An example of flow based covariance localisation in the EnKF
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Transcript of An example of flow based covariance localisation in the EnKF
An example of flow based covariance localisation in the EnKF
Dan Kuznetsov, Dick KachumaGRC, Total E&P UK
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Talk
Simple 2D example
Streamline-based localisation
Real field example
Conclusions
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Covariance in the EnKF
Update equation
is the key element
is estimated from the ensemble
Estimation from the ensemble of a practical size may lead to a spurious correlation between model parameters and predictions
fDfa HψdCHHCHCψψ
1TT
C
N
iiiN 1
T
1
1ψψψψC
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2D synthetic example
Single well
51x51x1
Permeability is a Gaussian field
Conditioned at the well location
Other parameters are constant
BHP is the observation data
Ensemble of 100 realisations
5 EnKF assimilation steps
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Initial ensemble
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BHP
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Consider data along the line
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Correlation coefficient and pressure along the line
Step 1, 100 members
Pressure
Corr Coef
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Correlation coefficient for bigger ensemble
Step 1, 1000 members
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Correlation coefficient and pressure along the line
Step 1, 100 members
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Covariance localisation
Covariance matrix estimated from the ensemble is modified in order to make it decreasing at some distance from the observation point
Multiplied by 1 within some zone and by 0 outside or multiplied by some coefficients derived, for example from streamline sensitivities
Distance based localisation Houtekamer and Mitchell 1998, Hamill et al 2001, Skjervheim et al 2006, … Truncation of the covariance for production data may not be physically based since
the well influence zone may have non trivial shape
Streamline based localisation Aroyyo-Negrete et al 2006, Devegowda et al 2007 Covariance is truncated with respect to the flow path and/or streamline based
sensitivities
dd CC ~
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SL based localisation for production data
Limit covariance to the zone that has been swept by the produced liquid
Reverse problem: if a producer had been an injector the swept zone would be limited by a front of the injected liquid
Propagation of the liquid front along streamlines can be described in terms of Time Of Flight
To find the localisation zone: Trace the streamlines based on the velocity field Calculate time of flight Limit drainage and injection zones accordingly to the current model time
dv
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SL based localisation
Trace streamline -> Select a zone -> Update within the zone
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Pressure TOF
Asymptotic solution to the pressure diffusivity equation Kulkarni, Datta-Gupta and Vasco, 2000
is the velocity of the pressure front
Pressure Time Of Flight
Pressure TOF is related to the observed time as
in 2D and in 3D
Track the pressure front and set covariance to 0 outside the influence zone
1 ptc
k
d
p
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2t
6
2t
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Localisation within the drainage zone. Step 1
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Localisation within the drainage zone. Step 5
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Correlation coefficient. Step 1
Pressure TOF
Pressure
Corr Coef
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Correlation coefficient. Step 1 after truncation
Pressure TOF
Pressure
Corr Coef
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Correlation coefficient. Step 5
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Correlation coefficient. Step 5 after truncation
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Match with and without localisation
No localisation SL TOF localisation
Initial
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Permeability along the line. Step 5Initial
SL TOF localisation
1000 realisation
No localisation
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Real field case example
70x49x34, 69290 active cells
Modified parameters:porosity,permeability,aquifer strength
NTG is different for each realisation but not updated
Initial ensemble for the fault permeability was generated separately and included into Kx and Ky
3 producers, 2 water injectors
900 days of production
Data to match: WBHP, WOPR, WWCT
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Algorithm for reservoir simulation
Forward step with a conventional finite-difference reservoir simulator
Streamline tracing using separate routine (Texas A&M)
Compute TOF and Pressure TOF Practically, for a full field reservoir simulation a full drainage zone is used for
localisation of the covariance with bottom hole pressure
Stack influence zones for each well and observation for the current time
EnKF update with modified covariance
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Perm X, realisation 1, Lay 8
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Drainage zones, lay 8
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Drainage zones, lay 8
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Drainage zones, lay 8
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Drainage zones, lay 8
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Drainage zones, lay 8
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Drainage zones, lay 8
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Drainage zones, lay 8
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Drainage zones, lay 8
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Drainage zones, lay 8
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Drainage zones, lay 8
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Covariance: perm X, lay 8 and OPR in P1
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Covariance: perm X, lay 8 and OPR in P1
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Covariance: perm X, lay 8 and OPR in P1
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Covariance: perm X, lay 8 and OPR in P1
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Covariance: perm X, lay 8 and OPR in P1
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Covariance: perm X, lay 8 and OPR in P1
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Covariance: perm X, lay 8 and OPR in P1
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Covariance: perm X, lay 8 and OPR in P1
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Covariance: perm X, lay 8 and OPR in P1
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Covariance: perm X, lay 8 and OPR in P1
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BHP Well 1Initial
No Localisation SL TOF Localisation
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BHP Well 2Initial
No Localisation SL TOF Localisation
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BHP Well 3
Initial
No Localisation SL TOF Localisation
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WOPR Well 1Initial
No Localisation SL TOF Localisation
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Initial
No Localisation SL TOF Localisation
WOPR Well 2
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WOPR Well 3Initial
No Localisation SL TOF Localisation
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WWCT Well 1Initial
No Localisation SL TOF Localisation
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Initial
No Localisation SL TOF Localisation
WWCT Well 2
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Initial
No Localisation SL TOF Localisation
WWCT Well 3
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Permeability, lay 4. No localisation
Initial No localisation
P3
P2
P1
I2
I1
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Permeability, lay 4. SL TOF based localisation
P3
P2
P1
I2
I1
Initial SL TOF localisation
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Estimation of permeability in cell 1
cell 1
No Localisation SL TOF Localisation
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Estimation of permeability in cell 2
cell 2
No Localisation SL TOF Localisation
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Summary
Significant covariance disturbance happens outside well influence zone
Streamline based covariance localisation helps to reduce spurious correlation and decrease non-data based perturbation of the model parameters
It’s not necessary to use a SL simulator when applying a SL based covariance localisation
SL based localisation does not necessarily improve the match;although it’s case dependant
Initial realisations are less modified when localisation is used