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

How to quantify uncertainty of models? Why quantify uncertainty? 2 Key issues SCRF 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 Sampling efficiency Key issues SCRF 2012 Rejection Sampler Markov Chain MC Reference d obs d predict Proposed model Forward modeling d predict

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

5 Creating a Markov chain: ISR SCRF 2012 Methodology

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 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

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

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

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

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

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

14SCRF Acoustic impedance: ASR Illustration

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

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 Seismogram Illustration

17SCRF Seismogram results using MDS projection Illustration

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

19 3. Finding facies not seen in well data Reference: facies Well data Wells Predicted seismic data SCRF 2012 Seismic data seismograms CDP 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

20SCRF 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

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

4. ASR as an optimizer 22SCRF 2012 Illustration

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 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

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

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

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 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

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

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

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

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 Multiple subsurface scenarios help to choose the most applicable setting for unknown reservoir modeling. SCRF 2012 Summary

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

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

33SCRF Multiple subsurface scenarios Ti1 = at the data location, Ti2 was , and Ti3 was 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

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

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

36SCRF 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

SEG Appendix III : Ti