History matching with EnKF

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History matching with EnKF

  1. 1. Using the Ensemble Kalman Filter for Reservoir Performance Forecasts Achieved by: Zyed BOUZARKOUNA Supervised by: Thomas SCHAAF Exploration Production Department E&P Seminar 2006 Scientific Support Division 19/ 06/ 2008
  2. 2. Outline Generalities • Reservoir Characterization using Geostatistical Simulations • History Matching Kalman Filtering • Basic Concept • Analysis Scheme • The Ensemble Kalman Filter (EnKF) The EnKF and History Matching • Concept • Algorithm The EnKF Applications: Results and Discussions Conclusions and Further Work E&P Seminar 2006 -2-
  3. 3. Outline Generalities • Reservoir Characterization using Geostatistical Simulations • History Matching Kalman Filtering • Basic Concept • Analysis Scheme • The Ensemble Kalman Filter (EnKF) The EnKF and History Matching • Concept • Algorithm The EnKF Applications: Results and Discussions Conclusions and Further Work E&P Seminar 2006 -3-
  4. 4. Geosatistics Geostatistics: A method used to determine the spatial distribution of reservoir parameters. Estimation Simulation Figure 1: Comparing kriging results (left) to two conditional simulation outcomes (right) E&P Seminar 2006 -4-
  5. 5. History Matching History matching: the act of reproducing a reservoir model until it closely reproduces the past behavior of a production history (relatively to a chosen criteria). without HM History with HM Prediction tcurrent time Figure 2: History matching and production forecasts E&P Seminar 2006 -5-
  6. 6. History Matching (Cont’d) Main challenges of History Matching: • Obtain a (set of) reservoir model(s) which gives more reliable future fluid flow performances • Dealing with many uncertainties (petrophysical reservoir description, data acquisition, etc.) • Working with many data (at different scales) E&P Seminar 2006 -6-
  7. 7. History Matching (Cont’d) Main approaches of History Matching: • Manual • (Semi) Automatic Gradient-based Methods: Minimization of a cost function Production Data Dobs 1 nobs 2 j 1 j  F θ    w j D obs  D simul θ  j 2 Simulation results Dsimul(θ) • The solution may be the local minimum • It supports only few parameters E&P Seminar 2006 -7-
  8. 8. Motivations of the Project A method: - Adapted to nonlinear problems Solution  local minimum - The gradient does not need integrate as many variables as we need to be calculated explicitly The Ensemble Kalman Filter (EnKF) E&P Seminar 2006 -8-
  9. 9. Outline Generalities • Reservoir Characterization using Geostatistical Simulations • History Matching Kalman Filtering • Basic Concept • Analysis Scheme • The Ensemble Kalman Filter (EnKF) The EnKF and History Matching • Concept • Algorithm The EnKF Applications: Results and Discussions Conclusions and Further Work E&P Seminar 2006 -9-
  10. 10. Basic Concept Figure 3: Typical Kalman Filer application E&P Seminar 2006 - 10 -
  11. 11. Analysis Scheme  f   t  p f    d  M   t t  : the true model; f : the model forecast or the first-guess estimate; d : the measurement of  ;t pf : the unknown error in the forecast;  : the unknown measurement error; M : the measurement matrix which relates the vector of measurements to the true state.  a   f  K (d  M  f ) C  ( I  KM )C a f where K  C M T ( MC M T  C )1 f f How can this concept be applied into oil reservoir monitoring? E&P Seminar 2006 - 11 -
  12. 12. Outline Generalities • Reservoir Characterization using Geostatistical Simulations • History Matching Kalman Filtering • Basic Concept • Analysis Scheme • The Ensemble Kalman Filter (EnKF) The EnKF and History Matching • Concept • Algorithm The EnKF Applications: Results and Discussions Conclusions and Further Work E&P Seminar 2006 - 12 -
  13. 13. Concept Figure 4: Description of the overall workflow of the EnKF E&P Seminar 2006 - 13 -
  14. 14. Algorithm The step-by-step process  The initialization step The ensemble of state variables • Geostatistical methods  m1 s . . . msN   1 N   (ti )   md . . . md   The forecast step:  d1 . . . dN  • Reservoir simulation (e.g. Eclipse)   • Applying the Kalman gain ms (ti )   ns static variables K i  Pi f M iT ( M i Pi f M iT  Ri ) 1 md (ti )   nd dynamic variables np d (ti )  production data  The update step: • Analysis equation  a   jf  K e (d j  M jf ) j E&P Seminar 2006 - 14 -
  15. 15. Outline Generalities • Reservoir Characterization using Geostatistical Simulations • History Matching Kalman Filtering • Basic Concept • Analysis Scheme • The Ensemble Kalman Filter (EnKF) The EnKF and History Matching • Concept • Algorithm The EnKF Applications: Results and Discussions Conclusions and Further Work E&P Seminar 2006 - 15 -
  16. 16. The 3-D Synthetic Reservoir A 3D-problem with: • 50 * 50 * 4 gridblocks • x  y  50 meters • z  20 meters • 10000 active cells • 2 production wells (oil) P1 and P2 • 1 injection well (water) I1. Figure 5: An overview of the synthetic 3-D reservoir E&P Seminar 2006 - 16 -
  17. 17. The Reference Property Fields Property Value Mean Porosity φ 0.25 Mean permeability kh 800 Porosity variance 0.001 Permeability variance 4000 Correlation coefficient 0.8 Variance reduction factor 1.0 Table 1: Geostatistical parameters Figure 6: The true rock property fields: (a): the porosity field  , (b): the permeability field kh E&P Seminar 2006 - 17 -
  18. 18. The Initial Ensemble Figure 7: Some realizations of porosity generated using SGcoSim E&P Seminar 2006 - 18 -
  19. 19. 1st Application: 4 Realizations with (φ, kh) and Constant Observations  The production history: 01/01/2007 to 01/01/2023 (16 years): • P1 and P2 (Production wells) are open from 01/01/2007 to 01/01/2023 • I1 (injection well) is open from 01/01/2009 to 01/01/2023  The parameters of inversion are: • 10000 porosity of each cell • 10000 horizontal permeability kh each cell of  The vector of observations: non perturbed Re  0  The observation data: the bottomhole pressure (BHP) and the watercut (WCT) of each well. E&P Seminar 2006 - 19 -
  20. 20. 1st Application: 4 Realizations with (φ, kh) and Constant Observations (Cont’d) Figure 8: BHP (a) and WCT (b) at well P1 using updated realizations at 16 years. Results from the reference model are in red dots. E&P Seminar 2006 - 20 -
  21. 21. 1st Application: 4 Realizations with (φ, kh) and Constant Observations (Cont’d) Figure 9: Zoom on the BHP at well P1 using updated realizations at 16 years. Results from the reference model are in red dots. • Size of the ensemble Main issues: • Observations non perturbed E&P Seminar 2006 - 21 -
  22. 22. 2nd Application: 20 Realizations with (φ, kh, ratio kv/kh) and Perturbed Observations  The parameters of inversion are: • 10000 porosity of each cell • 10000 horizontal permeability kh each cell of kv • the ratio (A Gaussian ensemble: mean = 0.1, coefficient of variation = 0.1) kh  The vector of observations: d per d obs d noise E&P Seminar 2006 - 22 -
  23. 23. 2nd Application: 20 Realizations with (φ, kh, ratio kv/kh) and Perturbed Observations (Cont’d) Figure 10: Production data at production wells (blue) simulated using the updated realizations at 16 years. Results from reference model are in red dots E&P Seminar 2006 - 23 -
  24. 24. 3rd Application: 25 Realizations with (φ, kh, ratio kv/kh, Multflt)  The production history: 01/01/2007 to 01/01/2023 (16 years): • P1 and P2 (Production wells) are open from 01/01/2007 to 01/01/2023 • I1 (injection well) is open from 01/01/2009 to 01/01/2023  The parameters of inversion are: • 10000 porosity of each cell • 10000 horizontal permeability kh each cell of kv • the ratio (A Gaussian ensemble: mean = 0.1, coef. of variation = 0.1) kh • The fault transmissibility Multflt (A Gaussian ensemble: mean = 1.2, coef. of variation = 0.1) E&P Seminar 2006 - 24 -
  25. 25. 3rd Application: 25 Realizations with (φ, kh, ratio kv/kh, Multflt) (Cont’d) Figure 11: Production data at production wells (red) simulated using the updated realizations at 16 years, compared to production data without EnKF (green). Results from reference model are in black dots E&P Seminar 2006 - 25 -
  26. 26. 4th Application: 60 Realizations with (φ, kh, ratio kv/kh, Multflt) Figure 12: Production data at production wells (red) simulated using the updated realizations at 16 years, compared to production data without EnKF (green). Results from reference model are in black dots E&P Seminar 2006 - 26 -
  27. 27. 4th Application: 60 Realizations with (φ, kh, ratio kv/kh, Multflt) (Cont’d) Figure 13: The evolution of the porosity field from t=0 to t=16 years: (a) through (q) for a member of the ensemble. The true model is represented in (r) E&P Seminar 2006 - 27 -
  28. 28. 4th Application: 60 Realizations with (φ, kh, ratio kv/kh, Multflt) (Cont’d) Figure 14: The ratio kv/kh versus the number of production data assimilated for 2 members of the ensemble. the true model is represented in red E&P Seminar 2006 - 28 -
  29. 29. 5th Application: 60 Realizations with (φ, kh, ratio kv/kh, Multflt) with a Different Initial Ensemble Figure 15: The initial ensemble generated using SGcosim E&P Seminar 2006 - 29 -
  30. 30. 5th Application: 60 Realizations with (φ, kh, ratio kv/kh, Multflt) with a Different Initial Ensemble (Cont’d) Figure 16: Production data at production wells (red) simulated using the updated realizations at 16 years, compared to production data without EnKF (green). Results from reference model are in black dots E&P Seminar 2006 - 30 -
  31. 31. Discussion (a): 25 realizations (b): 60 realizations (c): 60 realizations with the different initial ensemble Figure 17: The BHP at well P2 (red) simulated using the updated realizations at 16 years, compared to the BHP without EnKF (green). Results from reference model are in black dots E&P Seminar 2006 - 31 -
  32. 32. Production Forecasts Figure 18: The WCT at production wells ((a): at P1, (b) P2) (red) simulated using the updated realizations at 16 years and then predicted until t = 6845, compared to production data without EnKF (green). Results from reference model are in black dots E&P Seminar 2006 - 32 -
  33. 33. Outline Generalities • Reservoir Characterization using Geostatistical Simulations • History Matching Kalman Filtering • Basic Concept • Analysis Scheme • The Ensemble Kalman Filter (EnKF) The EnKF and History Matching • Concept • Algorithm The EnKF Applications: Results and Discussions Conclusions and Further Work E&P Seminar 2006 - 33 -
  34. 34. Conclusions • A small ensemble of realizations can't be representative of the full probability density function. • The use of perturbed observations is important in the EnKF to estimate the analysis-error covariances. • The choice of the initial ensemble must be adequate in order to have accurate predictions. • It is necessary to allow the updating of other variables than porosity and permeability fields in the assimilation using EnKF. E&P Seminar 2006 - 34 -
  35. 35. Suggestions for Further Work More applications (synthetic and real) to investigate: • The impact of the lack of observations on the robustness of the algorithm; • Non-Gaussian distributions; • The minimum number of realizations needed to reliably represent the uncertainty of the model. E&P Seminar 2006 - 35 -
  36. 36. Thank you for your attention E&P Seminar 2006 - 36 -
  37. 37. Using the Ensemble Kalman Filter for Reservoir Performance Forecasts Achieved by: Zyed BOUZARKOUNA Supervised by: Thomas SCHAAF Exploration Production Department E&P Seminar 2006 Scientific Support Division 19/ 06/ 2008

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