Integrating Production and Seismic Data into Gaussian and Pluri-Gaussian Models with EnKF(S) Yong Zhao Yudou Wang Gaoming Li Al Reynolds EnKF Workshop: Voss June 2008
Jan 25, 2016
Integrating Production and Seismic Data
into Gaussian and Pluri-Gaussian
Models with EnKF(S)
Yong Zhao
Yudou Wang
Gaoming Li
Al Reynolds
EnKF Workshop: Voss June 2008
Sequential Data Assimilation (Ensemble Kalman Filter)
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augmented data Or,
EnKF Analysis (Bayesian Updating and Sampling)
Critical Assumptions:1. Predictions of state vectors are Gaussian;2. Covariances can be represented by ensemble members;3. Gaussian noise in data; 4. Predicted data are a linear function of the state vector.
Or, with data augmented state vector1. Predictions of augmented vector are Gaussian;2. Gaussian noise in data;3. Covariances can be represented by ensemble members.
Potential Problems in EnKF
1. Each analyzed vector of model parameters is a linear combination of
initial ensemble.
2. Difficult to match large data sets, e.g., seismic data.
3. Non-Gaussianity.
4. Strong non-linearity.
5. Poor knowledge of measurement errors.
6. Modeling of modeling errors.
7. Sampling errors due to finite ensemble size.
8. Inconsistency: updated pressure and saturations are inconsistent with
the updated models (statistically different from those obtained by
simulating from time zero)
Rescaling for Different Types of Data
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Assimilating production data:
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Assimilating with rescaled data:
Better conditioned
Truncation of Singular Values, PUNQ, Est. Contact Depths
Truncated at 0.9999 Rescaled
Channel Model
2-D case2-D case– 100 X 100 grid, 100 X 100 grid, – 4 producers and 1 injector are located in the channel facies4 producers and 1 injector are located in the channel facies– 360 days of production with BHP and WCT measurements360 days of production with BHP and WCT measurements– 300 days of prediction300 days of prediction– 100 ensemble members100 ensemble members
Z1 Z1 Truncation Facies
Conditional Models and Sw
Facies En20
Sw from true model Sw En20
True facies
EnKF Predictions
Prior prediction Prediction from EnKF Rerun from time zero
Normal Score Transform
Sw
CDF
Sw
CDF
S’w
S’w
Before Analysis
After Analysis
Prediction Domain Analysis Domain
Normal Score Transform
Standard EnKF Global Transform Local Transform
Predictions From Transforms
No Transform Global LocalEnKF
Rerun
HIEnKF Method
If model changes significantly, updated primary field may be more inconsistent with the updated model.
When the change of model is significant, rerun from zero; otherwise, we use the EnKS.
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EnKF
Model changes significantly EnKF
Prod-1
Prod-2 Prod-3
Prod-4
Inj-1
20 40 60 80 100
20
40
60
80
100
0.20
0.37
0.55
0.73
0.90Y
X
Prod-1
Prod-2 Prod-3
Prod-4
Inj-1
20 40 60 80 100
20
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X
Y
0
25
50
75
100
Prod-1
Prod-2 Prod-3
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X
Y
0
25
50
75
100
Prod-1
Prod-2 Prod-3
Prod-4
Inj-1
20 40 60 80 100
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100
Y
X
Prod-1
Prod-2 Prod-3
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Inj-1
20 40 60 80 100
20
40
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Y
X
Prod-1
Prod-2 Prod-3
Prod-4
Inj-1
20 40 60 80 100
20
40
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100
0.20
0.37
0.55
0.73
0.90Y
X
EnKF HIEnKF
True
0 100 200 300 400 500 600 7000.0
0.2
0.4
0.6
0.8
1.0
Prod 3
Wat
er C
ut
TIME (Day)
EnKF HIEnKF
EnKF vs. HIEnKF
0 100 200 300 400 500 600 700
2000
3000
4000
5000Prod 3
BH
P (
psi)
TIME (Day)0 100 200 300 400 500 600 700
2000
3000
4000
5000Prod 3
BH
P (
psi)
TIME (Day)
Three-Facies Model
3-D case– 50 X 50X3 grid, – 4 producers and 1 injector– Total rate constraint for each well– Hard data: observed facies in well gridblocks– 360 days of production with BHP and WCT measurements (monthly)– 300 days of prediction– Seismic data (at time zero and 300 days)– 100 ensemble members– Fixed porosity and permeability
Permeability (11md, 100md, 528md); Porosity (0.06, 0.13, 0.21)
Layer 1 Layer 2 Layer 3
Assimilating Dynamic Data While Satisfying Hard Data, SPE 113990
If does not satisfy the hard data:If does not satisfy the hard data:
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Completely redo the assimilation step:
Expand data with pseudo data:
EnKF Predictions
Prior prediction Prediction from EnKF state Rerun from time zero
Acoustic Impedance
t = 0
Match seismic data at the time they are measuredMatch seismic data at the time they are measured
Prod-1
Prod-2 Prod-3
Prod-4
Inj-1
10 20 30 40 50
10
20
30
40
50
2.10E4
7.55E4
1.30E5
1.85E5
2.39E5Y
X
Prod-1
Prod-2 Prod-3
Prod-4
Inj-1
10 20 30 40 50
10
20
30
40
50
1.70E4
7.13E4
1.26E5
1.80E5
2.34E5Y
X
Prod-1
Prod-2 Prod-3
Prod-4
Inj-1
10 20 30 40 50
10
20
30
40
50
2.80E4
7.90E4
1.30E5
1.81E5
2.32E5Y
X
t = 300days
Prod-1
Prod-2 Prod-3
Prod-4
Inj-1
10 20 30 40 50
10
20
30
40
50
2.10E4
7.55E4
1.30E5
1.85E5
2.39E5Y
X
Prod-1
Prod-2 Prod-3
Prod-4
Inj-1
10 20 30 40 50
10
20
30
40
50
1.70E4
7.13E4
1.26E5
1.80E5
2.34E5Y
X
Prod-1
Prod-2 Prod-3
Prod-4
Inj-1
10 20 30 40 50
10
20
30
40
50
2.80E4
7.90E4
1.30E5
1.81E5
2.32E5Y
X
Matching Seismic Data:Local Analysis of EnKF
Local analysis:– Analyzed models are not constrained to the
sub-space spanned by the initial ensemble
– Undesired roughness can be introduced into the analyzed models
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NN NN NN NN NN
NN NN XX NN NN
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Projection Method for Local Analysis
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projection before :
A large ensemble with 1200 realizations of model
that honors the hard data (M0)– Use the first 200 eigenvectors
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Assimilate Seismic Data- Local Analysis (2 seismic + prod)
True Facies
No projectionEn20
Prod-1
Prod-2 Prod-3
Prod-4
Inj-1
10 20 30 40 50
10
20
30
40
50
Y
X
Prod-1
Prod-2 Prod-3
Prod-4
Inj-1
10 20 30 40 50
10
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50
Y
X
Prod-1
Prod-2 Prod-3
Prod-4
Inj-1
10 20 30 40 50
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Y
X
With projectionEn20
Prod-1
Prod-2 Prod-3
Prod-4
Inj-1
10 20 30 40 50
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30
40
50
Y
X
Prod-1
Prod-2 Prod-3
Prod-4
Inj-1
10 20 30 40 50
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50
Y
X
Prod-1
Prod-2 Prod-3
Prod-4
Inj-1
10 20 30 40 50
10
20
30
40
50
Y
X
Assimilate Seismic Data- Local Analysis With Projection
First Seismic OnlyContinue EnKF for production data
Rerun from time zero
100 200 300 400 500 600 7000.0
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Prod 1
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er C
ut
TIME (Day)
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ut
TIME (Day)
100 200 300 400 500 600 7000.0
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er C
ut
TIME (Day)
0 100 200 300 400 500 600 7000.0
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Wat
er C
ut
TIME (Day)
100 200 300 400 500 600 7000.0
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er C
ut
TIME (Day)
0 100 200 300 400 500 600 7000.0
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Prod 3
Wat
er C
utTIME (Day)
1st seismic
2nd seismic
1st seismic
2nd seismic
Structure Map of PUNQ-S3
grid.
Fault, gas cap, strong aquifer.
52819
Data: BHP GOR WCTMatch to 4032 days
Estimate the Depths of Fluid Contacts with EnKS
State Vector y
Model parameters m Primary variables p Production data d
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Pe
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Introduction to HIEnKS Method
If model changes significantly, updated primary field may be more inconsistent with the updated model.
Only use HIEnKS when the change of model is significant. otherwise, we use the EnKS.
1it it0t
aiai pm ,, ,am ,0
aim ,
1it
EnKS
Model changes significantly EnKS
Examples
Example A:• Prior mean of OWC shifted up 20 feet
• Prior mean of GOC shifted down 20 feet.
True GOC
True OWC
Prior Mean of GOC
Prior Mean of OWC
Example A, prior oil column too thin
True GOC
True OWC
Prior Mean of GOC
Prior Mean of OWC
Example B, prior contact depths too deep
Example B:• Prior mean of OWC shifted down 20 feet
• Prior mean of GOC shifted down 20 feet.
20ft
20ft
20ft
20ft
STD: 20 ft
Comparison of Estimates of Fluid Contacts Example A Example B
HIE
nK
SE
nK
S
En
KS
HIE
nK
S
Consistency of Prediction, Example A
EnKS HIEnKS
EnKS: Future predictions poor, inconsistent. HIEnKS: Data matches good, consistent.
Du
rin
g D
ata
Ass
imil
atio
nR
eru
n f
rom
Tim
e 0
Consistency of Prediction, Example A
EnKS
EnKS: Assimilation good, prediction poor, inconsistent. HIEnKS: Assimilation/Prediction good, roughly consistent.
Du
rin
g D
ata
Ass
imil
atio
nR
eru
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rom
Tim
e 0
HIEnKS
Rock Property Fields- 4th, 5th layersT
ruth
En
KS
Vertical Permeability
HIE
nK
S
Horizontal Permeability
Comments
Iteration can improve reliability of data match,
predictions and consistency between parameters
and dynamical variables but is expensive.
Scaling can be critical if SVD is used.
EnKF combined with pluri-Gaussian gives
reasonable results (3D - rock properties – hard
data).
Comments
Pluri-Gaussian inappropriate for fluvial systems
– Cosine transforms, MRFs, KPCA?
Seismic: local analysis with projection seems
feasible but is currently ad hoc.