1. Céline Scheidt and Jef Caers SCRF Affiliate Meeting– April 30, 2009

2.  Uncertainty in reservoir modeling is represented through a possibly large set of 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) 2 SCRF Affiliate Meeting – 04/30/09

3.  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) 3 SCRF Affiliate Meeting – 04/30/09

4. Distance Matrix D 1234 1  11  12  13  14 2  21  22  23  24 3  31  32  33  34 4  41  42  43  44 Model 1Model 2 Model 3 Model 4  12  13  24  34  32  14 F 2D projection of Feature Space 2D projection of Metric Space Apply Clustering in F P10,P50,P90 model selection M 2D projection of Metric Space MDS Kernels    Pre-image 4 SCRF Affiliate Meeting – 04/30/09 SCRF, 2008 SPE Journal, 2009

5.  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 5 SCRF Affiliate Meeting – 04/30/09

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

7.  : 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 (1 st estimate)  : bootstrap estimate of P10, P50 and P90 values  From ranking or DKM & parametric distribution (2 nd estimate) No additional flow simulations 7 SCRF Affiliate Meeting – 04/30/09

8. Model selection + flow simulation Proxy Values Application to model selection technique 8 Model selection + response evaluation Parametric Bootstrap Estimation of distributionGeneration of B samples from b = 1,…,B SCRF Affiliate Meeting – 04/30/09

9.  Distribution of the target and proxy responses:  Proposed bootstrap technique applied for several correlation scenarios between target and proxy responses ◦ Scenarios for :   xy = 1, 0.9, 0.8, 0.7, 0.6,0.5  = 0.9 9 SCRF Affiliate Meeting – 04/30/09

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

11.  For each of the B samples, a dimensionless error is defined to evaluate the accuracy of the estimated quantiles: 11 Error on bootstrap estimated quantiles:

12. 12  = 1.0  = 0.9  = 0.8  = 0.5  = 0.6  = 0.7 SCRF Affiliate Meeting – 04/30/09

13.  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) Courtesy of Chevron 1 mile 0.5 mile 800 feet 13 SCRF Affiliate Meeting – 04/30/09

14.  4 depositional facies ◦ Facies 1: Shale (55% of the reservoir) ◦ Facies 2: Poor quality sand #1 (d ebris flows or levees ) ◦ Facies 3: Poor quality sand #2 (d ebris 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 14 SCRF Affiliate Meeting – 04/30/09

15.  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 15 SCRF Affiliate Meeting – 04/30/09

16.  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:  X,Y) = 0.92 16 SCRF Affiliate Meeting – 04/30/09

17.  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 17 SCRF Affiliate Meeting – 04/30/09 True and proxy responses on the N Selected points

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

19.  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 19 SCRF Affiliate Meeting – 04/30/09

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

21. 21 5 simulations 10 simulations 15 simulations 20 simulations

22. 22  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. 23 N = 3 N = 8 3 simulations 8 simulations

24.  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 24 SCRF Affiliate Meeting – 04/30/09

25. 25 P10 P50 P90 P10 P50 P90 P10 P50 P90 P10 P50 P90