Uncertainty Quantification and Sensitivity Analysis of...
1
Global sensitivity analysis • GSA is based on the decomposition of variance of response . • First order sensitivity index quantifies the main effect of each model parameter to . = | /() • Total effect quantifies the effects of to including all the interactions =1− | ~ /() • For Monte Carlo samples and parameters, ( + 2) simulations are required Computationally expensive to apply to reservoir simulations. • is assumed to be univariate but reservoir responses are multivariate. Motivation • Uncertainty quantification is a key process for informed decision making in the development of oil/gas field. • Global sensitivity analysis (GSA, Satelli et al., 2008) is based on Monte Carlo sampling and has been widely used in a lot of fields of science and engineering. • Challenges to use GSA in reservoir forecasting includes multidimensionality of response (spatio-temporal) and large computations. • In the project the goal is to propose the workflow to quantify uncertainty and sensitivity of reservoir forecasts with high computational efficiency. Training data • Illustration of case study – Oil field in central northern Libya (Ahlbrandt, 2001) Methodology • High dimensionality is reduced by functional PCA (FPCA). • Regressions are performed to obtain a surrogate forward model of a full flow simulator. A boosting with regression trees is utilized. • The regressors are used to compute sensitivity indices of GSA. Uncertainty Quantification and Sensitivity Analysis of Reservoir Forecasts with Machine Learning Jihoon Park ([email protected]) Department of Energy Resources Engineering Name Type Boundary Condition I1 Injector Constant rate, 10,000 bbl/day I2 P2 Producer Constant bottom hole pressure, 1,000 psi P3 P4 Table 1. Well Plan Number Parameters Abbreviation Distribution 1 Oil-water contact owc U[-1076, -1061] 2 Transmissibility multiplier of fault 1 mflt1 U[0, 1] 3 Transmissibility multiplier of fault 2 mflt2 U[0, 1] 4 Transmissibility multiplier of fault 3 mflt3 U[0, 1] 5 Transmissibility multiplier of fault 4 mflt4 U[0, 1] 6 Residual oil saturation sor N[0.2, 0.05 2 ] 7 Connate water saturation swc N[0.2, 0.05 2 ] 8 Oil viscosity oilvis N[10,2 2 ] 9 Corey exponent of oil oilexp N[3,0.25 2 ] 10 Corey exponent of water watexp N[2,0.1 2 ] Fig. 1 Reservoir models for the case study -1.5 -1 -0.5 0 0.5 1 1.5 2 x 10 5 -2 -1.5 -1 -0.5 0 0.5 1 x 10 5 1st Comp. 2nd Comp. 2 4 6 8 10 12 14 16 18 65 70 75 80 85 90 95 100 # of comp. Var. explained (%) 2000 4000 6000 3000 4000 5000 6000 7000 P2 Time (days) OIL(stb/day) 2000 4000 6000 1000 1500 2000 2500 3000 P3 Time (days) OIL(stb/day) 2000 4000 6000 2000 3000 4000 5000 6000 P4 Time (days) OIL(stb/day) References Ahlbrandt, T. S., 2001. The Sirte Basin Province of Libya: Sirte-Zelten Total Petroleum System Saltelli et al., 2008, “Global sensitivity analysis: the primer”. Ramsay J.O. and Silverman B.W., 2012, “Functional Data Analysis”. • Distribution of uncertain parameters Table 2. List of uncertain model parameters Fig. 2 First two Principle components • Dimensionality reduction with FPCA Fig. 3 Cumulative variance explained • Uncertainty quantification with regression models Shows close approximation. Fig. 5 Forecast of oil rate at each well (gray: every model, blue: observed curves, red: curves from regression models, each three line represents P10,P50, P90) • Global sensitivity analysis 600,000 (n=50,000, k=10) forward runs can be performed rapidly with regression models. 0 0.2 0.4 0.6 0.8 mflt4 mflt1 mflt3 mflt2 oilexp watexp swc sor oilvis owc 0 0.2 0.4 0.6 0.8 mflt1 mflt3 mflt4 mflt2 oilexp watexp swc sor oilvis owc Fig. 6 The first order index (left) and total effect (right) -2 -1 0 1 2 3 x 10 5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 x 10 5 Training Data Predicted Score 1, Cor=1.00 0 50 100 150 200 250 300 350 400 450 500 0 0.5 1 1.5 2 2.5 3 x 10 9 # of trees Cr.Val.Error • Regression using boosting with regression trees The number of trees is determined by cross validation. Predictors are model parameters and responses are principle components. Fig. 4 Cross validation error w.r.t. the number of boosted trees Fig. 5 Training Data vs. Predicted values Results
Transcript of Uncertainty Quantification and Sensitivity Analysis of...
Global sensitivity analysis
• GSA is based on the decomposition of variance of response 𝑌.
• First order sensitivity index 𝑆𝑖 quantifies the main effect of each
model parameter 𝑋𝑖 to 𝑌.
𝑆𝑖 = 𝑉 𝐸 𝑌|𝑋𝑖 /𝑉(𝑌)
• Total effect 𝑆𝑇𝑖 quantifies the effects of 𝑋𝑖 to 𝑌 including all the
interactions
𝑆𝑖 = 1 − 𝑉 𝐸 𝑌|𝑋~𝑖 /𝑉(𝑌)
• For 𝑛 Monte Carlo samples and 𝑘 parameters, 𝑛(𝑘 + 2)simulations are required Computationally expensive to apply
to reservoir simulations.
• 𝑌 is assumed to be univariate but reservoir responses are
multivariate.
Motivation
• Uncertainty quantification is a key process for informed
decision making in the development of oil/gas field.
• Global sensitivity analysis (GSA, Satelli et al., 2008) is based
on Monte Carlo sampling and has been widely used in a lot of
fields of science and engineering.
• Challenges to use GSA in reservoir forecasting includes
multidimensionality of response (spatio-temporal) and large
computations.
• In the project the goal is to propose the workflow to quantify
uncertainty and sensitivity of reservoir forecasts with high
computational efficiency.
Training data
• Illustration of case study – Oil field in central northern
Libya (Ahlbrandt, 2001)
Methodology
• High dimensionality is reduced by functional PCA (FPCA).
• Regressions are performed to obtain a surrogate forward model
of a full flow simulator. A boosting with regression trees is
utilized.
• The regressors are used to compute sensitivity indices of GSA.
Uncertainty Quantification and Sensitivity Analysis of Reservoir Forecasts with Machine Learning
Jihoon Park ([email protected])
Department of Energy Resources Engineering
Name Type Boundary Condition
I1Injector
Constant rate, 10,000 bbl/dayI2
P2
ProducerConstant bottom hole
pressure, 1,000 psiP3
P4
Table 1. Well Plan
Number Parameters Abbreviation Distribution
1 Oil-water contact owc U[-1076, -1061]
2 Transmissibility multiplier of fault 1 mflt1 U[0, 1]
3 Transmissibility multiplier of fault 2 mflt2 U[0, 1]
4 Transmissibility multiplier of fault 3 mflt3 U[0, 1]
5 Transmissibility multiplier of fault 4 mflt4 U[0, 1]
6 Residual oil saturation sor N[0.2, 0.052]
7 Connate water saturation swc N[0.2, 0.052]
8 Oil viscosity oilvis N[10,22]
9 Corey exponent of oil oilexp N[3,0.252]
10 Corey exponent of water watexp N[2,0.12]
Fig. 1 Reservoir models for the case study
-1.5 -1 -0.5 0 0.5 1 1.5 2
x 105
-2
-1.5
-1
-0.5
0
0.5
1
x 105
1st Comp.
2nd C
om
p.
2 4 6 8 10 12 14 16 18
65
70
75
80
85
90
95
100
# of comp.
Va
r. e
xp
lain
ed (
%)
2000 4000 6000
3000
4000
5000
6000
7000
P2
Time (days)
OIL
(stb
/day)
2000 4000 6000
1000
1500
2000
2500
3000
P3
Time (days)
OIL
(stb
/day)
2000 4000 6000
2000
3000
4000
5000
6000
P4
Time (days)
OIL
(stb
/day)
ReferencesAhlbrandt, T. S., 2001. The Sirte Basin Province of Libya: Sirte-Zelten
Total Petroleum System
Saltelli et al., 2008, “Global sensitivity analysis: the primer”.
Ramsay J.O. and Silverman B.W., 2012, “Functional Data Analysis”.
• Distribution of uncertain parameters
Table 2. List of uncertain model parameters
Fig. 2 First two Principle components
• Dimensionality reduction with FPCA
Fig. 3 Cumulative variance explained
• Uncertainty quantification with regression models Shows close approximation.
Fig. 5 Forecast of oil rate at each well (gray: every model, blue: observed curves,
red: curves from regression models, each three line represents P10,P50, P90)
• Global sensitivity analysis 600,000 (n=50,000, k=10) forward runs can be performed rapidly
with regression models.
0 0.2 0.4 0.6 0.8
mflt4
mflt1
mflt3
mflt2
oilexp
watexp
swc
sor
oilvis
owc
0 0.2 0.4 0.6 0.8
mflt1
mflt3
mflt4
mflt2
oilexp
watexp
swc
sor
oilvis
owc
Fig. 6 The first order index (left) and total effect (right)
-2 -1 0 1 2 3
x 105
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
2.5
3x 10
5
Training Data
Pre
dic
ted
Score 1, Cor=1.00
0 50 100 150 200 250 300 350 400 450 5000
0.5
1
1.5
2
2.5
3x 10
9
# of trees
Cr.
Va
l.E
rro
r
• Regression using boosting with regression trees The number of trees is determined by cross validation.
Predictors are model parameters and responses are principle
components.
Fig. 4 Cross validation error w.r.t. the
number of boosted treesFig. 5 Training Data vs. Predicted values
Results
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