History matching with EnKF

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


Outline







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


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


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)


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


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


Outline






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Basic Concept

Figure 3: Typical Kalman Filer application

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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?

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Outline






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Concept

Figure 4: Description of the overall workflow of the EnKF

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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

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Outline






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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

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The Initial Ensemble

Figure 7: Some realizations of porosity generated using SGcoSim

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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



• 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


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

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3rd Application: 25 Realizations with (φ, kh, ratio kv/kh, Multflt)




• 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

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4th Application: 60 Realizations with (φ, kh, ratio kv/kh, Multflt)


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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

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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)


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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

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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

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Outline






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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.

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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

Technology

History matching with EnKF