# Multilevel modelling of an oil reservoir simulator - MUCM

Multilevel modelling of an oil reservoir simulator - MUCM

Multilevel modelling of an

oil reservoir simulator

Jonathan Cumming (Durham University)

John Paul Gosling (University of Leeds)

Jeremy Oakley (University of Sheffield)

Allan Seheult (Durham University)

Slide 1

The simulator

Wells

Monthly time steps from 1983 to 1999.

Overall, 25,090 model outputs to think about.

Slide 2

Multilevel models

5 minutes

≈ ≈

10 minutes

5 to 6 hours

The crudest version of the model was found

to be so far away from the most accurate

model that it added nothing to our analysis.

Slide 3

Initial data exploration

Slide 4

Water produced

Question of interest

Can we calibrate the model to match the

water production at the wells?

Well A9

Month of simulation

There are still about 5,000 model outputs

to consider and 23 accessible inputs.

Slide 5

Two approaches

General

difference

Difference in

refocussing

and history

matching

Difference in

emulator

fitting

Probabilistic Bayesian Bayes linear

Full specification of

probability distributions

for the quantities of

interest.

Computationally

expensive.

Identification through

calibration: provides

posterior credible

regions to concentrate

on.

Simple regression model

as prior for simulator

and let the GP term do

the work.

Specification of first- and

second-order moments

for the quantities of

interest.

Computationally

inexpensive.

Identification of poor

matches through

implausibility metric.

More effort put into

constructing a regression

model to bring GP term

closer to independent

error.

Slide 6

Dimension reduction (I)

Many of the outputs are zero over the

simulation period.

Select an “important” subset of outputs

using a principal variables approach.

Cumming & Wooff (2007). Dimension reduction via principal

variables. Comput. Statist. Data Anal..

Slide 7

Dimension reduction (I)

Each principal variable corresponds to one

month’s output from one well.

Slide 8

Cumulative percentage of variance explained

Dimension reduction (II)

In the probabilistic Bayesian camp, we

further reduced the output space using PCA.

PC1 ≈ 48%

PC2 ≈ 67%

PC3 ≈ 81%

PC4 ≈ 87%

PC5 ≈ 91%

PC6 ≈ 94%

PC7 ≈ 96%

Slide 9

Multilevel emulation

Bayes linear approach:

Linked through the prior specification for the accurate model

Probabilistic Bayesian approach:

Emulate the coarse model Emulate the difference between models

.

,

,

Slide 10

Implausibility measure:

Error model:

.

,

Slide 11

Implausibility

Implausibility

Input value

Slide 12

Implausibility

Slide 13

Implausibility

Slide 14

Credible regions

Slide 15

Comparison

Slide 16

Second set of training data

Slide 17

Second set of training data

Slide 18

Wave 2 implausibility

Implausibility

Input value

Slide 19

Wave 1 implausibility

Implausibility

Input value

Slide 20

Wave 2 credible regions

Slide 21

Wave 1 credible regions

Slide 22

Results and comparisons

Intervals have not

necessarily tightened

as a result of more

training data.

Bayes linear region

has a 96% posterior

probability.

Slide 23

Results and comparisons

We focussed on 14 outputs. How are we doing for the rest?

Water produced

Month of simulation

Slide 24

The poor model problem

Slide 25

The poor model problem

Slide 26

The poor model problem

Slide 27

The poor model problem

Slide 28

The poor model problem

Slide 29

References

Cumming, J.A. and Goldstein, M. (2010). Bayes linear

uncertainty analysis for oil reservoirs based on multiscale

computer experiments. In The Handbook of Applied

Bayesian Analysis, 241-70.

Kennedy, M.C. and O'Hagan, A. (2000). Predicting the

output from a complex computer code when fast

approximations are available. Biometrika, 87, 1-13.

www.mucm.ac.uk

Slide 30

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