Prediction of Oil Production With Confidence Intervals*

1. Prediction of Oil Production With Confidence Intervals* James Glimm1,2, Shuling Hou3, Yoon-ha Lee1, David H. Sharp3, Kenny Ye1 1. SUNY at Stony Brook 2. Brookhaven National Laboratory 3. Los Alamos National Laboratory *Supported in part by the Department of Energy and National Science Foundation

2. Application of Prediction Theory • Reservoir Development Choices, for example • Sizing of Production Equipment • Location of Infill Drilling • During Early Development Stages • Risk high • Payoff high

3. Basic Idea: I • History match with probability of error • Start with geostatistical probability model for permeability, etc. • Observe production rates, etc.

4. Basic Idea: II • Multiple simulations from ensemble • (Re)Assign probabilities based on data, degree of mismatch of simulation to history

5. Basic Idea: III Redefine probabilities and ensemble to be consistent with: (a) data (b) probable errors in simulation and data

6. Basic Idea: IV New ensemble of geologies = Posterior Prediction = sample from posterior Confidence intervals come from - posterior probabilities - errors in forward simulation

7. New Idea: Arrival Time Error Models • Formulate solution error model in terms of arrival times, rather than solution values • errors are equi-distributed relative to solution gradients, ie relative to changes in solution values:

8. Arrival Time Model is Simple, Robust Regard as the new independent variable and t as the new dependent variable. Thus and the error is equidistributed in units of This makes the error robust. We compare arrival time and solution based error models

9. Illustration:Posterior reduces choices and uncertainty

10. New Result: Predict outcomes and risk Risk is predicted quantitatively Risk prediction is based on - formal probabilities of errors in data and simulation - methods for simulation error analysis - Rapid simulation (upscale) allowing exploration of many scenarios

11. Problem Formulation Simulation study: Line drive, 2D reservoir Random permeability field log normal, random correlation length

12. Simple Reservoir Description in unit square constant

13. Ensemble 100 random permeability fields for each correlation length lnK gaussian, correlation length

14. Upscaling Solution from fine grid 100 x 100 grid Solution by upscaling 20 x 20, 10 x 10, 5 x 5 Upscaled grids

15. Upscaling References Upscaling by Wallstrom, Hou, Christie, Durlofsky, Sharp 1. Computational Geoscience 3:69-87 (1999) 2. SPE 51939 3. Transport in Porous Media (submitted)

16. Examples of Upscaled, Exact Oil Cut Curves Scale-up: Black (fine grid) Red (20x20) Blue (10x10) Green (5x5)

17. Design of Study Select one geology as exact. Observe production for Assign revised probabilities to all 500 geologies in ensemble based on: (a) coarse grid upscaled solutions (b) probabilities for coarse grid errors. Compared to data (from “exact” geology)

18. Bayes Theorem Permeability = geology Observation = past oil cut prior posterior;

19. Errors and Discrepancies Fine Coarse usually but implies geology geology

20. Example Fig. 1 Typical errors (lower, solid curves) and discrepancies (upper, dashed curves), plotted vs. PVI. The two families of curves are clearly distinguishable.

21. Mean error Sample covariance Precision Matrix Gaussian error model: has covariance C, mean

22. In Bayes Theorem, assume is exact. Then, is an error, probability

23. For arrival time error models, the formulation is identical, except that the independent variables s and t now denote the solution values, and not the time values, while the error e(s) denotes an error in the time of arrival of the solution value s.

24. Model Reduction: Limited data on solution errors Don’t over fit data Replace by finite matrix

25. Three Prediction Methods Prediction based on (a) Geostatistics only, no history match (prior). Average over full ensemble (b) History match with upscaled solutions (posterior). Bayesian weighted average over ensemble. (c) Window: select all fine grid solutions “close” to exact over past history. Average over restricted ensemble.

26. Comparing Prediction Methods • Window prediction is best, but not practical • -uses fine grid solutions for complete ensemble • -tests for inherent uncertainty • Prior prediction is worst • - makes no use of production data.

27. Error Reduction Prediction error reduction, as per cent of prior prediction choose present time to be oil cut of 0.6

28. Error Reduction Window based error reduction: 50% (fine grid: 100 x 100) Upscaled error reduction: 5 x 5 23% 10 x 10 32% 20 x 20 36%

29. Confidence Intervals 5% - 95% interval in future oil production Excludes extreme high-low values with 5% probability of occurrence Expressed as a per cent of predicted production

30. Confidence Intervals s0 = oil cut at present time. t0 = present time. Compute 5%--95% confidence intervals for future oil production, based on posterior and forward prediction using upscaled simulation. Result is a random variable. We express confidence intervals as a percent of predicted production, and take mean of this statistic.

31. Confidence Intervals Confidence intervals in percent for three values of present oil cut s0 and three levels of scaleup with fine grid values included. s0 100x100 20x20 10x10 5x5 0.8 [-13,22] [-21,36] [-24,35] [-27,34] 0.6 [-14,20] [-18,20] [-22,22] [-29,25] 0.4 [-14,17] [-18,18] [-24,21] [-33,23]

32. Arrival Time Error Analysis Error Model defined by 5 solution values: s = 1- (Breakthrough), 0.8, 0.6, 0.4, 0.2. Covariance is a 5 x 5 matrix, diagonally dominant, and neglecting diagonal terms, thus has 5 degrees of freedom. Thus it is simple. Covariance is basically independent of the geology correlation length. Thus it is robust.

33. Covariance Matrix: 10x10 Scaleup

34. Correlation matrix =

35. Histogram of off diagonal elements of the correlation matrices, 5x5, 10x10, 20x20 scaleup

36. Diagonal covariance matrix elements, three levels of scaleup, averaged over all correlation lengths

37. Error covariance for arrival time error model is proportional to the degree of scale up

38. Diagonal covariance matix elements for 10x10 scaleup, showing general lack of dependence on correlation length (except for s = 0.2 entry)

39. Covariance matrix diagonal entries for arrival time error model are independent of correlation length, except for final (s = 0.2) entry.

40. Confidence intervals for arrival time error model (%)

41. Arrival time error model vs. solution value model: confidence intervals (%) for s = 0.6 and 10x10 scaleup

42. Summary and Conclusions • New method to assess risk in prediction of future oil production • New methods to assess errors in simulations as probabilities • New upscaling allows consideration of ensemble of geology scenarios • Bayesian framework provides formal probabilities for risk and uncertainty

43. References • J. Glimm, S. Hou, H. Kim, D. H. Sharp, “A Probability Model for Errors in the Numerical Solutions of a Partial Differential Equation”. Computational Fluid Dynamics Journal, Vol. 9, 485-493 (2001). • J. Glimm, S. Hou, Y. Lee, D. H. Sharp, “Prediction of Oil Production with Confidence Intervals”, SPE reprint SPE66350 (2001). • J. Glimm, S. Hou, H. Kim, D. H. Sharp, K. Ye, W. Zhu, “Risk Management for Petroleum Reservoir Production”, J. Comp. Geosciences, to appear. • J. Glimm, Y. Lee, K. Ye, “A Simple Model for Scale Up Error” Cont. Math. 2002 (to appear).