Ensemble-based multi-scale history-matching device and method for reservoir characterization

1. A method for generating a multidimensional image of a subsurface of the earth, the method comprising:

receiving data related to the subsurface of the earth;

generating an ensemble of realizations associated with the subsurface based on the received data;

applying wavelet re-parameterization to spatial properties of members of the ensemble to calculate a set of wavelet coefficients;

reconstructing spatial properties of the ensemble based on a subset of the wavelet coefficients that correspond to a large scale;

applying a forward simulator to the reconstructed spatial properties of the ensemble for estimating one or more physical parameters of the subsurface;

applying an ensemble-based optimization method to update the subset of the wavelet coefficients; and

generating the multidimensional image of the subsurface based on the updated subset of the wavelet coefficients,

wherein the method further comprises:

refining the subset of the wavelet coefficients by adding non-included coefficients that belong to the set but not the subset, and

attenuating the non-included coefficients and progressively re-introducing the non-included attenuated coefficients during multi-scale iterations so that a mismatch reduced at first iterations when only large-scale coefficients are included is further reduced at subsequent iterations, when the non-included attenuated coefficients are added as fine scale coefficients such that perturbations of the fine scale coefficients are minimized, the non-included attenuated coefficients corresponding to a fine scale.

2. The method of claim 1 , further comprising: reconstructing the spatial properties for the ensemble based on the subset of the wavelet coefficients and the non-included attenuated coefficients.

3. The method of claim 1 , wherein the multi-scaling iterations are repeated until all the large scale coefficients are used and the non-included attenuated coefficients are progressively re-introduced from iteration to iteration.

4. The method of claim 1 , wherein the applying of the ensemble-based optimization method is repeated for each new larger subset.

5. The method of claim 1 , wherein the applying of the ensemble-based optimization method comprises:

calculating a Kalman gain for the ensemble;

applying a scale adaptive spatial localization to the Kalman gain; and

applying a wavelet localization to the Kalman gain.

6. The method of claim 1 , wherein the data includes at least one of seismic data and flow data associated with the subsurface.

7. The method of claim 1 , wherein at least one of the spatial properties of the ensemble is porosity or permeability.

8. The method of claim 1 , wherein the realizations are static, including petro-physical fields, and dynamic, including at least one of pressure and saturation evolution.

9. The method of claim 1 , wherein the wavelet re-parameterization includes a grid-adaptive second generation wavelet parameterization.

10. The method of claim 1 , wherein the ensemble-based optimization method is a Levenberg-Marquardt ensemble randomized maximum likelihood (LM-enRML) method.

11. The method of claim 10 , wherein the LM-enRML method calculates a global gradient that is used to simultaneously update the members of the ensemble.

12. The method of claim 1 , further comprising:

matching at least one of history production data and 4D seismic data with the one or more physical parameters of the subsurface.

13. A computing device for generating a multidimensional image of a subsurface of the earth, the computing device comprising:

an interface configured to receive data related to the subsurface of the earth; and

a processor connected to the interface and configured to,

generate an ensemble of realizations associated with the subsurface based on the received data,

apply wavelet re-parameterization to spatial properties of members of the ensemble to calculate a set of wavelet coefficients that correspond to a large scale,

reconstruct spatial properties of the ensemble based on a subset of the wavelet coefficients,

apply a forward simulator to the reconstructed spatial properties of the ensemble for estimating one or more physical parameters of the subsurface,

apply an ensemble-based optimization method to update the subset of the wavelet coefficients, and

generate the multidimensional image of the subsurface based on the updated subset of the wavelet coefficients

wherein the processor is further configured to:

refine the subset of the wavelet coefficients by adding non-included coefficients that belong to the set but not the subset, and

attenuate the non-included coefficients and progressively re-introducing the non-included attenuated coefficients during multi-scale iterations so that a mismatch reduced at first iterations when only large scale coefficients are included is further reduced at subsequent iterations, when the non-included attenuated coefficients are added as fine scale coefficients such that perturbations of the fine scale coefficients are minimized, wherein the non-included attenuated coefficients correspond to a fine scale.

14. The computing device of claim 13 , wherein the spatial properties of the ensemble are reconstructed based on the subset of the wavelet coefficients and the non-included attenuated coefficients.

15. The computing device of claim 13 , wherein the step of multi-scaling iterations are repeated until all the large scale coefficients are used, and the attenuated non-included coefficients are progressively re-introduced from iteration to iteration.

16. A non-transitory computer readable medium including computer executable instructions, wherein the instructions, when executed by a computer, implement a method for generating a multidimensional image of a subsurface of the earth, the method comprising:

receiving data related to the subsurface of the earth;

generating an ensemble of realizations associated with the subsurface based on the received data;

applying wavelet re-parameterization to spatial properties of members of the ensemble to calculate a set of wavelet coefficients;

reconstructing spatial properties of the ensemble based on a subset of the wavelet coefficients that correspond to a large scale;

applying a forward simulator to the reconstructed spatial properties of the ensemble for estimating one or more physical parameters of the subsurface;

applying an ensemble-based optimization method to update the subset of the wavelet coefficients; and

generating the multidimensional image of the subsurface based on the updated subset of the wavelet coefficients

wherein the method further comprises:

refining the subset of the wavelet coefficients by adding non-included coefficients that belong to the set but not the subset, and

attenuating the non-included coefficients and progressively re-introducing the non-included attenuated coefficients during multi-scale iterations so that a mismatch reduced at first iterations when only large-scale coefficients are included is further reduced at subsequent iterations, when the non-included attenuated coefficients are added as fine scale coefficients such that perturbations of the fine scale coefficients are minimized, the non-included attenuated coefficients corresponding to a fine scale.