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

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

7. EnMSF – Tree construction 2
- EIGENVALUE DECOMPOSITION

8. EnMSF – Tree construction 31 2 5 6 3 4 7 8 9 10 13 14 11 12 15 16

9. EnMSF – Tree construction 41 2 5 6 3 4 7 8 9 10 13 14 11 12 15 16

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

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

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

13. of the different measurement types and ensemble size
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 Tree id. [s] 11.75
EnMSF time [s]
1.7656
EnKF time [s]
RMSE EnMSF
1.0745
RMSE EnKF
1.1912

15. Theoretical example 3 50 replicates, finest scale Tree id. [s] 11.5469
EnMSF time [s]
EnKF time [s]
11.5
RMSE EnMSF
1.2704
RMSE EnKF
1.6769
Tree id. [s]
EnMSF time [s]
EnKF time [s]
RMSE EnMSF
1.3256
RMSE EnKF
1.3293
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. Practical example 2 ‘9 pixels’ ‘9 children’ 16 states on each node
9 states on the finest scale node
16 states on each coarser scale node

19. Practical example 3 RMSE EnKF 0.45404 ‘9 children’ 0.55402 ‘9 pixels’
The worst result – opposite diagonal numbering

20. Practical example 4 RMSE EnKF 0.45404 ‘9 children’ 0.47938 ‘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

25. 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.

26. 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:

27. 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.

28. Update and smoothing

29. More update and smoothing

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

31. 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

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

33. EnMSF – upward and downward sweeps 24 1 2 5 6 3 7 8 9 10 13 14 11 12 15 16

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

35. 16 15 1 2 5 6 3 4 7 8 9 10 13 14 11 12