Iterative ensemble smoothers and applications to
reservoir history matching
Yan Chen1 and Dean Oliver2
1
2
International Research Institute of Stavanger, Bergen, Norway
Uni Centre for Integrated Petroleum Research, Bergen, Norway
June 17, 2013
1/44
History matching: the Norne field
(Figures from Statoil reports)
I
Multi-phase flow is primarily controlled by heterogeneous
permeability and porosity
I
Faults with unknown sealing properties
2/44
History matching: data and unknown variables
(Figure from Statoil report)
I
I
Data are primarily limited to a small number of well locations
Variables to be estimated
1. Static variables: permeability and porosity of each simulation cell,
fault transmissibility, etc.
2. Dynamic variables: phase saturation and pressure, etc.
3/44
The Norne field drainage strategy
(Figure from Statoil report)
I
Water alternating gas injection
I
Use the iterative ES (ensemble smoother) to avoid updating
saturation with EnKF (ensemble Kalman filter)
4/44
Noisy data: production rates
Production since 1997. Full field data (through 2006) released in August
2012
B-2H
D-2H
7000
6000
6000
5000
5000
Rates
Rates
4000
4000
3000
3000
2000
2000
1000
1000
0
0
0
2
4
6
Time (year)
8
0
2
4
6
Time (year)
8
Oil, gas and water production rates. Unit is m3 /day for oil and water and
1000 m3/day for gas
5/44
Sampling from posterior pdf
Compute parameters that minimize a stochastic objective function:
Sum of squared data mismatch
z
1
Ji (x) = (g(x)
2
}|
doi )T CD 1 (g(x)
{
doi )
+
1
x
2|
xpr
i
T
CX 1 x
{z
xpr
.
i
}
Model parameter mismatch
where
x is vector of model variables
xpr
i is the ith sample from the prior distribution
do = g(xtr ) + ✏ and ✏ ⇠ N[0, CD ]
doi ⇠ N[do , CD ]
6/44
Gauss-Newton method
Solving rJi (x) = 0,
x`i =
(x`i
xpr
i )
where G` = rg(x` ).
⇣
⌘
T
CX GT
C
+
G
C
G
D
` X `
`
⇣
g(x`i )
1
doi
G` (x`i
g(xpr
i )
⌘
doi .
⌘
xpr
)
.
i
At the first iteration (` = 1), when x`i = xpr
i
x1i =
⇣
⌘
T
CX GT
C
+
G
C
G
1
D
X
1
1
1⇣
Depending on the amount of data, Gauss-Newton and quasi-Newton
methods have been used to obtain parameter estimates, from which state
estimates were obtained.
7/44
Generating samples from the posterior pdf
I
T
EnKF: Approximate CX GT
1 and G1 CX G1 from ensemble
x1i =
=
⇣
⌘ 1⇣
⌘
T
o
CX GT
g(xpr
)
d
1 CD + G1 CX G1
i
i
⇣
⌘ 1⇣
T
xpr dT
(N
1)C
+
d
d
g(xpr
e
1
D
1
1
i )
doi
⌘
8/44
Generating samples from the posterior pdf
I
T
EnKF: Approximate CX GT
1 and G1 CX G1 from ensemble
x1i =
=
I
⇣
⌘ 1⇣
⌘
T
o
CX GT
g(xpr
)
d
1 CD + G1 CX G1
i
i
⇣
⌘ 1⇣
T
xpr dT
(N
1)C
+
d
d
g(xpr
e
1
D
1
1
i )
doi
⌘
T
GN-EnRML: Approximate G` CX GT
` and CX G` and G` from
ensemble
x`i =
`
` (xi
xpr
i )
⇣
⌘ 1
T
T
C
G
C
+
G
C
G
` X `
D
` X `
⇣
g(x`i ) doi
G` (x`i
⌘
xpr
)
.
i
8/44
Five-spot waterflood example
500
0
0
500
1000
x
1500
2000
1500
1000
500
0
0
500
1000
x
1500
2000
0.121
0.112
0.103
0.094
0.085
0.076
0.067
0.058
0.049
0.040
0.031
0.022
0.013
0.003
-0.006
-0.015
-0.024
-0.033
-0.042
-0.051
-0.060
-0.069
-0.078
-0.087
-0.096
-0.105
2000
1500
1000
500
0
0
500
1000
x
1500
2000
1.597
1.512
1.427
1.341
1.256
1.171
1.085
1.000
0.915
0.830
0.744
0.659
0.574
0.489
0.403
0.318
0.233
0.147
0.062
-0.023
-0.108
-0.194
-0.279
-0.364
-0.449
-0.535
ensemble CX G T
2000
1500
y
1000
2000
y
y
Well P1
1500
adjoint CX G T
ensemble G
0.341
0.327
0.312
0.298
0.283
0.269
0.254
0.240
0.225
0.211
0.196
0.181
0.167
0.152
0.138
0.123
0.109
0.094
0.080
0.065
0.051
0.036
0.022
0.007
-0.008
-0.022
y
adjoint G
2000
1000
500
0
0
500
1000
x
1500
2000
4.005
3.774
3.542
3.310
3.079
2.847
2.615
2.383
2.152
1.920
1.688
1.457
1.225
0.993
0.761
0.530
0.298
0.066
-0.165
-0.397
-0.629
-0.861
-1.092
-1.324
-1.556
-1.787
9/44
Five-spot waterflood example
0
0
500
1000
x
1500
2000
1500
y
Well P3
2000
1000
500
0
0
500
1000
x
1500
2000
2.89
2.76
2.64
2.51
2.39
2.27
2.14
2.02
1.89
1.77
1.65
1.52
1.40
1.27
1.15
1.03
0.90
0.78
0.65
0.53
0.40
0.28
0.16
0.03
-0.09
-0.22
0
0
500
1000
x
1500
2000
1500
1000
500
0
0
500
1000
x
1500
2000
2000
29.348
27.749
26.149
24.549
22.949
21.350
19.750
18.150
16.550
14.951
13.351
11.751
10.151
8.552
6.952
5.352
3.753
2.153
0.553
-1.047
-2.646
-4.246
-5.846
-7.446
-9.045
-10.645
1000
500
0
0
500
1000
x
1500
2000
1500
1000
500
0
0
500
1000
x
1500
ensemble CX G T
2000
4.005
3.774
3.542
3.310
3.079
2.847
2.615
2.383
2.152
1.920
1.688
1.457
1.225
0.993
0.761
0.530
0.298
0.066
-0.165
-0.397
-0.629
-0.861
-1.092
-1.324
-1.556
-1.787
2000
22.927
21.739
20.551
19.363
18.174
16.986
15.798
14.609
13.421
12.233
11.045
9.856
8.668
7.480
6.291
5.103
3.915
2.727
1.538
0.350
-0.838
-2.027
-3.215
-4.403
-5.591
-6.780
2000
1500
y
2000
0.653
0.616
0.579
0.543
0.506
0.470
0.433
0.396
0.360
0.323
0.286
0.250
0.213
0.177
0.140
0.103
0.067
0.030
-0.007
-0.043
-0.080
-0.116
-0.153
-0.190
-0.226
-0.263
500
1500
y
2000
1000
1.597
1.512
1.427
1.341
1.256
1.171
1.085
1.000
0.915
0.830
0.744
0.659
0.574
0.489
0.403
0.318
0.233
0.147
0.062
-0.023
-0.108
-0.194
-0.279
-0.364
-0.449
-0.535
2000
1000
500
0
0
500
1000
x
1500
2000
1500
y
500
1500
y
1000
0.121
0.112
0.103
0.094
0.085
0.076
0.067
0.058
0.049
0.040
0.031
0.022
0.013
0.003
-0.006
-0.015
-0.024
-0.033
-0.042
-0.051
-0.060
-0.069
-0.078
-0.087
-0.096
-0.105
2000
y
y
Well P1
1500
adjoint CX G T
ensemble G
0.341
0.327
0.312
0.298
0.283
0.269
0.254
0.240
0.225
0.211
0.196
0.181
0.167
0.152
0.138
0.123
0.109
0.094
0.080
0.065
0.051
0.036
0.022
0.007
-0.008
-0.022
y
adjoint G
2000
1000
500
0
0
500
1000
x
1500
9/44
Regularized ensemble-based iterative updating
1) Levenberg-Marquardt regularization
x`i =
h
(1 +
` )P`
1
i
1
+ GT
C
G
`
` D
h
⇥ CX 1 (x`i
1
1
T
`
xpr
i ) + G` CD (g (xi )
2) Matrix inversion lemma
x`i =
h
(1 +
3) Set P` =
x`i =
h
x`
(1 +
` )P`
i 1
1
+ GT
C
G
CX 1 (x`i xpr
`
` D
i )
h
i 1
T
P` GT
(1
+
)C
+
G
P
G
(g (x`i )
`
D
` ` `
`
1
x` T /(Ne
` )P`
i
doi ) .
1
1), then LM-EnRML is
1
+ GT
` CD G`
h
x` dT
` (1 +
doi ) .
` )(Ne
i
1
CX 1 (x`i
1)CD +
d`
xpr
i )
i
dT
`
1
(g(x`i )
doi )
10/44
Levenberg-Marquardt form of iterative ensemble smoother
(LM-EnRML)
x`i =
I
I
I
I
h
(1 +
` )P`
1
1
+ GT
` CD G`
h
x` dT
` (1 +
` )(Ne
i
1
CX 1 (x`i
1)CD +
d`
xpr
i )
i
dT
`
1
(g(x`i )
doi )
First iteration is exactly the same as would be obtained with the
ensemble Kalman filter (except that CD ! (1 + )CD ).
The initial value for is typically quite large in reservoir flow
problems ( 1 ⇠ 104 ).
If the objective function decreases, then decrease .
Note that the gradient of the objective function was not modified —
only the approximation to the Hessian.
11/44
Various test problems using LM-EnRML
I
One-variable nonlinear (validated against true posterior pdf)
3
2
1
5.2
6.
6.4
One-dimensional flow (validated against MCMC)
LM-EnRML
Ê
6
Ê
Ê
MCMC (Emerick and Reynolds, 2013)
7
Ê
ÊÊ
Ê
Ê
Ê
Ê
Ê
5
Ê
Ê
Ê
Ê
Ê
6
Ê
Ê
ÊÊ Ê
ÊÊ
Ê
ÊÊ
lnk
7
lnk
I
5.6
Ê
Ê
Ê
Ê
ÊÊ
Ê
Ê
Ê
Ê
Ê
5
Ê
Ê
4
Ê
Ê
ÊÊ
Ê
Ê
Ê
ÊÊ Ê
ÊÊ
Ê
ÊÊ
Ê
4
Ê
Ê
Ê
Ê
5
10
15
Gridblock
20
25
Ê
Ê
Ê
30
3
5
10
15
20
25
30
Gridblock
12/44
Rate of convergence
I
Brugge benchmark case
Data mismatch: (g (x)
108
Data mismatch
107
106
do )T CD 1 (g (x)
do )
GN-EnRML (orig)
LM-EnRML
LM-EnRML (approx)
105
104
1000
100
1 3 5 7 9 11 13 15 17 19 21 23 25
Iteration
13/44
The Norne field simulation model
Layer 1
F3
10
B1B
C3
K3
D2
B4D
20
E3A
B3
B4
C4
E2
E2A
B4B
B1
D3B
D3A
B2
E3
E3C
E1
F1
D4A
F2
D4
D1
C2
D1C
C1
30
C4A
F4
E4A
40
20
40
60
80
one layer of simulation model
I
I
100
structure map (statoil report)
The dimension is 46 ⇥ 112 ⇥ 22, 44927 active cells
Historical rate from Nov 1997 to Dec 2006
I
RFT pressure at 14 time instances from 14 wells
I
Four main fault blocks and in total 60 faults
I
Five equilibration regions with di↵erent depth of initial fluid contacts
14/44
Initial oil saturation
Layer 1
F3
10
B1B
C3
K3
D2
B4D
20
E3A
B3
B4
C4
E2
E2A
B4B
B1
D3B
D3A
B2
E3
E3C
E1
F1
D4A
F2
D4
D1
C2
D1C
C1
30
C4A
F4
E4A
40
20
40
60
80
0.96
0.86
0.77
0.67
0.58
0.48
0.38
0.29
0.19
0.10
0.00
100
Layer 11
F3
10
B1B
C3
K3
D2
B4D
20
E3A
B3
B4
C4
E2
E2A
B4B
B1
D3B
D3A
B2
E3
E3C
E1
F1
D4A
F2
D4
D1
C2
D1C
C1
30
C4A
F4
E4A
40
20
40
60
80
0.84
0.76
0.67
0.59
0.50
0.42
0.34
0.25
0.17
0.08
0.00
100
15/44
The Norne field
Layer 1
F3
10
B1B
C3
K3
D2
B4D
20
E3A
B3
B4
C4
E2
E2A
B4B
B1
D3B
D3A
B2
E3
E3C
E1
F1
D4A
F2
D4
D1
C2
D1C
C1
30
C4A
F4
E4A
40
20
40
60
80
100
As of December 2006, 17 active wells: 11 oil producers, 3 water
injectors, 3 gas producers.
Total of 31 wells with production data (9 injectors, 22 producers) from
Nov 1997 to Dec 2006.
16/44
Vertical communication
Inactive
MULTZ = 0.05
MULTZ = 0.01
MULTZ = 0.05
MULTZ = 0.001
MULTZ = 0.0
MULTZ: vertical transmissibility multiplier
(Figure from Statoil report)
17/44
Manual history matching
(Figures from thesis of Eirik Morell, NTNU, 2010)
Selected field-wide vertical transmissibility multiplier values, then
modified them locally to try to match RFT and water rise.
18/44
Manual history matching
E!01!F3
10
C!10
20
C!04
C!08
C!08!Ile
30
G!09
40
20
40
60
80
100
Figure: main fault blocks and location of faults
I
Modified transmissibility between main fault blocks
I
Modified transmissibility across faults
19/44
Variables for history matching (iterative ES)
Porosity Grid-based property
Permeability Grid-based property, correlated with porosity
Net-to-Gross Grid-based property
MULTZ Grid-based property, at six layers
MULTFLT Fault transmissibility multipliers (53 parameters)
Rel perm End-point water and gas relative permeability of four zones
(8 parameters)
MULTREGT Transmissibility multiplier between a few fault blocks (3
parameters)
WOC Initial water-oil contacts (5 parameters)
I
Number of model parameters is about 150,000
I
Used 100 realizations in the iterative ensemble smoother
20/44
Production data and noise assumed in data
Producers are under reservoir volume constraint
I
Water injection rate (20 m3 /day except for C-1H)
I
Gas injection rate (20 m3 /day)
Other data:
I
Oil production rate (100 m3 /day)
I
Water production rate (200 m3 /day)
I
Gas production rate (20000 m3 /day)
I
RFT pressure (2 bar)
The number of e↵ective data is about 2000
21/44
Distance-based localization
B4B
F3
B-4BH
10
B1B
C3
K3
D2
B4D
E3A
B3
B4
C4
E2AE2
B4B
B1
D3B
D3A
B2
E3
E3C
E1
F1
D4A
F2
D4
20
D1
C2
D1C
C1
C4A
30
F4
E4A
40
20
40
60
80
100
C3
F3
10
B1B
C3
K3
C-3H
D2
B4D
E3A
B3
B4
C4
E2AE2
B4B
B1
D3B
D3A
B2
E3
E3C
E1
F1
D4A
F2
D4
20
D1
C2
D1C
C1
C4A
30
F4
E4A
40
20
40
60
80
100
22/44
Vertical localization by location of completions
(Figure from Statoil report)
23/44
Localization
Area of localization changes with layer and with time at which data were
obtained
I
Updates to permx, porosity, NTG localized to layers within zones of
completion
I
Shale barrier (gridblock vertical transmissibility) is updated when
well is completed in zones immediately above or below the barrier
I
Fault transmissibility is updated by data within the same fault block
(no distance discrimination)
I
Relative permeability endpoint of one zone is only updated by data
in the same zone
I
Water-oil contact of a fault block is only updated by data in the
same fault block
24/44
Generation of initial ensemble
F3
C3
10
20
C2
B1B
B3
B4 C4
K3
D2
B4D B2 B1
D4
D1
D1C
E1
F1
D4A
F2
C1
C4A
30
F4
E4A
40
20
40
60
80
F3
7.23
6.94
6.65
6.36
6.07
5.78
5.49
5.20
4.91
4.62
4.33
E3A
E2AE2 E3CE3
B4B
D3B
D3A
C3
10
20
C2
B1B
B3
B4 C4
K3
D2
B4D B2 B1
D4
D1
D1C
8.19
7.73
7.27
6.81
6.35
5.89
5.43
4.97
4.51
4.05
3.59
E3A
E2AE2 E3CE3
B4B
D3B
D3A
E1
F1
D4A
F2
C1
C4A
30
F4
E4A
40
100
20
40
kriged map
60
80
100
one realization
Prediction from the initial ensemble
6000
7000
2850
6000
2800
5000
4000
3000
2000
2000
2
Time HyearL
4
OPR
6
8
Ê
ÊÊ
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
2650
Ê
Ê
Ê
Ê
2550
0
0
Ê
Ê
2700
2600
1000
0
Ê
2750
4000
Depth
WPR Hm^3êdayL
OPR Hm^3êdayL
E-3BH
D-3AH
D-2H
8000
0
2
Time HyearL
4
WPR
6
8
150
200
250
Pressure
300
350
RFT
25/44
Data mismatch
Data mismatch, Log10
Od = (dsim
dobs )T CD 1 (dsim
dobs )
+
6.0 +
+
+
+
5.5 +
+
5.0
+
+
+ +
+ +
+
+
+
+
+ + +
+ +
4.5
1
3
5
7
9
11
13
15
Iteration
Dashed horizontal line shows Od of the manual history matched model. The number
of e↵ective data is about 2000.
26/44
Data match: oil production rate
B-4DH
B-1H
4000
3000
6000
OPR Hm^3êdayL
OPR Hm^3êdayL
8000
4000
2000
1000
0
0
0
2
Time HyearL
4
6
8
0
2
Time HyearL
4
6
8
6
8
D-2H
D-1H
8000
6000
6000
OPR Hm^3êdayL
OPR Hm^3êdayL
2000
4000
2000
4000
2000
0
0
0
2
Time HyearL
4
6
8
0
2
Time HyearL
4
27/44
Data match: oil production rate
D-4AH
E-1H
8000
6000
OPR Hm^3êdayL
OPR Hm^3êdayL
1000
500
4000
2000
0
0
0
2
Time HyearL
4
6
8
0
2
E-4AH
Time HyearL
4
6
8
6
8
E-3AH
3000
5000
2500
OPR Hm^3êdayL
OPR Hm^3êdayL
4000
3000
2000
1000
2000
1500
1000
500
0
0
0
2
Time HyearL
4
6
8
0
2
Time HyearL
4
28/44
Data match: water production rate
B-1BH
B-1H
3000
6000
2500
WPR Hm^3êdayL
WPR Hm^3êdayL
5000
2000
1500
1000
4000
3000
2000
500
1000
0
0
-500
0
2
Time HyearL
4
6
8
0
2
D-4H
Time HyearL
4
6
8
6
8
D-3AH
3000
1500
WPR Hm^3êdayL
WPR Hm^3êdayL
2500
1000
500
2000
1500
1000
500
0
0
0
2
Time HyearL
4
6
8
-500
0
2
Time HyearL
4
29/44
Data match: water production rate
D-1CH
E-1H
2000
6000
5000
WPR Hm^3êdayL
WPR Hm^3êdayL
1500
1000
500
4000
3000
2000
1000
0
0
0
2
Time HyearL
4
6
8
0
2
Time HyearL
4
6
8
6
8
E-3AH
E-2H
3000
2000
WPR Hm^3êdayL
WPR Hm^3êdayL
2500
2000
1500
1000
1500
1000
500
500
0
0
-500
0
2
Time HyearL
4
6
8
0
2
Time HyearL
4
30/44
Data match: gas production rate
B-1BH
B-1H
1 ¥ 106
800 000
GPR Hm^3êdayL
GPR Hm^3êdayL
1.5 ¥ 106
1.0 ¥ 106
500 000
600 000
400 000
200 000
0
0
0
2
Time HyearL
4
6
8
0
4
6
8
6
8
1.5 ¥ 106
6
1.5 ¥ 106
GPR Hm^3êdayL
GPR Hm^3êdayL
Time HyearL
E-1H
D-3AH
2.0 ¥ 10
2
1.0 ¥ 106
1.0 ¥ 106
500 000
500 000
0
0
0
2
Time HyearL
4
6
8
0
2
Time HyearL
4
31/44
Data match: RFT pressure
E-3H
C-3H
2850
2800
2850
0.83 year
2800
Ê
1.51 year
Ê
Ê
2750
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
2700
2650
Depth
Depth
2750
Ê
Ê
Ê
2600
Ê
2700
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
2650
2600
2550
200
250
Pressure
300
350
150
Ê
Ê
Ê
200
F-4H
3.66 year
2800
350
2700
Ê
Ê
Ê
7.40 year
Ê
2750
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Depth
Depth
300
2850
2750
2600
Ê
Ê
Ê
ÊÊ
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
Ê
2700
2650
2600
2550
150
250
Pressure
E-3BH
2850
2650
Ê
2550
150
2800
Ê
Ê
Ê
Ê
Ê
Ê
2550
200
250
Pressure
300
350
150
200
250
Pressure
300
350
32/44
Faults in Norne model
E!01!F3
10
C!10
20
C!04
C!08
C!08!Ile
30
G!09
40
20
40
60
80
100
33/44
1.5 !
!
1.0
!
!
!
!
!
"1.0 !
C-08
!
!
C-10
!
"1.5
0.5
!
!
"2.0
0.0
!
"0.5
!
"1.0
"1.5 !
!
Multiplier
Multiplier
Estimation: fault transmissibility multiplier
!
!
!
1
3
5
7
!
!
!
"2.5
!
!
!
!
9
11
!
!
!
!
!
13
!
!
!
"3.5
1
3
5
9
11
13
15
0.0
!
"0.5
Multiplier
Multiplier
7
Iteration
E-01-F3
"1.5
!
"3.0
!
!
!
15
Iteration
"1.0
!
"2.0
"1.0
"2.5
"1.5
"3.0
!
!
!
!
"3.5
"2.0
"4.0
"2.5
!
"4.5
1
3
5
7
9
Iteration
11
13
15
"3.0
G-09
1
3
5
7
9
11
13
15
Iteration
34/44
Estimation: oil-water contact
2700
+
2595
+
+
+
+
+
+
+
+
+
Depth
Depth
2695
2590
2690
2685
2585
2680
2580
2675
1
3
5
7
9
Iteration
11
13
15
2575
1
3
5
C&D – Garn
9
11
13
15
11
13
15
G – Garn
+
2640
7
Iteration
+
+
+
2705
2635
+
+
+
+
+
7
9
2700
Depth
Depth
2630
2625
2695
2620
2690
2615
2685
2610
1
3
5
7
9
Iteration
E – Garn
11
13
15
1
3
5
Iteration
C&D&E – Ile to Tilje
35/44
Estimation: mean gridblock permeability of Layer 1
Initial
10
C3
20
C2
B1B
B3
B4 C4
K3
D2
B1
B4D B2
D4
D1
D1C
F3
E3A
E2AE2 E3CE3
B4B
D3B
D3A
E1
F1
D4A
F2
C1
C4A
30
F4
E4A
40
20
Updated
10
C3
20
C2
40
B1B
B3
B4 C4
K3
D2
B4D B2 B1
D4
D1
D1C
60
F3
80
100
E3A
E2AE2 E3CE3
B4B
D3B
D3A
E1
F1
D4A
F2
C1
C4A
30
F4
E4A
40
20
40
60
80
7.15
6.88
6.60
6.32
6.04
5.76
5.49
5.21
4.93
4.65
4.37
8.44
7.99
7.53
7.08
6.62
6.17
5.71
5.26
4.80
4.35
3.89
100
36/44
Estimation: mean gridblock permeability of Layer 11
Initial
10
C3
20
C2
B1B
B3
B4 C4
K3
D2
B1
B4D B2
D4
D1
D1C
F3
E3A
E2AE2 E3CE3
B4B
D3B
D3A
E1
F1
D4A
F2
C1
C4A
30
F4
E4A
40
20
Updated
10
C3
20
C2
40
B1B
B3
B4 C4
K3
D2
B4D B2 B1
D4
D1
D1C
60
F3
80
100
E3A
E2AE2 E3CE3
B4B
D3B
D3A
E1
F1
D4A
F2
C1
C4A
30
F4
E4A
40
20
40
60
80
6.26
6.02
5.77
5.53
5.28
5.04
4.79
4.55
4.30
4.06
3.81
8.34
7.54
6.74
5.94
5.14
4.34
3.54
2.74
1.94
1.14
0.34
100
37/44
Reduction in uncertainty: gridblock permeability
Layer 1
10
C3
20
C2
B1B
B3
B4 C4
K3
D2
B1
B4D B2
D4
D1
D1C
F3
E3A
E2AE2 E3CE3
B4B
D3B
D3A
E1
F1
D4A
F2
C1
C4A
30
F4
E4A
40
20
Layer 11
10
C3
20
C2
40
B1B
B3
B4 C4
K3
D2
B4D B2 B1
D4
D1
D1C
60
F3
80
100
E3A
E2AE2 E3CE3
B4B
D3B
D3A
E1
F1
D4A
F2
C1
C4A
30
F4
E4A
40
20
40
60
80
0.69
0.62
0.55
0.49
0.42
0.35
0.28
0.21
0.14
0.07
0.00
0.85
0.77
0.68
0.60
0.51
0.43
0.34
0.26
0.17
0.09
0.00
100
38/44
Estimation: mean vertical transmissibility multiplier
Layer 8
10
C3
20
C2
B1B
B3
B4 C4
K3
D2
B1
B4D B2
D4
D1
D1C
F3
E3A
E2AE2 E3CE3
B4B
D3B
D3A
E1
F1
D4A
F2
C1
C4A
30
F4
E4A
40
20
Layer 12
10
C3
20
C2
40
B1B
B3
B4 C4
K3
D2
B4D B2 B1
D4
D1
D1C
60
F3
80
100
E3A
E2AE2 E3CE3
B4B
D3B
D3A
E1
F1
D4A
F2
C1
C4A
30
F4
E4A
40
20
40
60
80
0.00
-0.40
-0.80
-1.20
-1.60
-2.00
-2.40
-2.80
-3.20
-3.60
-4.00
-0.14
-0.53
-0.91
-1.30
-1.68
-2.07
-2.46
-2.84
-3.23
-3.61
-4.00
100
39/44
Summary
I
Fairly complex case (uncertainty in compartmentalization, point
properties, fluid contacts, rock-fluid properties) with great
nonlinearity handled almost routinely
I
Was not necessary to spend much time “improving the prior” (as
might be required for ES)
I
Method was quite robust with respect to parameters of minimization
(initial choice of , reduction rule, etc.)
I
In general, should not worry about “the danger of over
parameterization” if localization is done properly
I
Probably still missing some sources of uncertainty: structure, skin
distribution, fault transmissibility distribution, etc. (but then places
greater requirement on localization)
I
Localization needs simplification and improvement (some collapse of
variability of perm in C block)
40/44
Emerick, A. A. and A. C. Reynolds, Investigation of the sampling
performance of ensemble-based methods with a simple reservoir model,
Computat. Geosci., 2013.
44/44