Baehyun min pareto_optimality

UT-TAMU 교류전
University of Texas at Austin
Center for Petroleum & Geosystems Engineering
Pareto-Optimality in Multi-Objective Optimization
& Its Application to History Matching
Postdoctoral Fellow
Baehyun Min

CONTENTS
I. INTRODUCTION
II. METHODOLOGY
III. APPLICATION
IV.CONCLUSIONS
CONTENTS
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WHAT IS HISTORY MATCHING?
§ Goal
• Generate approximate geomodels for production estimation
§ Inverse Modeling
• Estimate uncertain reservoir parameters from certain static and dynamic parameters
ü Reservoir properties: permeability, PVT data
ü Static data: core, logging, seismic
ü Dynamic data: oil rate, water cut, BHP
§ Reason of Mismatch
• Reservoir uncertainties
• Ill-posedness of inverse modeling
I. INTRODUCTION
Mismatch of oil production
Mismatchofwatercut
Approximate geomodels
Earth
Mismatch
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§ Multi-Objective Minimization Problem
• Objective function = Discrepancy between the observed and the calculated production data
WHAT IS HISTORY MATCHING?
[ ],)(,),(,),(),,()( 11 xfxfxfxxfxfy MiN LLL ===
I. INTRODUCTION
Mi
DD
xf
obs
iN
j
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i
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cal
ji
i 1,...,0)(
2
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,,
="³
÷
÷
ø
ö
ç
ç
è
æ -
= å= s
x: vector of uncertain parameters
y: vector of individual objective functions
fi(x): individual objective functions
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§ Global Objective Function
where
SINGLE-OBJECTIVE OPTIMIZATION
Global optimum Local optimums
G
L
)(xF
x
å=
=
M
i
ii xfxF
1
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I. INTRODUCTION
0³iw Mi ,,1 L="
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MULTI-OBJECTIVE OPTIMIZATION
§ Example: Minimization Problem with Two Objective Functions
)0(2
1 ³= xxy )0()2( 2
1 ³-= xxy
2)1(2 2
21
+-=
+=
x
yyY
])2(,[],[ 22
21 -== xxyyy
Q. If both objective functions are desired to be minimized simultaneously,
is (x, Y)=(1, 2) the unique solution?
I. INTRODUCTION
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§ Example: Minimization Problem with Two Objective Functions
])2(,[],[ 22
21 -== xxyyy
MULTI-OBJECTIVE OPTIMIZATION
Set of Ideal Solutions
I. INTRODUCTION
(x, y) = (1, 2) = (y1, y2) = (1, 1)
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§ Optimal Allocation of Resources
• A state that no one can be made better off without making someone worse off
§ Pareto-optimal Front (POF)
• A set of Pareto-optimal solutions
§ Based on “Non-Domination”
• .
PARETO-OPTIMALITY
{ } { } )()(:M,...,1)()(:M,...,1 2121 xfxfjxfxfi jjii <Î$Ù£Î"
Both A and B are
Pareto-optimal solutions.
ü A dominates a and a’.
ü B dominates b and b’.
ü A is not dominated to B.
(= B is equivalent to A.)
A
B
a’
b'
a
b
minimize
minimize
Feasible
solution
domain
Infeasible
solution
domain
fj
fi
Pareto-optimal front
I. INTRODUCTION
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PERFORMANCE METRICS
§ Two Goals of Multi-Objective Optimization
minimize
minimize jf
if
m
1st goal : convergence
2nd goal : diversity
I. INTRODUCTION
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PERFORMANCE METRICS
§ Good Convergence, but Poor Distribution
jf
if
minimize
minimize
I. INTRODUCTION
10/30

PERFORMANCE METRICS
§ Poor Convergence, but Good Distribution
jf
if
minimize
minimize
I. INTRODUCTION
11/30

PERFORMANCE METRICS
§ Poor Convergence and Poor Distribution
jf
if
minimize
minimize
I. INTRODUCTION
12/30

LITERATURE REVIEW
§ Single-Objective History Matching
• Genetic Algorithm (GA)
ü Soleng, 1999; Ballester and Carter, 2007
• Evolution Strategy (ES)
ü Schulze-Riegert et al., 2002; Cheng et al., 2008
• Ensemble Kalman Filter (EnKF)
ü Nævdal et al., 2005; Arroyo-Negrete et al., 2006; Park and Choe, 2006
§ Multi-Objective History Matching
• Evolutionary multi-objective optimization (EMO) algorithm
ü Schulze-Riegert et al., 2007: history matching on realistic North Sea field using SPEA
ü Han et al., 2011: waterflood history matching using NSGA-II
ü Hajizadeh et al., 2011: uncertainty quantification in recovery of PUNQ-S3 model using DEMOPR
ü Mohamed et al., 2011: history matching of ICFM model using particle swarm optimization
ü King et al., 2013: handling conflicting multiple objectives in history matching using Pareto-based
evolutionary algorithm
I. INTRODUCTION
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LITERATURE REVIEW
§ Remedies for “Curse of Dimensionality” in Multi-Objective Problems
☞ Preference-Ordering Approach
• Assumption: no redundant objective
• Goal: finding good solutions with small loss in diversity-preservation
ü Reducing the number of non-dominated points: Sato et al., 2007
ü Assigning different ranks to non-dominated points: Drechsler et al., 2001;
Corne and Knowles, 2007; Kukkonen and Lampinen, 2007; Köppen and Yoshida, 2007
ü Scalarizing functions: Hughes, 2005; Ishibuchi, 2006
ü Indicator functions: Zitzler and Künzli, 2004; Ishibuchi et al., 2007; Wagner et al., 2007
ü Decision makers' preference: Branke and Deb, 2004; Thiele et al., 2009
☞ Objective-Reduction Approach
• Assumption: redundant objectives
• Goal: identifying essential m objectives (m < M)
ü Feature selection: Deb and Saxena, 2005; Singh et al., 2011; Saxena et al., 2013;
I. INTRODUCTION
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OBJECTIVE
§ Objective
• Production estimation from Pareto-optimal reservoir models
§ Originality
• Development of an advanced multi-objective evolutionary model
§ Generalization
• Verification of the model’s accuracy for multi-objective problems
§ Application
• Validation of the model’s applicability to history matching
I. INTRODUCTION
15/30

CONTENTS
I. INTRODUCTION
II. METHODOLOGY
III. APPLICATION
IV.CONCLUSIONS
CONTENTS
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NEW MODEL: DS-MOGA
§ This study developed DGP & SLOR and integrated them with MOGA (Multi-
Objective Genetic Algorithm) to find reservoir models that represent the POF.
• NSGA-II (Non-dominated Sorting Genetic Algorithm-II) (Deb et al., 2002)
ü Generate non-dominated geomodels
• DGP (Dynamic Goal Programming)
ü Prioritize the qualified geomodels
• SLOR (Successive Linear Objective Reduction)
ü Reduce the dimension of objective space by excluding insignificant and/or redundant objectives
II. METHODOLOGY
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NEW MODEL: DS-MOGA
§ Flow Chart of MOGA (NSGA-II)
II. METHODOLOGY
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NEW MODEL: DS-MOGA
§ Flow Chart of D-MOGA
II. METHODOLOGY
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NEW MODEL: DS-MOGA
§ Flow Chart of DS-MOGA
II. METHODOLOGY
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SNU_IRMS
§ Production Management Program
• Programming language: Borland C++
• Algorithms: single- and multi-objective optimization
II. METHODOLOGY
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SNU_IRMS
§ Production Management Program
• Programming language: Visual C#
• User-friendly GUI
II. METHODOLOGY
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CONTENTS
I. INTRODUCTION
II. METHODOLOGY
III. APPLICATION
IV.CONCLUSIONS
CONTENTS
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§ Sector Model of “H” Field in Athabasca, Canada
• Clean sandstone
• Average porosity: 29.0%
• Average permeability: 6,500 md
• Oil gravity: 15.0 ˚API
History Matching Prediction
Permeability Map Production History
HEAYY OIL HISTORY MATCHING
III. APPLICATION
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DS-MOGA NSGA-II
§ Production Profiles
• Cumulative Oil Production of Field
III. APPLICATION
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DS-MOGA SOGA
§ Range of Uncertainty: P90–P10
• DS-MOGA: P90 ≈ actual cumulative oil production
• SOGA: actual cumulative oil production < minimum
III. APPLICATION
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DISCUSSION
§ Performance Comparison
Method Advantage Disadvantage
DS-MOGA
· Stable convergence when M ≥ 4
· Diversity preservation
· Decreasing efficiency in proportion
to the number of objectives
NSGA-II · Efficient for finding trade-off solutions · Inefficient when M > 3
SOGA · Fast convergence on single optimum · Hard to preserve diversity of solutions
III. APPLICATION
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CONTENTS
I. INTRODUCTION
II. METHODOLOGY
III. APPLICATION
IV.CONCLUSIONS
CONTENTS
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CONCLUSIONS
§ The developed evolutionary algorithm, DS-MOGA, contributed to
production estimation from diversity-preserved reservoir models.
§ The integration of preference-ordering and objective-reduction
approach improved the efficiency of multi-objective optimization.
§ The method relieved the divergence problem in multi-objective
optimization as well as the scale-dependency problem in single-
objective optimization.
IV. CONCLUSIONS
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CONCLUSIONS
“A general-purpose universal optimization strategy is
theoretically impossible, and the only way one strategy
can outperform another is if it is specialized to the
specific problem under consideration.”
Ho and Pepyne, 2002
IV. CONCLUSIONS
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Baehyun min pareto_optimality

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