Towards automatic and adaptive localization for

Towards automatic and adaptive localization for ensemble-based history matching Xiaodong Luo and Tuhin Bhakta, IRIS, a group company of NORCE, Norway

Outline • Background and motivations

• Correlation-based automatic and adaptive localization • Application examples • Discussion and conclusion

Ensemble-based history matching methods Reservoir models

Observational data

History matching (data assimilation) to update reservoir models

✓Ensemble-based history matching methods provide a means of uncertainty quantification (UQ) for the estimated petrophysical parameters (inputs)

Poor UQ performance due to ensemble collapse Desired scenario

Reality: ensemble collapse

Estimates Truth ❑ Ensemble collapse: a phenomenon in which estimated reservoir models become almost identical with very few varieties

Kalman-gain-type localization to tackle ensemble collapse 𝑓

𝑚𝑖𝑢 = 𝑚𝑖 + σ𝑗 𝐾𝑖𝑗 ∆𝑦𝑗

(update formula without localization)

𝑓

𝑚𝑖𝑢 = 𝑚𝑖 + σ𝑗 𝜉𝑖𝑗 𝐾𝑖𝑗 ∆𝑦𝑗 (update formula with localization) The tapering coefficient 𝜉𝑖𝑗 ∈ [0,1] depends on the specific localization method

Correlation-based adaptive localization in previous work (1/2) The tapering rule is thus as follows* 𝑁

𝑁

1, 𝑖𝑓 𝜌𝑖𝑗𝑒 ≥ 𝜆𝐺𝑖𝑒𝑗 𝑁𝑒 𝑁𝑒 𝜉𝑖𝑗 = 𝐼 𝜌𝑖𝑗 ≥ 𝜆𝐺𝑖𝑗 ≡ ൝ 0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 𝑁 where 𝜌𝑖𝑗𝑒 is the sample correlation coefficient (with 𝑁𝑒 samples ) 𝑁

between 𝑚𝑖 and the jth simulated observation element 𝑦𝑗 , 𝜆𝐺𝑖𝑒𝑗 is a threshold value with respect to a group of model variables (indexed by 𝐺𝑖 ). *Luo X., Bhakta T. and Nævdal, G. Correlation-based adaptive localization with applications to ensemblebased 4D-seismic history matching, SPE Journal, vol. 23, pp. 396-427, 2018. SPE-185936-PA.

Correlation-based adaptive localization in previous work (2/2) Overcomes/mitigates some noticed issues in convectional distance-based localization* Dependence on physical distance

Usability/re-usability

Effect of ensemble size

Non-local observations

ISSUES

Time-lapse observations

Different types of model-data pairs

*Luo, X, Lorentzen, R., Valestrand, R. & Evensen, G. (2018). Correlation-based adaptive localization for ensemble-based history matching: Applied to the Norne field case study. SPE Reservoir Evaluation & Engineering, in press. SPE-191305-PA

Outline • Background and motivations

• Correlation-based automatic and adaptive localization • Application examples • Discussion and conclusion

Choosing threshold values 𝑁

The choice of threshold values 𝜆𝐺𝑖𝑒𝑗 is crucial to the performance of correlation-based localization (CBL). Here is our proposed approach to the choice of 𝑁

𝑁

𝑁𝑒 𝜆𝐺𝑖𝑗

:

𝑒 𝝆𝐺𝑖𝑒𝑗 = 𝝆∞ + 𝝐 𝐺𝑖 𝑗 𝐺𝑖 𝑗 (decomposition of sample correlation fields)

𝑁 𝑁𝑒 correlation fields with anoninfinite sizesample size 𝑁𝑒 Sampling errors (noise) tosample the finite 𝜆𝐺𝑖𝑒𝑗True are then determined based 𝝐due 𝐺𝑖 𝑗 *:

𝑁

𝝐𝐺𝑖𝑒𝑗

𝑁

Noise levels of 𝝐𝐺𝑖𝑒𝑗

𝑁

Thresholds 𝜆𝐺𝑖𝑒𝑗

*Luo X., Bhakta T. and Nævdal, G. Correlation-based adaptive localization with applications to ensemblebased 4D-seismic history matching, SPE Journal, vol. 23, pp. 396-427, 2018. SPE-185936-PA.

Approach to obtaining sampling errors in previous work Sampling errors 𝑁

𝑁 𝝐𝐺𝑖𝑒𝑗

Noise levels of

𝑁𝑒 𝝐𝐺𝑖 𝑗

𝑁

Thresholds 𝜆𝐺𝑖𝑒𝑗

𝑁

𝑒 𝝆𝐺𝑖𝑒𝑗 = 𝝆∞ + 𝝐 𝐺𝑖 𝑗 𝐺𝑖 𝑗 (decomposition of sample correlation fields)

𝑁

𝐿 An automatic approach (Diff): Approximate 𝝆∞ 𝐺𝑖 𝑗 by another sample correlation filed 𝝆𝐺𝑖 𝑗

𝑁

𝑁

𝑁

with a relatively large ensemble/sample size 𝑁𝐿 , hence 𝝐𝐺𝑖𝑒𝑗 ≈ 𝝆𝐺𝑖𝑒𝑗 − 𝝆𝐺𝑖𝐿𝑗 . Disadvantage: Computationally expensive/impractical to run a large number of forward reservoir simulations

A workflow for automatic noise-level estimation (1/3) 𝑁𝑒 𝝆𝐺𝑖 𝑗

=

𝝆∞ 𝐺𝑖 𝑗

𝑁𝑒 + 𝝐𝐺𝑖 𝑗

(decomposition of sample correlation fields) 𝑁

We consider to find substitute sampling errors 𝝐෤ 𝐺𝑖𝑒𝑗 , and

𝑁

estimate noise levels/compute threshold values based on 𝝐෤ 𝐺𝑖𝑒𝑗

A workflow for automatic noise-level estimation (2/3) 𝑁𝑒 𝝆𝐺𝑖 𝑗

=

𝝆∞ 𝐺𝑖 𝑗

𝑁𝑒 + 𝝐𝐺𝑖 𝑗

(decomposition of sample correlation fields) General idea

𝑁

• We know 𝝆𝐺𝑖𝑒𝑗 is the sample correlation coefficient between 𝒎𝒊 and the jth simulated observation element 𝑦𝑗 ෥ 𝑖 which is independent of 𝑦𝑗 , then we have • If there is another 𝒎 𝑁 𝑁 𝑁 ෥∞ ෥𝐺 𝑒𝑗 = 𝝆 ෥∞ ෤ 𝐺 𝑒𝑗 = 𝝐෤ 𝐺 𝑒𝑗 𝝆 𝐺𝑖 𝑗 = 0; hence 𝝆 𝐺𝑖 𝑗 + 𝝐 𝑖

𝑖

𝑖

𝑁

෥𝐺 𝑒𝑗 can be used as substitute • This means that the sample correlation field 𝝆 𝑖 𝑁

sampling errors for 𝝐𝐺𝑖𝑒𝑗 for the calculation of noise levels (hence correlation thresholds) 𝑁 ෥𝐺𝑖𝑒𝑗 𝝆

𝑁

෥𝐺𝑖𝑒𝑗 Noise levels of 𝝆

𝑁

Thresholds 𝜆𝐺𝑖𝑒𝑗

A workflow for automatic noise-level estimation (3/3) Implementation example Simulated data set 1 (𝒚𝟏 )

Simulated data set 2 (𝒚𝟐 ) Simulated data set 3 (𝒚𝟑 )

…... I.I.D model 1 (𝒎𝟏 )

I.I.D model 2 (𝒎𝟐 )

I.I.D model 3 (𝒎𝟑 )

A workflow for automatic noise-level estimation (3/3) Implementation example

𝑁𝑒 {𝒎𝑖 }𝑖=1 :

𝑀≡ The ensemble of reservoir models 𝑁

𝑒 𝐷 ≡ {𝑌𝑖 }𝑖=1 : The corresponding ensemble of simulated data

Original sample correlation 𝑁 fields 𝝆𝐺𝑖𝑒𝑗

A workflow for automatic noise-level estimation (3/3) Implementation example Procedures to get substitute sampling errors for the computation of threshold values No change to the ensemble of simulated data

Simulated data set 1 (𝒚𝟏 )

Simulated data set 2 (𝒚𝟐 ) Simulated data set 3 (𝒚𝟑 )

…...

A workflow for automatic noise-level estimation (3/3) Implementation example Randomly shuffle the order of reservoir models

I.I.D model 1 (𝒎𝟏 )

I.I.D model 2 (𝒎𝟐 )

I.I.D model 3 (𝒎𝟑 )

…...

A workflow for automatic noise-level estimation (3/3) Implementation example I.I.D model 3 (𝒎𝟑 )

I.I.D model 1 (𝒎𝟏 )

I.I.D model 2 (𝒎𝟐 )

…... 𝑁

This results in a new ensemble 𝑀′ ≡ {𝒎𝑖 ′ : 𝒎𝑖 ′ ≠ 𝒎𝑖 , 𝑖 ′ = 𝑖}𝑖 ′𝑒=1 of reservoir models, which differs from the original ensemble 𝑁𝑒 𝑀 = {𝒎𝑖 }𝑖=1 only in the order of the reservoir models

A workflow for automatic noise-level estimation (3/3) Implementation example

Simple, efficient, and automatic without manual tuning

𝑁

෥𝐺𝑖𝑒𝑗 as the desired Sample correlations 𝝆 substitute sampling errors

Demonstration in a 2D example (1/3) Sample correlation maps between porosity (PORO) and an observation

Ens-100 Sampling errors obtained by

Diff (Ens-100 – Ens-3000)

Ens-3000

Random shuffle (RndShfl)

Demonstration in a 2D example (3/3) Estimated noise levels for two example observations

Diff (Ens-100 – Ens-3000)

RndShfl (mean ± STD w.r.t 20 experiments)

Obs-1

0.1120

0.0946 ± 0.0089

Obs-2

0.1039

0.0982 ± 0.0080

Approach

Outline • Background and motivations

• Correlation-based automatic and adaptive localization • Application examples • Discussion and conclusion

Synthetic 2D Norne field case Information summary of the 2D Norne case study* Model dimension

39 × 26 (1014 gridblocks), with 739 out of 1014 being active cells

Parameters to estimate

X-directional permeability (PERMX) and porosity (PORO) on active gridblocks. Total number: 2 × 739 = 1478

Seismic data

Leading wavelet coefficients of amplitude-versusangle (AVA) data, as in Scenario (S2) of Luo et al. (2018)*. Total number: 3757

History matching method

Iterative ensemble smoother (Luo et al., 2015) with an ensemble of 100 reservoir models, together with correlation-based localization

*Luo X., Bhakta T. and Nævdal, G. Correlation-based adaptive Localization for ensemblebased history matching methods, SPE Journal, vol. 23, pp. 396-427, 2018. SPE-185936-PA.

Synthetic 2D Norne field case RMSE (Diff-induced CBL)

RMSE (RndShfl-induced CBL)

PERMX

PORO

Iteration step

Iteration step

3D Brugge benchmark case Information summary of the Brugge benchmark case study* Model dimension

139 × 48 × 9, with 44550 out of 60048 being active cells

Parameters to estimate PORO, PERMX, PERMY, PERMZ. Total number is 4 × 44550 = 178200 Seismic data

Leading wavelet coefficients of the AVA data. Total number: 3293

History matching method

Iterative ensemble smoother (Luo et al., 2015) with an ensemble of 103 reservoir models

*Luo X., Bhakta T. and Nævdal, G. Correlation-based adaptive Localization for ensemblebased history matching methods, SPE Journal, vol. 23, pp. 396-427, 2018. SPE-185936-PA.

3D Brugge benchmark case Diff-induced CBL* is not considered in this case, due to the potentially high computational cost. Results below are w.r.t RndShfl-induced CBL Data mismatch

RMSE (PERMX)

RMSE (PORO)

*Luo X., Bhakta T. and Nævdal, G. Correlation-based adaptive Localization for ensemblebased history matching methods, SPE Journal, vol. 23, pp. 396-427, 2018. SPE-185936-PA.

Outline • Background and motivations

• Correlation-based automatic and adaptive localization • Application examples • Discussion and conclusion

Discussion and conclusion • Correlation-based localization (CBL) helps overcome/mitigate some longstanding issues in conventional distance-based localization • The current work proposes an automatic workflow to choose threshold values for CBL; In turn, this forms the basis of developing a (correlation-based) automatic and adaptive localization (AutoAdaLoc) scheme • AutoAdaLoc is simple to implement, computationally very efficient, and does not involve any manual tuning. It achieves good history matching performance in both case studies • Tests on real field case studies are of interest

Acknowledgements / Thank You / Questions XL acknowledges partial financial supports from the CIPR/IRIS cooperative research project “4D Seismic History Matching” which is funded by industry partners Eni Norge AS, Petrobras, and Total EP Norge, as well as the Research Council of Norway (PETROMAKS). Both authors acknowledge the Research Council of Norway and the industry partners – ConocoPhillips Skandinavia AS, Aker BP ASA, Eni Norge AS, Total E&P Norge AS, Equinor ASA, Neptune Energy Norge AS, Lundin Norway AS, Halliburton AS, Schlumberger Norge AS, Wintershall Norge AS, and DEA Norge AS – of The National IOR Centre of Norway for partial financial supports.