1. Brookhaven Science Associates U.S. Department of Energy Center for Data Intensive Computing 1 Chaos, Fluid Mixing, Uncertainty James Glimm Department of Applied Mathematics & Statistics State University of New York Stony Brook, NY 11794-3600 Center for Data Intensive Computing Brookhaven National Laboratory Upton, NY 11973-5000 - with - D. Sharp, B. Cheng, X.L. Li, H. Jin, D. Saltz, Z. Xu, F. Tangerman, M. Laforest, A. Marchese, E. George, Y. Zhang, S. Dutta, H. J. Kim, Y. Lee, K. Ye, S. Hou

2. Brookhaven Science Associates U.S. Department of Energy 2 Center for Data Intensive Computing Chaos and Prediction Chaos:Sensitive dependence on data, for example, initial conditions Result:Predictability lost Cure:Predict averages, statistics, probabilities Result:Predictability regained

3. Brookhaven Science Associates U.S. Department of Energy 3 Center for Data Intensive Computing Examples 1.Fluid Mixing -acceleration of density contrasting layers -unknown and sensitive initial data 2.Flow in Porous Media -unknown heterogeneous geology

4. Brookhaven Science Associates U.S. Department of Energy 4 Center for Data Intensive Computing Prediction for Multiscale Chaos -Fine scale features influence macroscopic flow -Multiscale defines frontier problems in many areas of science

5. Brookhaven Science Associates U.S. Department of Energy 5 Center for Data Intensive Computing Basic Methods I.Fine Scale Science -Study single unstable mode -Study mode coupling and interactions -Analytic methods -DNS = direct numerical simulation

6. Brookhaven Science Associates U.S. Department of Energy 6 Center for Data Intensive Computing Basic Methods II.Micro-Macro Coupling -Statistical models of mode interactions -Statistical analysis of DNS III.Macro Theories -Averaged equations -Validated by comparison to I, II -Validated by comparison to experiment -Mathematical analysis of averaged equations

7. Brookhaven Science Associates U.S. Department of Energy 7 Center for Data Intensive Computing Basic Methods IV.Prediction and Uncertainty -Errors in micro-macro approximation -Errors in numerics and experiment -Observational/exp. data: reduce uncertainty -Statistical inference: quantify uncertainty

8. Brookhaven Science Associates U.S. Department of Energy 8 Center for Data Intensive Computing Fluid Mixing Simulation Early time FronTier simulation ofLate time FronTier simulation of a3D RT mixing layer.

9. Brookhaven Science Associates U.S. Department of Energy 9 Center for Data Intensive Computing Comparison to Laboratory Experiments penetration distance of light fluid into the heavy fluid (bubbles)

10. Brookhaven Science Associates U.S. Department of Energy 10 Center for Data Intensive Computing FronTier and TVD Simulations Compared with Experiment  b = 0.06-0.07 FronTier (above)  b = 0.03-0.04 TVD (below)  b = 0.05 Experiment (Dimonte)  b = 0.06 - 0.07 (Youngs)

11. Brookhaven Science Associates U.S. Department of Energy 11 Center for Data Intensive Computing Comparison of Simulations FT nondiffusive across interface TVD diffusive across interface Mass diffusion appears to cause ~ 50% error in mixing rate.

12. Brookhaven Science Associates U.S. Department of Energy 12 Center for Data Intensive Computing Cross Section & Mass Diffusion: Comparison of TVD and FronTier TVD FronTier

13. Brookhaven Science Associates U.S. Department of Energy 13 Center for Data Intensive Computing

14. Brookhaven Science Associates U.S. Department of Energy 14 Center for Data Intensive Computing Simulation of two immersed bubbles

15. Brookhaven Science Associates U.S. Department of Energy 15 Center for Data Intensive Computing Simulation in Spherical Geometry Cross-sectional view of the growth of instability in a randomly perturbed axisymmetric sphere driven by an imploding shock wave in air. The shock Mack number M is 1.2 and the Atwood number A is 2/3.

16. Brookhaven Science Associates U.S. Department of Energy 16 Center for Data Intensive Computing Spherical Mix, Later Time Later time in the instability evolution.

17. Brookhaven Science Associates U.S. Department of Energy 17 Center for Data Intensive Computing Analytic Models Statistical Models of Interacting Bubbles Bubble Merger Models Bubble velocity = single mode velocity + envelope velocity Advanced bubble Retarded bubble Advanced bubble

18. Brookhaven Science Associates U.S. Department of Energy 18 Center for Data Intensive Computing Analytic Models Statistical Models of Interacting Bubbles Bubble Merger Models Bubble velocity = single mode velocity + envelope velocity Advanced bubble Retarded bubble Advanced bubble

19. Brookhaven Science Associates U.S. Department of Energy 19 Center for Data Intensive Computing Bubble Merger Criterion Envelope velocity > 0advanced bubble Envelope velocity < 0retarded bubble Remove bubble from ensemble where velocity = 0:  single mode velocity  =  envelope velocity 

20. Brookhaven Science Associates U.S. Department of Energy 20 Center for Data Intensive Computing Statistical Physics Model of Mixing Rate Solve statistical bubble model at renormalization group fixed point B. Cheng, J. Glimm, D. Sharp  b  0.05 - 0.06

21. Brookhaven Science Associates U.S. Department of Energy 21 Center for Data Intensive Computing COM Hypothesis Stationary Center of Mass (COM) (approx. valid unless A  1) h s = penetration of heavy fluid into light h s =  s Agt 2 (RT) COM   s /  b = solution of quadratic equation  s =  s (  b ) B. Cheng, J. Glimm, D. Saltz, D. Sharp

22. Brookhaven Science Associates U.S. Department of Energy 22 Center for Data Intensive Computing Mixing Zone Edge Models Z b,s (t)= mixing zone edge =h b,s in RT case =  bs Agt 2 in RT case Buoyancy Drag equation for Z b,s (t): Determine C b,s from FT theory above ODE valid for arbitrary acceleration

23. Brookhaven Science Associates U.S. Department of Energy 23 Center for Data Intensive Computing Chunk Mix Model Complete fluid variables for each fluid --Mathematically stable equations Improved physics model for mix --Pressure difference forces ~ drag Thermodynamics is process independent New closure proposed and tested --Zero parameters (incompressible flow) Analytic solution for incompressible case

24. Brookhaven Science Associates U.S. Department of Energy Center for Data Intensive Computing 2424 Multiphase Averaged Equations Microphysics: Macrophysics: Closure Problem: Determine F ren

25. Brookhaven Science Associates U.S. Department of Energy 25 Center for Data Intensive Computing Ensemble Averages Assume two fluids, labeled k=1 (light) and k=2 (heavy). Define

26. Brookhaven Science Associates U.S. Department of Energy 26 Center for Data Intensive Computing Closure Assume: v * depends on v 1 and v 2 and spatially dimensionless quantities only. Assume: regularity of v*. Theorem: (convex combination) and related relations for p* and (pv)* Assume: all  ’s depend on  k and t only.

27. Brookhaven Science Associates U.S. Department of Energy 27 Center for Data Intensive Computing Explicit Model:Zero Parameters (incompressible) One Parameter (compressible) Assume are fractional linear in  k. Then with k’ denoting the other fluid index and With the mixing zone boundaries Z k (t), and velocities V k (t), for incompressible flow. Boundary accelerations must be must be supplied externally to this model. Drag + buoyancy

28. Brookhaven Science Associates U.S. Department of Energy 28 Center for Data Intensive Computing Analytic Solution: Incompressible Case Let Then

29. Brookhaven Science Associates U.S. Department of Energy 29 Center for Data Intensive Computing Pressure versus distance

30. Brookhaven Science Associates U.S. Department of Energy 30 Center for Data Intensive Computing

31. Brookhaven Science Associates U.S. Department of Energy 31 Center for Data Intensive Computing

32. Brookhaven Science Associates U.S. Department of Energy 32 Center for Data Intensive Computing Asymptotic Expansion in Powers of M = Mach Number 0 th order =incompressible v,  1 st order=correction v,  2 nd order=incompressible p 1, p 2 + v,  correction 2 nd order p 1, p 2 =incompressible p 1, p 2  constraint: “missing” incompressible pressure equation

33. Brookhaven Science Associates U.S. Department of Energy 33 Center for Data Intensive Computing Summary: Multiscale Science Multiple Methods to Solve Chaotic Mix Problem -- Analytic Methods -- Microscopic Simulation (DNS) -- Edge Motion Models -- Averaged Solutions Closed Form Solutions Asymptotics Numerical Solutions

34. Brookhaven Science Associates U.S. Department of Energy 34 Center for Data Intensive Computing Summary: Multiscale Science Scientific Understanding: Consistent theory, experiment, simulation Still Needed: Other mix/chaos problems Transients, shock waves Validation, comparison to other closures FronTier Lite Tech transfer to other codes

35. Brookhaven Science Associates U.S. Department of Energy 35 Center for Data Intensive Computing Prediction and Uncertainty for Chaotic Flow n Prediction of Oil Reservoir Production n Confidence Intervals Allows Evaluation of Risk in Decision Making n Reservoir Development Choices Sizing of Production Equipment Location of New Wells

36. Brookhaven Science Associates U.S. Department of Energy 36 Center for Data Intensive Computing Basic Idea: I n Match geology to past oil production datai with probability of error n Start with geostatistical probability model for geology (permeability, etc). n Observe production rates, etc.

37. Brookhaven Science Associates U.S. Department of Energy 37 Center for Data Intensive Computing Basic Idea: II n Multiple simulations from ensemble n (Re)Assign probabilities based on data, degree of mismatch of simulation to history

38. Brookhaven Science Associates U.S. Department of Energy 38 Center for Data Intensive Computing Basic Idea: III Redefine probabilities and ensemble to be consistent with: (a) data (b) probable errors in simulation and data

39. Brookhaven Science Associates U.S. Department of Energy 39 Center for Data Intensive Computing Basic Idea: IV New ensemble of geologies = Posterior Prediction = sample from posterior Confidence intervals come from - posterior probabilities - errors in forward simulation

40. Brookhaven Science Associates U.S. Department of Energy 40 Center for Data Intensive Computing Illustration: Posterior reduces choices and uncertainty

41. Brookhaven Science Associates U.S. Department of Energy 41 Center for Data Intensive Computing 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

42. Brookhaven Science Associates U.S. Department of Energy 42 Center for Data Intensive Computing Problem Formulation Simulation study: Line drive, 2D reservoir Random permeability field log normal, random correlation length

43. Brookhaven Science Associates U.S. Department of Energy 43 Center for Data Intensive Computing Simple Reservoir Description in unit square constant

44. Brookhaven Science Associates U.S. Department of Energy 44 Center for Data Intensive Computing Ensemble 100 random permeability fields for each correlation length lnK gaussian, correlation length

45. Brookhaven Science Associates U.S. Department of Energy 45 Center for Data Intensive Computing Upscaling Solution from fine grid 100 x 100 grid Solution by upscaling 20 x 20, 10 x 10, 5 x 5 Upscaled grids

46. Brookhaven Science Associates U.S. Department of Energy 46 Center for Data Intensive Computing Upscaling by Wallstrom, Hou, Christie, Durlofsky, Sharp 1. Computational Geoscience 3:69-87 (1999) 2. SPE 51939 3. Transport in Porous Media (submitted) Upscaling References

47. Brookhaven Science Associates U.S. Department of Energy 47 Center for Data Intensive Computing Examples of Upscaled, Exact Oil Cut Curves Scale-up: Black (fine grid), Red (20x20), Blue (10x10), Green (5x5)

48. Brookhaven Science Associates U.S. Department of Energy 48 Center for Data Intensive Computing 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) Design of Study

49. Brookhaven Science Associates U.S. Department of Energy 49 Center for Data Intensive Computing Bayes Theorem Permeability = geology Observation = past oil cut posterior; prior

50. Brookhaven Science Associates U.S. Department of Energy 50 Center for Data Intensive Computing Errors and Discrepancies usually but implies geology FineCoarse

51. Brookhaven Science Associates U.S. Department of Energy 51 Center for Data Intensive Computing Example: Fig. 1 Typical errors (lower, solid curves) and discrepancies (upper, dashed curves), plotted vs. PVI. The two families of curves are clearly distinguishable.

52. Brookhaven Science Associates U.S. Department of Energy 52 Center for Data Intensive Computing Mean error Sample covariance Precision Matrix Gaussian error model: has covariance C, mean

53. Brookhaven Science Associates U.S. Department of Energy 53 Center for Data Intensive Computing In Bayes Theorem, assume is exact. Then, is an error, and probability

54. Brookhaven Science Associates U.S. Department of Energy 54 Center for Data Intensive Computing Model Reduction: Limited data on solution errors Don’t over fit data Replace by finite matrix

55. Brookhaven Science Associates U.S. Department of Energy 55 Center for Data Intensive Computing 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. Three Prediction Methods

56. Brookhaven Science Associates U.S. Department of Energy 56 Center for Data Intensive Computing Window prediction is best, but not practical -uses fine grid solutions for complete ensemble Prior prediction is worst - makes no use of production data. Comparing Prediction Methods

57. Brookhaven Science Associates U.S. Department of Energy 57 Center for Data Intensive Computing Prediction error reduction, as per cent of prior prediction choose present time to be oil cut of 0.6 Error Reduction

58. Brookhaven Science Associates U.S. Department of Energy 58 Center for Data Intensive Computing Window based error reduction:50% (fine grid:100 x 100) Upscaled error reduction: 5 x 523% 10 x 1032% 20 x 2036% Error Reduction

59. Brookhaven Science Associates U.S. Department of Energy 59 Center for Data Intensive Computing 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

60. Brookhaven Science Associates U.S. Department of Energy 60 Center for Data Intensive Computing Predicted future production confidence interval prediction + 13% - 28% Prediction depends on reservoir  % depends on reservoir  % averaged over choice of reservoir Confidence Intervals

61. Brookhaven Science Associates U.S. Department of Energy 61 Center for Data Intensive Computing Summary and Conclusions n New method to assess risk in prediction of future oil production n New methods to assess errors in simulations as probabilities n New upscaling allows consideration of ensemble of geology scenarios n Bayesian framework provides formal probabilities for risk and uncertainty