history matching with enkf
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E&P Seminar 2006
Using the Ensemble Kalman Filter for Reservoir Performance Forecasts
Achieved by: Zyed BOUZARKOUNA
Supervised by: Thomas SCHAAF
Exploration Production Department
Scientific Support Division 19/ 06/ 2008
E&P Seminar 2006 - 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 - 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 - 4 -
Geosatistics
Geostatistics: A method used to determine the spatial distribution of reservoir parameters.
Figure 1: Comparing kriging results (left) to two conditional simulation outcomes (right)
Estimation Simulation
E&P Seminar 2006 - 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).
timetcurrent
with HM
without HM
Prediction
History
Figure 2: History matching and production forecasts
E&P Seminar 2006 - 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)
E&P Seminar 2006 - 7 -
History Matching (Cont’d)
Main approaches of History Matching:
• Manual
• (Semi) Automatic
Simulation results Dsimul(θ)
Production Data Dobs
nobs
j
simulj
obsjj θDDwθF
1
2
21
Gradient-based Methods: Minimization of a cost function
• The solution may be the local minimum
• It supports only few parameters
E&P Seminar 2006 - 8 -
Motivations of the Project
A method:
The Ensemble Kalman Filter (EnKF)
Solution local minimum- Adapted to nonlinear problems
- The gradient does not need to be calculated explicitly
integrate as many variables as we need
E&P Seminar 2006 - 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 - 10 -
Basic Concept
Figure 3: Typical Kalman Filer application
E&P Seminar 2006 - 11 -
Analysis Schemef t f
t
pd M
: the true model; t
: the model forecast or the first-guess estimate;fd : 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.
where 1( ) f fT TK C M MC M C
( )f fa K d M
( ) faC I KM C
How can this concept be applied into oil reservoir monitoring?
E&P Seminar 2006 - 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 - 13 -
Concept
Figure 4: Description of the overall workflow of the EnKF
E&P Seminar 2006 - 14 -
Algorithm
The ensemble of state variables
1
1
1
. . .( ) . . .
. . .
Ns s
Ni d d
N
m mt m m
d d
static variables( ) sns im t
( ) dnd im t
( ) pnid t
dynamic variables
production data
The initialization step • Geostatistical methods
The forecast step: • Reservoir simulation (e.g. Eclipse)• Applying the Kalman gain
1( )f T f Ti i i i i i iK P M M P M R
The update step: • Analysis equation
The step-by-step process
( ) a f fj j e j jK d M
E&P Seminar 2006 - 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 - 16 -
The 3-D Synthetic Reservoir
• 50 * 50 * 4 gridblocks
• meters
• meters
• 10000 active cells
• 2 production wells (oil) P1 and P2
• 1 injection well (water) I1.
50x y
20z
Figure 5: An overview of the synthetic 3-D reservoir
A 3D-problem with:
E&P Seminar 2006 - 17 -
The Reference Property Fields
Figure 6: The true rock property fields: (a): the porosity field , (b): the permeability field kh
Property Value
Mean Porosity φMean permeability khPorosity variancePermeability variance Correlation coefficientVariance reduction factor
0.258000.00140000.81.0
Table 1: Geostatistical parameters
E&P Seminar 2006 - 18 -
The Initial Ensemble
Figure 7: Some realizations of porosity generated using SGcoSim
E&P Seminar 2006 - 19 -
1st Application: 4 Realizations with (φ, kh) and Constant Observations
0eR
The observation data: the bottomhole pressure (BHP) and the watercut (WCT) of each well.
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 vector of observations: non perturbed
The parameters of inversion are:
• 10000 porosity of each cell
• 10000 horizontal permeability of each cell
hk
E&P Seminar 2006 - 20 -
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.
1st Application: 4 Realizations with (φ, kh) and Constant Observations (Cont’d)
E&P Seminar 2006 - 21 -
1st Application: 4 Realizations with (φ, kh) and Constant Observations (Cont’d)
Main issues:
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
• Observations non perturbed
E&P Seminar 2006 - 22 -
2nd Application: 20 Realizations with (φ, kh, ratio kv/kh) and Perturbed Observations
• 10000 porosity of each cell
• 10000 horizontal permeability of each cell
• the ratio (A Gaussian ensemble: mean = 0.1, coefficient of variation = 0.1)
hk
v
h
kk
The vector of observations:
per obs noised d d
The parameters of inversion are:
E&P Seminar 2006 - 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
E&P Seminar 2006 - 24 -
3rd Application: 25 Realizations with (φ, kh, ratio kv/kh, Multflt)
The parameters of inversion are:
• 10000 porosity of each cell
• 10000 horizontal permeability of each cell
• the ratio (A Gaussian ensemble: mean = 0.1, coef. of variation = 0.1)
• The fault transmissibility Multflt (A Gaussian ensemble: mean = 1.2, coef. of variation = 0.1)
hk
v
h
kk
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
E&P Seminar 2006 - 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
E&P Seminar 2006 - 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 - 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)
E&P Seminar 2006 - 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 theensemble. the true model is represented in red
E&P Seminar 2006 - 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
E&P Seminar 2006 - 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
E&P Seminar 2006 - 31 -
Discussion
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
(a): 25 realizations (b): 60 realizations
(c): 60 realizations with the different initial ensemble
E&P Seminar 2006 - 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 - 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 - 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 - 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.
E&P Seminar 2006 - 36 -
Thank you for your attention
E&P Seminar 2006
Using the Ensemble Kalman Filter for Reservoir Performance Forecasts
Achieved by: Zyed BOUZARKOUNA
Supervised by: Thomas SCHAAF
19/ 06/ 2008
Exploration Production Department
Scientific Support Division
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