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

Adaptive multi-scale ensemble- based history matching using wavelets “EnKF workshop “ Bergen May 2013 Théophile Gentilhomme , Dean Oliver, Trond Mannesth, Remi Moyen, Guillaume Caumon 6/4/2013

I - 2/21 Motivations Match the data and preserve the prior models           6/4/2013

I - 2/21 Motivations Match the data and preserve the prior models      Optimization      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 Motivations Match the data and preserve the prior models  Multi-scale approach:   Helps to avoid local minima  Stabilizes the inversion  Modifies low resolution first   Prior model Seismic  Informative power   Resolution 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   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 - 3/21 Multi-scale parameterization: wavelets  Parameterization localized both in space and frequency Frequency Coarse version Frequency Original signal From [Xiang-Yang, 2008] 6/4/2013

I - 3/21 Multi-scale parameterization: wavelets  Parameterization localized both in space and frequency Frequency Coarse version Frequency Original signal From [Xiang-Yang, 2008] 6/4/2013

I - 3/21 Multi-scale parameterization: wavelets  Parameterization localized both in space and frequency Frequency Coarse version Frequency Original signal From [Xiang-Yang, 2008] 6/4/2013

I - 3/21 Multi-scale parameterization: wavelets  Parameterization localized both in space and frequency Frequency Coarse version Frequency Original signal From [Xiang-Yang, 2008] 6/4/2013

I - 3/21 Multi-scale parameterization: wavelets  Parameterization localized both in space and frequency Frequency Coarse version Frequency Original signal From [Xiang-Yang, 2008] 6/4/2013

I - 3/21 Multi-scale parameterization: wavelets  Parameterization localized both in space and frequency Frequency Coarse version Frequency Original signal From [Xiang-Yang, 2008] 6/4/2013

I - 3/21 Multi-scale parameterization: wavelets  Parameterization localized both in space and frequency Frequency Coarse version Frequency Original signal From [Xiang-Yang, 2008] 6/4/2013

I - 4/21 Multi-scale parameterization: wavelets Sparse basis: only few coefficients are needed to characterize  most significant features: Initial 3D property  Second generation wavelets  Much more flexible: can be used on stratigraphical grids 6/4/2013

I - 4/21 Multi-scale parameterization: wavelets Sparse basis: only few coefficients are needed to characterize  most significant features: Property reconstructed using Initial 3D property 1% of the wavelets coefficients  Second generation wavelets  Much more flexible: can be used on stratigraphical grids 6/4/2013

I – 5/21 Adaptive multi-scale ensemble based inversion First parameters to optimize 6/4/2013

I – 5/21 Adaptive multi-scale ensemble based inversion R2 R3 R2 R2 R4 First R3 R3 parameters to optimize R4 R4 Wavelet decomposition 6/4/2013

I – 5/21 Adaptive multi-scale ensemble based inversion R2 R3 R2 R2 R4 First R3 R3 parameters to optimize R4 R4 Wavelet decomposition 6/4/2013

I – 5/21 Adaptive multi-scale ensemble based inversion R2 R3 R2 R2 R4 First R3 R3 parameters to optimize R4 R4 Wavelet decomposition Reversible smoothing assists first optimizations 6/4/2013

I – 5/21 Adaptive multi-scale ensemble based inversion R2 R3 R2 R2 R4 First R3 R3 parameters to optimize Ensemble- R4 R4 based Optimization Wavelet decomposition 6/4/2013

I – 5/21 Adaptive multi-scale ensemble based inversion R2 R3 R2 R2 R4 First R3 R3 parameters to optimize Ensemble- R4 R4 based Optimization Wavelet decomposition Coarse update 6/4/2013

I – 5/21 Adaptive multi-scale ensemble based inversion R2 R3 R2 R2 R4 First R3 R3 parameters to optimize Ensemble- R4 R4 based Optimization Wavelet decomposition Sensitivity analysis Resolution 0 Adaptive localization and refinement Re-introduction of smoothed frequencies 6/4/2013

I – 5/21 Adaptive multi-scale ensemble based inversion R2 R3 R2 R2 R4 R3 R3 Ensemble- R4 R4 based Optimization Wavelet decomposition Sensitivity analysis Resolution 0 Resolution 1 0,2 1 Adaptive localization and refinement Re-introduction of smoothed frequencies 6/4/2013

I – 5/21 Adaptive multi-scale ensemble based inversion R2 R3 R2 R2 R4 R3 R3 Ensemble- R4 R4 based Optimization Wavelet decomposition Sensitivity analysis Resolution 0 Resolution 1 0,2 1 Adaptive localization and refinement Re-introduction of smoothed frequencies 6/4/2013

I – 5/21 Adaptive multi-scale ensemble based inversion R2 R3 R2 R2 R4 R3 R3 Ensemble- R4 R4 based Optimization Wavelet decomposition Sensitivity analysis Resolution 0 Resolution 2 Resolution 1 0,2 1 Adaptive localization and refinement Re-introduction of smoothed frequencies 6/4/2013

I – 5/21 Adaptive multi-scale ensemble based inversion R2 R3 R2 R2 R4 R3 R3 Ensemble- R4 R4 based Optimization Wavelet decomposition Sensitivity analysis Resolution 3 Resolution 0 Resolution 2 Resolution 1 0,2 1 Adaptive localization and refinement Re-introduction of smoothed frequencies 6/4/2013

I – 5/21 Adaptive multi-scale ensemble based inversion R2 R3 R2 R2 R4 R3 R3 Ensemble- R4 R4 based Optimization Wavelet decomposition Sensitivity analysis Resolution 4 Resolution 3 Resolution 0 Resolution 1 Resolution 2 0,2 1 Adaptive localization and refinement Re-introduction of smoothed frequencies 6/4/2013

I – 6/21 Iterative LM-enRML using wavelet parameterization Levenberg-Marquadt optimization:  1 𝜇+1 (𝜀𝛅 𝑞𝑠 + 𝑳(𝜇). 𝑯. 𝜀𝛅 𝑞𝑠 − 𝑳(𝜇). 𝜀𝒆 δ𝛅 opt = − Data mismatch term Prior constraint term 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 6/4/2013

I - 8/21 Key points of the method             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         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  optimization 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  optimization 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 - 9/21 Synthetic 2D case 0,29 7,5 PORO LOG PERMX 0,03 2,5  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