Integrating Production and Seismic Data into Gaussian and Pluri-Gaussian Models with EnKF(S)

1. Integrating Production and Seismic Data into Gaussian and Pluri-Gaussian Models with EnKF(S) Yong Zhao Yudou Wang Gaoming Li Al Reynolds EnKF Workshop: Voss June 2008

2. Sequential Data Assimilation (Ensemble Kalman Filter) Update

3. EnKF Analysis (Bayesian Updating and Sampling) • Critical Assumptions: • Predictions of state vectors are Gaussian; • Covariances can be represented by ensemble members; • Gaussian noise in data; • Predicted data are a linear function of the state vector. • Or, with data augmented state vector • Predictions of augmented vector are Gaussian; • Gaussian noise in data; • Covariances can be represented by ensemble members.

4. Potential Problems in EnKF • Each analyzed vector of model parameters is a linear combination of initial ensemble. • Difficult to match large data sets, e.g., seismic data. • Non-Gaussianity. • Strong non-linearity. • Poor knowledge of measurement errors. • Modeling of modeling errors. • Sampling errors due to finite ensemble size. • Inconsistency: updated pressure and saturations are inconsistent with the updated models (statistically different from those obtained by simulating from time zero)

5. Rescaling for Different Types of Data Assimilating production data: Assimilating with rescaled data: Better conditioned

6. Truncation of Singular Values, PUNQ, Est. Contact Depths Rescaled Truncated at 0.9999

7. 2-D case 100 X 100 grid, 4 producers and 1 injector are located in the channel facies 360 days of production with BHP and WCT measurements 300 days of prediction 100 ensemble members Channel Model Truncation Facies Z1 Z1

8. Conditional Models and Sw Facies En20 True facies Sw from true model Sw En20

9. EnKF Predictions Rerun from time zero Prediction from EnKF Prior prediction

10. Normal Score Transform PDF PDF S’w Sw Before Analysis CDF CDF Sw S’w After Analysis Prediction Domain Analysis Domain

11. Normal Score Transform Global Transform Local Transform Standard EnKF

12. Predictions From Transforms Local No Transform Global EnKF Rerun

13. HIEnKF Method • If model changes significantly, updated primary field may be more inconsistent with the updated model. • When the change of model is significant, rerun from zero; otherwise, we use the EnKS. EnKF Model changes significantly EnKF

14. EnKF HIEnKF True

15. EnKF vs. HIEnKF HIEnKF EnKF

16. 3-D case 50 X 50X3 grid, 4 producers and 1 injector Total rate constraint for each well Hard data: observed facies in well gridblocks 360 days of production with BHP and WCT measurements (monthly) 300 days of prediction Seismic data (at time zero and 300 days) 100 ensemble members Fixed porosity and permeability Permeability (11md, 100md, 528md); Porosity (0.06, 0.13, 0.21) Three-Facies Model Layer 2 Layer 3 Layer 1

17. Assimilating Dynamic Data While Satisfying Hard Data, SPE 113990 • If does not satisfy the hard data: Expand data with pseudo data: Completely redo the assimilation step:

18. EnKF Predictions Prediction from EnKF state Rerun from time zero Prior prediction

19. Acoustic Impedance • Match seismic data at the time they are measured t = 0 t = 300days

20. Matching Seismic Data:Local Analysis of EnKF • Local analysis: • Analyzed models are not constrained to the sub-space spanned by the initial ensemble • Undesired roughness can be introduced into the analyzed models

21. A large ensemble with 1200 realizations of model that honors the hard data (M0) Use the first 200 eigenvectors Projection Method for Local Analysis

22. Assimilate Seismic Data- Local Analysis (2 seismic + prod) True Facies No projection En20 With projection En20

23. Assimilate Seismic Data- Local Analysis With Projection Continue EnKF for production data First Seismic Only Rerun from time zero 1st seismic 2nd seismic 1st seismic 2nd seismic

24. Structure Map of PUNQ-S3 Fault, gas cap, strong aquifer. grid. Data: BHP GOR WCT Match to 4032 days

25. Estimate the Depths of Fluid Contacts with EnKS State Vector y Model parameters m Production data d Primary variables p BHP WCT GOR Porosity Pressure Permeability Gas saturation Water saturation Solution gas-oil ratio Fluid contact depths

26. Introduction to HIEnKS Method • If model changes significantly, updated primary field may be more inconsistent with the updated model. • Only use HIEnKS when the change of model is significant. otherwise, we use the EnKS. EnKS Model changes significantly EnKS

27. Examples • Example A: • Prior mean of OWC shifted up 20 feet • Prior mean of GOC shifted down 20 feet. • Example B: • Prior mean of OWC shifted down 20 feet • Prior mean of GOC shifted down 20 feet. STD: 20 ft True GOC True GOC 20ft 20ft Prior Mean of GOC Prior Mean of GOC Prior Mean of OWC True OWC 20ft 20ft True OWC Prior Mean of OWC Example A, prior oil column too thin Example B, prior contact depths too deep

28. Comparison of Estimates of Fluid Contacts Example A Example B EnKS EnKS HIEnKS HIEnKS

29. Consistency of Prediction, Example A EnKS HIEnKS During Data Assimilation Rerun from Time 0 • EnKS: Future predictions poor, inconsistent. • HIEnKS: Data matches good, consistent.

30. Consistency of Prediction, Example A EnKS HIEnKS During Data Assimilation Rerun from Time 0 • EnKS: Assimilation good, prediction poor, inconsistent. • HIEnKS: Assimilation/Prediction good, roughly consistent.

31. Rock Property Fields- 4th, 5th layers Vertical Permeability Horizontal Permeability Truth EnKS HIEnKS

32. Comments • Iteration can improve reliability of data match, predictions and consistency between parameters and dynamical variables but is expensive. • Scaling can be critical if SVD is used. • EnKF combined with pluri-Gaussian gives reasonable results (3D - rock properties – hard data).

33. Comments • Pluri-Gaussian inappropriate for fluvial systems – Cosine transforms, MRFs, KPCA? • Seismic: local analysis with projection seems feasible but is currently ad hoc.