1. Adaptive Spatial Resampling as a McMC Method for Uncertainty Quantification in Seismic Reservoir Modeling Cheolkyun Jeong*, Tapan Mukerji, and Gregoire Mariethoz Stanford Center for Reservoir Forecasting

2. How to quantify uncertainty of models? Why quantify uncertainty? 2 Key issues SCRF 2012 1.We make decisions under uncertainty 2.Modeling subsurface reservoir is a uncertain process 1.In a Bayesian framework, sampling posterior distribution can quantify the uncertainty 2.Rejection sampler is a theoretically perfect method but inefficient A critical issue is to sample posteriors efficiently : A Markov chain Monte Carlo method as an equivalent posterior sampler

3. 3 Sampling efficiency Key issues SCRF 2012 Rejection Sampler Markov Chain MC Reference d obs d predict Proposed model Forward modeling d predict

4. 4SCRF 2012 a b c d e e Creating a Markov chain: Iterative Spatial Resampling (ISR) Methodology

5. 5 Creating a Markov chain: ISR SCRF 2012 Methodology

6. 6 ASR algorithm in acoustic impedance Randomly sampled subset points Adaptively sampled subset points Randomly sampled subset points Adaptively sampled subset points SCRF 2012 Methodology – Adaptive Spatial Resampling spatial error map

7. 7 ASR algorithm in seismic section SCRF 2012 Methodology – Adaptive Spatial Resampling Seismogram: obtained data Seismogram: predicted model Cross correlation coefficient in each trace time correlation coefficient CDP Higher correlation Higher chance Lower correlation More perturbation subset

8. Reference: facies Reference Iterative Spatial Resampling Adaptive Spatial Resampling SCRF 2012 ASR algorithm in acoustic impedance Methodology – Adaptive Spatial Resampling Log 10 RMSE Iteration

9. 9 1. Fraction rate in ASR SCRF 2012 Methodology – Parameter sensitivity Log 10 RMSE Iterations

10. 10 2. Number of traces in seismic section SCRF 2012 Methodology – Parameter sensitivity Log 10 SSE Iterations

11. 11 1. Acoustic impedance for lithofacies characterization Reference: facies Well data Wells Predicted seismic data SCRF 2012 Illustration Seismic data acoustic impedance CDP 25 125 MRayls Vp Bivariate pdf Rockphysics

12. Reference: facies Etype of priors Etype of sampled posteriors (RS) Variance of sampled posteriors (RS) 100,000 priors 125 posteriors 12SCRF 2012 1. Acoustic impedance: Rejection Sampler Illustration

13. Reference: facies 1. Acoustic impedance: Results 13 Etype Variance 125 posteriors (100,000 eval.) 21 posteriors (500 eval.) 94 posteriors (500 eval.) Rejection sampling Iterative Spatial Resampling Adaptive Spatial Resampling SCRF 2012 Illustration

14. 14SCRF 2012 1. Acoustic impedance: ASR Illustration

15. 15 2. Seismograms for facies characterization Reference: facies Well data Wells Predicted seismic data SCRF 2012 Seismic data seismograms CDP 25 125 Vp Bivariate pdf Rockphysics Illustration

16. Reference: facies 2. Seismogram: Results 16 Etype Variance 140 posteriors (100,000 eval.) 29 posteriors (500 eval.) 51 posteriors (500 eval.) Rejection sampling Iterative Spatial Resampling Adaptive Spatial Resampling SCRF 2012 2. Seismogram Illustration

17. 17SCRF 2012 2. Seismogram results using MDS projection Illustration

18. 18 1 st principal coordinate 2 nd principal coordinate SCRF 2012 3. Verification using MDS projection Illustration

19. 19 3. Finding facies not seen in well data Reference: facies Well data Wells Predicted seismic data SCRF 2012 Seismic data seismograms CDP 25 125 Vp Bivariate pdf Rockphysics Oil sand Brine sand Shale *Not detected oilsand distribution is generated by Gassmann’s equation Facies Actual Logs Vp One model in priors One model in posteriors Illustration

20. 20SCRF 2012 24 posteriors (50,000 eval.) 43 posteriors (1000 eval.) 3. Finding facies not seen in well data Probability of Oil Sand CDP Probability Oil sand Brine sand Shale Rejection sampling Adaptive Spatial Resampling Reference: facies Illustration

21. 4. ASR as an optimizer 21 Log 10 RMSE Iterations Reference: facies SCRF 2012 Illustration

22. 4. ASR as an optimizer 22SCRF 2012 Illustration

23. 2. The adaptive spatial resampling (ASR) is a good approximation of rejection sampler as a posterior sampler, and it’s more efficient. 1. Sampling posteriors in seismic inverse modeling can be a good uncertainty quantification tool for decision making. 23 3. Depending on the acceptation/rejection criterion, it is possible to obtain a chain for sampling posterior or calibrating the most likely earth model. SCRF 2012 Summary

24. 1. Application in actual dataset: West Africa dataset 24SCRF 2012 Ongoing and Future work 3 wells, Near and Far offset seismic data Geological Observation Rockphysics model (Dutta, 2009) Facies 1: Channel Deposition Facies 2: Near channel levees Facies 3: Medial-distal levees What we have

25. 1. Application in actual dataset: West Africa dataset 25SCRF 2012 Ongoing and Future work 2D slice : Acoustic Impedance Geological Observation Build Training images 3D study What we need

26. 2. The adaptive spatial resampling (ASR) is a good approximation of rejection sampler as a posterior sampler, and it’s more efficient. 1. Sampling posteriors in seismic inverse modeling can be a good uncertainty quantification tool for decision making. 26 3. Depending on the acceptation/rejection criterion, it is possible to obtain a chain for sampling posterior or calibrating the most likely earth model. SCRF 2012 Summary

27. 1. Application in actual dataset: West Africa dataset 27SCRF 2012 Ongoing and Future work Multiple subsurface scenarios

28. 1. P(Tis | Seismogram) using pattern validation Geologist (1) Geologist (2) Geologist (3) Pattern Validation for finding distances between seismogram images Generate priors, m Forward model, g(m) Multiple subsurface scenarios

29. 2. P(RPs | Seismic data) using pattern validation Rockphysics (1) Pattern Validation for finding distances between seismogram Forward model, g(m) 3. Multiple subsurface scenarios Rockphysics (2) Generate priors, m

30. 2. The adaptive spatial resampling (ASR) is a good approximation of rejection sampler as a posterior sampler, and it’s more efficient. 1. Sampling posteriors in seismic inverse modeling can be a good uncertainty quantification tool for decision making. 30 3. Multiple subsurface scenarios help to choose the most applicable setting for unknown reservoir modeling. SCRF 2012 Summary

31. 31SCRF 2012 3. Multiple subsurface scenarios 1. P(Ti | Seismic data) using pattern validation

32. 32SCRF 2012 3. Multiple subsurface scenarios [301x301] 1. P(Ti | Seismic data) using pattern validation

33. 33SCRF 2012 3. Multiple subsurface scenarios Ti1 = 0.0014 at the data location, Ti2 was 2.4498, and Ti3 was 5.6447 According to Bayesian theorem and Park(2011), P(Ti2|data) = 30% and P(Ti3|data) = 70% Ti2 Ti3 1. P(Ti | Seismic data) using pattern validation

34. 2. P(RPs | Seismic data) using pattern validation Rockphysics (1) Pattern Validation for finding distances between seismogram Forward model, g(m) 3. Multiple subsurface scenarios Rockphysics (2) Generate priors, m

35. 35SCRF 2012 3. Multiple subsurface scenarios 2. P(RPs | Seismic data) using pattern validation

36. 36SCRF 2012 24 posteriors (50,000 eval.) 43 posteriors (1000 eval.) 3. Finding facies not seen in well data Probability of Shale Probability of Brine Sand Probability of Oil Sand CDP Probability Oil sand Brine sand Shale Rejection sampling Adaptive Spatial Resampling Illustration

37. SEG 201137 Appendix III : Ti