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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

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An ensemble-based kernel le learning framework to handle data assimilation problems wit ith im imperfect forw rward simulators

Xiaodong Luo, NORCE

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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
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Seismic survey for hydrocarbon reservoir monitoring and management

Source: https://oilnow.gy/ More advanced techniques available e.g., Ocean Bottom Cable (OBC) or even Permanent Reservoir Monitoring (PRM) system

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Output: seismic data Input: petrophysical parameters Forward seismic simulator History matching algorithm

Seismic history matching (S (SHM)

SHM involves using (3D or 4D) seismic data to estimate properties of reservoir formations

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Prone to model errors

Rock physics model

Impedance (π‘€π‘ž, 𝑀𝑑, 𝜍) Impedance (π‘€π‘ž, 𝑀𝑑, 𝜍) Petrophysical parameters Forward simulation Petrophysical parameters Saturation Pressure

Reservoir simulation

Inversion

Forw rward seismic simulation and inversion

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Motivation

Develop a workflow to account for model errors in rock physics models (RPM)

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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
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Supervised learning (1 (1/3)

…

𝑦1 𝑧1 𝑦2 𝑧2 𝑦3 𝑧3 𝑦𝑂𝑑 𝑧𝑂𝑑 Inputs (x) Outputs (y)

  • We have a set of inputs π‘Œ ≑ {𝑦𝑗}𝑗=1

𝑂𝑑 with 𝑂𝑑

samples; and a corresponding set of noisy

  • utputs 𝑍 ≑ {𝑧𝑗}𝑗=1

𝑂𝑑

  • We want to learn a function β„Ž so that

β„Ž 𝑦𝑗 match 𝑧𝑗 to a good extent, for 𝑗 = 1, 2, … , 𝑂𝑑

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Supervised learning (2 (2/3)

To this end, we solve a functional optimization (known as empirical risk minimization, ERM) problem to find the optimal β„Žβˆ— β„Žβˆ— = argmin

β„Ž

1 𝑂𝑑 βˆ‘

𝑗

𝑧𝑗 βˆ’ β„Ž 𝑦𝑗

2 + 𝛿 𝑆(||β„Ž||)

  • 𝛿:

regularization parameter

  • R:

regularization functional to avoid overfitting, e.g., 𝑆 𝑦 = 𝑦2

  • ||β„Ž||:

functional norm in a certain function space

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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., β„Žβˆ— = argmin

β„Ž

1 𝑂𝑑 βˆ‘

𝑗

𝑧𝑗 βˆ’ β„Ž 𝑦𝑗

2 + 𝛿 𝑆(||β„Ž||)

πœ„βˆ— = argmin

πœ„

1 𝑂𝑑 βˆ‘

𝑗

𝑧𝑗 βˆ’ β„Ž πœ„; 𝑦𝑗

2 + 𝛿 𝑆 πœ„ ;

Examples of parametric model for functional approximation: generalized linear models support vector machines (SVM) (shallow or deep) neural networks β„Ž πœ„; 𝑦𝑗 β‰ˆ β„Ž 𝑦𝑗 β„Ž πœ„; 𝑦𝑗 β‰ˆ β„Ž 𝑦𝑗

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Ensemble-based supervised learning (1 (1/2)

πœ„βˆ— = argmin

πœ„

1 𝑂𝑑 βˆ‘

𝑗

𝑧𝑗 βˆ’ β„Ž πœ„; 𝑦𝑗

2 + 𝛿 𝑆 πœ„

πœ„βˆ— = argmin

πœ„

𝑍 βˆ’ 𝐼 πœ„; π‘Œ

2 + 𝛿 𝑆 πœ„

vectorize Naturally, in light of the developments of ensemble based data assimilation methods, instead of estimating a single set πœ„ of parameters, we can estimate an ensemble Θ ≑ {πœ„π‘˜}π‘˜=1

𝑂𝑓

  • f such parameters

Similar to a Variational Data Assimilation (Var) problem

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Ensemble-based supervised learning (2 (2/2)

πœ„βˆ— = argmin

πœ„

𝑍 βˆ’ 𝐼 πœ„; π‘Œ

2 + 𝛿 𝑆 πœ„

Ξ˜βˆ— = argmin

Θ={πœ„π‘˜}π‘˜=1

𝑂𝑓

1 𝑂𝑓 { βˆ‘ π‘˜

𝑍 βˆ’ 𝐼 πœ„

π‘˜; π‘Œ 2

+ 𝛿 𝑆 πœ„

π‘˜ }

ensemblize We will obtain all the benefits in using ensemble based methods:

  • Adjoint free
  • Uncertainty quantification
  • Fast implementation

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.

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Kernel method for fu functional approximation

β„Ž 𝑦; πœ„ = ෍

𝑙

𝑑𝑙 𝐿( 𝑦 βˆ’ 𝑦𝑙

π‘‘π‘ž

; 𝛾𝑙) πœ„ = 𝑑1, 𝑑2, … π‘‘π‘‚π‘‘π‘ž; 𝛾1, 𝛾2, … π›Ύπ‘‚π‘‘π‘ž

T

for a set of β€œcenter points” 𝑦𝑙

π‘‘π‘ž (𝑙 = 1, 2, … , π‘‚π‘‘π‘ž), where

  • 𝑑𝑙 and 𝛾𝑙 are parameters associated with the k-th center point
  • K is a certain kernel function. Here we use Gaussian kernel

𝐿 𝑦 βˆ’ 𝑦𝑙

π‘‘π‘ž

; 𝛾𝑙 = π‘“βˆ’π›Ύπ‘™

2 π‘¦βˆ’π‘¦π‘™ π‘‘π‘ž 2

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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
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Problem statement (1 (1/3)

  • Problem in consideration:

𝒛𝑝 = π’ˆ π’šπ‘’π‘  + 𝝑 where

  • 𝒛𝑝:
  • bserved output (observation)
  • π’šπ‘’π‘ :

underlying true model variables that generate 𝒛𝒑 through the true forward simulator π’ˆ

  • π’ˆ:

true (but unknown) forward simulator

  • 𝝑:
  • bservation noise. 𝝑~𝑢(𝟏, 𝑫𝒆)
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Problem statement (2 (2/3)

  • In history matching (data assimilation), we may use the

following forward simulation system 𝒛𝑑𝑗𝑛 = 𝒉 π’š where

  • 𝒛𝑑𝑗𝑛:

simulated observation

  • π’š:

model variables to be estimated

  • 𝒉:

imperfect forward simulator

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Problem statement (3 (3/3)

𝒛𝒑= 𝒉 π’š + 𝒛𝒑 βˆ’ 𝒉 π’š β‰ˆ 𝒉 π’š + 𝒔(π’š, 𝜾) Kernel methods (or other machine learning models) can be used to reparametrize/approximate the residual term* 𝒔 π’š, 𝜾 ≑ 𝒔(π’š, 𝜾; 𝒛𝒑, 𝒛𝒅𝒒

𝒑 , π’šπ’…π’’)

so instead of trying to find an optimal functional form for 𝒔, we

  • ptimize/estimate a set 𝜾 of parameters (as well as π’š) instead.
  • 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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Ensembled-based data assim imilation wit ith kernel approximation to the residual term

Ξ˜βˆ— = argmin

Θ={[π’šπ’Œ;πœΎπ’Œ]}π‘˜=1

𝑂𝑓

βˆ‘

π‘˜

𝒛𝒑 βˆ’ 𝒉 π’šπ’Œ βˆ’ 𝒔 π’šπ’Œ, πœΎπ’Œ

π‘ˆ

𝐷𝑒

βˆ’1 𝒛𝒑 βˆ’ 𝒉 π’šπ’Œ

βˆ’ 𝒔 π’šπ’Œ, πœΎπ’Œ + 𝛿 𝑆 [π’šπ’Œ; πœΎπ’Œ]

  • This optimization problem can still be solved through an iterative ensemble smoother
  • We need to jointly estimate/update π’šπ’Œ and πœΎπ’Œ
  • In implementation, it just means that we augment π’šπ’Œ and πœΎπ’Œ into model variable

vectors that will be updated

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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
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Synthetic example 1: : superv rvised learning

Blue: Ensemble of predicted functions Red (dashed): reference function Green (dashed): biased function Cyan (solid): Ensemble mean Red (dashed): reference function Green (dashed): biased function Initial ensemble Final ensemble

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Synthetic example 2: : data assimilation

Truth Mean of initial ensemble Mean of final ensemble (no model error correction) Mean of final ensemble (with model error correction)

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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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Real l fie field ld ap appli licatio ion: : ac accountin ing for

  • r roc
  • ck-physics-

model l im imperfection in in his istory matchin ing se seis ismic ic data fr from No Norne fie field ld In In coll

  • llaboratio

ion wit ith my col

  • lle

leagues Rolf lf Lo Lorentzen, Tuhin in Bhakta

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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

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Experimental settings

Rock physics model

Impedance (π‘€π‘ž, 𝑀𝑑, 𝜍) Impedance (π‘€π‘ž, 𝑀𝑑, 𝜍) Petrophysical parameters Petrophysical parameters Saturation Pressure

Reservoir simulation

Inversion Forward simulation

The setting without model error correction (MEC)

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Experimental settings

Rock physics model Kernel-based residual model

Impedance (π‘€π‘ž, 𝑀𝑑, 𝜍) Impedance (π‘€π‘ž, 𝑀𝑑, 𝜍) Petrophysical parameters Petrophysical parameters Saturation Pressure

Reservoir simulation

Inversion Forward simulation

The setting with model error correction (MEC)

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Experimental settings

Kernel-based residual model (inputs/output) at each active gridblock

Input 5: Water Saturation Input 4: Gas Saturation Input 3: Pressure

NTG Pressure PORO Gas Saturation Water Saturation

Output: Impedance

Input 1: PORO Input 2: NTG

Total number of kernel parameters: 120,000 (with 20,000 center points)

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Experimental settings

  • Only seismic data are used history matching
  • Production data are reserved for cross-validation
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Experimental results: data mismatch

Results without MEC Results with MEC Seismic data mismatch (history matching) Production data mismatch (cross validation)

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Experimental results: mismatch reduction

Reductions of average production data mismatch with respect to the initial ensemble

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Experimental results: forecast

Initial ensemble Final ensemble (no MEC) Final ensemble (with MEC) Predicted water production rates (WPR) at well D-1H

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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
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Discussion and conclusion

  • We show similarities between supervised learning and data

assimilation; As such, it becomes natural for us to develop an ensemble-based framework for supervised learning problems

  • With minor modifications, ensemble-based learning can also

be extended to handle data assimilation problems in the presence of model errors

  • The integrated data assimilation framework appears to be

useful for improving DA performance in both synthetical and real-world problems presented here

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Acknowledgement

XL acknowledges financial supports from the β€œDIGIRES” project (RCN no. 280473) funded by indutry partners: AkerBP ASA, DEA Norge AS, EquiNor ASA, Lundin Norway AS, Neptun Energy Norge AS, Petrobras and VΓ₯r Energi AS, as well as the Research Council

  • f Norway
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Q&A