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Bootstrap Confidence Intervals for Reservoir Model Selection Techniques

Bootstrap Confidence Intervals for Reservoir Model Selection Techniques. Céline Scheidt and Jef Caers. SCRF Affiliate Meeting– April 30, 2009. Model Selection Techniques. Uncertainty in reservoir modeling is represented through a possibly large set of reservoir models

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Bootstrap Confidence Intervals for Reservoir Model Selection Techniques

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  1. Bootstrap Confidence Intervals for Reservoir Model Selection Techniques CélineScheidt and JefCaers SCRF Affiliate Meeting– April 30, 2009

  2. Model Selection Techniques • Uncertaintyin reservoir modeling is represented through a possibly large setof reservoir models • Generated by varying several input parameters • High CPU demand for flow simulations requires the use of model selection techniques • Evaluate uncertainty on a subset of models • Model selection techniques select a subset of representative realizations which should preserve the statistics of the entire set of realizations • Eg.: Ranking, Distance-Kernel Method (DKM) SCRF Affiliate Meeting – 04/30/09

  3. Goal • If we select N realizations, perform flow simulation, and quantify uncertainty: • How do we know if the results are accurate? • Can we be confident with the results? • Should we do more simulations? • We use of bootstrap methodology to evaluate the accuracy of the uncertainty quantification • Applicable to standard ranking or new distance-kernel method (DKM) SCRF Affiliate Meeting – 04/30/09

  4. Model 1 Model 2 d12 d14 d13 d24 d32 d34 Model 4 Model 3 Distance Kernel Method (DKM) SCRF, 2008 SPE Journal, 2009 2D projection of Metric Space Distance Matrix D MDS M j Kernels P10,P50,P90 model selection 2D projection of Feature Space 2D projection of Metric Space Apply Clustering in F j-1 Pre-image F SCRF Affiliate Meeting – 04/30/09

  5. Traditional Ranking Techinque • Generate a proxy response for each L realizations (ranking measure) • Should be strongly correlated to the actual response • Select N realizations for flow simulations • Traditionally, N=3 • Realizations equally spaced according to the ranking measure • Estimation of the distribution of the response using the N simulations • Compute P10, P50 and P90 statistics SCRF Affiliate Meeting – 04/30/09

  6. Review: Parametric Bootstrap – Simple Example ? B bootstrap estimates of the mean and variance 1st estimate b = 1,..,B 2nd estimate SCRF Affiliate Meeting – 04/30/09

  7. Notations • : Proxy response (ranking measure) • Eg. Streamline simulations • : True response • Eg. Eclipse simulations • : Selected realizations by model selection • : estimate of P10, P50 and P90 values • From ranking or DKM & flow simulation (1st estimate) • : bootstrap estimate of P10, P50 and P90 values • From ranking or DKM & parametric distribution (2nd estimate) No additional flow simulations SCRF Affiliate Meeting – 04/30/09

  8. Proposed Bootstrap Methodology Application to model selection technique Proxy Values Model selection + flow simulation b = 1,…,B Model selection + response evaluation SCRF Affiliate Meeting – 04/30/09

  9. Illustration: Bivariate Gaussian Distribution • Distribution of the target and proxy responses: • Proposed bootstrap technique applied for several correlation scenarios between target and proxy responses • Scenarios for: • rxy= 1, 0.9, 0.8, 0.7, 0.6,0.5 r = 0.9 SCRF Affiliate Meeting – 04/30/09

  10. Histograms of the Estimated Quantiles • L= 100, r = 0.9 • Selection of 15 realizations using DKM • Number of bootstrap samples: B = 1000 Bootstrap estimated P90 Bootstrap estimated P50 Bootstrap estimated P10 Estimated P50 Estimated P10 Estimated P90 SCRF Affiliate Meeting – 04/30/09

  11. Definition of the Error on the Bootstrap Estimated Quantiles • For each of the B samples, a dimensionless error is defined to evaluate the accuracy of the estimated quantiles: Error on bootstrap estimated quantiles:

  12. Bivariate Gaussian distribution Confidence Intervals for different correlations scenarios r = 0.8 r = 0.9 r = 1.0 r = 0.5 r = 0.7 r = 0.6 SCRF Affiliate Meeting – 04/30/09

  13. West Coast African Reservoir (WCA) Courtesy of Chevron • WCA is a deepwater turbidite offshore reservoir located in a slope valley • Dimensions of the reservoir model • 78 x 59 x 116 gridblocks • 100,000 active gridblocks • 28 wells • 20 production wells (red) • 8 injection wells (blue) 1 mile 0.5 mile 800 feet SCRF Affiliate Meeting – 04/30/09

  14. West Coast African Reservoir • 4 depositional facies • Facies 1: Shale (55% of the reservoir) • Facies 2: Poor quality sand #1 (debris flows or levees) • Facies 3: Poor quality sand #2 (debris flows or levees) • Facies 4: Good quality channels (28 %) • Porosity for each facies determined by SGS conditioned to well data • Vshale for each facies modeled by SGS correlated to porosity • Permeability calculated analytically from Vshale SCRF Affiliate Meeting – 04/30/09

  15. Uncertainty in Reservoir Description • Uncertainty exists for: • Depositional environment • Modeled using 12 training images (TI) & snesim • Facies proportions • Modeled with 3 different probability cubes • Probability cubes come from seismic • 2 realizations were generated for each combination of TI and facies probability cube • 72 possible realizations of the WCA reservoir SCRF Affiliate Meeting – 04/30/09

  16. Definition of the Responses • True response X: • Cumulative oil production after 1200 days of production (evaluated by full flow simulation) • Proxy response Y: • Cumulative oil production after 1215 days of production (evaluated by fast streamline simulation) • Correlation coefficient: r(X,Y) = 0.92 SCRF Affiliate Meeting – 04/30/09

  17. Definition of Distribution Function • Parametric bootstrap requires an assumption of the bivariate distribution function ( ) • Not known a priori in real case (contrary to previous example) • Use of a smoothing technique to obtain the distribution of the N selected bivariate samples True and proxy responses on the N Selected points SCRF Affiliate Meeting – 04/30/09

  18. Generation of Bootstrap Samples Proxy measure (Streamline) Flow simulations (Chears) on N selected realizations 1stModel Selection to select N real. Sampling to generate new bivariate bootstrap datasets Smoothing on N selected realizations Bivariate response B times 2ndModel Selection to select N real. SCRF Affiliate Meeting – 04/30/09

  19. Application of Bootstrap to WCA • Distance (for DKM only) • Difference in proxy response for every pair of realizations • Comparison between 3 model selection methods: • DKM, ranking and random selection • Selection of N realizations: N = 3,5,8,10,15,20 • The set of selected realizations are different for each N • Number of new bootstrap data sets generated: • B = 1000 SCRF Affiliate Meeting – 04/30/09

  20. Bootstrap Estimated P10, P50 and P90 Quantiles SCRF Affiliate Meeting – 04/30/09

  21. Error on Bootstrap Quantiles Estimations 10 simulations 5 simulations 20 simulations 15 simulations SCRF Affiliate Meeting – 04/30/09

  22. Error on Bootstrap Quantiles Estimations • N = 8 or 10 simulations should be sufficient to obtain an accurate uncertainty quantification • Previous work (SCRF 2008) showed that with 7 simulations, uncertainty quantification on cumulative oil production was very accurate SCRF Affiliate Meeting – 04/30/09

  23. Comparison of the Results to the “Truth” N = 8 N = 3 8 simulations 3 simulations

  24. Conclusion • We have established a workflow to construct confidence intervals for quantile estimations • Workflow uses any model selection technique and parametric bootstrap procedure • DKM provides more robust results and outperforms ranking • The magnitude of the confidence intervals can show if more simulations are required for a better uncertainty quantification • Does not suggest how many more, only if sufficiently accurate SCRF Affiliate Meeting – 04/30/09

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