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 (jhpark3@stanford.edu)

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