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) Two types of data: 5 points Large scale

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

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

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

EnMSF – Tree construction 2 - EIGENVALUE DECOMPOSITION

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

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

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

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

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

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

Theoretical example 2 50 replicates, st. dev = 9 Tree id. [s] 11.75 EnMSF time [s] 1.7656 EnKF time [s] 586.0625 RMSE EnMSF 1.0745 RMSE EnKF 1.1912

Theoretical example 3 50 replicates, finest scale Tree id. [s] 11.5469 EnMSF time [s] 0.59375 EnKF time [s] 11.5 RMSE EnMSF 1.2704 RMSE EnKF 1.6769 Tree id. [s] 11.2188 EnMSF time [s] 0.48438 EnKF time [s] 11.3281 RMSE EnMSF 1.3256 RMSE EnKF 1.3293 Divergent EnKF

Theoretical example 4 With channel No channel

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

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

Practical example 3 RMSE EnKF 0.45404 ‘9 children’ 0.55402 ‘9 pixels’ 0.65588 The worst result – opposite diagonal numbering

Practical example 4 RMSE EnKF 0.45404 ‘9 children’ 0.47938 ‘9 pixels’ 0.46337 The best result – square-like numbering

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

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

Thank you

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.

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:

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.

Update and smoothing

More update and smoothing

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

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

EnKF 1 model mean error covariance Kalman gain analyzed ensemble

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

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

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