1 / 35

Multiscale Ensemble Filtering in Reservoir Engineering Applications

Multiscale Ensemble Filtering in Reservoir Engineering Applications. Wiktoria Lawniczak Technical University in Delft. Content. Problem statement Introduction to multiscale ensemble filter Applications Conclusions. Problem statement. Estimating permeability given pressure rates (model)

inez
Download Presentation

Multiscale Ensemble Filtering in Reservoir Engineering Applications

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Multiscale Ensemble Filtering in Reservoir Engineering Applications Wiktoria Lawniczak Technical University in Delft

  2. Content • Problem statement • Introduction to multiscale ensemble filter • Applications • Conclusions

  3. Problem statement • Estimating permeability given pressure rates (model) • Two types of data: • 5 points • Large scale

  4. Multiscale ensemble filter THREE STEPS: • Tree construction • Upward sweep (update) • Downward sweep (smoothing)

  5. 1 1 1 1 1 1 1 2 2 2 2 2 2 2 5 5 5 5 5 5 5 6 6 6 6 6 6 6 3 3 3 3 3 3 3 4 4 4 4 4 4 4 7 7 7 7 7 7 7 8 8 8 8 8 8 8 9 9 9 9 9 9 9 10 10 10 10 10 10 10 13 13 13 13 13 13 13 14 14 14 14 14 14 14 11 11 11 11 11 11 11 12 12 12 12 12 12 12 15 15 15 15 15 15 15 16 16 16 16 16 16 16 Ensemble

  6. 1 2 5 6 3 4 7 8 9 10 13 14 11 12 15 16 EnMSF – Tree construction 1 SCALE 0 SCALE 1 SCALE 2=M

  7. EnMSF – Tree construction 2 - EIGENVALUE DECOMPOSITION

  8. 1 2 5 6 3 4 7 8 9 10 13 14 11 12 15 16 EnMSF – Tree construction 3

  9. 1 2 5 6 3 4 7 8 9 10 13 14 11 12 15 16 EnMSF – Tree construction 4

  10. 1 2 5 6 3 4 7 8 9 10 13 14 11 12 15 16 EnMSF – upward and downward sweeps 1

  11. EnMSF – upward and downward sweeps 2 Upward sweep - update Downward sweep - smoothing

  12. EnMSF summary A way to represent the covariance matrix with the tree structure measurements EnMSF updated ensemble ensemble

  13. Theoretical example 1 • replicates of size 64x64 • updating permeability with permeability 16 16 16 cells -check the influence of the different measurement types and ensemble size

  14. Theoretical example 2 50 replicates, st. dev = 9

  15. Theoretical example 3 50 replicates, finest scale Divergent EnKF

  16. Theoretical example 4 With channel No channel

  17. Practical example 1 • replicates of size 48x48 • updating permeability with rates • 94 members of ensemble • measurements from 5 wells • Tested: • 2 types of trees • different numbering schemes • correlation represented by the tree

  18. ‘9 pixels’ ‘9 children’ Practical example 2 • 16 states on each node • 9 states on the finest scale node • 16 states on each coarser scale node

  19. Practical example 3 ‘9 children’ RMSE ‘9 pixels’ The worst result – opposite diagonal numbering

  20. Practical example 4 ‘9 children’ RMSE ‘9 pixels’ The best result – square-like numbering

  21. Practical example 4 - correlation ‘9 pixels’ opposite diagonal numbering PRODUCT MOMENT CORRELATION

  22. Conclusions • EnMSF is a good tool to assimilate large scale data • Only one update step can already give a good representation of the truth • It gives a possibility to include prior knowledge about the field, numbering and tree topology can preserve important dependencies • Small ensemble can already give informative results • Still needs research on the proper use of the parameters from the tree construction step

  23. Thank you

  24. Downward recursion equation Upward recursion equation Search for a set of V(s) that provides the Markov property (the forecast covariance is well approximated). For simplicity V(s) is block diagonal.

  25. Predictive efficiency Computing all conditional cross-cov would be expensive -> predictive efficiency. It picks Vi(s) which minimizes the departure of optimality of the estimate: It was proved that they are given by the first rows of:

  26. Ui(s) Ui(s) contains the column eigenvectors in decreasing order of: zic(s) can be constrained by the neighborhood notion to ease the computations.

  27. Update and smoothing

  28. More update and smoothing

  29. 1 2 5 6 3 4 7 8 9 10 13 14 11 12 15 16 EnMSF – Tree construction 1 SCALE 0 SCALE 1 SCALE 2=M

  30. EnMSF – Tree construction SCALE 0 SCALE 1 SCALE 2 SCALE 3=M 1 2 5 6 3 4 7 8 9 10 13 14 11 12 15 16

  31. EnKF 1 model mean error covariance Kalman gain analyzed ensemble

  32. 1 2 5 6 3 4 7 8 9 10 13 14 11 12 15 16 EnMSF – upward and downward sweeps 2

  33. MEASUREMENTS TIME PROPAGATION (MODEL) UPDATE t EnKF 2 t-1

  34. 1 1 1 1 1 1 2 2 2 2 2 2 5 5 5 5 5 5 6 6 6 6 6 6 3 3 3 3 3 3 4 4 4 4 4 4 7 7 7 7 7 7 8 8 8 8 8 8 9 9 9 9 9 9 10 10 10 10 10 10 13 13 13 13 13 13 14 14 14 14 14 14 11 11 11 11 11 11 12 12 12 12 12 12 15 15 15 15 15 15 16 16 16 16 16 16

More Related