1. Using the Ensemble Kalman Filter for
Reservoir Performance Forecasts
Achieved by: Zyed BOUZARKOUNA
Supervised by: Thomas SCHAAF
Exploration Production Department
E&P Seminar 2006
Scientific Support Division 19/ 06/ 2008

2. Outline
Generalities
• Reservoir Characterization using Geostatistical Simulations
• History Matching
Kalman Filtering
• Basic Concept
• Analysis Scheme
• The Ensemble Kalman Filter (EnKF)
The EnKF and History Matching
• Concept
• Algorithm
The EnKF Applications: Results and Discussions
Conclusions and Further Work
E&P Seminar 2006 -2-

3. Outline
Generalities
• Reservoir Characterization using Geostatistical Simulations
• History Matching
Kalman Filtering
• Basic Concept
• Analysis Scheme
• The Ensemble Kalman Filter (EnKF)
The EnKF and History Matching
• Concept
• Algorithm
The EnKF Applications: Results and Discussions
Conclusions and Further Work
E&P Seminar 2006 -3-

4. Geosatistics
Geostatistics: A method used to determine the spatial distribution of
reservoir parameters.
Estimation Simulation
Figure 1: Comparing kriging results (left) to two conditional simulation outcomes (right)
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5. History Matching
History matching: the act of reproducing a reservoir model until it closely reproduces the
past behavior of a production history (relatively to a chosen criteria).
without HM
History
with HM
Prediction
tcurrent time
Figure 2: History matching and production forecasts
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6. History Matching (Cont’d)
Main challenges of History Matching:
• Obtain a (set of) reservoir model(s) which gives more reliable future fluid flow
performances
• Dealing with many uncertainties (petrophysical reservoir description, data acquisition,
etc.)
• Working with many data (at different scales)
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7. History Matching (Cont’d)
Main approaches of History Matching:
• Manual
• (Semi) Automatic
Gradient-based Methods: Minimization of a cost function
Production Data Dobs
1 nobs
2 j 1
j 
F θ    w j D obs  D simul θ 
j 2
Simulation results Dsimul(θ)
• The solution may be the local minimum
• It supports only few parameters
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8. Motivations of the Project
A method:
- Adapted to nonlinear problems Solution  local minimum
- The gradient does not need integrate as many variables as we need
to be calculated explicitly
The Ensemble Kalman Filter (EnKF)
E&P Seminar 2006 -8-

9. Outline
Generalities
• Reservoir Characterization using Geostatistical Simulations
• History Matching
Kalman Filtering
• Basic Concept
• Analysis Scheme
• The Ensemble Kalman Filter (EnKF)
The EnKF and History Matching
• Concept
• Algorithm
The EnKF Applications: Results and Discussions
Conclusions and Further Work
E&P Seminar 2006 -9-

10. Basic Concept
Figure 3: Typical Kalman Filer application
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11. Analysis Scheme
 f   t  p f


 d  M  
t
t

: the true model;
f : the model forecast or the first-guess estimate;
d : the measurement of  ;t
pf : the unknown error in the forecast;
 : the unknown measurement error;
M : the measurement matrix which relates the vector of measurements to the true state.
 a   f  K (d  M  f )
C  ( I  KM )C
a f
where K  C M T ( MC M T  C )1
f f
How can this concept be applied into oil reservoir monitoring?
E&P Seminar 2006 - 11 -

12. Outline
Generalities
• Reservoir Characterization using Geostatistical Simulations
• History Matching
Kalman Filtering
• Basic Concept
• Analysis Scheme
• The Ensemble Kalman Filter (EnKF)
The EnKF and History Matching
• Concept
• Algorithm
The EnKF Applications: Results and Discussions
Conclusions and Further Work
E&P Seminar 2006 - 12 -

13. Concept
Figure 4: Description of the overall workflow of the EnKF
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14. Algorithm
The step-by-step process
 The initialization step
The ensemble of state variables • Geostatistical methods
 m1
s . . . msN 
 1 N 
 (ti )   md . . . md   The forecast step:
 d1 . . . dN  • Reservoir simulation (e.g. Eclipse)
  • Applying the Kalman gain
ms (ti )   ns static variables K i  Pi f M iT ( M i Pi f M iT  Ri ) 1
md (ti )   nd dynamic variables
np
d (ti )  production data
 The update step:
• Analysis equation
 a   jf  K e (d j  M jf )
j
E&P Seminar 2006 - 14 -

15. Outline
Generalities
• Reservoir Characterization using Geostatistical Simulations
• History Matching
Kalman Filtering
• Basic Concept
• Analysis Scheme
• The Ensemble Kalman Filter (EnKF)
The EnKF and History Matching
• Concept
• Algorithm
The EnKF Applications: Results and Discussions
Conclusions and Further Work
E&P Seminar 2006 - 15 -

16. The 3-D Synthetic Reservoir
A 3D-problem with:
• 50 * 50 * 4 gridblocks
• x  y  50 meters
• z  20 meters
• 10000 active cells
• 2 production wells (oil) P1 and P2
• 1 injection well (water) I1.
Figure 5: An overview of the synthetic 3-D reservoir
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17. The Reference Property Fields
Property Value
Mean Porosity φ 0.25
Mean permeability kh 800
Porosity variance 0.001
Permeability variance 4000
Correlation coefficient 0.8
Variance reduction factor 1.0
Table 1: Geostatistical parameters
Figure 6: The true rock property fields: (a): the porosity field 
, (b): the permeability
field kh
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18. The Initial Ensemble
Figure 7: Some realizations of porosity generated using SGcoSim
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19. 1st Application: 4 Realizations with (φ, kh) and Constant
Observations
 The production history: 01/01/2007 to 01/01/2023 (16 years):
• P1 and P2 (Production wells) are open from 01/01/2007 to 01/01/2023
• I1 (injection well) is open from 01/01/2009 to 01/01/2023
 The parameters of inversion are:
• 10000 porosity of each cell
• 10000 horizontal permeability kh each cell
of
 The vector of observations: non perturbed Re  0
 The observation data: the bottomhole pressure (BHP) and the watercut (WCT) of each well.
E&P Seminar 2006 - 19 -

20. 1st Application: 4 Realizations with (φ, kh) and Constant
Observations (Cont’d)
Figure 8: BHP (a) and WCT (b) at well P1 using updated realizations at 16
years. Results from the reference model are in red dots.
E&P Seminar 2006 - 20 -

21. 1st Application: 4 Realizations with (φ, kh) and Constant
Observations (Cont’d)
Figure 9: Zoom on the BHP at well P1 using updated realizations at 16 years. Results
from the reference model are in red dots.
• Size of the ensemble
Main issues:
• Observations non perturbed
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22. 2nd Application: 20 Realizations with (φ, kh, ratio kv/kh)
and Perturbed Observations
 The parameters of inversion are:
• 10000 porosity of each cell
• 10000 horizontal permeability kh each cell
of
kv
• the ratio (A Gaussian ensemble: mean = 0.1, coefficient of variation = 0.1)
kh
 The vector of observations:
d per
d obs
d noise
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23. 2nd Application: 20 Realizations with (φ, kh, ratio kv/kh)
and Perturbed Observations (Cont’d)
Figure 10: Production data at production wells (blue) simulated using the updated realizations at 16 years.
Results from reference model are in red dots
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24. 3rd Application: 25 Realizations with (φ, kh, ratio kv/kh, Multflt)
 The production history: 01/01/2007 to 01/01/2023 (16 years):
• P1 and P2 (Production wells) are open from 01/01/2007 to 01/01/2023
• I1 (injection well) is open from 01/01/2009 to 01/01/2023
 The parameters of inversion are:
• 10000 porosity of each cell
• 10000 horizontal permeability kh each cell
of
kv
• the ratio (A Gaussian ensemble: mean = 0.1, coef. of variation = 0.1)
kh
• The fault transmissibility Multflt (A Gaussian ensemble: mean = 1.2, coef. of variation = 0.1)
E&P Seminar 2006 - 24 -

25. 3rd Application: 25 Realizations with (φ, kh, ratio kv/kh, Multflt)
(Cont’d)
Figure 11: Production data at production wells (red) simulated using the updated realizations at 16 years,
compared to production data without EnKF (green). Results from reference model are in black dots
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26. 4th Application: 60 Realizations with (φ, kh, ratio kv/kh, Multflt)
Figure 12: Production data at production wells (red) simulated using the updated realizations at 16 years,
compared to production data without EnKF (green). Results from reference model are in black dots
E&P Seminar 2006 - 26 -

27. 4th Application: 60 Realizations with (φ, kh, ratio kv/kh, Multflt)
(Cont’d)
Figure 13: The evolution of the porosity field from t=0 to t=16 years: (a) through (q) for a member of the ensemble.
The true model is represented in (r)
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28. 4th Application: 60 Realizations with (φ, kh, ratio kv/kh, Multflt)
(Cont’d)
Figure 14: The ratio kv/kh versus the number of production data assimilated for 2 members of the
ensemble. the true model is represented in red
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29. 5th Application: 60 Realizations with (φ, kh, ratio kv/kh, Multflt)
with a Different Initial Ensemble
Figure 15: The initial ensemble generated using SGcosim
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30. 5th Application: 60 Realizations with (φ, kh, ratio kv/kh, Multflt)
with a Different Initial Ensemble (Cont’d)
Figure 16: Production data at production wells (red) simulated using the updated realizations at 16 years,
compared to production data without EnKF (green). Results from reference model are in black dots
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31. Discussion
(a): 25 realizations (b): 60 realizations
(c): 60 realizations with the different initial ensemble
Figure 17: The BHP at well P2 (red) simulated using the updated realizations at 16 years, compared to the BHP
without EnKF (green). Results from reference model are in black dots
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32. Production Forecasts
Figure 18: The WCT at production wells ((a): at P1, (b) P2) (red) simulated using the updated realizations at 16
years and then predicted until t = 6845, compared to production data without EnKF (green). Results from
reference model are in black dots
E&P Seminar 2006 - 32 -

33. Outline
Generalities
• Reservoir Characterization using Geostatistical Simulations
• History Matching
Kalman Filtering
• Basic Concept
• Analysis Scheme
• The Ensemble Kalman Filter (EnKF)
The EnKF and History Matching
• Concept
• Algorithm
The EnKF Applications: Results and Discussions
Conclusions and Further Work
E&P Seminar 2006 - 33 -

34. Conclusions
• A small ensemble of realizations can't be representative of the full probability density function.
• The use of perturbed observations is important in the EnKF to estimate the analysis-error covariances.
• The choice of the initial ensemble must be adequate in order to have accurate predictions.
• It is necessary to allow the updating of other variables than porosity and permeability fields in the
assimilation using EnKF.
E&P Seminar 2006 - 34 -

35. Suggestions for Further Work
More applications (synthetic and real) to investigate:
• The impact of the lack of observations on the robustness of the algorithm;
• Non-Gaussian distributions;
• The minimum number of realizations needed to reliably represent the uncertainty of the model.
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36. Thank you for your attention
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37. Using the Ensemble Kalman Filter for
Reservoir Performance Forecasts
Achieved by: Zyed BOUZARKOUNA
Supervised by: Thomas SCHAAF
Exploration Production Department
E&P Seminar 2006
Scientific Support Division 19/ 06/ 2008