Upload
buithuan
View
225
Download
0
Embed Size (px)
Citation preview
Project MotivationReservoir Simulator
Accounting for input error in 4D-VarResults
Summary
Importance of input error in reservoir
simulations
Joanna S. Pelc Remus G. HaneaArnold W. Heemink Jan-Dirk Jansen
Delft University of Technology
The Eighth International Workshop on Adjoint Model Applicationsin Dynamic Meteorology, Tannersville 18-22 May 2009
J.S. Pelc, R.G. Hanea, A.W. Heemink, J.D. Dirk Input error in reservoir simulations
Project MotivationReservoir Simulator
Accounting for input error in 4D-VarResults
Summary
Outline
1 Project Motivation
2 Reservoir Simulator
3 Accounting for input error in 4D-Var
4 Results
5 Summary
J.S. Pelc, R.G. Hanea, A.W. Heemink, J.D. Dirk Input error in reservoir simulations
Project MotivationReservoir Simulator
Accounting for input error in 4D-VarResults
Summary
How is oil produced?
With primary recoveryoil production is usuallyless than 30%.
J.S. Pelc, R.G. Hanea, A.W. Heemink, J.D. Dirk Input error in reservoir simulations
Project MotivationReservoir Simulator
Accounting for input error in 4D-VarResults
Summary
How is oil produced?
Water is injected into the reservoir field to force more oil intoproduction wells
J.S. Pelc, R.G. Hanea, A.W. Heemink, J.D. Dirk Input error in reservoir simulations
Project MotivationReservoir Simulator
Accounting for input error in 4D-VarResults
Summary
What mathematics has to do with oil?
A suitable reservoir rock must be porousand permeable
J.S. Pelc, R.G. Hanea, A.W. Heemink, J.D. Dirk Input error in reservoir simulations
Project MotivationReservoir Simulator
Accounting for input error in 4D-VarResults
Summary
Reservoir Simulator
State vectorPressure at time 451.454 days
5 10 15 20
2
4
6
8
10
12
14
16
18
20 2.55
2.6
2.65
2.7
2.75
2.8
2.85
2.9
2.95
3
x 107
pressure
Saturation at time 451.454 days
5 10 15 20
2
4
6
8
10
12
14
16
18
20 0.25
0.3
0.35
0.4
0.45
0.5
0.55
0.6
0.65
0.7
0.75
saturation
ParametersPermeability (log scale)
5 10 15 20
2
4
6
8
10
12
14
16
18
20−31
−30.5
−30
−29.5
−29
−28.5
permeability
xi = Mi(xi−1) where xi = [pi si ]T
inputs: prescribed bottom holepressures and prescribed flow rates
J.S. Pelc, R.G. Hanea, A.W. Heemink, J.D. Dirk Input error in reservoir simulations
Project MotivationReservoir Simulator
Accounting for input error in 4D-VarResults
Summary
4D Variational Data Assimilation
Cost function in 4D-Var
J(p) =n∑
i=1
(Hi (xi) − yi)T
︸ ︷︷ ︸Distances
R−1i (Hi (xi ) − yi )︸ ︷︷ ︸
Distances
+ (p − pb)TB−1(p − pb)︸ ︷︷ ︸
Background termn∑
i=1
(Mi (xi−1) − xi)T Q−1
i (Mi (xi−1) − xi )︸ ︷︷ ︸Model error term
minimization withconstraints:
xi = Mi(xi−1) + ε
Control variables:
parameters p (permeabilities)
model error ε
J.S. Pelc, R.G. Hanea, A.W. Heemink, J.D. Dirk Input error in reservoir simulations
Project MotivationReservoir Simulator
Accounting for input error in 4D-VarResults
Summary
Twin Experiment
The Truth
One model simulation is generated, saved and called the
Truth.
Observations
Observations are generated with a chosen measurement errorfrom the Truth.
J.S. Pelc, R.G. Hanea, A.W. Heemink, J.D. Dirk Input error in reservoir simulations
Project MotivationReservoir Simulator
Accounting for input error in 4D-VarResults
Summary
Imperfect truth construction
In reservoir simulator perturbing grid pressure is not effective andunrealistic
White noise
0 500 1000 15002.45
2.5
2.55
2.6
2.65
2.7
2.75x 10
7
Time [days]
Grid
Pre
ssur
e at
the
wel
ls
NW − 1SW − 21NE − 421SE − 441Deterministic
Red noise
0 500 1000 15002.4
2.45
2.5
2.55
2.6
2.65
2.7
2.75x 10
7
Time [days]
Grid
Pre
ssur
e at
the
wel
ls
NW − 1SW − 21NE − 421SE − 441Deterministic
Perturbed grid pressure at the production wellsJ.S. Pelc, R.G. Hanea, A.W. Heemink, J.D. Dirk Input error in reservoir simulations
Project MotivationReservoir Simulator
Accounting for input error in 4D-VarResults
Summary
Imperfect truth construction
The Idea: perturb inputs instead of the state vector
Red noise
0 500 1000 15002.3
2.4
2.5
2.6
2.7
2.8x 10
7
Time [days]
Bot
tom
hol
e pr
essu
res
121421441Prior
Bias
0 50 100 150 200 250
2.2
2.25
2.3
2.35
2.4
2.45
2.5
2.55
2.6
2.65
x 107
Time [days]
Pre
scrib
ed b
hp
121421441Prior
Perturbed prescribed bottom hole pressures at the producers
J.S. Pelc, R.G. Hanea, A.W. Heemink, J.D. Dirk Input error in reservoir simulations
Project MotivationReservoir Simulator
Accounting for input error in 4D-VarResults
Summary
Imperfect truth construction
Obtained model perturbation looks realistic
4D-Var can be applied adding the inputs to thecontrol variable
Red noise
0 500 1000 15002.52
2.54
2.56
2.58
2.6
2.62
2.64
2.66
2.68
2.7
2.72x 10
7
Time [days]
pres
sure
s in
wel
l grid
blo
cks
SW − 421
PerturbedDeterministic
Bias
0 500 1000 15002.35
2.4
2.45
2.5
2.55
2.6
2.65
2.7
2.75x 10
7
Time [days]
Grid
pre
ssur
e
SW − 421
DeterministicPerturbed
Effect of different input perturbations on the grid pressureat the producers
J.S. Pelc, R.G. Hanea, A.W. Heemink, J.D. Dirk Input error in reservoir simulations
Project MotivationReservoir Simulator
Accounting for input error in 4D-VarResults
Summary
Accounting for input error in 4D-Var
Cost function in 4D-Var
J(p, u) =n∑
i=1
(Hi (xi) − yi)T
︸ ︷︷ ︸Distances
R−1i (Hi(xi ) − yi )︸ ︷︷ ︸
Distances
+ (p − pb)T B−1(p − pb)︸ ︷︷ ︸
Background term
minimization withconstraints:
xi = Mi(xi−1)
Control variables:
parameters p (permeabilities)
inputs u
J.S. Pelc, R.G. Hanea, A.W. Heemink, J.D. Dirk Input error in reservoir simulations
Project MotivationReservoir Simulator
Accounting for input error in 4D-VarResults
Summary
Accounting for input bias - Results
PriorPRIOR permeability (log scale)
2 4 6 8 10 12 14 16 18 20
2
4
6
8
10
12
14
16
18
20
−31
−30.5
−30
−29.5
−29
−28.5
TruthTrue permeability (log scale)
2 4 6 8 10 12 14 16 18 20
2
4
6
8
10
12
14
16
18
20
−31
−30.5
−30
−29.5
−29
−28.5
With InputsRESULTING PERMABILITY RMSE = 0.683751
2 4 6 8 10 12 14 16 18 20
2
4
6
8
10
12
14
16
18
20
−31
−30.5
−30
−29.5
−29
−28.5
Without InputsRESULTING PERMABILITY RMSE = 2.84442
2 4 6 8 10 12 14 16 18 20
2
4
6
8
10
12
14
16
18
20
−31
−30.5
−30
−29.5
−29
−28.5
0 500 1000 15002.2
2.25
2.3
2.35
2.4
2.45
2.5
2.55
2.6
2.65
2.7x 10
7
Time [days]
Pre
scrib
ed b
otto
m h
ole
pres
sure
4D−Var (solid line), Truth (dashed line), Prior (dash−dotted line)
Prod: NWProd: NEProd: SWProd: SEPrior
very good permeability estimationwith accounting for input bias
very bad permeability estimationwithout accounting for input bias
very good estimation of inputs
J.S. Pelc, R.G. Hanea, A.W. Heemink, J.D. Dirk Input error in reservoir simulations
Project MotivationReservoir Simulator
Accounting for input error in 4D-VarResults
Summary
Accounting for input bias - Results
Prior4D−Var permeability estimation with inputs
2 4 6 8 10 12 14 16 18 20
2
4
6
8
10
12
14
16
18
20−34
−33
−32
−31
−30
−29
−28
−27
−26
TruthTrue permeability (log scale)
2 4 6 8 10 12 14 16 18 20
2
4
6
8
10
12
14
16
18
20
−31
−30
−29
−28
−27
−26
With Inputs4D−Var permeability estimation with inputs
2 4 6 8 10 12 14 16 18 20
2
4
6
8
10
12
14
16
18
20
−31
−30
−29
−28
−27
−26
Without InputsRESULTING PERMABILITY RMSE = 2.84442
2 4 6 8 10 12 14 16 18 20
2
4
6
8
10
12
14
16
18
20
−31
−30
−29
−28
−27
−26
0 500 1000 15002.2
2.25
2.3
2.35
2.4
2.45
2.5
2.55
2.6
2.65
2.7x 10
7
Time [days]
Pre
scrib
ed b
otto
m h
ole
pres
sure
4D−Var (solid line), Truth (dashed line), Prior (dash−dotted line)
Prod: NWProd: NEProd: SWProd: SEPrior
very good permeability estimationwith accounting for input bias
very bad permeability estimationwithout accounting for input bias
very good estimation of inputs
J.S. Pelc, R.G. Hanea, A.W. Heemink, J.D. Dirk Input error in reservoir simulations
Project MotivationReservoir Simulator
Accounting for input error in 4D-VarResults
Summary
Accounting for input bias - Results
Total flow rate
0 200 400 600 800 1000 1200−10
−8
−6
−4
−2
x 10−4
Time [days]
Tot
al fl
ow r
ate
in th
e w
ells
Production well: NW
0 200 400 600 800 1000 1200
−10
−8
−6
−4
−2
x 10−4
Time [days]
Tot
al fl
ow r
ate
in th
e w
ells
Production well: NE
0 200 400 600 800 1000 1200−2
−1.8
−1.6
−1.4
−1.2
−1
−0.8
−0.6
−0.4
x 10−3
Time [days]
Tot
al fl
ow r
ate
in th
e w
ells
Production well: SW
0 200 400 600 800 1000 1200−1.8
−1.6
−1.4
−1.2
−1
−0.8
−0.6
−0.4
x 10−3
Time [days]
Tot
al fl
ow r
ate
in th
e w
ells
Production well: SE
TruthPrior4D−Var With Inputs4D−Var Without InputsObs 5%
TruthPrior4D−Var With Inputs4D−Var Without InputsObs 5%
TruthPrior4D−Var With Inputs4D−Var Without InputsObs 5%
TruthPrior4D−Var With Inputs4D−Var Without InputsObs 5%
RESULTING PERMABILITY RMSE = 2.84442
2 4 6 8 10 12 14 16 18 20
2
4
6
8
10
12
14
16
18
20
−31
−30
−29
−28
−27
−26
0 500 1000 15002.2
2.25
2.3
2.35
2.4
2.45
2.5
2.55
2.6
2.65
2.7x 10
7
Time [days]
Pre
scrib
ed b
otto
m h
ole
pres
sure
4D−Var (solid line), Truth (dashed line), Prior (dash−dotted line)
Prod: NWProd: NEProd: SWProd: SEPrior
J.S. Pelc, R.G. Hanea, A.W. Heemink, J.D. Dirk Input error in reservoir simulations
Project MotivationReservoir Simulator
Accounting for input error in 4D-VarResults
Summary
Accounting for input bias - Results
Water break trough time prediction:
Water saturation
0 500 1000 15000.2
0.4
0.6
0.8
1NW − 1
satu
ratio
ns in
wel
l grid
blo
cks
Time [days]
0 500 1000 15000.2
0.4
0.6
0.8
1SW − 21
satu
ratio
ns in
wel
l grid
blo
cks
Time [days]
0 500 1000 15000.2
0.4
0.6
0.8
1NE − 421
satu
ratio
ns in
wel
l grid
blo
cks
Time [days]
0 500 1000 15000.2
0.4
0.6
0.8
1SE − 441
satu
ratio
ns in
wel
l grid
blo
cks
Time [days]
TruthPrior4D−Var
u
4D−Var strong
J.S. Pelc, R.G. Hanea, A.W. Heemink, J.D. Dirk Input error in reservoir simulations
Project MotivationReservoir Simulator
Accounting for input error in 4D-VarResults
Summary
Accounting for input bias - Results
PriorPRIOR permeability (log scale)
2 4 6 8 10 12 14 16 18 20
2
4
6
8
10
12
14
16
18
20
−31
−30.5
−30
−29.5
−29
−28.5
TruthTrue permeability (log scale)
2 4 6 8 10 12 14 16 18 20
2
4
6
8
10
12
14
16
18
20
−31
−30.5
−30
−29.5
−29
−28.5
With InputsRESULTING PERMABILITY RMSE = 0.683751
2 4 6 8 10 12 14 16 18 20
2
4
6
8
10
12
14
16
18
20
−31
−30.5
−30
−29.5
−29
−28.5
Without InputsRESULTING PERMABILITY RMSE = 2.84442
2 4 6 8 10 12 14 16 18 20
2
4
6
8
10
12
14
16
18
20
−31
−30.5
−30
−29.5
−29
−28.5
PRIOR permeability (log scale)
2 4 6 8 10 12 14 16 18 20
2
4
6
8
10
12
14
16
18
20
−31
−30.5
−30
−29.5
−29
−28.5
TRUE PERMABILITY (log scale)
2 4 6 8 10 12 14 16 18 20
2
4
6
8
10
12
14
16
18
20−31.5
−31
−30.5
−30
−29.5
−29
−28.5
RESULTING PERMABILITY RMSE = 0.945085
2 4 6 8 10 12 14 16 18 20
2
4
6
8
10
12
14
16
18
20−31.5
−31
−30.5
−30
−29.5
−29
−28.5
RESULTING PERMABILITY RMSE = 1.06337
2 4 6 8 10 12 14 16 18 20
2
4
6
8
10
12
14
16
18
20−31.5
−31
−30.5
−30
−29.5
−29
−28.5
PRIOR permeability (log scale)
2 4 6 8 10 12 14 16 18 20
2
4
6
8
10
12
14
16
18
20
−31
−30.5
−30
−29.5
−29
−28.5
TRUE PERMABILITY (log scale)
2 4 6 8 10 12 14 16 18 20
2
4
6
8
10
12
14
16
18
20
−31
−30.5
−30
−29.5
−29
−28.5
RESULTING PERMABILITY RMSE = 0.66807
2 4 6 8 10 12 14 16 18 20
2
4
6
8
10
12
14
16
18
20
−31
−30.5
−30
−29.5
−29
−28.5
RESULTING PERMABILITY RMSE = 1.98559
2 4 6 8 10 12 14 16 18 20
2
4
6
8
10
12
14
16
18
20
−31
−30.5
−30
−29.5
−29
−28.5
J.S. Pelc, R.G. Hanea, A.W. Heemink, J.D. Dirk Input error in reservoir simulations
Project MotivationReservoir Simulator
Accounting for input error in 4D-VarResults
Summary
Accounting for input bias - Results
Prior4D−Var permeability estimation with inputs
2 4 6 8 10 12 14 16 18 20
2
4
6
8
10
12
14
16
18
20−34
−33
−32
−31
−30
−29
−28
−27
−26
TruthTrue permeability (log scale)
2 4 6 8 10 12 14 16 18 20
2
4
6
8
10
12
14
16
18
20
−31
−30
−29
−28
−27
−26
With Inputs4D−Var permeability estimation with inputs
2 4 6 8 10 12 14 16 18 20
2
4
6
8
10
12
14
16
18
20
−31
−30
−29
−28
−27
−26
Without InputsRESULTING PERMABILITY RMSE = 2.84442
2 4 6 8 10 12 14 16 18 20
2
4
6
8
10
12
14
16
18
20
−31
−30
−29
−28
−27
−26
4D−Var permeability estimation with inputs
2 4 6 8 10 12 14 16 18 20
2
4
6
8
10
12
14
16
18
20−34
−33
−32
−31
−30
−29
−28
−27
−26
True permeability (log scale)
2 4 6 8 10 12 14 16 18 20
2
4
6
8
10
12
14
16
18
20 −31.5
−31
−30.5
−30
−29.5
−29
−28.5
−28
−27.5
4D−Var permeability estimation with inputs
2 4 6 8 10 12 14 16 18 20
2
4
6
8
10
12
14
16
18
20 −31.5
−31
−30.5
−30
−29.5
−29
−28.5
−28
−27.5
4D−Var permeability estimation without inputs
2 4 6 8 10 12 14 16 18 20
2
4
6
8
10
12
14
16
18
20 −31.5
−31
−30.5
−30
−29.5
−29
−28.5
−28
−27.5
4D−Var permeability estimation with inputs
2 4 6 8 10 12 14 16 18 20
2
4
6
8
10
12
14
16
18
20−34
−33
−32
−31
−30
−29
−28
−27
−26
True permeability (log scale)
2 4 6 8 10 12 14 16 18 20
2
4
6
8
10
12
14
16
18
20−34
−33
−32
−31
−30
−29
−28
−27
−26
4D−Var permeability estimation with inputs
2 4 6 8 10 12 14 16 18 20
2
4
6
8
10
12
14
16
18
20−34
−33
−32
−31
−30
−29
−28
−27
−26
4D−Var permeability estimation without inputs
2 4 6 8 10 12 14 16 18 20
2
4
6
8
10
12
14
16
18
20−34
−33
−32
−31
−30
−29
−28
−27
−26
J.S. Pelc, R.G. Hanea, A.W. Heemink, J.D. Dirk Input error in reservoir simulations
Project MotivationReservoir Simulator
Accounting for input error in 4D-VarResults
Summary
Summary
Perturbing the state vector in reservoir simulations is noteffective neither realistic
Model inputs have a big impact on the reservoir behavior
Accounting for the input bias as control variable in 4D-Vargives very good estimations of the permeability, the outputsof the model, as well as the inputs themselves
J.S. Pelc, R.G. Hanea, A.W. Heemink, J.D. Dirk Input error in reservoir simulations
Project MotivationReservoir Simulator
Accounting for input error in 4D-VarResults
Summary
Thank you for your attention!
Questions?
J.S. Pelc, R.G. Hanea, A.W. Heemink, J.D. Dirk Input error in reservoir simulations