Use of Model Error With the EnKF
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Transcript of Use of Model Error With the EnKF
Centre for Integrated Petroleum Research
University of Bergen, Norway
Use of Model Error With the EnKF
S. I. Aanonsen
Voss Workshop 18 – 20 June 2008
Centre for Integrated Petroleum Research
University of Bergen, Norway
Motivation
• An unkown model error is present in all real problems
• Originally, the Kalman filter and the EnKF was developed to update state variables for models with error
• Within the petroleum applications focus has mainly been on EnKF as a history-matching tool. Model error is normally not included(?) – However, real-time reservoir management also requires
short term predictions based on a very accurate data match
Centre for Integrated Petroleum Research
University of Bergen, Norway
Questions
• Is it possible to obtain good short term predictions and better estimates of unknown parameters by including a model error term?
• Can a model error be estimated as a part of the EnKF updating process?
Centre for Integrated Petroleum Research
University of Bergen, Norway
Model Problem
• Single-phase flow:
on0,0)0(
0,],0[],0[
)())((
n
ptp
tLLx
xQpxKt
p
Centre for Integrated Petroleum Research
University of Bergen, Norway
True permeability field Fine grid (20x20)
Centre for Integrated Petroleum Research
University of Bergen, Norway
Estimated mean permeabilityFine Scale EnKF. No model error
Log-normal Prior model:
Mean log(K) = 2 (100mD)
SD [log(K)] = 0.7
Spherical variogram with correlation length = L/4
Ensemble size: 40
Centre for Integrated Petroleum Research
University of Bergen, Norway
EnKF update and ensemble predictions Well pressures vs time
Measurement error: 0.3
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Rerun from time zero Well pressures vs time
Centre for Integrated Petroleum Research
University of Bergen, Norway
EnKF update and ensemble predictionsCoarse grid (10x10). No model error
Same prior model as in fine-scale EnKF, except that the correlation length is L/2
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University of Bergen, Norway
Mean estimated permeability coarse modelNo model error
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University of Bergen, Norway
Including model error
• Alternative 1. Uncorrelated error (in time):
,...2,1,)( 11 ncwugu nan
fn
where
,...2,1,0,),0(~ nCNw wn
Cw has the same correlation structure as the prior permeability field
and same variance as the measurement error. c = 1.
No error added to the permeability.
Centre for Integrated Petroleum Research
University of Bergen, Norway
Including model error
• Alternative 2. Correlated error (from Geir Evensen’s book). Model error is added to the state vector and updated “as usual”.
Forecast step:
fn
an
fn
nan
fn
cqugu
wqq
)(
)1(
1
11
where
1
),0(~
,...2,1,0,),0(~
0
w
wn
CNq
nCNw
Centre for Integrated Petroleum Research
University of Bergen, Norway
EnKF update and ensemble predictionsCoarse grid (10x10). Uncorrelated model error
Centre for Integrated Petroleum Research
University of Bergen, Norway
EnKF update and ensemble predictionsCoarse grid (10x10). Correlated model error, = 1
Centre for Integrated Petroleum Research
University of Bergen, Norway
Mean estimated permeability coarse model With model error
Uncorrelated error Correlated error, = 1
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University of Bergen, Norway
Rerun from time zeroCoarse model. No model error
Centre for Integrated Petroleum Research
University of Bergen, Norway
Rerun from time zeroCorrelated model error
Centre for Integrated Petroleum Research
University of Bergen, Norway
Rerun from time zeroUncorrelated model error
Centre for Integrated Petroleum Research
University of Bergen, Norway
Conclusions
• A typical model error may result in EnKF not being able to match data, ensemble collapse and poor predictions and parameter estimates.
• Adding an “unknown” model error may improve all of this and provide reasonably good predictions, at least short term.