𝑦 − 𝑦 𝑙 ; 𝛾 𝑙 ) 𝑙 T 𝜄 = 𝑑 1 , 𝑑 2 , … 𝑑 𝑂 𝑡𝑞 ; 𝛾 1 , 𝛾 2 , … 𝛾 𝑂 𝑡𝑞 𝑑𝑞 ( 𝑙 = 1, 2, … , 𝑂 𝑡𝑞 ), where for a set of “center points” 𝑦 𝑙 • 𝑑 𝑙 and 𝛾 𝑙 are parameters associated with the k-th center point • K is a certain kernel function. Here we use Gaussian kernel 𝑑𝑞 2 2 𝑦−𝑦 𝑙 𝑑𝑞 = 𝑓 −𝛾 𝑙 𝐿 𝑦 − 𝑦 𝑙 ; 𝛾 𝑙
Outline • Background and motivation • An ensemble-based kernel algorithm for supervised learning • From supervised learning to data assimilation with model errors • Synthetical examples and a real field application • Discussion and conclusion
Problem statement (1 (1/3) • Problem in consideration: 𝒛 𝑝 = 𝒈 𝒚 𝑢𝑠 + 𝝑 where • 𝒛 𝑝 : observed output (observation) • 𝒚 𝑢𝑠 : underlying true model variables that generate 𝒛 𝒑 through the true forward simulator 𝒈 • 𝒈 : true (but unknown) forward simulator • 𝝑 : observation noise. 𝝑~𝑶(𝟏, 𝑫 𝒆 )
Problem statement (2 (2/3) • In history matching (data assimilation), we may use the following forward simulation system 𝒛 𝑡𝑗𝑛 = 𝒉 𝒚 where • 𝒛 𝑡𝑗𝑛 : simulated observation • 𝒚 : model variables to be estimated • 𝒉 : imperfect forward simulator
Problem statement (3 (3/3) 𝒛 𝒑 = 𝒉 𝒚 + 𝒛 𝒑 − 𝒉 𝒚 ≈ 𝒉 𝒚 + 𝒔(𝒚, 𝜾) Kernel methods (or other machine learning models) can be used to reparametrize/approximate the residual term* 𝒑 , 𝒚 𝒅𝒒 ) ≡ 𝒔(𝒚, 𝜾; 𝒛 𝒑 , 𝒛 𝒅𝒒 𝒔 𝒚, 𝜾 so instead of trying to find an optimal functional form for 𝒔 , we optimize/estimate a set 𝜾 of parameters (as well as 𝒚 ) instead. X. Luo, 2019. Ensemble-based kernel learning for a class of data assimilation problems with imperfect forward simulators. Available from arXiv:1901.10758
Ensembled-based data assim imilation wit ith kernel approximation to the residual term 𝑈 −1 𝒛 𝒑 − 𝒉 𝒚 𝒌 Θ ∗ = 𝒛 𝒑 − 𝒉 𝒚 𝒌 argmin ∑ − 𝒔 𝒚 𝒌 , 𝜾 𝒌 𝐷 𝑒 − 𝒔 𝒚 𝒌 , 𝜾 𝒌 + 𝛿 𝑆 [𝒚 𝒌 ; 𝜾 𝒌 ] 𝑂𝑓 𝑘 Θ={[𝒚 𝒌 ;𝜾 𝒌 ]} 𝑘=1 • This optimization problem can still be solved through an iterative ensemble smoother • We need to jointly estimate/update 𝒚 𝒌 and 𝜾 𝒌 • In implementation, it just means that we augment 𝒚 𝒌 and 𝜾 𝒌 into model variable vectors that will be updated
Outline • Background and motivation • An ensemble-based kernel algorithm for supervised learning • From supervised learning to data assimilation with model errors • Synthetical examples and a real field application • Discussion and conclusion
Synthetic example 1: : superv rvised learning Blue: Ensemble of predicted functions Cyan (solid): Ensemble mean Red (dashed): reference function Red (dashed): reference function Green (dashed): biased function Green (dashed): biased function Initial ensemble Final ensemble
Synthetic example 2: : data assimilation Truth Mean of initial ensemble Mean of final ensemble Mean of final ensemble (with model error correction) (no model error correction)
More information and results of both synthetical examples (supervised learning and data assimilation) can be found in the preprint X. Luo, 2019. Ensemble-based kernel learning for a class of data assimilation problems with imperfect forward simulators. Available from arXiv:1901.10758
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Experimental settings Types of settings Values/Info Reservoir model size: 46 x 112 x 22 Seismic data (four surveys) Acoustic impedance on each active gridblock Total number: 453,376; reduced to 24,232 through wavelet-based sparse representation* Production data (1997 - 2006) WOPRH, WGPRH, WWPRH Total number: 5,038 Model variables to estimate PERM, PORO, NTG etc. Total number: 148,183 History matching algorithm Iterative ES (Luo et al. 2015) + correlation-based adaptive localization (Luo et al. 2018, 2019) *X Luo, T Bhakta, M Jakobsen, G Nævdal, 2017. An ensemble 4D-seismic history-matching framework with sparse representation based on wavelet multiresolution analysis. SPE Journal , 22, 985 - 1,010
Experimental settings The setting without model error correction (MEC) Impedance Impedance ( 𝑤 𝑞 , 𝑤 𝑡 , 𝜍 ) ( 𝑤 𝑞 , 𝑤 𝑡 , 𝜍 ) Forward simulation Rock physics Inversion model Saturation Pressure Reservoir simulation Petrophysical Petrophysical parameters parameters
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