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# Baehyun min pareto_optimality

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Pareto-optimality, multi-objective optimization. history matching

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### Baehyun min pareto_optimality

1. 1. 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 Center for Petroleum & Geosystems Engineering University of Texas at Austin Baehyun Min
2. 2. CONTENTS I. INTRODUCTION II. METHODOLOGY III. APPLICATION IV.CONCLUSIONS University of Texas at Austin Center for Petroleum & Geosystems Engineering CONTENTS 2/30
3. 3. 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 3/30
4. 4. University of Texas at Austin Center for Petroleum & Geosystems Engineering § 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 obs i obs ji cal ji i 1,...,0)( 2 1 ,, ="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 4/30
5. 5. University of Texas at Austin Center for Petroleum & Geosystems Engineering § Global Objective Function where SINGLE-OBJECTIVE OPTIMIZATION Global optimum Local optimums G L )(xF x å= = M i ii xfxF 1 )()( w I. INTRODUCTION 0³iw Mi ,,1 L=" 5/30
6. 6. University of Texas at Austin Center for Petroleum & Geosystems Engineering 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 6/30
7. 7. University of Texas at Austin Center for Petroleum & Geosystems Engineering § 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) 7/30
8. 8. University of Texas at Austin Center for Petroleum & Geosystems Engineering § 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 8/30
9. 9. University of Texas at Austin Center for Petroleum & Geosystems Engineering PERFORMANCE METRICS § Two Goals of Multi-Objective Optimization minimize minimize jf if Pareto-optimal front m 1st goal : convergence 2nd goal : diversity I. INTRODUCTION 9/30
10. 10. University of Texas at Austin Center for Petroleum & Geosystems Engineering PERFORMANCE METRICS § Good Convergence, but Poor Distribution jf if Pareto-optimal front minimize minimize I. INTRODUCTION 10/30
11. 11. University of Texas at Austin Center for Petroleum & Geosystems Engineering PERFORMANCE METRICS § Poor Convergence, but Good Distribution jf if Pareto-optimal front minimize minimize I. INTRODUCTION 11/30
12. 12. University of Texas at Austin Center for Petroleum & Geosystems Engineering PERFORMANCE METRICS § Poor Convergence and Poor Distribution jf if Pareto-optimal front minimize minimize I. INTRODUCTION 12/30
13. 13. University of Texas at Austin Center for Petroleum & Geosystems Engineering 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 13/30
14. 14. University of Texas at Austin Center for Petroleum & Geosystems Engineering 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 14/30
15. 15. University of Texas at Austin Center for Petroleum & Geosystems Engineering 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
16. 16. CONTENTS I. INTRODUCTION II. METHODOLOGY III. APPLICATION IV.CONCLUSIONS University of Texas at Austin Center for Petroleum & Geosystems Engineering CONTENTS 16/30
17. 17. University of Texas at Austin Center for Petroleum & Geosystems Engineering 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 17/30
18. 18. University of Texas at Austin Center for Petroleum & Geosystems Engineering NEW MODEL: DS-MOGA § Flow Chart of MOGA (NSGA-II) II. METHODOLOGY 18/30
19. 19. University of Texas at Austin Center for Petroleum & Geosystems Engineering NEW MODEL: DS-MOGA § Flow Chart of D-MOGA II. METHODOLOGY 19/30
20. 20. University of Texas at Austin Center for Petroleum & Geosystems Engineering NEW MODEL: DS-MOGA § Flow Chart of DS-MOGA II. METHODOLOGY 20/30
21. 21. University of Texas at Austin Center for Petroleum & Geosystems Engineering SNU_IRMS § Production Management Program • Programming language: Borland C++ • Algorithms: single- and multi-objective optimization II. METHODOLOGY 21/30
22. 22. University of Texas at Austin Center for Petroleum & Geosystems Engineering SNU_IRMS § Production Management Program • Programming language: Visual C# • User-friendly GUI II. METHODOLOGY 22/30
23. 23. CONTENTS I. INTRODUCTION II. METHODOLOGY III. APPLICATION IV.CONCLUSIONS University of Texas at Austin Center for Petroleum & Geosystems Engineering CONTENTS 23/30
24. 24. University of Texas at Austin Center for Petroleum & Geosystems Engineering § 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 24/30
25. 25. University of Texas at Austin Center for Petroleum & Geosystems Engineering DS-MOGA NSGA-II § Production Profiles • Cumulative Oil Production of Field HEAYY OIL HISTORY MATCHING III. APPLICATION 25/30
26. 26. University of Texas at Austin Center for Petroleum & Geosystems Engineering DS-MOGA SOGA § Range of Uncertainty: P90–P10 • DS-MOGA: P90 ≈ actual cumulative oil production • SOGA: actual cumulative oil production < minimum HEAYY OIL HISTORY MATCHING III. APPLICATION 26/30
27. 27. University of Texas at Austin Center for Petroleum & Geosystems Engineering 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 27/30
28. 28. CONTENTS I. INTRODUCTION II. METHODOLOGY III. APPLICATION IV.CONCLUSIONS University of Texas at Austin Center for Petroleum & Geosystems Engineering CONTENTS 28/30
29. 29. University of Texas at Austin Center for Petroleum & Geosystems Engineering 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 29/30
30. 30. University of Texas at Austin Center for Petroleum & Geosystems Engineering 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 30/30