based history matching using wavelets EnKF workshop Bergen May - - PowerPoint PPT Presentation

Théophile Gentilhomme, Dean Oliver, Trond Mannesth, Remi Moyen, Guillaume Caumon

Adaptive multi-scale ensemble- based history matching using wavelets

6/4/2013

“EnKF workshop “ Bergen May 2013

Motivations

• Match the data and preserve the prior models

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6/4/2013 I - 2/21

Motivations

• Match the data and preserve the prior models

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6/4/2013 I - 2/21

Optimization

Motivations

• Match the data and preserve the prior models
• Multi-scale approach:
• Helps to avoid local minima
• Stabilizes the inversion
• Modifies low resolution first

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6/4/2013 I - 2/21

Motivations

• Match the data and preserve the prior models
• Multi-scale approach:
• Helps to avoid local minima
• Stabilizes the inversion
• Modifies low resolution first

    

6/4/2013 I - 2/21

Seismic Prior model

Informative power Resolution

Motivations

• Match the data and preserve the prior models
• Multi-scale approach:
• Helps to avoid local minima
• Stabilizes the inversion
• Modifies low resolution first
• Adaptive localization
• Identify important parameters
• Preservation of the prior: modify only where needed

 

6/4/2013 I - 2/21

Motivations

• Match the data and preserve the prior models
• Multi-scale approach:
• Helps to avoid local minima
• Stabilizes the inversion
• Modifies low resolution first
• Adaptive localization
• Identify important parameters
• Preservation of the prior: modify only where needed

 

6/4/2013 I - 2/21

Motivations

• Match the data and preserve the prior models
• Multi-scale approach:
• Helps to avoid local minima
• Stabilizes the inversion
• Modifies low resolution first
• Adaptive localization
• Identify important parameters
• Preservation of the prior: modify only where needed
• Ensemble based method:
• Use of any parameterization

6/4/2013 I - 2/21

Multi-scale parameterization: wavelets

• Parameterization localized both in space and frequency

6/4/2013 I - 3/21

Frequency Frequency From [Xiang-Yang, 2008] Coarse version Original signal

Multi-scale parameterization: wavelets

• Parameterization localized both in space and frequency

6/4/2013 I - 3/21

Frequency Frequency From [Xiang-Yang, 2008] Coarse version Original signal

Multi-scale parameterization: wavelets

• Parameterization localized both in space and frequency

6/4/2013 I - 3/21

Frequency Frequency From [Xiang-Yang, 2008] Coarse version Original signal

Multi-scale parameterization: wavelets

• Parameterization localized both in space and frequency

6/4/2013 I - 3/21

Frequency Frequency From [Xiang-Yang, 2008] Coarse version Original signal

Multi-scale parameterization: wavelets

• Parameterization localized both in space and frequency

6/4/2013 I - 3/21

Frequency Frequency From [Xiang-Yang, 2008] Coarse version Original signal

Multi-scale parameterization: wavelets

• Parameterization localized both in space and frequency

6/4/2013 I - 3/21

Frequency Frequency From [Xiang-Yang, 2008] Coarse version Original signal

Multi-scale parameterization: wavelets

• Parameterization localized both in space and frequency

6/4/2013 I - 3/21

Frequency Frequency From [Xiang-Yang, 2008] Coarse version Original signal

Multi-scale parameterization: wavelets

• Sparse basis: only few coefficients are needed to characterize

most significant features:

• Second generation wavelets
• Much more flexible: can be used on stratigraphical grids

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Initial 3D property

Multi-scale parameterization: wavelets

• Sparse basis: only few coefficients are needed to characterize

most significant features:

• Second generation wavelets
• Much more flexible: can be used on stratigraphical grids

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Initial 3D property Property reconstructed using 1% of the wavelets coefficients

Adaptive multi-scale ensemble based inversion

First parameters to

• ptimize

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Adaptive multi-scale ensemble based inversion

First parameters to

• ptimize

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Wavelet decomposition R4 R4 R4

R3 R3 R3

R2 R2 R2

Adaptive multi-scale ensemble based inversion

First parameters to

• ptimize

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Wavelet decomposition R4 R4 R4

R3 R3 R3

R2 R2 R2

Adaptive multi-scale ensemble based inversion

First parameters to

• ptimize

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Reversible smoothing assists first

• ptimizations

Wavelet decomposition R4 R4 R4

R3 R3 R3

R2 R2 R2

Adaptive multi-scale ensemble based inversion

First parameters to

• ptimize

Ensemble- based Optimization

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Wavelet decomposition R4 R4 R4

R3 R3 R3

R2 R2 R2

Adaptive multi-scale ensemble based inversion

First parameters to

• ptimize

Ensemble- based Optimization

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Coarse update Wavelet decomposition R4 R4 R4

R3 R3 R3

R2 R2 R2

Adaptive multi-scale ensemble based inversion

First parameters to

• ptimize

Ensemble- based Optimization Adaptive localization and refinement Resolution 0

Sensitivity analysis

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Re-introduction of smoothed frequencies

Wavelet decomposition R4 R4 R4

R3 R3 R3

R2 R2 R2

1 0,2

Adaptive multi-scale ensemble based inversion

Ensemble- based Optimization Adaptive localization and refinement Resolution 0 Resolution 1

Sensitivity analysis

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Re-introduction of smoothed frequencies

Wavelet decomposition R4 R4 R4

R3 R3 R3

R2 R2 R2

1 0,2

Adaptive multi-scale ensemble based inversion

Ensemble- based Optimization Adaptive localization and refinement Resolution 0 Resolution 1

Sensitivity analysis

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Re-introduction of smoothed frequencies

Wavelet decomposition R4 R4 R4

R3 R3 R3

R2 R2 R2

1 0,2

Adaptive multi-scale ensemble based inversion

Ensemble- based Optimization Adaptive localization and refinement Resolution 0 Resolution 1 Resolution 2

Sensitivity analysis

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Re-introduction of smoothed frequencies

Wavelet decomposition R4 R4 R4

R3 R3 R3

R2 R2 R2

1 0,2

Adaptive multi-scale ensemble based inversion

Ensemble- based Optimization Adaptive localization and refinement Resolution 0 Resolution 1 Resolution 2

Sensitivity analysis

Resolution 3

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Re-introduction of smoothed frequencies

Wavelet decomposition R4 R4 R4

R3 R3 R3

R2 R2 R2

1 0,2

Adaptive multi-scale ensemble based inversion

Ensemble- based Optimization Adaptive localization and refinement Resolution 0 Resolution 1 Resolution 2

Sensitivity analysis

Resolution 3 Resolution 4

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Re-introduction of smoothed frequencies

Wavelet decomposition R4 R4 R4

R3 R3 R3

R2 R2 R2

Iterative LM-enRML using wavelet parameterization

• Levenberg-Marquadt optimization:

δ𝛅opt = −

1 𝜇+1 (𝜀𝛅𝑞𝑠 + 𝑳(𝜇). 𝑯. 𝜀𝛅𝑞𝑠 − 𝑳(𝜇). 𝜀𝒆 where 𝛅 :{vector of wavelet coefficients}, 𝜇:{LM damping factor}, 𝑳:{similar to Kalman gain}, 𝑯 :{Sensitivity matrix}, 𝜀𝒆 :{data mismatch}

• Prior constraint term dominates in insensitive areas
• Data mismatch term dominates in sensitive areas
• Global sensitivity matrix G computed from an ensemble
• Sensitivity matrix is used to automatically compute the

localization vector

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Prior constraint term Data mismatch term

Key points of the method

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Key points of the method

• Initial smoothing:
• Automatically done by dividing wavelets coefficients
• Easily reversible
• Minimize the effects of high frequencies on flow response
• Preserve the initial main features

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6/4/2013 I - 8/21

Key points of the method

• Initial smoothing:
• Automatically done by dividing wavelets coefficients
• Easily reversible
• Minimize the effects of high frequencies on flow response
• Preserve the initial main features
• Multi-scale approach
• The optimization of the low frequencies does not destroying main

features

• The mismatch is significantly decreased when starting the
• ptimization of the high frequencies

   

6/4/2013 I - 8/21

Key points of the method

• Initial smoothing:
• Automatically done by dividing wavelets coefficients
• Easily reversible
• Minimize the effects of high frequencies on flow response
• Preserve the initial main features
• Multi-scale approach
• The optimization of the low frequencies does not destroying main

features

• The mismatch is significantly decreased when starting the
• ptimization of the high frequencies
• Multi-scale Adaptive localization
• Automatic and dynamic: compute from the current sensitivity

matrix

• Allows large scale updates
• Good preservation of the prior in insensitive areas

6/4/2013 I - 8/21

Synthetic 2D case

• Grid with 3400 active cells
• 4 injectors (injection rate constraint) and 9 producers

(Oil recovery constraint)

• 7,5 years of history: Gas-Oil-Ratio (GOR), water cut

(WWCT), pressure (WBHP)

6/4/2013 I - 9/21

2,5 7,5 LOG PERMX 0,03 0,29 PORO

Optimizations

• Case A: Adaptive multi-scale LM-enRML
• Case B: LM-enRML with prior term
• Case C: LM-enRML without prior term
• No a prior localization
• About 350 data points
• Ensemble of 60 realizations generated using object-

based modeling

• 15 LM-enRML iterations
• Use the same LM-enRML control parameter 𝜇

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Average data mismatches

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R0 R1 R2 R3 R4

Average data mismatches

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R0 R1 R2 R3 R4 Adaptive localization

WBHP PRO 6

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Prior ens. …. Data Truth Case A Mean +/- std Case B Case C Time (days) Adaptive multi-scale LM-enRML

WBHP PRO 6

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Prior ens. …. Data Truth Case A Mean +/- std Case B Case C Time (days) LM-enRML with prior constraint

WBHP PRO 6

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Prior ens. …. Data Truth Case A Mean +/- std Case B Case C Time (days) LM-enRML without prior constraint

WWCT PRO 6

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Prior ens. …. Data Truth Case A Mean +/- std Case B Case C Time (days) Adaptive multi-scale LM-enRML

WWCT PRO 6

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Prior ens. …. Data Truth Case A Mean +/- std Case B Case C Time (days) LM-enRML with prior constraint

WWCT PRO 6

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Prior ens. …. Data Truth Case A Mean +/- std Case B Case C Time (days) LM-enRML without prior constraint

Ensemble averages

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TRUE PORO TRUE LOG-PERM Prior Case A Case B Case C Case A Case B Case C Prior 2,5 7,5 LOG PERMX 0,03 0,29 PORO

PORO realizations

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PRIOR Case A Case B Case C 0,03 0,29

LOG-PERM realizations

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PRIOR Case A Case B Case C 2,5 7,5

Average deviation from prior

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Case A Case B Case C PORO LOG-PERM

Conclusions

• The wavelet parameterization permits to work both in space and

frequency

• The adaptive multi-scale method stabilizes the inversion:
• Manage to get a good match while minimizing the changes
• Avoid addition of noise, better preserve the prior and avoid ensemble

collapse

• Three keys points of the method:
• Simplification of the problem (initial smoothing) helps to improve the

estimation of G for the large scale coefficients

• Multi-scale approach: allows a significant reduction of the mismatch by
• nly modifying large scale parameters
• Adaptive localization: is dynamic, automatic and allows global updates of

the field

6/4/2013 I - 21/21

Conclusions

• The wavelet parameterization permits to work both in space and

frequency

• The adaptive multi-scale method stabilizes the inversion:
• Manage to get a good match while minimizing the changes
• Avoid addition of noise, better preserve the prior and avoid ensemble

collapse

• Three keys points of the method:
• Simplification of the problem (initial smoothing) helps to improve the

estimation of G for the large scale coefficients

• Multi-scale approach: allows a significant reduction of the mismatch by
• nly modifying large scale parameters
• Adaptive localization: is dynamic, automatic and allows global updates of

the field

6/4/2013 I - 21/21