ADVANCED TECHNIQUES FOR CLOSED-LOOP …qc661yn3508/PhDthesis_main...ADVANCED TECHNIQUES FOR...

ADVANCED TECHNIQUES FOR CLOSED-LOOP RESERVOIR

OPTIMIZATION UNDER UNCERTAINTY

A DISSERTATION

SUBMITTED TO THE DEPARTMENT OF

ENERGY RESOURCES ENGINEERING

AND THE COMMITTEE ON GRADUATE STUDIES

OF STANFORD UNIVERSITY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

Mehrdad Gharib Shirangi

April 2017

http://creativecommons.org/licenses/by-nc/3.0/us/

This dissertation is online at: http://purl.stanford.edu/qc661yn3508

© 2017 by Mehrdad Gharib Shirangi. All Rights Reserved.

Re-distributed by Stanford University under license with the author.

This work is licensed under a Creative Commons Attribution-Noncommercial 3.0 United States License.

ii



http://purl.stanford.edu/qc661yn3508

I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.

Louis Durlofsky, Primary Adviser


Tapan Mukerji


Oleg Volkov

Approved for the Stanford University Committee on Graduate Studies.

Patricia J. Gumport, Vice Provost for Graduate Education

This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file inUniversity Archives.

iii

To the memory of my grandfather

Seyed Khalil Hosseini Ghasemi (1933-2015)

v

Abstract

In this work, we introduce and apply several new techniques for oil/gas reservoir op-

timization under uncertainty. As the first contribution, we develop a general method-

ology for optimal closed-loop field development (CLFD) under geological uncertainty.

CLFD involves three major steps: optimizing the field development plan based on

current geological knowledge, drilling new wells and collecting hard (well) data and

production data, and updating multiple geological models based on all of the avail-

able data. In the optimization step, the number, type, locations and controls for

new wells (and future controls for existing wells) are optimized using a hybrid Par-

ticle Swarm Optimization – Mesh Adaptive Direct Search algorithm. The objective

in the examples presented is to maximize expected (over multiple realizations) net

present value (NPV) of the overall project. History matching is accomplished using an

adjoint-gradient-based randomized maximum likelihood (RML) procedure. Different

treatments are presented for history matching Gaussian and channelized models.

Because the CLFD history matching component is fast relative to the optimiza-

tion component, we generate a relatively large number of history matched models.

Optimization is then performed using a representative subset of these realizations.

We introduce a systematic optimization with sample validation (OSV) procedure, in

which the number of realizations used for optimization is increased if a validation crite-

rion is not satisfied. The CLFD methodology is applied to two- and three-dimensional

example cases. Results show that the use of CLFD increases the NPV for the ‘true’

(synthetic) model by 10% –70% relative to that achieved by optimizing over a large

number of prior realizations.

The CLFD framework includes several components, and different approaches for

vii

history matching, optimization, model selection and economic evaluation can be ap-

plied. In our second contribution, we address the problem of selecting a subset of

representative geological realizations from a large set. Towards this goal, we intro-

duce a general framework, based on clustering, for selecting a representative subset of

realizations for use in simulations involving ‘new’ sets of decision parameters. Prior

to clustering, each realization is represented by a low-dimensional feature vector that

contains a combination of permeability-based and flow-based quantities. Calculation

of flow-based features requires the specification of a (base) flow problem and simula-

tion over the full set of realizations. Permeability information is captured concisely

through use of principal component analysis. By computing the difference between

the flow response for the subset and the full set, we quantify the performance of var-

ious realization-selection methods. The impact of different weightings for flow and

permeability information in the cluster-based selection procedure is assessed for a

range of examples involving different types of decision parameters. These decision

parameters are generated either randomly, in a manner that is consistent with the

solutions proposed in global stochastic optimization procedures such as GA and PSO,

or through perturbation around a base case, consistent with the solutions considered

in pattern search optimization. We find that flow-based clustering is preferable for

problems involving new well settings (e.g., time-varying well bottom-hole pressures)

or small changes in well configuration, while both permeability-based and flow-based

clustering provide similar results for (new) random multiwell configurations. We also

investigate the use of efficient tracer-type simulations for obtaining flow-based fea-

tures, and demonstrate that this treatment performs nearly as well as full-physics

simulations for the cases considered. The various procedures are applied to select

realizations for use in production optimization under uncertainty, which greatly ac-

celerates the optimization computations. Optimization performance is shown to be

consistent with the realization-selection results for cases involving new decision pa-

rameters.

In the third contribution, we introduce a methodology for the joint optimization of

economic project life and well controls. We present a nested formulation for this joint

optimization problem where we maximize NPV, subject to the constraint that the rate

of return of operations is greater than the minimum attractive rate of return (MARR)

viii

or hurdle rate. The methodology provides the optimal project life and the optimal

well controls such that the maximum NPV is obtained at the end of the project

life, and the rate of return of the project is essentially equal to MARR. Application

of this procedure, enables avoiding situations where NPV increases slowly in time,

but the benefit relative to the capital employed is extremely low. We demonstrate

the successful application of this treatment for production optimization for two- and

three-dimensional reservoir models.

ix

Acknowledgments

First and foremost, I would like to thank God who has always helped and guided me

throughout my life and academic career.

I would like to express my sincere appreciation to my adviser, Prof. Louis Durlof-

sky, for his incredible support, patience, guidance and encouragement throughout my

PhD study. Working with him has been a wonderful journey to learn and grow in

many ways, including research and communication skills, critical thinking, concise

and coherent writing, and professionalism. I am particularly indebted to him for his

confidence in me and for providing me the freedom and opportunity to pursue some

of my own ideas, while watching over my progress and directing the research towards

a coherent set of contributions. I consider myself very fortunate for having him as

my PhD adviser. I also would like to thank Dr. Oleg Volkov for his help during

my PhD, and for serving on my PhD committee. My acknowledgements extend to

Profs. Tapan Mukerji, Roland Horne, and Peter Kitanidis for serving on my defense

committee.

I would like to extend my thanks to all other faculty and staff in the Energy

Resources Engineering Department. In particular, I want to thank Profs. Khalid

Aziz, Hamdi Tchelepi, Jef Caers, Anthony Kovscek, Kurt House and Marco Thiele,

with whom I have met on occasion for various discussions on research. I am grateful

to Dr. Obiajulu Isebor for providing the PSO-MADS code and for many useful dis-

cussions. Special thanks also to Profs. Carlo Alberto Magni (University of Modena),

Michael Saunders and Trevor Hastie (Stanford University), Jo Eidsvik (NTNU), Hadi

Hajibeygi and Denis Voskov (Delft University), Dr. David Echeverrıa Ciaurri (IBM),

and Drs. Mohammad Karimi-Fard and Celine Scheidt (Stanford University) for help-

ful discussions. I also thank Eiko Rutherford and Joanna Sun, our administrative

xi

associates, for their support on various occasions.

I would like to thank the industrial affiliates of the Stanford Smart Fields Con-

sortium for financial support, and the Stanford Center for Computational Earth &

Environmental Science (CEES) for providing the computational resources used in

this work. Special thanks to Dennis Michael who has always been of great help in

facilitating issues with CEES.

I have been fortunate to have great friends at Stanford University. I would like

to express my special thanks to my friends in the ERE department with whom I

had discussions about research, Jacob Englander, Mohammad S. Masnadi, Charles

Kang and Philip Brodrick. I would like to thank other friends in the ERE De-

partment, Yashar Mehmani, Sara Farshidi, Amir Salehi, Amir Delgoshaie, Moham-

mad Bazargan, Alireza Iranshahr, Mohammad Shahvali, Yongduk Shin, Elnur Aliyev,

Karine Levonyan, Julia Foster, Wen Song, Morgan Ames, Forest Jiang, Sumeet Tre-

han, Matthieu Rousset and Hai Vo. I also thank my Stanford friends outside the

department, Michael Albert, Long Do, Robert Shields, Rall Walsh, Ali Shariati, Ali

Shahmoradi and Asieh Tarami.

I want to thank my family who supported me during my PhD study. I especially

thank my little nephews Sajjad, Mohammad Arvin, and Ryan, whose presence gave

me hope and energy during the past years. I dedicate this dissertation to the memory

of my grandfather, Seyed Khalil Hosseini Ghasemi.

xii

Contents

Abstract vii

Acknowledgments xi

1 Introduction 1

1.1 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.1.1 Optimization approaches for oil field operations . . . . . . . . 2

1.1.2 Optimization under uncertainty and selection of representative

models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.1.3 Economic measures for reservoir performance . . . . . . . . . 10

1.1.4 History matching of production data . . . . . . . . . . . . . . 11

1.1.5 Closed-loop reservoir management . . . . . . . . . . . . . . . . 12

1.2 Scope of Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.3 Dissertation Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2 Closed-Loop Field Development 19

2.1 CLFD workflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2 CLFD optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3 Optimization with sample validation . . . . . . . . . . . . . . . . . . 25

2.4 CLFD history matching . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.5 Computational results . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.5.1 Example 2.1: Simultaneous versus sequential optimization of

field development . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.5.2 Example 2.2: CLFD for a two-dimensional reservoir model . . 33

2.5.3 Example 2.3: CLFD for a three-dimensional reservoir model . 49

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2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3 Selection of Representative Models 55

3.1 Assessment of flow-response statistics . . . . . . . . . . . . . . . . . . 56

3.2 Unsupervised Learning for Model Selection . . . . . . . . . . . . . . . 60

3.2.1 Feature selection . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.2.2 Clustering for selection of representative realizations . . . . . . 64


3.3.1 Example 3.1: new well settings in channelized models . . . . . 66

3.3.2 Example 3.2: new well configurations . . . . . . . . . . . . . . 74

3.3.3 Summary of realization-selection results . . . . . . . . . . . . 80

3.4 production optimization under uncertainty . . . . . . . . . . . . . . . 83

3.4.1 Optimization of well controls with representative realizations . 83

3.4.2 Example 3.3: production optimization under uncertainty . . . 85

3.4.3 Additional observations . . . . . . . . . . . . . . . . . . . . . . 88

3.4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

4 Optimization of Economic Project Life 91

4.1 Economic measures and production optimization . . . . . . . . . . . . 92

4.1.1 Net present value computation . . . . . . . . . . . . . . . . . . 92

4.1.2 Modified internal rate of return and economic project life . . . 93

4.1.3 Optimization problem statement . . . . . . . . . . . . . . . . 96


4.2.1 Example 4.1: 2D bimodal reservoir . . . . . . . . . . . . . . . 98

4.2.2 Example 4.2: 3D binary reservoir . . . . . . . . . . . . . . . . 111

4.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

5 Summary, Conclusions and Future Work 117

5.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

5.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

Nomenclature 123

Bibliography 127

xiv

Appendices 145

A CLFD for Channelized Models 147

A.1 History matching for channelized models . . . . . . . . . . . . . . . . 147

A.2 Example A1: CLFD for a channelized model . . . . . . . . . . . . . . 149

B Representative Models for a Binary System 157

B.1 Example B1: new well settings . . . . . . . . . . . . . . . . . . . . . . 157

B.1.1 Representative realizations for random well controls . . . . . . 158

B.1.2 Representative realizations for small changes in well controls . 159

B.2 Example B2: new well configurations . . . . . . . . . . . . . . . . . . 161

B.2.1 Representative realizations for random well configurations . . 162

B.2.2 Representative realizations for small changes in well locations 163

B.2.3 Summary of realization-selection results . . . . . . . . . . . . 166

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List of Tables

2.1 Optimization parameters for all examples . . . . . . . . . . . . . . . . 31

2.2 Final NPVs ($ MM) for three runs for sequential (well-by-well) and

simultaneous optimization (Example 2.1) . . . . . . . . . . . . . . . . 32

2.3 NPV values ($ MM) from optimization over 50 prior realizations and

from CLFD optimization, for three different true models (Example 2.2) 45

2.4 Initial and final numbers of representative realizations (determined us-

ing OSV) and the corresponding relative improvement values at each

CLFD step (Example 2.3) . . . . . . . . . . . . . . . . . . . . . . . . 52

3.1 Median (mD) values of Dα=0, Dα=0.5, Dα=1 and Drand, for 100 random

well control vectors, for different nr. Average mD values and average

ranking are also provided (Example 3.1). . . . . . . . . . . . . . . . . 69

3.2 Median values (mD) of Dα=0,prx and Dα=0.5,prx for 100 random well

control vectors, for different nr. Average mD values are also provided

(Example 3.1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.3 Results for various permeability-based clustering treatments (α = 1 in

all cases). Median (mD) values of Dl=99, Dfull PCA and Dfull perm, for

100 random well control vectors, for different nr. Average mD values

are also provided (Example 3.1). . . . . . . . . . . . . . . . . . . . . . 72

3.4 Results using k-medoids clustering. Median (mD) values of Dα=0,

Dα=0.5 and Dα=1, for 100 random well control vectors, for different

nr. Last column shows computational time for selection with α = 0.

Average values are also provided (Example 3.1). . . . . . . . . . . . . 73

xvii

3.5 Median (mD) values of Dα=0, Dα=0.5 and Dα=1, for 100 well control

vectors corresponding to pattern search mesh points, for different nr.

Average mD values and average ranking are also provided (Example 3.1). 75

3.6 Median values (mD) of Dα=0,prx and Dα=0.5,prx for 100 well control


Average mD values are also provided (Example 3.1). . . . . . . . . . 75

3.7 Median (mD) values of Dα=0, Dα=0.5, Dα=1 and Drand, for 100 random

well configurations, for different nr. Average mD values and average

ranking are also provided (Example 3.2). . . . . . . . . . . . . . . . 77

3.8 Median values (mD) of Dα=0,prx and Dα=0.5,prx for 100 random well

configurations, for different nr. Average mD values are also provided

(Example 3.2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

3.9 Median (mD) values of Dα=0, Dα=0.5 and Dα=1, for 40 well configu-

rations corresponding to pattern search mesh points, for different nr.

Average mD values and average ranking are also provided (Example 3.2). 81

3.10 Median values (mD) ofDα=0,prx andDα=0.5,prx for 40 well configurations

corresponding to pattern search mesh points, for different nr. Average

mD values are also provided (Example 3.2). . . . . . . . . . . . . . . 81

3.11 Summary of results: average mD values for nr = 3, . . . , 15 for all cases.

The smallest value for each case is indicated in bold. . . . . . . . . . 83

3.12 Economic parameters and bounds for Example 3.3 . . . . . . . . . . . 85

3.13 Improvement in expected objective (in $106) for the full set of 200

realizations, J(xopt,Mfull)− J(x0,Mfull), evaluated using xopt obtained

from optimization runs with nr = 3 (Example 3.3). . . . . . . . . . . 87

3.14 Improvement in expected objective (in $106) for the full set of 200

realizations, J(xopt,Mfull)− J(x0,Mfull), evaluated using xopt obtained

from optimization runs with nr = 6 (Example 3.3). . . . . . . . . . . 88

4.1 Economic parameters and BHP ranges for all examples . . . . . . . . 100

4.2 NPV ($ MM), corresponding MIRR and fluid production/injection

(MM STB) for optimal controls with different project life. The so-

lution with T ∗ = 2340 days represents an optimum (Example 4.1). . . 105

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4.3 Optimal NPV, corresponding MIRR and fluid production/injection

(MM STB) from optimal controls with different initial-guess BHPs

with T = 2340 days (Example 4.1). . . . . . . . . . . . . . . . . . . . 108

4.4 Optimal NPV and the corresponding MIRR for different (specified)

project life (Example 4.1). . . . . . . . . . . . . . . . . . . . . . . . . 109

4.5 Optimal NPV and the corresponding MIRR and EPL (T ∗) from opti-

mizations with different discount rates (Example 4.1). . . . . . . . . . 110

A.1 NPV values ($ MM) from optimization over prior realizations with nr =

5 and by use of OSV (where nr is increased to satisfy RI > 0.5) and

from CLFD by use of OSV, for five different true models (Example A1) 156

A.2 NPV values ($ MM) from optimization over prior realizations by use

of OSV (where nr is increased to satisfy RI > 0.5) and from CLFD by

use of OSV, for true model 4 (Example A1) . . . . . . . . . . . . . . 156

B.1 Median values (mD) of Dα=0, Dα=0.5, Dα=1 and Drand, for 300 random

well control vectors, for different nr. Average mD values and average

ranking are also provided (Example B1). . . . . . . . . . . . . . . . 160

B.2 Median (mD) values of Dα=0, Dα=0.5 and Dα=1, for 126 well control


Average mD values and average ranking are also provided (Example B1).162

B.3 Median (mD) values of Dα=0, Dα=0.5, Dα=1 and Drand, for 300 random

well configurations, for different nr. Average mD values and average

ranking are also provided (Example B2). . . . . . . . . . . . . . . . . 164

B.4 Median (mD) values of Dα=0, Dα=0.5 and Dα=1, for 40 well configu-

rations corresponding to pattern search mesh points, for different nr.

Average mD values and average ranking are also provided (Example B2).165

B.5 Summary of results: average mD values for nr = 3, . . . , 15 for all cases.

The smallest value for each case is indicated in bold. . . . . . . . . . 166

xix

List of Figures

1.1 Schematic of closed-loop reservoir management. . . . . . . . . . . . . 13

1.2 Schematic of closed-loop field development (CLFD). . . . . . . . . . . 14

2.1 Schematic and notation for the closed-loop field development optimiza-

tion procedure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.2 True log-permeability field for Examples 2.1 and 2.2. Permeability is

in mD. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.3 Oil and water relative permeability curves for Example 2.1. . . . . . 31

2.4 Final oil saturation (at 3000 days) from optimal solutions for the two

approaches (Example 2.1). Well locations are also shown, with red

denoting producer, blue denoting injector, and the well numbers indi-

cating the drilling sequence. . . . . . . . . . . . . . . . . . . . . . . . 33

2.5 Oil and water relative permeability curves for Examples 2 and 3. . . 34

2.6 Well configuration from deterministic optimization (using mtrue), with

red denoting producer, blue denoting injector, and the well numbers

indicating the drilling sequence. Background shows final oil saturation

(Example 2.2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.7 Ordered NPV plot for NR = 50 prior realizations based on initial guess

for decision variables x0. The three selected realizations are shown in

red (Example 2.2, nr = 3). . . . . . . . . . . . . . . . . . . . . . . . . 37

2.8 Three representative prior realizations of log-permeability, along with

the initial-guess well configuration (Example 2.2). Red (outlined) cir-

cles denote producers, blue injectors, and the well numbers indicate

the drilling sequence. . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

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2.9 Evolution of expected NPV (J(x,M1rep)) for optimization over nr = 3

prior realizations (Example 2.2). . . . . . . . . . . . . . . . . . . . . . 38

2.10 Optimal well configuration and drilling sequence at t1 and t2. Solid red

and blue circles denote producers and injectors (drilled or in the process

of being drilled), and outlined red and blue circles denote planned

producers and injectors. Numbers indicate the drilling sequence and

background shows log-permeability for one realization (Example 2.2,

nr = 3). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.11 Optimal expected NPV, J(xi,M irep), and the expected NPV for the

corresponding initial guess, J(xi−1,M irep), versus CLFD step (Exam-

ple 2.2, nr = 3). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.12 Optimal expected NPV, J(xi,M irep), and the corresponding NPV for

the true model, J(xi,mtrue), versus CLFD step. The star shows the

final true NPV from CLFD (Example 2.2, nr = 3). . . . . . . . . . . 40

2.13 NPV for the nr = 3 (representative) realizations, and the correspond-

ing NPV for the true model, versus CLFD step (Example 2.2). . . . . 42

2.14 Optimal expected NPV, and the corresponding NPV for the true model,

versus CLFD step, for different numbers of representative realizations.

The star shows the final true NPV from CLFD (Example 2.2). . . . . 44

2.15 NPV for different numbers of representative realizations, and the cor-

responding NPV for the true model, versus CLFD step (Example 2.2). 45

2.16 P10, P50, P90 NPVs evaluated for the entire set of 50 realizations,

along with the expected NPV for the representative set, versus CLFD

step (Example 2.2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46


versus CLFD step. The number of realizations at each CLFD step is

determined using OSV. The star shows the final true NPV from CLFD

(Example 2.2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

xxii

2.18 Evolution of two RML realizations (log-permeability is shown) for dif-

ferent CLFD steps. Current optimal well configuration and drilling

sequence is also depicted. Solid red and blue circles denote producers

and injectors (drilled or in the process of being drilled), and outlined

red and blue circles denote planned producers and injectors. Numbers

indicate the drilling sequence (Example 2.2). . . . . . . . . . . . . . . 48

2.19 True log-permeability field with initial guess for well configuration,

which includes three horizontal producers and three vertical injectors

(Example 2.3). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50




(Example 2.3). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

2.21 Final optimal solution from CLFD with OSV. Horizontal producers

are shown in red and vertical injectors in blue (Example 2.3). . . . . . 53

3.1 Illustration of some of the components of qkj (for well k = 1). The

reservoir life is 3000 days, which is divided into nt = 3 intervals. . . . 57

3.2 Oil and water relative permeability curves for all examples. . . . . . . 66

3.3 Three conditional realizations of log-permeability field for bimodal

channelized model. Fixed well configuration is also shown – circles

denote producers and triangles indicate injectors (Examples 3.1 and

3.3). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.4 Injector BHP profiles corresponding to a random well-control vector

xnew (Example 3.1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.5 Box plots of Dα=0, Dα=0.5 and Dα=1 for 100 random well control vec-

tors. The red line within each box corresponds to the median, and

the bottom and top of each box correspond to the 25th and 75th per-

centiles. The lines above and below the boxes correspond to the 2nd

and 98th percentiles (Example 3.1). . . . . . . . . . . . . . . . . . . . 70

xxiii

3.6 Box plots of Dα=0, Dα=0.5 and Dα=1 for 100 well control vectors corre-

sponding to pattern search mesh points. The red line within each box

corresponds to the median, and the bottom and top of each box cor-

respond to the 25th and 75th percentiles. The lines above and below

the boxes correspond to the 2nd and 98th percentiles (Example 3.1). 74

3.7 Three realizations of log-permeability field and three base well config-

urations for computing flow-based features used in clustering. Circles

denote producers and triangles indicate injectors (Example 3.2). . . . 75

3.8 Three (out of 100) random well configurations, xnew, for computing flow

responses. Circles indicate producers and triangles denote injectors

(Example 3.2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

3.9 Box plots of Dα=0, Dα=0.5 and Dα=1 for 100 random well configura-

tions. The red line within each box corresponds to the median, and



and 98th percentiles (Example 3.2). . . . . . . . . . . . . . . . . . . . 78

3.10 Base-case well configuration, and two (out of 40) new well configu-

rations corresponding to pattern search mesh points. Circles denote

producers and triangles indicate injectors (Example 3.2). . . . . . . . 79

3.11 Box plots of Dα=0, Dα=0.5 and Dα=1 for 40 well configurations corre-




the boxes correspond to the 2nd and 98th percentiles (Example 3.2). 80

4.1 Example cash flow stream for a production optimization problem. Cash

flows are computed for each 90-day control step. . . . . . . . . . . . . 95

4.2 MIRR trajectory corresponding to cash flow stream in Fig. 4.1. The

dashed vertical line shows the time where the rate of return becomes

smaller than the specified MARR of 0.15. . . . . . . . . . . . . . . . . 96

4.3 Log-permeability field with well locations. Circles denote producers

and triangles denote injectors (Example 4.1). . . . . . . . . . . . . . . 99

xxiv

4.4 NPV trajectory from inner optimization with (a) initial guess T , and

(b) T for which NPV in (a) is the maximum. The dashed vertical line in

(a) shows the time where the maximum NPV is obtained (Example 4.1).101

4.5 Cash flow stream for the optimal controls with T = 3240 days (Exam-

ple 4.1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

4.6 (a) Cash flow percentage, computed yearly, versus time for optimal

controls with T = 3240 days, and (b) magnification for the last three

years (Example 4.1). . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

4.7 MIRR trajectory for optimal controls with T = 3240 days, computed

for the period (0, t) (Example 4.1). . . . . . . . . . . . . . . . . . . . 104

4.8 (a) NPV trajectory for the optimal solution with T ∗ = 2340 days, and

(b) magnification of NPV trajectory for the period of (2000, 2400) days

from solutions for T ∗ = 2340 days (which is the optimal solution) and

T = 3240 days (Example 4.1). . . . . . . . . . . . . . . . . . . . . . . 105

4.9 MIRR trajectory for the optimal solution (T ∗ = 2340 days) computed

for the period (0, t). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

4.10 Optimal controls x∗ for three producers and three injectors correspond-

ing to T ∗ = 2340 days (Example 4.1). . . . . . . . . . . . . . . . . . . 106

4.11 Final Sw distribution from optimal controls with different T . The well

configuration is also shown, with red circles denoting producers and

blue circles indicating injectors (Example 4.1). . . . . . . . . . . . . . 107

4.12 Initial-guess BHP profiles for three producer wells (T = 2340 days). . 108

4.13 Relationship between MIRR and optimal NPV for optimizations with

different specified project life. Only the solution corresponding to

MIRR= 0.202, NPV=$311.6 MM, is optimal in the sense of Eq. 4.7

(Example 4.1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

4.14 Binary permeability field, with red indicating sand facies (permeability

of 500 mD), and blue indicating shale facies (permeability of 10 mD).

The well configuration, which includes five horizontal producers, de-

noted by circles and lines, and six vertical injectors, denoted by trian-

gles, is also shown (Example 4.2). . . . . . . . . . . . . . . . . . . . . 112

4.15 NPV trajectory for optimal controls with T = 4950 days (Example 4.2).113

xxv

4.16 MIRR trajectory for optimal controls (T = 4950 days) computed for

the period (0, t). Dashed horizontal line shows the value of MARR

(Example 4.2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

4.17 MIRR trajectory for optimal solution (T ∗ = 3060 days) computed for

the period (0, t) (Example 4.2). . . . . . . . . . . . . . . . . . . . . . 114

4.18 Final water saturation maps from optimal solution (x∗ with T ∗ =

3060 days). (Example 4.2). . . . . . . . . . . . . . . . . . . . . . . . . 114

A.1 True permeability field, with red indicating sand facies (permeability of

500 mD), and blue indicating shale facies (permeability of 10 mD). The

initial well configuration is also shown, with circles denoting producers

and triangles denoting injectors (Example A1). . . . . . . . . . . . . 150

A.2 Three prior realizations of the permeability field, with red indicating

sand facies (permeability of 500 mD), and blue indicating shale facies

(permeability of 10 mD). . . . . . . . . . . . . . . . . . . . . . . . . . 150

A.3 Well configuration from deterministic optimization (using mtrue), with

red denoting producer, blue denoting injector, and the well numbers

indicating the drilling sequence. Background shows final oil saturation.

Note that two wells are drilled at a time (Example A1). . . . . . . . 151

A.4 Optimal expected NPV, and the corresponding NPV for the true model,



(Example A1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

A.5 P10, P50, P90 NPVs evaluated for the entire set of 50 realizations,

along with the expected NPV for the representative subset, versus

CLFD step (Example A1). . . . . . . . . . . . . . . . . . . . . . . . . 153

A.6 Evolution of two RML realizations for different CLFD steps, with red

indicating sand facies (permeability of 500 mD), and blue indicating

shale facies (permeability of 10 mD). Current optimal well configu-

ration and drilling sequence is also depicted. Solid white circles and

triangles denote producers and injectors (drilled or in the process of be-

ing drilled), and yellow circles and triangles denote planned producers

and injectors. Numbers indicate the drilling sequence (Example A1). 154

xxvi

A.7 Evolution of mean of (NR = 50) prior realizations (conditioned to hard

data) and mean of (NR = 50) posterior realizations of facies distribu-

tion, for different CLFD steps. Current optimal well configuration and

drilling sequence is also depicted. White circles and triangles denote

producers and injectors, respectively. Wells with colored (red or blue)

numbers are drilled, while outlined red circles and blue triangles de-

note planned producers and injectors. For the prior model (a-d) only

the drilled wells are shown. Numbers indicate the drilling sequence

(Example A1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

B.1 Three unconditional realizations of binary channelized model. Red in-

dicates sand facies (permeability of 500 mD) while blue shows non-sand

facies (permeability of 10 mD). Fixed well configuration is also shown

– circles denote producers and triangles indicate injectors (Example B1).158

B.2 Injector BHPs corresponding to a random well-control vector xnew (Ex-

ample B1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

B.3 Box plots of Dα=0, Dα=0.5 and Dα=1 for 300 random well control vec-

tors. The red line within each box corresponds to the median, and



and 98th percentiles (Example B1). . . . . . . . . . . . . . . . . . . . 160

B.4 Box plots of Dα=0, Dα=0.5 and Dα=1 for 126 well control vectors corre-




the boxes correspond to the 2nd and 98th percentiles (Example B1). . 161

B.5 Three realizations and three base well configurations for computing

flow-based features used in clustering. Circles denote producers and

triangles indicate injectors (Example B2). . . . . . . . . . . . . . . . 162

B.6 Three (out of 300) random well configurations, xnew, for computing flow

responses. Circles indicate producers and triangles denote injectors

(Example B2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

xxvii

B.7 Box plots of Dα=0, Dα=0.5 and Dα=1 for 300 random well configura-

tions. The red line within each box corresponds to the median, and



and 98th percentiles (Example B2). . . . . . . . . . . . . . . . . . . . 164

B.8 Base-case well configuration, and two (out of 40) new well configu-

rations corresponding to pattern search mesh points. Circles denote

producers and triangles indicate injectors (Example B2). . . . . . . . 165

B.9 Box plots of Dα=0, Dα=0.5 and Dα=1 for 40 well configurations corre-




the boxes correspond to the 2nd and 98th percentiles (Example B2). . 166

xxviii

Chapter 1

Introduction

Optimization is encountered in essentially all engineering disciplines. An optimization

problem is typically defined by a set of decision parameters, an objective function to

be minimized or maximized, and a set of constraints. Determining decision param-

eters such as the locations of new wells and operational settings of existing wells is

of primary importance in oil reservoir management, where the goal is to maximize

oil recovery or an economic measure of the project such as net present value (NPV).

These optimizations are computationally intensive because the objective function is

evaluated through a numerical simulation, which may take hours. A significant chal-

lenge in reservoir performance optimization is to appropriately account for geological

uncertainty. This is usually accomplished by considering multiple realizations of the

geological model. Optimization then involves, for example, maximizing the expected

NPV. In this case, each function evaluation performed during optimization technically

requires evaluating flow simulation results for all realizations employed, which could

be extremely expensive. Computational cost can be reduced, however, by selecting a

few representative realizations.

Our goal in this work is to develop and apply computational procedures for closed-

loop reservoir optimization under geological uncertainty. Toward this end, in our

first contribution, we develop a general framework for closed-loop field development

1

2 CHAPTER 1. INTRODUCTION

(CLFD) under uncertainty. CLFD is a comprehensive reservoir management frame-

work that includes optimization and history matching steps that are performed re-

peatedly throughout the development process. The impact of this research is poten-

tially significant as drilling new wells is one of the most expensive parts of reservoir

operations.

Our second contribution concerns the realization selection problem in optimization

under uncertainty. Specifically, we develop and test a general methodology to select a

small number of representative models from a large set of geological realizations for use

in optimization. In our third contribution, we consider the problem of determining the

economic project life (EPL) for optimal operation of existing wells. In optimization of

reservoir operations, the project life is typically specified heuristically. We introduce

a method for the joint determination of optimal well controls and EPL. Our approach

involves the application of financial metrics such as modified internal rate of return

(MIRR) and minimum attractive rate of return (MARR) to reservoir optimization

problems.

1.1 Literature Review

In this section we discuss relevant work in the areas of optimization of well control,

field development optimization, optimization under uncertainty, economic measures,

history matching, closed-loop optimization of existing wells, and selection of repre-

sentative geological realizations. There is extensive literature in many of these areas,

and we limit our discussion to the papers that are most relevant to this study.

1.1.1 Optimization approaches for oil field operations

Historically, optimization approaches were investigated separately for decisions in-

volving (1) the operation of existing wells, and (2) field development planning. In

the first case, the continuous operational settings (time-varying well rate or bottom-

hole pressure settings) of existing wells are optimized. In field development planning,

which represents a much more complex optimization problem, decision parameters

can include the number of new wells, well type (producer or injector), well locations,

drilling sequence, and well settings. Most papers considered the optimization of only

1.1. LITERATURE REVIEW 3

a subset of these parameters, mainly the location of new wells (referred to as the well

placement problem). In recent work, however, various researchers have investigated

more general problems. We first review approaches for optimization of continuous

settings of existing wells, and we then discuss optimization of field development plan-

ning.

Well control optimization

The well control problem, often referred to as production optimization, involves the

optimization of the continuous (time-varying) operational settings of existing wells

to maximize an economic objective such as oil recovery or NPV. Early research in

this subject was performed for simplified treatments of EOR operations. This in-

cludes the work of Fathi et al. [42], who considered surfactant flooding, and Wei

et al. [145], who considered steam flooding. However, most of the work in the past

decade has targeted optimization of water-flooding operations. Both gradient-based

and derivative-free optimization methods have been applied for this problem. The

gradient-based methods are typically more efficient, though they converge to a locally

optimal solution. Derivative-free approaches are noninvasive methods as they do not

require implementation of an adjoint code, but they often require many more func-

tion evaluations (though these are typically performed in parallel). Here we consider

gradient-based and derivative-free approaches in turn.

Various gradient-based approaches have been applied for production optimiza-

tion, including the steepest descent algorithm [154, 22], sparse nonlinear optimizer

(SNOPT) [144, 83], sequential quadratic programming (SQP) [112, 142], and sequen-

tial convex programming (SCP) solver based on the method of moving asymptotes

(MMA) [24]. In gradient-based production optimization, the adjoint method is typi-

cally implemented to compute the gradient through a backward simulation run. See

Kourounis et al. [83] and references therein for discussion of the adjoint formulation.

The computational time for an adjoint solution is roughly equivalent to 1/2 (or less)

of a typical forward simulation run.

In the absence of adjoint-based gradients, derivative-free methods and numerical

gradient methods are viable alternatives. In the context of well control optimization,


pattern search procedures [36, 72] and stochastic search methods such as genetic al-

gorithms (GAs) [36] and particle swarm optimization (PSO) [73] have been applied.

A number of researchers also investigated the application of gradient-based optimiza-

tion methods by computing an approximate gradient through noninvasive approaches.

Yeten et al. [152] applied the nonlinear conjugate gradient approach with the gradient

computed through numerical finite difference (FD). A similar approach was consid-

ered by Yeten et al. [151] and Aitokhuehi and Durlofsky [2]. Echeverrıa Ciaurri et al.

[36] applied a SQP algorithm with the gradient computed through FD. The computa-

tional time of the FD approach scales linearly with the number of control parameters

(when computation is based on a single compute node), though the FD gradient can

be computed efficiently through parallelized computation given access to an adequate

number of computing nodes. An approximate gradient can also be computed through

the SPSA approach [128, 127, 34] or by use of an ensemble-based method [30, 33].

Most approaches in production optimization are based on optimizing long-term

reservoir performance through maximization of NPV. As optimization of this long-

term objective may reduce the short-term oil production/NPV, several researchers

have investigated approaches for simultaneous long-term and short-term production

optimization. In particular, van Essen et al. [136] proposed a hierarchical approach

where, after optimizing the long-term NPV, they used the redundant degrees of free-

dom to optimize the short-term NPV. Their approach, however, is computationally

expensive as it requires multiple computations of the Hessian and its null space during

optimization iterations. Fonseca et al. [43] proposed a modified hierarchical approach

by use of ensemble-based optimization (EnOpt), which is a gradient-free approach.

They also suggested the use of BFGS to obtain an approximation of the Hessian.

Chen et al. [29] applied a sequential approach where they first optimized the long-term

NPV, and then maximized the short-term NPV with an optimization constraint to

ensure that the long-term NPV did not change significantly. Isebor and Durlofsky [69]

developed and applied a hybrid global-local derivative-free approach for bi-objective

optimization of short-term and long-term NPV. They considered the more complex

problem of generalized field development.


Well placement optimization

In well placement problems, decision parameters typically correspond to locations of

new wells, which are treated as quasi-integer variables. For fully-penetrating vertical

wells, each well is defined by its (i, j) location on the grid. For partially penetrat-

ing wells, horizontal wells and multilaterals, additional parameters are required to

fully define the well trajectory [124, 150]. The well location problem can also be

treated with continuous variables, which must then be rounded since well locations

are generally specified based on a discrete grid. Many of the methods developed

for this problem entail stochastic (global) search methods such as GAs [56, 150, 23],

PSO [99, 11, 9, 7], evolution strategy with covariance matrix adaptation (CMA-

ES) [16, 17, 20], improved harmony search (IHS) [1], and differential evolution (DE)

[18, 96, 47]. Local optimization methods such as SPSA [74, 77], pattern search tech-

niques [146, 27] and gradient-based methods [155] have also been applied for well

placement optimization.

Several studies have considered the optimization of nonconventional wells. Yeten

et al. [150] developed a GA-based approach for optimizing multilateral wells (including

the number of laterals), while Artus et al. [12] applied a GA with a statistical proxy

for optimization of monobore and dual-lateral wells. Onwunalu and Durlofsky [100]

applied a PSO algorithm for the optimization of vertical, deviated, and dual-lateral

wells. They found that PSO outperformed GA for the problems (and algorithmic

implementations) considered. In later work, they applied PSO for large-scale field

development optimization [99].

Constraint handling has also been addressed in a number of studies. In particular,

Emerick et al. [39] and Jesmani et al. [76] applied a stochastic search algorithm with

a penalty method for handling geometrical location constraints. Isebor et al. [70]

introduced a filter-based approach for general nonlinear constraint handling (such

as maximum field liquid production and well-distance constraints). This approach

entails minimizing the aggregate constraint violation, along with (say) maximizing

NPV, as the optimization proceeds.


Joint optimization of well locations and controls

Most papers on well placement optimization considered optimizing well locations with

their controls (or a simple control strategy) specified a priori. Recent work, however,

has indicated that a sequential approach for the optimization of well location and

control, in which well locations are determined first (with an assumed well control

strategy), and well controls second, often yields suboptimal solutions compared to

the joint/simultaneous optimization of well locations and controls [87, 65, 46, 70, 44].

These studies applied a variety of optimization techniques, and in most examples

considered, joint optimization yielded higher objective function values than sequen-

tial optimization. We note, however, that Humphries et al. [65] provided counter

examples, where their sequential approach outperformed joint optimization.

Bellout et al. [19] introduced a nested approach for the joint optimization problem

in which the outer well placement optimization is solved using a pattern search (PS)

optimization method, while the inner well control problem is solved using gradient-

based SQP. Li and Jafarpour [86] presented an alternating iterative solution of the

decoupled well placement and control subproblems, where each subproblem is solved

in turn. They applied a SPSA-type algorithm for optimization of well locations, and

a gradient-based approach for optimizing well controls.

Isebor et al. [70, 71] developed a formulation based on a hybrid of PSO, a global

stochastic search algorithm, and mesh adaptive direct search (MADS), a local pattern

search method. This PSO-MADS procedure can simultaneously optimize the number

and type (e.g., injector or producer) of new wells and the drilling sequence, in addition

to well locations and controls. This algorithm will be used in this work for the CLFD

optimization step. The PSO-MADS algorithm has also been extended for bi-objective

optimization [69]. Other applications of this algorithm include optimization of energy

systems [78, 21, 79] and shale gas field development [31].

Due to its complexity, the joint optimization problem typically requires many

function evaluations. Proxy-based approaches such as streamline simulation [64, 68,

67], reduced-order models [135], response surface techniques [8], and upscaled models

[3, 110] can be used to accelerate the computations. Recently, Aliyev and Durlofsky [3]

introduced a multilevel (multifidelity) optimization approach to significantly reduce

the computational effort required for the joint optimization problem. In the multilevel


approach [3, 5], the optimization is performed over a sequence of upscaled models of

increasing fidelity. After convergence of the optimizer at a given level, the optimal

solution is used as the initial guess for the next (finer) level. The multilevel approach

has been extended to optimization under geological uncertainty [4].

1.1.2 Optimization under uncertainty and selection of repre-

sentative models

Because subsurface geology is always uncertain, in any optimization the evaluation of

a given set of decision parameters is best made by considering flow simulation results

over an ensemble of realizations intended to capture the current state of geological

knowledge. Technically, this requires computing flow responses, for each set of pa-

rameters considered, over a large number of realizations. Because computational cost

scales directly with the number of realizations employed, it is preferable to use as

few realizations as possible. If too few realizations are considered, however, results

may not represent the response from the full set, because geological uncertainty is

not properly modeled. Therefore, in order to achieve the optimal balance between

cost and ‘representivity,’ the subset of geological realizations used for flow simulation

must be selected carefully.

In commonly-used derivative-free algorithms, such as GA and PSO, each iteration

may involve, say, 100 function evaluations (meaning the population or the swarm

size is about 100). However, in order to optimize expected reservoir performance

over a set of NR realizations, a single function evaluation requires flow simulation

to be performed over all of the realizations considered. If an optimization requires

(say) 1000 iterations, this corresponds to 105 × NR flow simulations. If we take

NR to be 100 (which is a typical value), a total of 107 simulations will be required.

However, if we can find nr representative realizations (with nr � NR) that can

approximate the expected flow performance of the full set of NR realizations, then we

will achieve computational savings of a factor of NR/nr, which can be very substantial.

Consistent with this, it is of interest to develop a general framework that can be used

to appropriately select a representative set of nr realizations for use in optimization or

decision making. Because the amount of computation required in optimization is so

large, it is cost-effective to perform some number of flow simulations in determining


the nr representative realizations.

Various approaches, within different contexts, for the selection of representative

realizations from a large set of models have been presented in previous work. In

the context of uncertainty assessment for future reservoir production, Scheidt and

Caers [117] introduced a realization-selection method using kernel k-means cluster-

ing and streamline simulation. With this method, a few representative realizations

are selected for flow simulation, with the goal that results for particular statistics

characterizing future oil production are similar to those for the entire set. Scheidt

and Caers [118] also proposed a distance kernel method to select a subset of reservoir

models that provide an uncertainty range for a particular production response (such

as cumulative oil production versus time) in agreement with that of the full set for a

base operating scenario.

Yeh et al. [149] applied a similar approach using flow-based features from stream-

line simulation. Meira et al. [93] and Rahim et al. [104] introduced optimization-based

methods for selecting a subset of realizations that are intended to be representative

of the full set in terms of NPV distribution and simulation results. These approaches

were applied for a particular well configuration and set of well controls. Armstrong

et al. [10] presented a multistage programming with recourse procedure for selecting

a representative subset of realizations in a mineral deposit problem.

We now discuss previous work on optimization under uncertainty. In robust op-

timization, geological uncertainty is accounted for by optimizing over multiple real-

izations, and the objective function typically involves maximizing or minimizing an

expectation. Robust optimization has been investigated for various subsurface flow

problems, such as well control optimization [123, 127, 137, 45], well placement op-

timization [150, 12, 124], and optimization of well location and rate in groundwater

management [17, 133].

Ozdogan and Horne [101] studied the problem of optimizing the locations of a

sequence of wells in a water-flooding project under uncertainty. For optimization,

they used a hybrid GA algorithm with a kriging proxy. The formulation of Ozdo-

gan and Horne [101] optimizes the locations of all new wells simultaneously, but it

does not include the optimization of well types and controls. This work is distinct

from the previous work on well placement optimization under uncertainty, as each


function evaluation in their optimization involves a history matching step. Further,

they introduced the concept of pseudo-history, which is the ‘probable’ production his-

tory of the next well to be drilled. They defined the pseudo-history to be the future

production data for the realization corresponding to the P50 of the final NPV. Each

function evaluation involves history matching the pseudo-history for all realizations.

In their optimization, they maximized the utility, which is a function of both the

expected NPV and a risk term. They showed that maximizing the utility by inte-

grating pseudo-history in the optimization reduced the risk in terms of the standard

deviation of the final NPV.

Various strategies have been applied to select a representative subset of realizations

for use in optimization. For well control optimization, Shirangi and Mukerji [127]

selected representative realizations by applying k-medoids clustering using some flow-

based features, while Yasari et al. [148] selected realizations based on the ranking of

NPVs obtained from an initial control strategy. For well placement optimization,

Wang et al. [143] applied k-means clustering, using a few static and simulation-based

quantities. Torrado et al. [134] applied a similar approach using only static features.

Yang et al. [147] selected realizations for the robust optimization of well locations in

SAGD operations by ranking models in terms of NPV for a base well location and

control strategy, and then selecting nine realizations corresponding to P10, P20, . . . ,

P90 of the NPV distribution (here P10, P20 and P90 denote the 10th, 20th and 90th

percentiles). Bayer et al. [17, 18] developed a stack ordering approach for identifying

critical realizations for optimization of well locations in groundwater management

problems.

Most of the studies noted above used a single set of realizations throughout the

optimization. These realizations were selected according to their flow response for

a problem involving a particular (base-case) well configuration and control strategy.

Wang et al. [143], however, modified the set of representative realizations during the

course of the optimization based (in part) on the evolving flow response. Previous

studies did not provide procedures to assess whether the selected realizations ade-

quately represented the entire set during the course of the optimization. In addition,


there does not appear to have been much study of the impact of the realization-

selection procedure on optimization results. These issues will be addressed in this the-

sis.

1.1.3 Economic measures for reservoir performance

In the oil/gas reservoir optimization literature, the optimization objective is typically

NPV or cumulative oil recovery. In fact, we are not aware of previous work in this

area that used other financial measures such as rate of return. In investment science

and engineering economics, however, various financial measures besides NPV are con-

sidered in project evaluation. These include internal rate of return (IRR), modified

internal rate of return (MIRR), the benefit-to-cost ratio, payback period, and prof-

itability index (see, e.g., Luenberger [89], Higgins [62] and Magni [91]). Among these

measures, NPV and IRR are the two most popular, though each has advantages and

shortcomings. While NPV is sensitive to the project time-line, it does not reflect the

benefit-to-cost ratio. In addition, NPV is not informative regarding the trajectory

of the cash flow stream. IRR, by contrast, strongly depends on the properties of

the cash flow stream. Another key difference is that IRR does not depend on the

prevailing interest rate, while NPV is sensitive to the discount (interest) rate.

The cash flow stream in water-flooding projects is typically negative in early pe-

riods due to capital investment, and it is positive thereafter. In economics, this is

referred to as a conventional cash flow stream, for which a unique IRR exists [60, 58].

The IRR is, formally, the interest rate such that the present value of costs becomes

equal to the present value of returns. Equivalently, the IRR is the discount rate for

which NPV becomes zero. The implicit assumption in computing IRR is that inter-

mediate income is to be reinvested at the internal rate of return, while the funds for

intermediate costs are shifted from other investments earning the same internal rate

[88]. In addition, as discussed by Magni [91], the IRR computation is insensitive to

the economic life of the project when the late-time cash flows are small. Therefore,

when a project is continued with small positive cash flow, the IRR of the project

continues to increase, although with a negligible rate.

The modified internal rate of return (MIRR) [88] was initially introduced as a

robust financial measure for selection among multiple investment projects. There has


been growing attention in the use of MIRR for project evaluation (see, e.g., Ryan

and Ryan [109], Satyasai et al. [115], Park et al. [102] and Hurley et al. [66]). Balyeat

et al. [14] and Kierulff [80] argue that MIRR is a better measure of the rate of

return for a project than IRR. This is because, in computing MIRR, one must specify

the reinvestment rate for intermediate cash flows. As opposed to IRR, MIRR is an

appropriate measure for determining the economic project duration. As we will see,

this is particularly useful for determining economic project lifetime in water-flooding

operations.

Finally, it is worth mentioning that, more recently, the average internal rate of

return (AIRR) has been introduced as an improved measure for computing the rate

of return of a project [90, 6, 91]. Computing the AIRR, however, requires the spec-

ification of the capital value of the project throughout the project timeline, which

introduces additional complexity.

1.1.4 History matching of production data

The goal of history matching (also referred to as data assimilation, model calibration

or model updating) is to generate one or more geological models that are consistent

with prior geological information and provide flow simulation results that match (to

within some tolerance) observed production data, i.e., the rates or BHP measurements

[49, 48]. To enable the assessment of uncertainty, multiple history matched models are

generated. In this case all realizations (essentially) match production history, though

each realization provides a different prediction for future reservoir performance. In the

context of inverse problem theory [132], generating multiple history matched models

is equivalent to sampling the posterior probability density function (pdf). In this

work, we use the randomized maximum likelihood (RML) method [81, 98, 126, 121]

for generating multiple realizations in the CLFD history matching step.

In the context of RML, a sample from the posterior pdf is generated by minimizing

an objective function that quantifies the mismatch between observed and simulated

data. This objective function also has a model mismatch term to preserve the prior

geological information. This minimization is a challenging optimization problem as

history matching is usually ill-posed and the number of unknown model parameters

can be very large. The ill-posedness of history matching can be mitigated by reducing


the number of parameters through an appropriate parameterization such as TSVD

[122, 126], PCA [114, 139], and kernel PCA [114, 141]. Vo and Durlofsky [140, 139]

presented a differentiable PCA-based parameterization that enables application of

efficient gradient-based approaches for history matching complex channelized models.

This optimization-based principal component analysis (O-PCA) approach is used in

the computational results presented in Appendix A of this thesis.

Various optimization methods have been applied for solving the minimization

problem in history matching, including gradient-based methods such as BFGS [124,

51, 50], Gauss-Newton [126], Levenberg-Marquardt algorithm [126, 122, 120], and

SNOPT [139, 140]. Shirangi [122] presented efficient and scalable procedures for

generating multiple realizations of permeability and porosity fields of large-scale three-

dimensional reservoir models. He also introduced the ensemble-based regularization

approach for the efficient computation of the model mismatch term and its derivative

in history matching. Shirangi and Emerick [126] introduced an efficient TSVD-based

Levenberg-Marquardt algorithm for history matching and showed this algorithm to be

more reliable than the Gauss-Newton approach for solving nonlinear inverse problems.

History matching can also be accomplished by use of a Kalman filter [52, 85, 82],

sparsity-based approaches [41, 54], or by ensemble-based data assimilation methods

[30, 97], which do not require the computation of gradients. Joint history matching of

production and seismic data (when available) can result in more uncertainty reduction

than can be achieved using only production data. Recent procedures that use 4D

seismic and production data include those presented by Suman et al. [129], Lee [84],

Echeverrıa Ciaurri et al. [37] and Bukshtynov et al. [24].

1.1.5 Closed-loop reservoir management

The optimal continuous operation of existing wells, often referred to as closed-loop

reservoir management (CLRM), has been the subject of significant research in recent

years [95, 113, 75]. CLRM, depicted in Fig. 1.1, entails optimizing well settings based

on current geological knowledge, operating the reservoir and collecting data over some

time period, and performing data assimilation (history matching) to update the geo-

logical description for consistency with observed data. This procedure, repeated over

the reservoir life, can provide improved performance relative to heuristic approaches


Update Models

CollectReservoir Data

OptimizeWell Settings

Set Well Controls & Operate

Figure 1.1: Schematic of closed-loop reservoir management.

for reservoir management.

Most papers on CLRM investigated the application of particular history match-

ing and optimization approaches for water-flooding operations. Jansen et al. [75]

applied an adjoint-based steepest descent algorithm for production optimization and

the ensemble Kalman filter for data assimilation. They investigated the impact of

optimization frequency on NPV improvement. Aitokhuehi and Durlofsky [2] applied

a nonlinear conjugate gradient approach for optimization and a probability perturba-

tion approach [25] for history matching. They investigated the effect of the number of

history-matched realizations used in optimization on final true-model NPV. A greater

NPV was obtained when multiple realizations were used in optimization than when a

single realization was used. Sarma et al. [112] applied PCA parameterization for his-

tory matching and used the SQP approach for production optimization. Chen et al.

[30] introduced an ensemble-based CLRM implementation in which ensemble-based

methods were applied for both history matching and production optimization.

Bukshtynov et al. [24] introduced a comprehensive adjoint-gradient-based frame-

work for CLRM where automatic differentiation was used in constructing the ad-

joint formulation. They investigated the use of SCP-MMA optimization in CLRM

and showed that this approach outperformed SNOPT for production optimization,

though SNOPT was still preferred for history matching. CLRM has also been applied


Update Models

Drill New Well

CollectReservoir Data

Optimize Field Development

Figure 1.2: Schematic of closed-loop field development (CLFD).

to SAGD operations [106, 119], and for the management of geological carbon storage

operations [26].

1.2 Scope of Work

As discussed in Section 1.1.5, closed-loop reservoir management has been investigated

extensively. The treatments used in CLRM, however, are not typically applied for

field development decisions. If computational optimization is even used for field

development, key decisions, such as the optimal number of wells, well types and

locations, are often determined a priori, by optimizing the expected objective (e.g.,

net present value, cumulative oil recovery) over a set of prior geological realizations.

In this work, we develop and apply a general methodology for optimal closed-loop

field development under uncertainty. This new CLFD framework, depicted in Fig. 1.2,

includes (1) solving an optimization problem for the well number, type, location and

controls based on current geological knowledge, (2) sequentially drilling new wells and

collecting data, and (3) performing history matching based on all currently available

data. This process is repeated until the optimal number of wells has been drilled. It

is important to emphasize that the full development plan is (re-)optimized at each

CLFD step; i.e., the location of the next well is determined based on the fact that it

1.2. SCOPE OF WORK 15

is one in a sequence of wells.

Various treatments are also considered for the history matching step in CLFD.

Specifically, procedures are presented for treating both two-point geostatistical (Gaus-

sian) models and channelized reservoir models described by multipoint geostatistics

(MPS). For each case, methods for the integration of hard data and production data

are described.

To accomplish field development optimization over a potentially large number of

geological realizations, we develop a new treatment, which we refer to as optimization

with sample validation (OSV). This approach shares some similarities with the retro-

spective optimization (RO) procedure introduced by Wang et al. [143] for optimizing

over multiple realizations. In RO, instead of optimizing the expected value of the

objective over the entire set of realizations, a sequence of optimization subproblems,

which contain increasing numbers of realizations, is solved. In OSV, the optimization

at each CLFD step is performed over a subset of realizations that are selected to be

representative. Following this optimization, a sample validation procedure is applied

to assess whether these realizations are indeed sufficiently representative of the entire

set. If the subset is found not to be representative, a larger number of realizations is

selected and the optimization step is repeated.

The CLFD framework is very general and various procedures for history matching,

optimization, model selection and economic evaluation can be applied. Following our

development and evaluation of the general CLFD methodology, we focus on two key

components within the overall framework. Specifically, we investigate in detail the

selection of representative models and the application of new economic measures for

reservoir operations. Our intent is to study these two topics as standalone subjects,

as opposed to applying them within the CLFD framework. Integration of these new

treatments into CLFD could be topics for future work.

The OSV procedure developed in the context of CLFD requires an approach to

select a representative subset of models from a large set. While various approaches

have been presented for this, as described earlier, we are not aware of any previous

studies that assessed the performance of different selection methods. In addition, it

is important to recognize that the appropriate selection method may be different in

different contexts. For example, a selected subset that is the most representative for


a well control optimization problem (with fixed well locations) may not be the best

choice for a well placement optimization problem, as these two problems are sensitive

to different geological details.

To address these issues, we devise a procedure to quantitatively assess different

selection approaches. We introduce a general clustering method for this purpose.

In this clustering, each realization is represented by a feature vector composed of a

weighted combination of flow-based and geological quantities. Principal component

analysis (PCA) is used to express the geology (permeability field) in terms of a small

number of features, while flow-based features are obtained by solving one or more

base-case flow problems. The use of both full-physics simulations and efficient tracer-

type simulations for obtaining these flow-based features will be considered. We also

investigate the performance of different feature weightings in optimization problems.

In previous work on the optimization of reservoir operations, the project life is

(almost always) specified a priori. The economic project life (EPL) for operation of

existing wells, however, depends on the specific problem and the way in which the

reservoir is operated. Therefore, the project life should be treated as a variable, and

the EPL and well controls should be determined through a joint optimization (that

includes an appropriate definition of rate of return). In this work, we introduce a

nested formulation for the joint optimization of EPL and well controls. In particular,

we show that the use of the modified internal rate of return (MIRR) and the minimum

attractive rate of return (MARR) enables us to jointly determine optimal EPL and

well controls. Our treatments are quite useful in avoiding situations where the NPV

continues to increase in time, but the cash flow is negligible compared to the capital

value of the project.

Consistent with the discussion above, the key research objectives of this work are

as follows:

• Develop a general framework for optimal closed-loop field development op-

timization under uncertainty. Efficient approaches for the optimization and

history matching steps will be implemented. CLFD will be applied to both

Gaussian and channelized reservoir models, which entail different treatments

for history matching.

• Introduce an approach for efficient optimization with model uncertainty (which

1.3. DISSERTATION OUTLINE 17

involves the use of multiple realizations). Toward this goal, optimization with

sample validation is developed and incorporated into the CLFD optimization

step.

• Devise a general methodology for selecting representative models from a large

set of realizations for decision making and optimization under uncertainty. This

includes the introduction of a statistical procedure for comparing various ap-

proaches for this selection.

• Develop a technique to jointly determine the optimal economic project life and

optimal well controls within the context of production optimization.

1.3 Dissertation Outline

In Chapter 2, we describe the CLFD methodology and present extensive numerical re-

sults using this framework. We first introduce the CLFD workflow, and then describe

the optimization and history matching steps. Next, the optimization with sample val-

idation approach is discussed. Computational results are then presented for two- and

three-dimensional Gaussian reservoir models. CLFD results for channelized models

are presented in Appendix A.

In Chapter 3, we investigate the problem of selecting a representative subset of

realizations from a large set. We first define a flow-response vector to assess the qual-

ity of different realization selection approaches. We then present a general method,

based on unsupervised learning techniques, for selecting a representative subset. Var-

ious selection procedures are then assessed for a range of problems, including cases

involving new well locations, new well controls, and production optimization. Differ-

ent algorithmic treatments are considered, as is the use of tracer-type simulations in

place of full-physics simulations. Additional results for binary channelized reservoir

models are presented in Appendix B.

In Chapter 4, we develop and apply a new approach, based on the computation

of modified internal rate of return, for the joint optimization of well controls and

economic project life. We first discuss the computation of modified internal rate of


return for water-flooding problems, and then present our procedure for the joint op-

timization. Examples are presented for two- and three-dimensional reservoir models.

Chapter 5 includes a summary, conclusions, and recommendations for future work

in closed-loop field development and related areas.

The new CLFD framework, along with the optimization with sample validation

treatment (described in Chapter 2), has already appeared in an SPE Journal arti-

cle [124]. Our general method for selecting representative realizations, described in

Chapter 3, has been published in Computers & Geosciences [123]. Our new method-

ology for optimizing EPL, described in Chapter 4, has been presented at the 2017

SPE Reservoir Simulation Conference [125].

Chapter 2

Closed-Loop Field Development

Optimization Under Uncertainty

In this chapter, we first describe the closed-loop field development (CLFD) optimiza-

tion framework, including the procedures used for optimization and history matching.

The selection of representative realizations to use for optimization, and optimization

with sample validation, are then discussed. We next present computational results for

two- and three-dimensional examples that demonstrate the application and potential

benefits of CLFD. We end this chapter with a brief summary.

2.1 CLFD workflow

We consider the field development optimization problem in which decisions such as

the number, types, drilling sequence, locations and time-varying controls of new wells

are to be determined. The term ‘decision variables’ is used to refer to the associated

optimization variables. The reservoir is initially described by a set of prior geological

realizations. New wells are to be drilled sequentially, one (or a few) at a time, as is

commonly the case in practice.

The CLFD optimization workflow entails solving an optimization problem to de-

termine the decision variables for new and existing wells (future well control variables

are determined for existing wells), ‘drilling’ a new well and collecting the associated

data, and performing history matching based on all available data. At each step of

19

20 CHAPTER 2. CLOSED-LOOP FIELD DEVELOPMENT

the procedure, the overall future development/operating plan is optimized; i.e., the

location for the next well is determined with the knowledge that it is one well within

a sequence of wells. Optimization is performed over multiple realizations to account

for uncertainty. While we use a particular set of methods in the optimization and

history matching steps of CLFD, the framework is general and different methods can

be employed.

Similar to closed-loop reservoir management (CLRM), the CLFD workflow in-

cludes optimization and history matching steps. The optimization step in CLFD is,

however, much more complex as it involves categorical (number/type and drilling

schedule for new wells), quasi-integer (well location), and continuous (well control)

variables. This is in contrast to CLRM, which entails only well control variables. The

CLFD history matching step is also slightly different from that in CLRM. This is be-

cause, in CLFD, the observed data include both production data and hard data (e.g.,

measured values of porosity or permeability) at well locations. In CLRM, hard data

are known a priori and are thus treated differently in the history matching procedure.

In all of our computational experiments, ‘observed’ production data are generated by

simulating a model selected randomly from the prior distribution. This model, which

also provides hard data at well locations, is referred to as the ‘true model.’

We now describe the CLFD framework for the case where one well is drilled at

a time. Extension to more general cases in which two or more wells are drilled at

a time is straightforward. Let t1, t2, . . . ti, . . . tn denote a discrete time series, where

ti indicates the time (in days) that production/injection for Well i is started. We

assume that the location and type (injector or producer) of Well i are determined at

time ti−1, and that Well i is drilled during the period (ti−1, ti), which corresponds to

control step i − 1. It is also possible that, depending on the problem definition, the

optimal solution may be not to drill Well i.

Our approach for CLFD is illustrated in Fig. 2.1. The notation used in the figure

and throughout this chapter is as follows. The vector x defines the decision variables

for all wells. The maximum number of wells is denoted by nw. There can be fewer

than nw wells since in the optimization some wells may be determined to be of type ‘do

not drill’ (rather than injector or producer). A current (time ti) geological realization

is denoted by mij, with j = 1, . . . , NR, where NR is the number of realizations. For

2.1. CLFD WORKFLOW 21

the case where the geological model is fully described by a single value in each grid

block, mij will be of dimension Nb, where Nb is the number of grid blocks in the model.

Each of the current realizations is inserted (as a column) into the matrix M i, which

is thus of dimensions Nb × NR. The optimization problem entails maximization of

expected net present value (NPV) of the field development project. Expected NPV

is denoted by J .

CLFD proceeds as illustrated in Fig. 2.1. The set of initial (prior) realizations is

designated M1. Using these models, and x0 as the initial guess for the field develop-

ment decision variables (which could be user provided or generated randomly), the

optimization problem is solved to provide x1. The PSO-MADS hybrid algorithm is

used for this optimization. The optimal solution x1 defines the location and type of

Wells 1 and 2. Production from Well 1 starts at t1 (using the optimal setting) and

Well 2 is drilled during the period (t1, t2).

The first CLFD history matching is performed at t2 (Fig. 2.1). In this compu-

tation, we use production data from Well 1 and hard data from Well 1 and (newly

drilled) Well 2 to determine a new set of conditioned realizations M2. The next

optimization is performed using realizations M2, with x1 as the initial guess. This

computation provides the new optimal solution x2. As prescribed by x2, Well 3 is

then drilled, and the controls of Wells 1 and 2 are set for the next control step. This

procedure is repeated until a maximum of nw wells has been drilled. It is important

to emphasize that, at each CLFD step, the full field development plan is updated; i.e.,

the locations, types and controls for all future wells are determined. Thus, when each

new well is drilled, the fact that additional (planned) wells will be drilled is taken

into account. The locations and types of the planned wells are, however, updated in

subsequent CLFD steps.

In the history matching performed at time ti, the current realizations (conditioned

to all data up to time ti−1) are used as the initial guesses. The total history matching

period is from t1 to ti. After performing this history matching step, the simulation

state variables (phase pressures and saturations) are saved to restart files. During

optimization, all reservoir simulations start at time ti and proceed until time T (the

end of the reservoir life). This avoids repeating the simulation for the time period

before ti and thus enhances overall computational efficiency.


History Matching History Matching

Optimization Optimization

Productionfrom Well 1

Production / Injectionfrom Wells 1 & 2

Drilling Well 2 Drilling Well 3

Figure 2.1: Schematic and notation for the closed-loop field development optimizationprocedure.

In the computational results, the ‘true’ (synthetic) model, designated mtrue, is

known. Therefore, we can evaluate the ‘true NPV’ for each xi (Fig. 2.1) by performing

flow simulation with mtrue. We will use the evolution of the true NPV, designated

J(xi,mtrue), to evaluate the performance of CLFD.

In the next two sections, we describe the optimization and history matching CLFD

components in more detail.

2.2 CLFD optimization

In the optimization step of CLFD, the expected NPV computed for a set of (current)

geological realizations is maximized. For a current (time ti) realization of the reservoir

model mij (where j is the realization index), the NPV for the field development

2.2. CLFD OPTIMIZATION 23

optimization problem is computed as

J(x,mij) =

Nl∑l=1

[NP∑k=1

(poqlo,k − cwpq

lw,k)−

NI∑k=1

cwiqlwi,k

]∆tl

(1 + rd)tl/365

−NP+NI∑i=1

cwell(1 + rd)ti/365

, (2.1)

where x is the vector of decision parameters, Nl is the number of simulation time

steps, NP and NI denote the number of producers and injectors, respectively, and po,

cwp and cwi indicate the oil price and the cost of produced and injected water (all in

$/STB). Variables qlo,k and qlw,k denote oil and water production rates for producer k

at the simulation time step l, while qlwi,k denotes the water injection rate of injector

k (all in STBD), rd is the annual discount rate, cwell is the cost of drilling a well, and

∆tl is the size of time step l. The expected NPV, J , is defined as

J(x,M irep) =

1

nr

nr∑j=1

J(x,mij), (2.2)

where M irep is a matrix containing nr representative realizations of the current reser-

voir model (selection of the representative realizations will be discussed later), i.e.,

M irep = [mi

1 mi2 . . . mi

nr]. (2.3)

The robust field development optimization problem is defined as

maximize J(x,M irep)

subject to ck(x,mij) ≤ 0, k = 1, · · · , nic,

xl ≤ x ≤ xu,

(2.4)

where nic denotes the number of inequality constraints, ck denotes an inequality

constraint, and xl and xu denote the vectors of lower and upper bounds on the

decision variables. Nonlinear constraints considered in this work include well distance

constraints and limits on maximum rates for each well. The well distance constraint

is handled using the filter method [70], and the rate constraints are handled within


the forward simulation. The optimal solution computed at ti is denoted by xi, and

the corresponding optimal value is J(xi,M irep). The initial guess for optimization

at time ti is the optimal solution at ti−1 (i.e., xi−1). During optimization at ti, all

parameters corresponding to decisions prior to ti are held as constant.

All optimizations in this work are performed using PSO-MADS. See Isebor et al.

[70, 71] for full details on the procedure and for a number of examples that demon-

strate algorithmic performance. For a concise description of the individual PSO and

MADS algorithms, and the PSO-MADS hybrid, see Aliyev and Durlofsky [3]. PSO is a

stochastic population-based cooperative-search algorithm that provides some amount

of global exploration but is not guaranteed to converge to a (local or global) min-

imum. Each PSO ‘particle’ represents a potential solution (e.g., well configuration

and controls), which moves through the search space based on a set of ‘velocity’ com-

ponents. MADS, by contrast, is a pattern search method that provides convergence,

in many cases, to a local minimum. By combining these two procedures, we achieve

global search along with local convergence (it is important to note, however, that the

PSO-MADS algorithm does not in general find the global optimum). Examples in

Isebor et al. [70] highlight the advantages of the PSO-MADS hybrid over standalone

PSO and MADS.

The method requires a large number of function evaluations, each of which entails

nr flow simulations. Because both PSO and MADS parallelize naturally, elapsed

times, which in this work correspond to the time required to perform hundreds to

a few thousand simulations, are not overly excessive assuming simulation run times

are on the order of minutes. PSO-MADS termination criteria can be based on a

minimum improvement in the objective function, a minimum stencil size in MADS, or

a maximum number of function evaluations. At early steps of CLFD, the algorithm

typically terminates when a maximum number of function evaluations is reached,

while at later steps, termination is usually due to one of the other criteria.

2.3. OPTIMIZATION WITH SAMPLE VALIDATION 25

2.3 Optimization with sample validation and de-

termination of representative realizations

The number of simulations that must be performed in CLFD is directly proportional

to the number of realizations considered (as is evident from Eq. 2.2). This means it

is beneficial to use as few realizations as possible. It is also important to note that

the CLFD optimization usually requires many more flow simulations than the history

matching component. Thus, it is relatively inexpensive to generate ‘excess’ history

matched models.

These observations motivate the optimization with sample validation (OSV) pro-

cedure used in this work. We first describe the sample validation method and then the

overall procedure. Assume that a large number (NR) of realizations have been gener-

ated in the most recent history matching step. We wish to perform optimization using

a subset (nr) of ‘representative’ realizations. We will discuss later how representative

realizations are selected. With Mrep denoting a set of nr representative realizations

and xi−1 indicating the initial guess for the current optimization, the PSO-MADS

algorithm provides an optimum set of decision variables xi by optimizing over the

Mrep realizations. The increase in J achieved in this optimization step is given by

J(xi,Mrep)− J(xi−1,Mrep).

Our interest, however, is in maximizing the expected NPV over the entire cur-

rent set of NR realizations, denoted by M . We assess this quantity by computing

J(xi,M) − J(xi−1,M) after the optimization over Mrep has been completed. The

validation procedure requires that the ratio of the increase in J for NR realizations

relative to the increase in J for nr realizations exceeds a threshold parameter θ. We

call this ratio the relative improvement RI and express the validation criterion as:

RI =J(xi,M)− J(xi−1,M)

J(xi,Mrep)− J(xi−1,Mrep)≥ θ. (2.5)

In this work we use θ = 0.5, though of course other values could be used if we wish

to be more or less stringent. If Eq. 2.5 is not satisfied, we choose a larger number

of representative realizations (by increasing nr) and repeat the optimization (i.e.,

proceed to the next subproblem) to provide a new xi. This process terminates when


RI ≥ θ, or after a maximum number of optimizations (ns) have been performed.

This approach assures that, in general, most (half or more) of the benefit obtained

by optimizing over a subset of models is achieved for the full set of NR models.

The number of realizations for each subproblem, Nk, k = 1, . . . ns, is specified

by the user. The OSV procedure, however, requires a method for choosing the Nk

representative realizations out of the total set of NR realizations. Our approach for

determining these realizations is discussed next. The OSV procedure is detailed in

Algorithm 2.1. We note that other validation criteria, such as out-of-sample validation

and cross-validation, which are used for model selection in statistical learning [59] and

stochastic programming [111], could also be tested within the CLFD framework.

Our specific selection procedure is as follows. At step k of OSV (recall that all OSV

steps are at a single CLFD step), the NPV for each of the NR realizations is computed

based on xk−1. The NPVs are then scaled from 0 and 100, and Nk realizations are

selected. This is performed such that two of the realizations correspond to scaled

NPVs of 10 and 90, and the remaining Nk − 2 realizations correspond to equally

spaced NPVs between 10 and 90. For each value, we select the realization with

scaled NPV closest to the desired value. This selection procedure is referred to as the

“objective-function-variation approach.” The issue of model selection is considered

in detail in Chapter 3, and any of the methods described there could be applied

within OSV. Regardless of the method used, it is essential that the representative

realizations be “reselected” at each CLFD step.

2.4 CLFD history matching for two-point geosta-

tistical models

The history matching procedure in CLFD entails the minimization of an objective

function that quantifies the mismatch between observed and simulated data. This

objective function includes a regularization term to account for prior geological knowl-

edge regarding the spatial distribution of rock properties. The problem can be for-

mulated within a Bayesian framework, in which case the history matching problem

entails maximizing the posterior pdf of the model m given observed data dobs. This

2.4. CLFD HISTORY MATCHING 27

Algorithm 2.1 Optimization with sample validation

Specify initial guess x0 (x0 = xi−1)for k = 1, 2, . . . , ns do

Select a set of Nk representative realizations, Mkrep = [m1 m2 . . . mNk

], usingthe objective-function-variation approachObtain xk as argmaxxJ(x,Mk

rep) using xk−1 as the initial guessEvaluate RI in Eq. 2.5 (by replacing Mrep by Mk

rep and xi by xk)if the sample validation criterion is satisfied (RI ≥ θ), then

set xi = xk and terminate the loopend ifif k = ns then

set xi = argmaxxk,k=1,...ns(J(xk,M))

end ifend for

can be expressed as [132]

f(m|dobs) = a exp(−S(m)), (2.6)

where a is the normalizing constant and S(m), referred to as the total objective

function, is given by

S(m) =1

2(m−mprior)

TC−1m (m−mprior) +

1

2(gp(m)− dpobs)

TC−1d,p(gp(m)− dpobs)

+1

2(gh(m)− dhobs)

TC−1d,h(gh(m)− dhobs)

= Sm(m) + Spd(m) + Shd (m).

(2.7)

Here Sm is the model mismatch term, Spd is the production data mismatch term, Shd is

the hard data mismatch term, mprior is the mean of the prior model, Cm denotes the

prior covariance matrix, dpobs and dhobs, of dimensions Npd and Nh

d indicate the vectors

of observed production data and hard data, gp(m) and gh(m) denote the vectors

of predicted production data and hard data, Cd,p indicates the Npd×N

pd (diagonal)

covariance matrix for the measurement error in production data, and Cd,h is the

Nhd×Nh

d (diagonal) covariance matrix for the measurement error in hard data.

The maximum a posteriori (MAP) estimate is the model that maximizes Eq. 2.6,


or equivalently minimizes Eq. 2.7. The MAP estimate represents the mode of the

posterior pdf, though in many cases it is a smooth model that does not reflect the

heterogeneity in the underlying geology. As our goal is to characterize the uncer-

tainty in the geological description, multiple geological realizations are generated by

sampling the posterior pdf. To achieve this goal, we use the randomized maximum

likelihood (RML) method [98], which is an approximate sampling procedure. To gen-

erate a realization with RML, an unconditional realization, muc, is generated from

the prior pdf, N(mprior, Cm), along with perturbed observation vectors, dpuc and dhuc,

which are generated from N(dpobs, Cd,p) and N(dhobs, Cd,h). Then, Eq. 2.7 is modified

by replacing mprior by muc, dpobs by dpuc, and dhobs by dhuc. Generating NR realizations

using RML involves minimizing NR objective functions. The minimization of S(m)

is usually accomplished using a gradient-based optimization algorithm with gradients

provided by an adjoint procedure [112, 122, 126, 120].

In this work, we use the LBFGS method with a damping procedure for the mini-

mization problem [51]. Damping is used because large changes in model parameters

at early iterations of the optimization adds roughness to the model that is difficult

to remove at later iterations. This can result in convergence to a model that does

not provide acceptable agreement in production data [126]. In order to prevent large

changes in model parameters at early iterations, a damped objective function is in-

stead minimized, i.e., Eq. 2.7 is modified to

S(m) = Sm(m) + γ(Spd(m) + Shd (m)), (2.8)

where γ is a damping factor calculated as described in [51] (γ < 1 at early iterations

and γ = 1 at later iterations). After calculating γ, the damped objective function

(with this value of γ) is minimized until convergence of the LBFGS algorithm or

until a maximum number of iterations (specified to be 15 in this work) is performed.

At this point, the damping factor γ is recalculated and the new objective function is

minimized until convergence. The convergence criteria for history matching are based

on the requirements that both the relative change in the model and the relative change

in the objective function become smaller than a prescribed value.

In our computational results, a history matched RML realization is ‘accepted’ if

its normalized total objective function, SN = S(m)/Nd, where Nd = Npd +Nh

d , is less

2.5. COMPUTATIONAL RESULTS 29

than a specified value (here we require SN ≤ 5). Otherwise, the realization is not

considered to be a sample of the posterior pdf. Relatively few (e.g., < 5%) of the

generated realizations are discarded based on this requirement.

Because the RML runs are independent of one another, each RML realization

can be generated on a separate compute node using distributed computing. We

use Stanford’s Automatic Differentiation-based General Purpose Research Simulator,

AD-GPRS [157] for our computational experiments. The existing OpenMP-based

parallelized version of AD-GPRS [156] allows us to run each simulation on a compu-

tational node with 16 cores. This gives an average speedup of about a factor of 10 for

each simulation. The gradient for history matching is generated using the automatic

differentiation framework [83, 24]. For the CLFD results presented in this work, at

each history matching step, we generate NR = 50 posterior RML realizations plus

the MAP estimate. These models are all generated simultaneously using 51 compute

nodes.

2.5 Computational results

We now apply the CLFD procedure to three example cases, all involving oil-water

systems. In Example 2.1, simultaneous and sequential (well-by-well) optimization of

a field development scenario are compared for a deterministic reservoir description.

In Examples 2 and 3, the CLFD optimization framework is applied to two- and three-

dimensional reservoir models. The effectiveness of the sample validation procedure

is illustrated in these examples. An example involving a channelized reservoir is

presented in Appendix A.

2.5.1 Example 2.1: Simultaneous versus sequential optimiza-

tion of field development

In CLFD optimization, the goal is to determine near-term decision variables in light

of the overall field development plan (i.e., with recognition that the next well is one

well in a sequence). Optimization of immediate decisions without considering future

decisions is essentially a ‘greedy’ approach, which will in general result in a suboptimal


X

Y

10 20 30 40 50 60

10

20

30

40

50

60

0

2

4

6

8

Figure 2.2: True log-permeability field for Examples 2.1 and 2.2. Permeability is inmD.

solution. In this example, we compare results for well-by-well and simultaneous field

development optimization for a problem with a deterministic reservoir description

(meaning the history matching component in CLFD is not applied).

The two-dimensional (x − y) reservoir model is defined on a 60 × 60 uniform

grid. The log-permeability field (all permeabilities in this work are in mD) is shown

in Fig. 2.2. In this and subsequent examples in this chapter, porosity is constant

and equal to 0.2 in all grid blocks. The log-permeability field is characterized by

a spherical variogram, with a maximum range of 35 grid blocks in the northwest-

southeast direction and a minimum range of 15 blocks in the northeast-southwest

direction. The mean and standard deviation of log-permeability are 4.6 and 1.5,

respectively. The grid block dimensions are ∆x = ∆y = 100 ft, ∆z = 15 ft. The

initial reservoir pressure is 4500 psi. Initially the reservoir contains oil and connate

water (irreducible water saturation, Swc, is 0.18). The oil and water viscosities are

specified as 3 cp and 1 cp, respectively. The formation volume factors for both oil and

water are set to 1, and rock compressibility is specified as 10−3 bar−1. The relative

permeability curves used for this example are shown in Fig. 2.3.

The objective is to optimize the location, type, time-varying controls and drilling

sequence of four wells, with each new well being drilled every 210 days. Well type


0.1 0.3 0.5 0.7 0.90

0.2

0.4

0.6

0.8

1

Sw

k r

krw

kro

Figure 2.3: Oil and water relative permeability curves for Example 2.1.

Table 2.1: Optimization parameters for all examplesParameter Value

cwell $25 MMpo $90 STBcwp $10 STBcwi $10 STBProd. BHP range 1000− 4100 psiInj. BHP range 4600− 7000 psi

(injector or producer) is defined by a binary categorical variable, as described in [71].

The total reservoir life is 3000 days. The optimization consists of four control steps,

each of length 210 days, followed by a final control step of length 2160 days. Wells

are operated using BHP control with a maximum oil rate constraint of 20,000 STBD

and a maximum water injection rate constraint of 10,000 STBD. Table 2.1 shows the

optimization parameters for this example. The discount rate is specified to be zero.

The simultaneous optimization problem contains four categorical variables, eight

quasi-integer variables, and 14 continuous control variables, for a total of 26 optimiza-

tion variables. The number of control variables is not simply the maximum number of

wells times the number of control steps, because not all wells exist at all control steps.

Specifically, in the first control step only one well appears in the model, so only one


Table 2.2: Final NPVs ($ MM) for three runs for sequential (well-by-well) and simul-taneous optimization (Example 2.1)

Case Well-by-well Simultaneous

Best 655 716Intermediate 625 713Worst 567 709Average 616 713

control is computed. Similarly, two and three wells, respectively, exist in the second

and third control steps. In the fourth and fifth control steps, all four wells appear in

the model. Hence, there are a total of 1 + 2 + 3 + 4 + 4 = 14 control variables. For

this case, we use 20 particles in PSO.

For the well-by-well optimization, we first optimize the location and type of Well 1.

Then, given the optimal location and type of Well 1, the location and type of Well 2

(drilled at 210 days) are optimized. This procedure is continued to determine the

location and type for all four wells. For this optimization, the number of optimization

variables varies from eight (for Well 1) to 12 (for Well 3). We use 12 PSO particles for

the well-by-well optimizations. In each optimization, future controls for all existing

wells are also optimized.

The optimizations were run three times, using different initial guesses, for each

approach. For all three runs, the simultaneous optimization converged to a solution

with three producers and one injector, while the well-by-well optimization provided a

solution with two producers and two injectors. The optimal well controls for all cases

correspond to fully open wells (i.e., BHPs at the bounds). Optimal NPVs for all runs

are shown in Table 2.2. The average NPV from simultaneous optimization is about

16% higher than that from the well-by-well approach. The final oil saturation maps,

which also show the optimal well locations, type and sequence, are shown in Fig. 2.4.

These results correspond to the best solutions in Table 2.2. It is evident that the

simultaneous optimization provides better sweep than the well-by-well solution.

The results of this example demonstrate that the simultaneous optimization of

all wells leads to a better solution than that achieved using a sequential approach in

which each well is optimized independently. This is as would be expected, assuming


2

3

1

4

20 40 60

10

20

30

40

50

600.2

0.3

0.4

0.5

0.6

0.7

0.8

(a) Well-by-well optimization

2

1

3

4

20 40 60

10

20

30

40

50

600.2

0.3

0.4

0.5

0.6

0.7

0.8

(b) Simultaneous optimization

Figure 2.4: Final oil saturation (at 3000 days) from optimal solutions for the twoapproaches (Example 2.1). Well locations are also shown, with red denoting producer,blue denoting injector, and the well numbers indicating the drilling sequence.

the optimization method provides a sufficiently complete search. More specifically, if

the greedy (well-by-well) solution was indeed optimal, the simultaneous optimization

procedure could ‘find’ it, but the opposite is not generally true. This justifies the use

of simultaneous optimization within the CLFD framework.

2.5.2 Example 2.2: CLFD for a two-dimensional reservoir

model

This example involves a two-dimensional horizontal reservoir represented on a 60×60

uniform grid. The log-permeability field for the true model (Fig. 2.2) and optimization

parameters (Table 2.1) are the same as in Example 2.1. However, some simulation

quantities, such as the relative permeability curves (shown in Fig. 2.5), are different

between the two examples. Initially, the reservoir contains oil and connate water

(Swc = 0.1). The permeability distribution is uncertain, while fluid properties and

variogram parameters are assumed to be known.

The reservoir in this example is developed with a maximum of nw = 8 wells,

where drilling each well takes 210 days. Well type is described by a ternary categor-

ical variable (-1 for injector, 1 for producer, 0 for do not drill), as discussed in [71].


0 0.5 10

0.2

0.4

0.6

0.8

1

Sw

k r

krw

kro

Figure 2.5: Oil and water relative permeability curves for Examples 2 and 3.

The optimization consists of eight control steps, each of length 210 days, followed by

a final control step of duration 1320 days. The times, ti, i = 1, 2 . . . 8, that Well i

begins producing or injecting are given by {0, 210, 420 . . . 1470}. During optimiza-

tion, the wells are operated on BHP control, with a maximum oil rate constraint of

25,000 STBD and maximum water injection rate constraint of 12,500 STBD. A pro-

ducer is shut in if water cut exceeds the economic limit, i.e., when the cost of handling

produced water from the well exceeds the revenue from oil production. This strategy

is used when evaluating each trial point for each realization during optimization. The

objective is again to maximize NPV.

Decision parameters (x) consist of eight categorical variables (for well types),

16 quasi-integer variables (for well locations) and 40 continuous variables (for well

controls). In this example, we consider four cases, corresponding to using different

numbers of realizations (nr) in the optimizations. In the first three cases, we specify

a fixed value for nr, while in the fourth case we apply the OSV procedure. The final

CLFD solution for these cases will be compared with the solution from deterministic

optimization (in which the reservoir geology is known), and with optimization over

prior realizations. Because the number of decision parameters is smaller for later

optimization steps in CLFD, we vary the number of PSO particles used in these runs.

Specifically, the number of particles is taken to be the minimum of 60 and the number


of decision parameters.

For the history matching step at time ti, the observed data include production

quantities measured at 30-day intervals from the first i− 1 wells, and hard data from

all existing wells including the most recent well (Well i). Synthetic observed data

are generated by adding Gaussian random noise to the true data, where the true

data are generated by running the simulator with the true model. Production data

include injection rates for existing injectors, and water and oil production rates from

existing producers, from time zero to ti. When the true data are generated, some of

the wells switch to rate control due to the constraints. In this case, the observed data

correspond to BHPs if the well is an injector, and BHPs and phase rates if the well is

a producer. The standard deviation of measurement error is 3% of rates for rate data

and 3 psi for BHP data. The minimum and maximum measurement errors for rates

are specified to be 3 STBD and 30 STBD. The standard deviation of measurement

error for the observed hard data (log-permeability of well blocks) is 0.2. At each

history matching step, 50 RML realizations (i.e., NR = 50) and the MAP estimate

are generated.

We first consider the optimal solution when the true model is known. These

optimization results were generated using 60 PSO particles. The optimal solution

over the true model (mtrue) entails five wells (out of a maximum of eight), with the

types, locations and drilling sequence indicated in Fig. 2.6. The final oil saturation

is also shown in the figure. The optimal NPV in this case is $730 MM.

Case 1: nr = 3

We first apply the CLFD optimization using only three realizations for all of the

optimizations. The initial field development plan is determined by optimization over a

set of prior geological realizations. A set of NR = 50 prior realizations are considered,

from which nr = 3 representative realizations are selected with the objective-function-

variation approach. This procedure is illustrated in Fig. 2.7. For computing the NPV

values required in the objective-function-variation approach, all realizations are run

using the initial guess (x0), which corresponds to a line drive configuration with four

producers and four injectors. The three representative realizations thus selected, along


2

4

5

1

3

20 40 60

10

20

30

40

50

60 0.2

0.3

0.4

0.5

0.6

0.7

0.8

Figure 2.6: Well configuration from deterministic optimization (using mtrue), withred denoting producer, blue denoting injector, and the well numbers indicating thedrilling sequence. Background shows final oil saturation (Example 2.2).

with the well configuration, are shown in Fig. 2.8. Using these three realizations, PSO-

MADS is applied to optimize the expected NPV. After optimization, the expected

NPV improves from an initial value of $297 MM to $576 MM, as shown in Fig. 2.9.

The total number of simulations here is 30,000. The number of equivalent simulation

runs which corresponds to the number of times a batch of simulation runs is submitted

to the compute cluster (with a maximum of 400 available cores for optimization runs),

is 288. The optimal solution based on the three prior models, designated x1, entails

four producers and two injectors, which are depicted in Fig. 2.10(a).

After obtaining the solution from optimization over three prior realizations, Well 1

(which is a producer) is drilled at its optimal location. At time zero (t1), Well 1 starts

producing, and the drilling of Well 2 (at its optimal location) also commences. At

210 days, the drilling of Well 2 is completed, and hard data for this well become

available. The first history matching is then performed to update the NR = 50 RML

realizations and the MAP estimate. Measured data at this point are production data

from Well 1 and hard data from Wells 1 and 2.


0 10 20 30 40 50

1

2

3

4

5x 10

8

$ N

PV

Ranked Realization Index

0 10 20 30 40 500

50

100

All RealizationsSelected Realizations

Figure 2.7: Ordered NPV plot for NR = 50 prior realizations based on initial guess fordecision variables x0. The three selected realizations are shown in red (Example 2.2,nr = 3).

X

Y

1

2

3

4

5

6

7

8

20 40 60

10

20

30

40

50

600

2

4

6

8

(a) Realization 1

X

Y

1

2

3

4

5

6

7

8

20 40 60

10

20

30

40

50

600

2

4

6

8

(b) Realization 2

X

Y

1

2

3

4

5

6

7

8

20 40 60

10

20

30

40

50

600

2

4

6

8

(c) Realization 3

Figure 2.8: Three representative prior realizations of log-permeability, along with theinitial-guess well configuration (Example 2.2). Red (outlined) circles denote produc-ers, blue injectors, and the well numbers indicate the drilling sequence.


0 0.5 1 1.5 2 2.5 3x 10

4

2

3

4

5

6x 108

Number of simulations

E[N

PV

] ($)

Initital E[NPV]

Optimal E[NPV]

Figure 2.9: Evolution of expected NPV (J(x,M1rep)) for optimization over nr = 3

prior realizations (Example 2.2).

Following the history matching step, any realization that does not provide a suffi-

ciently close match with the observed data is discarded (this typically involves few if

any realizations). The objective-function-variation approach is again used to choose

nr = 3 (new) representative realizations, and the next optimization is performed

over these realizations. The optimal solution (optimal well configuration and drilling

sequence) from the optimizations at t1 and t2, designated x1 and x2, are shown in

Fig. 2.10 (the background corresponds to log-permeability for one of the realizations).

In this and subsequent figures, wells depicted with solid colors (red or blue) have al-

ready been drilled or are in the process of being drilled, while wells that are outlined

in red or blue are planned and may change in later CLFD steps. It is evident that,

as we proceed from x1 to x2, an additional well has appeared, and Well 3 is in the

process of being drilled. The effect of history matching can also be seen by comparing

the areas around Wells 1 and 2 in Fig. 2.10(b) with those in Fig. 2.10(a).

The procedure described above is continued until the last optimization step (at

time t7), where the location and type of Well 8 is determined. The evolution of the

optimal expected NPV, J(xi,M irep), and the value of expected NPV for the initial

guess J(xi−1,M irep) at each CLFD step, are shown in Fig. 2.11. The initial and op-

timal expected NPV at t1 = 0 correspond to those indicated in Fig. 2.9. As noted


X

Y

1

23

4

5

6

20 40 60

10

20

30

40

50

600

2

4

6

8

(a) Optimal configuration at t1

X

Y

1

2

3

4

5

6

7

20 40 60

10

20

30

40

50

600

2

4

6

8

(b) Optimal configuration at t2

Figure 2.10: Optimal well configuration and drilling sequence at t1 and t2. Solid redand blue circles denote producers and injectors (drilled or in the process of beingdrilled), and outlined red and blue circles denote planned producers and injectors.Numbers indicate the drilling sequence and background shows log-permeability forone realization (Example 2.2, nr = 3).

in Fig. 2.11, each optimization (at ti) is performed over a (different) set of represen-

tative realizations based on current geological knowledge (M i). These realizations

are conditioned to all data available up to time ti. Fig. 2.11 demonstrates that, at

each ti, the PSO-MADS optimization improves the NPV, often significantly, over the

NPV corresponding to the initial guess. The initial guess at each ti corresponds to

the optimal solution from the previous step (xi−1), and it is apparent that there are

large jumps in NPV as we proceed from, e.g., M1 to M2. This occurs because the

geological models have changed substantially as a result of the history matching step.

At later CLFD steps the increases in NPV due to optimization are smaller, in part

because the geological models change less from step to step.

The results considered thus far demonstrate that our CLFD procedure is very ef-

fective in terms of increasing expected NPV. However, in actual practice, the quantity

of interest is not expected NPV but rather the NPV for the true reservoir. We now

consider how this quantity is improved through use of CLFD. Fig. 2.12 shows the

optimal expected NPV and the corresponding true NPV, J(xi,mtrue), versus CLFD

step (with nr = 3). True NPV is computed by evaluating NPV for the true model


0 210 420 630 840 1050 12602

3

4

5

6x 108

Time (Days)

NP

V (

$)

M1 M2 M3 M4 M5 M6 M7

J(xi−1,M i

rep)

J(xi,M i

rep)

Figure 2.11: Optimal expected NPV, J(xi,M irep), and the expected NPV for the

corresponding initial guess, J(xi−1,M irep), versus CLFD step (Example 2.2, nr = 3).

0 210 420 630 840 1050 12601

2

3

4

5

6

7

x 108

Time (Days)

NP

V (

$)

J(xi,M i

rep)

J(xi,mtrue)

DeterministicFinal NPV$ 398 MM

Figure 2.12: Optimal expected NPV, J(xi,M irep), and the corresponding NPV for the

true model, J(xi,mtrue), versus CLFD step. The star shows the final true NPV fromCLFD (Example 2.2, nr = 3).


(mtrue) and well scenario xi. This corresponds to the NPV that would be obtained

at that point if the field development was performed based on xi. As can be seen

in Fig. 2.12, at initial time, the expected NPV for x1 is $576 MM, while the corre-

sponding NPV for the true model is $163 MM, which is significantly lower. After

the second optimization (at t2), the expected NPV is $510 MM, which is much closer

to the corresponding true NPV of $376 MM. Of most importance is the final true

NPV achieved by performing CLFD optimization. This NPV is $398 MM, which is

a factor of 2.4 greater than what would be achieved by simply optimizing over three

‘representative’ prior realizations. The result from the deterministic optimization

(optimal NPV of $730 MM) is shown for comparative purposes as the horizontal line

in Fig. 2.12 (and in subsequent figures).

Additional observations can be made from the results in Fig. 2.12. The difference

between the optimal expected NPV, J(xi,M irep), and the corresponding true NPV,

J(xi,mtrue), is the greatest for optimization at t1 = 0 (over prior realizations) and

displays a generally decreasing trend in time. At steps later than t4 = 630 days,

the difference between J(xi,M irep) and J(xi,mtrue) stays quite small. Although in

general we do not expect a strict monotonic reduction in |J(xi,M irep)− J(xi,mtrue)|

with CLFD step, this general behavior would be expected because the reservoir model

becomes less uncertain (since more data are available) and because fewer decision vari-

ables enter the optimizations, which means that the well scenarios are not changing

drastically from step to step.

The large difference between J(xi,M irep) and J(xi,mtrue) at initial time suggests

that the three realizations used to characterize the system differ significantly (in terms

of flow) from mtrue. To illustrate this, in Fig. 2.13 we present the NPVs corresponding

to each of the nr = 3 models used at each CLFD step. We see that the true NPV does

not fall within the spread of NPVs for the representative realizations until 630 days.

This indicates that the three realizations do not adequately capture the uncertainty

in the underlying geology. This motivates the use of larger values for nr, as well as

the OSV procedure. We now present results using these approaches.


0 210 420 630 840 1050 12601

2

3

4

5

6

7

x 108

Time (days)

NP

V (

$)

J(xi,mtrue)

J(xi, M i

rep)

Figure 2.13: NPV for the nr = 3 (representative) realizations, and the correspondingNPV for the true model, versus CLFD step (Example 2.2).

Cases 2 and 3: nr = 5 and nr = 10

We now consider CLFD results using five and ten representative realizations. Results

for optimal expected NPV and the corresponding true NPV are shown in Fig. 2.14.

The results for nr = 5 are quite similar to those for nr = 3 (Fig. 2.12) and, consistent

with the discussion above, suggest that the five realizations are not sufficiently repre-

sentative of the true model. We see, however, that the final CLFD NPV is higher for

nr = 5 than for nr = 3 ($454 MM versus $398 MM). Results for nr = 10, however, are

quite different. These results display an even higher final NPV ($599 MM), and there

is close correspondence between results for optimal expected NPV and true NPV. In

addition, optimization over the prior models (at initial time) in this case provides an

NPV that is much higher than in the case of nr = 5 or nr = 3.

The results in Fig. 2.14 can be further understood by considering the NPVs for

the representative realizations at each CLFD step, which are presented in Fig. 2.15.

These results confirm our expectations; namely, that the use of nr = 10 leads to a

set of realizations that adequately ‘capture’ the response of the true model. This

is evident from the fact that the true model response falls within the spread of the

representative prior realizations. There is of course no reason the true model should

fall near the middle of the representative realizations (which it does in this case) —


the key point is that the true realization does not appear as an outlier (as it does for

nr = 3 and nr = 5). We note finally that, even though the nr = 10 prior realizations

are representative of the true model, the use of CLFD still leads to an 18% increase

in NPV compared to that achieved by simply optimizing over the prior realizations.

The reduction in uncertainty using CLFD, as well as the ‘representivity’ of the

selected realizations, can be further assessed by evaluating the NPVs for the entire set

of (50) current realizations with the current optimal solution xi. Fig. 2.16 presents

the P10–P50–P90 results for NPV, determined by simulating all 50 realizations and

then constructing the cdf, at each control step for nr = 5 and nr = 10. The expected

NPV based on the current representative set is also displayed. It is evident that for

nr = 5, the optimal expected NPV for the representative set falls outside of the P10–

P90 range at the first three CLFD steps (Fig. 2.16(a)). For nr = 10, by contrast,

the optimal expected NPV for the representative set always falls within the P10–P90

range. The generally decreasing trend in the P10–P90 range with increasing CLFD

step, apparent for both nr = 5 and nr = 10, highlights the reduction in uncertainty

achieved through application of CLFD.

In order to better quantify the performance of CLFD, we performed two additional

runs, with different mtrue, using both nr = 5 and nr = 10. For the mtrue considered

above, and for these two new cases, we also optimized over all 50 prior models and

then applied the optimal well configuration and controls to the true model. This

computation essentially represents the ‘most robust’ optimization we can perform

using only (the 50) prior models. Results for all of these computations are presented

in Table 2.3 (the first row corresponds to the true model considered above). From

the table, we see that the use of CLFD in all cases (even using nr = 5) provides

higher NPVs than does optimizing over 50 prior realizations. In addition, CLFD

with nr = 10 consistently outperforms CLFD with nr = 5. In fact, CLFD with

nr = 10 leads to improvements in NPV ranging from 45% to 71% over those achieved

by optimizing over 50 prior realizations.

Case 4: Use of optimization with sample validation

The results thus far suggest that the use of nr = 10 in CLFD is adequate, but nr =

3 or 5 is insufficient. In general, however, the appropriate value to use for nr is not


0 210 420 630 840 1050 12601

2

3

4

5

6

7x 10

8

Time (Days)

NP

V (

$)

J(xi,M i

rep)

J(xi,mtrue)

DeterministicFinal NPV$ 454 MM

(a) nr = 5

0 210 420 630 840 1050 12601

2

3

4

5

6

7x 10

8

Time (Days)

NP

V (

$)

J(xi,M i

rep)

J(xi,mtrue)

Deterministic

Final NPV$ 599 MM

(b) nr = 10

Figure 2.14: Optimal expected NPV, and the corresponding NPV for the true model,versus CLFD step, for different numbers of representative realizations. The star showsthe final true NPV from CLFD (Example 2.2).


0 210 420 630 840 1050 12601

2

3

4

5

6

7

8x 108

Time (Days)

NP

V (

$)

J(xi,mtrue)

J(xi, M i

rep)

(a) nr = 5

0 210 420 630 840 1050 12601

2

3

4

5

6

7

x 108

Time (Days)

NP

V (

$)

J(xi,mtrue)

J(xi, M i

rep)

(b) nr = 10

Figure 2.15: NPV for different numbers of representative realizations, and the corre-sponding NPV for the true model, versus CLFD step (Example 2.2).

Table 2.3: NPV values ($ MM) from optimization over 50 prior realizations and fromCLFD optimization, for three different true models (Example 2.2)

True model nr = 50 (prior) nr = 5 (CLFD) nr = 10 (CLFD)

1 350 454 5992 487 664 7103 432 577 628


0 210 420 630 840 1050 12601

2

3

4

5

6

7

x 108

Time (Days)

NP

V (

$)

J(xi,M irep)

P10-P50-P90

(a) nr = 5

0 210 420 630 840 1050 12601

2

3

4

5

6

7x 10

8

Time (Days)

NP

V (

$)

J(xi,M irep)

P10-P50-P90

(b) nr = 10

Figure 2.16: P10, P50, P90 NPVs evaluated for the entire set of 50 realizations, alongwith the expected NPV for the representative set, versus CLFD step (Example 2.2).


0 210 420 630 840 1050 12601

2

3

4

5

6

7

x 108

Time (Days)

NP

V (

$)

J(xi,M i

rep)

J(xi,mtrue)

Final NPV$ 586 MM

Deterministic

Figure 2.17: Optimal expected NPV, and the corresponding NPV for the true model,versus CLFD step. The number of realizations at each CLFD step is determinedusing OSV. The star shows the final true NPV from CLFD (Example 2.2).

obvious. The OSV procedure described earlier addresses this issue.

We now present CLFD results using OSV (for true model 1). We use a maximum

of three subproblems, with the number of realizations for each problem specified to be

{6, 10, 20}. This means that optimization is first performed over nr = 6 realizations,

and if the validation criterion (Eq. 2.5) is not satisfied, nr is increased to 10, etc.

The progression of true NPV, together with the optimal expected NPV, is shown in

Fig. 2.17. The final true NPV from CLFD with OSV is $586 MM, which is greater

than the final true NPV from CLFD with nr = 3 or 5, but slightly less than the final

NPV from CLFD with nr = 10 ($599 MM). OSV, however, provides a systematic

means for selecting nr at each CLFD step, and for this reason we view it as the

preferred strategy.

In Fig. 2.18 we present the evolution of the well configuration and the geological

model for two realizations. These results were generated using CLFD with OSV. It

is evident that the well scenario involves seven wells at t1, but only six wells at later

times. It is also apparent that the permeability fields continue to show differences

even at late time, though they are similar in the vicinity of wells that have been

drilled.


X

Y

1

2

3

45

67

20 40 60

10

20

30

40

50

600

2

4

6

8

(a) Realization 1 at t1 with x1

X

Y

1

2

3

4

5

6

20 40 60

10

20

30

40

50

600

2

4

6

8

(b) Realization 1 at t4 with x4

X

Y

1

2

3

4

5

6

20 40 60

10

20

30

40

50

600

2

4

6

8

(c) Realization 1 at t7 with x7

X

Y

1

2

3

45

67

20 40 60

10

20

30

40

50

600

2

4

6

8

(d) Realization 2 at t1 with x1

X

Y

1

2

3

4

5

6

20 40 60

10

20

30

40

50

600

2

4

6

8

(e) Realization 2 at t4 with x4

X

Y

1

2

3

4

5

6

20 40 60

10

20

30

40

50

600

2

4

6

8

(f) Realization 2 at t7 with x7

Figure 2.18: Evolution of two RML realizations (log-permeability is shown) for dif-ferent CLFD steps. Current optimal well configuration and drilling sequence is alsodepicted. Solid red and blue circles denote producers and injectors (drilled or in theprocess of being drilled), and outlined red and blue circles denote planned producersand injectors. Numbers indicate the drilling sequence (Example 2.2).


We now provide the computational requirements for CLFD with OSV and with

different nr values. With OSV, the procedure requires a total of about 220,000 flow

simulations, which corresponds to about 1000 equivalent simulations (calls to the

compute cluster, on which we can access a maximum of 400 cores). This corresponds

to an elapsed time of about 18 hours. Using nr = 10, about 350,000 flow simulations

(1700 equivalent simulations, or 30.5 hours of elapsed time) are required, and for

nr = 5, about 180,000 flow simulations (850 equivalent simulations, or 15.3 hours of

elapsed time) are required. Thus, OSV provides substantial computational savings

over the use of nr = 10. Although these computational requirements are large, it is

important to recognize that, in practice, these runs would be performed over many

months; i.e., at each CLFD step. When viewed in this way, and assuming that a large

number of cores and a parallelized flow simulator are available, the CLFD procedure

should be tractable for realistic field cases. We note finally that optimization over

50 prior models entails about 300,000 simulations (1500 equivalent simulations, or

27 hours of elapsed time). This approach is thus more computationally demanding

than CLFD with OSV (and it leads to lower NPV).

2.5.3 Example 2.3: CLFD for a three-dimensional reservoir

model

We now consider a reservoir model described on a uniform grid of dimensions 30 ×30 × 5. Grid blocks are of size ∆x = ∆y = 100 ft, ∆z = 15 ft. The true horizontal

log-permeability field, along with an initial guess for the well locations (x0), is shown

in Fig. 2.19. The log-permeability field is described by a spherical variogram, with a

range of 25 grid blocks in the northwest-southeast direction, a range of 8 grid blocks

in the northeast-southwest direction, and a range of 5 grid blocks in the vertical

direction. The mean and standard deviation of log-permeability are, respectively, 4

and 1. Optimization parameters are as specified in Table 2.1. Porosity is constant

and equal to 0.2.

For this case we specify that the development plan is to include three horizontal

producers and three vertical injectors, with a specified sequence of {P1, I1, P2, I2,

P3, I3}. Drilling each well takes 210 days. We apply CLFD with OSV, with the

maximum number of OSV subproblems set to four. The numbers of realizations for


P1

I1

P2

I2

P3

I3

10 20 30

5

10

15

20

25

30

(a) Layer 1

I1 I2 I3

10 20 30

5

10

15

20

25

30

(b) Layer 2

I1 I2 I3

10 20 30

5

10

15

20

25

30

(c) Layer 3

I1 I2 I3

10 20 30

5

10

15

20

25

30

(d) Layer 4

I1 I2 I3

10 20 30

5

10

15

20

25

30 1

2

3

4

5

6

7

(e) Layer 5

Figure 2.19: True log-permeability field with initial guess for well configuration, whichincludes three horizontal producers and three vertical injectors (Example 2.3).

these subproblems are specified to be {3, 6, 10, 20}. Decision parameters (x) include

the optimal location and controls for new wells and future controls for existing wells.

In addition to the drilling cost of $25 MM per well, we specify a perforation cost of

$2 MM per grid block for horizontal wells and $0.5 MM per grid block for vertical

wells. The optimization entails six CLFD steps, each of length 210 days, followed by

a final control step of duration 740 days. As in Example 2.2, during optimization a

producer is shut in if the water production cost exceeds the oil revenue from the well.

Wells in this case are defined in terms of four location variables, as in [71]. Vertical

wells are described by integer (grid) variables defining their areal (x, y) positions and

their upper (z1) and lower (z2) completion locations. Horizontal wells are required

to extend in the y-direction and lie in a single layer. They are defined by their heel

and toe locations, which in this case are integer variables corresponding to (x1, y1, z1)

and (x1, y2, z1), respectively. The decision variables thus include a total of 24 integer

variables for well locations and 27 continuous variables for well controls.

Observed data for each CLFD history matching step include production data


0 210 420 630 8407

7.5

8

8.5

9

9.5x 108

Time (Days)

NP

V (

$)

J(xi,M i

rep)

J(xi,mtrue)

Deterministic

Final NPV$ 899 MM

Figure 2.20: Optimal expected NPV, and the corresponding NPV for the true model,versus CLFD step. The number of realizations at each CLFD step is determinedusing OSV. The star shows the final true NPV from CLFD (Example 2.3).

measured at 30-day intervals and hard data from all existing wells. Observed data

types and measurement errors are analogous to those for Example 2.2. We assume

that hard data are available for all perforated grid blocks. Model parameters for

history matching include horizontal log-permeabilities for all grid blocks. The ratio

of vertical permeability to horizontal permeability (kz/kh) is specified to be 0.2 and

is assumed to be known.

We first apply deterministic optimization (using the true model) for this problem,

which gives an optimal NPV of $941 MM. The progress of optimal expected NPV and

true NPV, as a function of CLFD step, is shown in Fig. 2.20. The deterministic result

appears as the horizontal line. CLFD performs quite well in this case, and achieves

a final NPV of $899 MM, which is close to the deterministic result. Although there

is an offset in Fig. 2.20 between true and expected NPV in the first two CLFD steps,

the value of this offset is not large (note the relatively small NPV range on the y-axis

in Fig. 2.20). The application of CLFD with OSV provides about a 20% improvement

in the true NPV over the NPV achieved from optimization over the prior realizations.

For this example, we again optimize over all 50 prior realizations in order to

compare CLFD results with those from robust optimization over prior models. The

expected NPV over the 50 prior realizations is $878 MM, but the NPV for the true


Table 2.4: Initial and final numbers of representative realizations (determined us-ing OSV) and the corresponding relative improvement values at each CLFD step(Example 2.3)

Time nr RI

0 3 -0.060 6 0.52210 3 -0.65210 20 0.95420 3 0.25420 20 0.68630 3 0.57840 3 0.05840 10 0.52

model is only $814 MM. Thus CLFD (with a final NPV of $899 MM) provides about

a 10% improvement over robust optimization with prior models for this example.

Table 2.4 presents the initial and final numbers of realizations and the relative

improvement (RI) at each CLFD step. Recall that the number of realizations is

increased by the OSV procedure if RI is less than 0.5. We see that the number of

realizations is increased in four of the five optimization steps. It is also apparent that

the RI value corresponding to the final set of realizations at each CLFD step is indeed

greater than 0.5, which means that the expected NPV for the full set of realizations

achieves at least half of the benefit attained by the representative set of realizations.

The final well configuration determined using CLFD is shown in Fig. 2.21. The

horizontal producers have all been placed in different layers, and the injectors are far

apart. This well arrangement differs significantly from the initial guess configuration.

In terms of computational requirements for this case, CLFD with OSV requires a

total of about 851,800 flow simulations. This corresponds to about 3789 equivalent

simulations and a total elapsed time of about 46 hours. We note again that, in

a practical setting, these computations would be performed over an extended time

period.

2.6. SUMMARY 53

010

2030

010

2030

0

2

4

XY

Z

Figure 2.21: Final optimal solution from CLFD with OSV. Horizontal producers areshown in red and vertical injectors in blue (Example 2.3).

2.6 Summary

In this chapter, we applied CLFD to three example cases. The first example demon-

strated that the simultaneous optimization of all wells provided superior results to

those from a well-by-well optimization, where each well is optimized independently.

This result motivates the use of a simultaneous optimization approach within CLFD.

In the second example, we applied CLFD optimization with fixed numbers of (rep-

resentative) realizations and with the OSV procedure. The final NPV for the true

model achieved using CLFD was in all cases considerably larger (18% or more, de-

pending on the case) than the true-model NPV obtained by optimizing over prior

realizations. Results also showed that using too few realizations in the CLFD op-

timization steps can lead to lower true-model NPVs. The use of CLFD with OSV

provided a true-model NPV that was comparable to the best achieved with a fixed

(relatively large) number of realizations. OSV is thus quite useful, since it represents

a systematic strategy for selecting the required number of realizations to use in opti-

mization. In the third example, which involved a three-dimensional reservoir model,

CLFD using OSV was shown to provide a final true-model NPV that represented a

10% improvement over the NPV achieved by optimizing over all 50 prior realizations.

It is worth mentioning that Morosov and Schiozer [94] recently applied the CLFD


framework to a realistic example. They used different reservoir modeling and dif-

ferent history matching and optimization approaches in their CLFD application. In

particular, they applied ensemble-based data assimilation (ES-MDA) [40] for history

matching. In their optimization step, nine representative models were selected, and

each model was optimized independently, resulting in nine different optimal solutions,

from which the solution that provided the highest expected NPV for the full set was

applied. Due to the complexity of the reservoir model, the history matching proce-

dure was not effective in providing appropriate uncertainty quantification, and this

caused the true model NPV to fall outside of the uncertainty range. The expected

NPV, however, increased by 29%.

The need for representative realizations in the CLFD optimization step motivated

us to further pursue the subject of selection of representative models. In this chapter,

representative models were selected based on a ranking of NPVs. In Chapter 3,

we will investigate more sophisticated procedures that consider a combination of

flow quantities and static (e.g., permeability field) information for the selection of

representative models.

We note finally that, in the CLFD framework presented in this chapter, the project

life was specified a priori. This is the usual approach taken in the reservoir optimiza-

tion literature. In Chapter 4 we will demonstrate that project life should also be

treated as an optimization variable. There, we investigate the joint optimization of

economic project life and well controls for a set of existing wells. Determination of

optimal project life for the field development optimization problem in CLFD is left

as future work.

Chapter 3

Selection of Representative Models

for Decision Making and

Optimization Under Uncertainty

In this chapter, we address the realization-selection problem systematically. Toward

this end, we first describe a procedure for quantitatively assessing different approaches

using a flow-response variable. We then introduce a general clustering method for

the selection of representative realizations. In this clustering, each realization is

represented by a feature vector composed of a weighted combination of flow-based

and geological quantities. Principal component analysis (PCA) is used to express

the geology (permeability field) in terms of a small number of features, while flow-

based features are obtained by solving one or more flow problems. The use of both

full-physics simulations and efficient tracer-type simulations for obtaining these flow-

based features is considered. We then investigate the performance of different feature

weightings for problems involving new well locations or new well controls, with the

goal of identifying the selection method that provides the ‘best’ subsets of realizations

(meaning that the flow response computed for a subset is in close agreement with that

for the full set). Finally, the performance of various methods will be assessed for a

well control optimization problem. Additional results, involving binary channelized

models, are presented in Appendix B.

55

56 CHAPTER 3. SELECTION OF REPRESENTATIVE MODELS

3.1 Assessment of flow-response statistics for a rep-

resentative subset

In this section, we define the flow-response vector, r, and we then describe an ap-

proach for comparing different selection methods based on flow responses. We let

the Nm-dimensional vector mj designate a geological realization of the model, which

contains as its elements the grid block log-permeability values (where j is the real-

ization index). The matrix Mfull designates the full set of geological models, Mfull =

[m1 m2 . . . mNR]. The NR realizations of mj typically correspond to a sampling

from a probability distribution. Our interest is to define a low-dimensional vector r

that captures the flow response of realization mj for a given x, where x defines the

well configuration (here we consider x−y spatial locations) and control (time-varying

bottom-hole pressure or BHP) parameters. We also refer to x as the well-parameter

vector or the vector of decision parameters.

For realization mj and well-parameter vector x, we define rj as

rj(x,mj) = [q1j ,q

2j , . . .q

nwj ]T , (3.1)

where qkj , k = 1, . . . , nw is a row vector of incremental phase production/injection

data for well k, and nw designates the number of wells. The vector qkj is referred to

as the well flow response. To compute qkj , the reservoir life is divided into a relatively

small number of time intervals, nt. By specifying a small number of time intervals,

the continuous flow response for a realization can be captured with a low-dimensional

vector. For a production well, qkj contains the total production for each phase (oil

and water in this chapter) at each time interval for the well, while for an injection

well, qkj includes the total water injected at each time interval. With nt = 3, for

example, qkj for a producer well is expressed as

qkj = [Qko,1, Q

ko,2, Q

ko,3, Q

kw,1, Q

kw,2, Q

kw,3], (3.2)

where Qko,1 and Qk

w,1, respectively, denote the total oil and water production for well

k in realization j during the first time interval, Qko,2 and Qk

w,2 are the corresponding

quantities for the second time interval, etc. Fig. 3.1 defines Qko components for a

3.1. ASSESSMENT OF FLOW-RESPONSE STATISTICS 57

0 500 1000 1500 2000 2500 30000

1

2

3

4

5x 104

Time (Days)

Oil

Pro

duct

ion

(ST

B)

Qo,11

Qo,21

Qo,31

Figure 3.1: Illustration of some of the components of qkj (for well k = 1). The reservoirlife is 3000 days, which is divided into nt = 3 intervals.

particular well. For an injection well, qkj contains only total water injection at each

time interval:

qk = [Qkw,1, Q

kw,2, Q

kw,3]. (3.3)

We now discuss the computation of flow-response vectors for a representative

subset of realizations, Mrep = [mr1 mr2 . . . mrnr], where nr (nr � NR) is the

number of realizations in Mrep (note that in Chapter 2, we used the notation Mrep

instead of Mrep). When flow information is used in the selection of a representative

subset, we must define a base-case well-parameter vector, x0, which specifies the well

locations and settings used for flow simulations. We reiterate that our goal here is to

find a subset of realizations that is representative in terms of the flow responses of

the full set of NR realizations for new well-parameter vectors that differ from x0.

We let xnew denote one such ‘new’ well-parameter vector. If flow simulation is

performed for all realizations in Mfull using xnew, the set of flow-response vectors,

denoted by Sfull, can be expressed as Sfull(xnew) = {rj,mj ∈ Mfull}. A set of flow-

response vectors can also be computed for a representative subset, which we denote

as Srep(xnew) = {rj,mj ∈ Mrep}. We quantify the difference between Srep and Sfull

in terms of a normalized Euclidean distance, d(rrep, rfull), between the mean flow-

response vectors:

d(rrep, rfull) =1√w

∥∥Σ−1full (rrep − rfull)

∥∥2, (3.4)


where w is the dimension of the flow-response vector and Σfull is a diagonal matrix

with each diagonal entry equal to the standard deviation of the corresponding element

of the flow-response vector computed from the full set of realizations. A smaller value

of d(rrep, rfull), which we will refer to as ‘dissimilarity,’ is desirable as it corresponds to

a smaller difference between the mean flow response for the full set and that for the

representative subset of realizations. We note that other measures of dissimilarity

could also be used if particular flow-response statistics (e.g., P25) are of specific

interest.

We can now state the problem of realization selection more formally as follows.

Given a reference set Mfull of NR realizations, we seek a subset Mrep of size nr whose

mean flow-response vector rrep is as close as possible to the mean flow-response vector

of the full set, rfull, for a new well-parameter vector, xnew. In our case there are usually

many possible xnew, and these are often unknown a priori, which introduces additional

complication.

There are essentially an infinite number of possible new decision vectors xnew

that could be considered, so it is necessary to limit their range in our assessments.

In this regard, we first assume that our realization selection is being performed in

the context of a particular optimization problem and/or decision-making framework.

This will specify, or at least constrain, parameters such as the number and types of

wells, minimum well-to-well distances, bounds for well rates or BHPs, etc. To further

focus the range of xnew, we additionally consider the types of searches commonly used

in reservoir optimization algorithms. As discussed in Chapter 1, these optimizations

typically employ search steps that are either global and stochastic (e.g., GA and PSO

algorithms) or local and deterministic (pattern search or gradient-based algorithms).

The PSO–MADS algorithm used in all of the optimizations in Chapter 2 combines

both types of searches.

We thus consider well-parameter vectors xnew that correspond to these two types

of ‘moves.’ Specifically, we will assess the performance of various realization-selection

methods for sets of xnew that correspond to (1) large (random) global shifts, and (2)

small local shifts in well parameters relative to x0. In PSO and GA well placement

optimization algorithms, for example, many (in fact, nearly all at early iterations) of

the configurations considered are essentially random, perhaps subject to a minimum

3.1. ASSESSMENT OF FLOW-RESPONSE STATISTICS 59

well-to-well distance constraint. Thus, although they may appear unrealistic in some

instances, random xnew represent test cases that are fully consistent with the types of

solutions proposed by global stochastic optimization algorithms. In our assessments,

well location and well control parameters will be considered separately, though it

is of course also possible to consider them in combination. We note finally that,

although the xnew considered here are modeled on the moves applied in optimization

algorithms, in some cases they also correspond to the sorts of perturbations (relative

to x0) considered in heuristic reservoir engineering evaluations.

Our goal is to use the quantities and approaches described thus far to compare

different methods for selecting representative subsets of realizations. As an example,

consider two different selection methods, A and B. For a specified nr, each method

will in general find a different representative subset, Mrep,A and Mrep,B. Our approach

for comparing selection methods A and B is as follows. Given Mrep,A and Mrep,B, we

generate a relatively large number n of new well-parameter vectors xnew, and for each

xknew, k = 1, . . . , n, we compute d(rkrep,A, rkfull) and d(rkrep,B, r

kfull). We will thus have

n dissimilarity values for each selected subset. In other words, for each subset, we

obtain a distribution of d values. We let DA and DB designate the sets of d values,

i.e., DA = {d(rkrep, rkfull), k = 1, . . . , n}, and similarly for DB.

As our interest is to identify the method with the smallest d value, we simply

compare the median values, designated mD,A and mD,B, for DA and DB. By varying

nr over a range of possible values, we can investigate which selection method is, in

an overall sense, more accurate. Our detailed approach is as follows:

1. Compute flow-response vectors for the full set, Mfull, using base well-parameter

vector(s) (x0). This gives Sfull(x0) = {rj(mj,x0),mj ∈ Mfull}. These flow

responses will be used to select a representative subset (as described in the next

section).

2. Generate n relevant new well-parameter vectors. Compute the flow responses

for the full set, Sfull(xknew) = {rj(mj,x

knew),mj ∈ Mfull}, and the mean flow

response, rfull(xknew), for k = 1, . . . , n.

3. For various values of nr, select a representative subset, Mrep, using each of the

selection methods under consideration.


4. For each Mrep, compute the mean flow response for each of the n new well-

parameter vectors, and then compute d values.

5. Report the set of dissimilarity values, D = {d(rkrep, rkfull), k = 1, . . . , n}, and its

median, mD, for each Mrep.

6. For each nr, rank the selection methods, with the best method corresponding

to that with the smallest mD value.

3.2 Unsupervised learning for selecting a repre-

sentative subset

In this work, we consider uncertainty in only the permeability field. All other sys-

tem parameters are considered to be known, though our framework could be readily

generalized to treat uncertainty in other quantities. Our goal is to select nr repre-

sentative realizations from a large set of NR models. The selected realizations will

then be used to provide ‘representative’ flow simulation results for a wide range of

(currently unknown) new decision parameter vectors. Our selection method involves

a two-stage clustering approach. In the first stage, k-means clustering is applied to

divide the NR realizations into nr clusters, such that realizations in each cluster cor-

respond to similar features. In the second stage, k-medoids clustering is applied to

select one representative realization from each cluster (an alternate approach involv-

ing only k-medoids clustering will also be assessed). Prior to clustering, a feature

matrix is constructed, with each realization represented by a feature vector. We first

discuss our approach for generating the feature matrix (which contains the NR feature

vectors as columns), followed by a description of the detailed selection algorithm.

3.2.1 Feature selection

Selection of appropriate features/attributes is essential to the success of the over-

all methodology, and different features may be more or less relevant depending on

the problem. In the clustering procedure, an NR × NR (symmetric) distance ma-

trix is constructed from the feature vectors. Because each attribute influences the

3.2. UNSUPERVISED LEARNING FOR MODEL SELECTION 61

realization-to-realization distance equally, the inclusion of a large number of less

relevant attributes may hinder the performance of the selection procedure. In our

approach, each realization is represented in the feature matrix by a combination of

low-dimensional flow-based and permeability-based vectors, as we now describe.

For computing flow-based features for the NR realizations, one or multiple flow

problems are solved using base well-parameter vectors xi0, i = 1, . . . , b. As discussed

above, these xi0 are selected consistent with the problem under study. The flow-based

feature matrix Zf is then constructed as

Zf =

r1(x

10) . . . rNR

(x10)

.... . .

...

r1(xb0) . . . rNR

(xb0)

. (3.5)

The first row of Zf corresponds to the flow-response quantities obtained by performing

flow simulation using x10, and the last row corresponds to those obtained with xb0. In

this work, we will consider cases with b = 1 or 3.

When it is computationally feasible, flow-based features should be computed using

full-physics flow simulations for the full set of NR realizations. For large models, or for

cases involving very large NR, performing full-physics flow simulations may be overly

demanding computationally. In such cases, flow-based features can be computed using

reduced-physics ‘proxy’ simulations. The precise form of these proxy simulations will

depend on the key features in the full-physics problem. Here we are interested in

nearly incompressible oil-water problems, with oil-water viscosity ratio of around 3.

A natural surrogate in this case is a tracer-type constant-mobility model, in which

we specify krw = Sw, kro = 1 − Sw, and µw = µo, where krw and kro are relative

permeabilities to water and oil, Sw is water saturation, and µw and µo are water and

oil viscosities. In this case the total mobility, which appears in the oil-water pressure

equation, does not vary with Sw. This means that the pressure solution (and thus

the Darcy velocity) does not vary in time, so the pressure equation need only be

solved once. The saturation (transport) equation, however, is solved at every time

step. Our current implementation uses Stanford’s Automatic-Differentiation-based

General Purpose Research Simulator (AD-GPRS) [157], which solves the pressure

equation at every time step. Thus it does not provide as much speedup as would be


achieved with a specialized treatment such as that provided by a streamline simulator,

though this is not a problem here since our goal is to assess the performance of proxy

flow information.

We note that tracer-type proxy simulations have been used in a number of previous

applications. For example, Durlofsky et al. [35] used these computations as a fast flow

diagnostic within the context of flow-based gridding and upscaling. In a setting closer

to that considered here, Scheidt and Caers [117] used tracer simulation for obtaining

flow-based features for realization selection (though as noted earlier, they did not

consider a range of new decision parameters, as is done in this study).

In addition to flow information, either full-physics or proxy, it is also reasonable to

include the underlying geological parameters, i.e., porosity and permeability (which

are often correlated), in the feature vector. Here we use only permeability param-

eters, though porosity could also be incorporated if necessary. In order to obtain

concise feature vectors, we use PCA (principal component analysis) representations

of permeability fields.

PCA parameterization is often applied to generate a new model m from a low-

dimensional random (standard normal) vector ξ of length l. The PCA parameteriza-

tion, described in detail by many investigators (see, e.g., Vo and Durlofsky [139]), first

entails the generation of a large number (L) of geostatistical realizations conditioned

to hard data (if available). Any geostatistical software package [92] can be used to

construct these models. The L realizations are centered, by subtracting the sample

mean m, and are then inserted as columns in a matrix X. A truncated SVD of X,

expressed as UlΛlVTl , is then constructed, with l < L. A new realization, consistent

with the covariance and hard data that characterize the original L realizations, can

then be generated through application of m = m + UlΛlξ.

In this work, our interest is in expressing an Nm-dimensional realization m in

terms of its corresponding l-dimensional ξ vector. The relevant expression for this

mapping is ξ = Λ−1l UT

l (m − m), where we use the fact that Ul is orthonormal. In

this study we set L = 1000. Various procedures have been considered to determine

an appropriate value for l. Defining the total variation as the sum of the principal

values (which lie on the diagonal of Λ), a common approach is to choose l such that

a prescribed fraction of this total variation is captured [153]. By ordering principal

3.2. UNSUPERVISED LEARNING FOR MODEL SELECTION 63

components in order of decreasing principal values, a given fraction can be captured

with the smallest value of l. In this work, we select l such that 65% of the total

variation is captured, except where otherwise indicated.

The permeability-based feature matrix can now be represented as

Zp =[ξ1 . . . ξNR

], (3.6)

where ξj, j = 1, . . . , NR, is the PCA representation of realization j. Each ξj is of

length l. The resulting feature matrix, containing both flow-based and permeability-

based features, is given by

Z =

[Zp

Zf

]. (3.7)

Both the Zp and Zf matrices have NR columns, but they typically have different

numbers of rows. In our examples, the number of rows of Zp (which is equal to l) is

on the order of 100, while the number of rows in Zf is between 80 and 240, though in

general it will depend on the number of wells and the value of nt (here we use nt = 5).

To arrive at a feature matrix that is not (artificially) dominated by either category

of attributes, both Zp and Zf are normalized by their corresponding number of rows.

Each row of Zp and Zf is then normalized by its standard deviation, which assures

that the distance between feature vectors will not be dominated by extreme values.

After applying these normalizations, the matrices are designated Zp and Zf.

We now define a general feature matrix in which the flow-based and permeability-

based features can be assigned a user-defined weighting α. In our numerical tests,

we will consider different weightings with the goal of determining the best α for a

particular problem. The general feature matrix is expressed as

Z =

[αZp

(1− α)Zf

], (3.8)

where 0 ≤ α ≤ 1, with α = 0 corresponding to a purely flow-based feature matrix,

α = 1 to a purely permeability-based feature matrix, and α = 0.5 to an equal

weighting of both permeability-based and flow-based features.


In addition to permeability, other simulation quantities, such as relative perme-

ability or fluid property parameters, may also be uncertain. In such cases, these

parameters could be added as additional rows to Zp. Detailed treatments, as well as

the performance of our methods with these additional parameters, should be assessed

in future work.

3.2.2 Clustering for selection of representative realizations

Once the feature matrix Z is generated (which requires specifying α), we apply a k-

means clustering algorithm [57] to divide the NR realizations into nr clusters. In the k-

means algorithm, a distance matrix is computed based on Euclidean distances between

feature vectors. The algorithm then divides the realizations into nr clusters such that

the within-cluster sum of square distances is minimized. Here we use the k-means

implementation from R Core Team [103], which involves some tuning parameters

such as the maximum number of iterations and the number of starting points. We

specify large values for both of these parameters (10,000 maximum iterations and

1000 starting points) to ensure appropriate performance. The computational time

for the k-means clustering for the cases considered here is on the order of seconds.

After application of the k-means algorithm, we have identified the nr clusters,

each typically containing multiple members. We then apply a k-medoids algorithm

[55] to the realizations in each cluster to find the centroid (representative) realization

for the cluster. For k-medoids clustering, we use the implementation of Hornik [63].

The overall method is outlined in Algorithm 3.1. In one of the examples below,

we will compare this two-stage procedure to a single-stage treatment that uses only

k-medoids clustering. Another approach for dividing realizations into nr clusters is

to apply hierarchical clustering [32] instead of the k-means procedure. The k-means

clustering, however, is more accurate at finding realizations with similar features. For

details on clustering methods, refer to Hastie et al. [59, Ch. 14].

We note finally that Scheidt and Caers [117] previously applied k-medoids clus-

tering for selecting a subset of realizations. The goal in that work, however, was to

identify realizations that provide statistics in a particular production response (such

as cumulative oil production versus time) that are close to those for the full set of

realizations, for a specified well configuration and set of controls. Our goal here, by


contrast, is to select a subset of realizations that is representative (relative to the full

set) for many possible new well configurations or control specifications.

Algorithm 3.1 Clustering for selection of representative realizations

Specify the number of representative realizations, nrConstruct the feature matrix ZApply k-means clustering to divide the NR realizations into nr clustersfor k = 1, . . . , nr do

Apply k-medoids clustering to realizations in cluster k to obtain the cluster cen-troid

end forReport the cluster centroids as the nr representative realizations

3.3 Computational results: representative realiza-

tions for new controls and configurations

In this section we apply clustering with various weightings for selecting representative

subsets. We consider clustering with flow-based features only (α = 0), a combination

of flow and permeability-based features (α = 0.5), and permeability-based features

only (α = 1). These selection methods will be compared with random sampling in

some cases. Other α values could of course also be used, and the best value for a

particular problem may indeed be different than the three values considered.

The selection methods will be applied in two different contexts – when the decision

vector x defines new well operational settings (for a fixed configuration), and when x

contains new well locations (with fixed well controls). For each case, consistent with

the discussion in Section 3.1, both random well-parameter vectors (which correspond

to solutions proposed by global stochastic search algorithms such as GA and PSO),

and small local changes in a base well-parameter vector (which correspond to solutions

proposed by pattern search methods) will be considered. Algorithmic treatments,

such as the use of PCA rather than the full permeability field in clustering, and the

use of the two-stage k-means/k-medoids clustering, will be assessed in Example 3.1.

Channelized models involving a bimodal log-permeability distribution are used

in all examples in this chapter. Results for binary channelized systems appear in


0 0.5 10

0.2

0.4

0.6

0.8

1

Sw

k r

krw

kro

Figure 3.2: Oil and water relative permeability curves for all examples.

Appendix B.

All examples involve two-phase oil-water flow. Oil and water viscosities are speci-

fied as 3 cp and 1 cp, respectively, and both fluids are considered to be incompressible

(formation volume factors are set to 1). The rock compressibility is set to 10−3 bar−1.

Relative permeability curves are shown in Fig. 3.2. The grid block dimensions in all

cases are ∆x = ∆y = 100ft, ∆z = 15ft. The initial reservoir pressure is 4500 psi. The

reservoir initially contains oil and connate water, with irreducible water saturation of

Swc = 0.1. All flow simulations are performed using Stanford’s AD-GPRS [157].

3.3.1 Example 3.1: new well settings in channelized models

The reservoir model is two-dimensional and contains 100 × 100 grid blocks. The

isotropic log-permeability field displays a bimodal distribution. The two modes are

of mean (in log-permeability) 8 and 3, with corresponding variances 0.4 and 0.8.

Porosity is uniform and equal to 0.1. The permeability realizations, generated using

a cookie-cutter approach [28], are all conditioned to hard data at well locations. Three

realizations are shown in Fig. 3.3. There are four injection wells and six production

wells in the reservoir.

The reservoir life is specified to be 10 years. This period is divided into nt = 5

equal time intervals (of length two years) in the construction of the flow-response

vectors. The number of control steps used when generating new control vectors x,

designated ncs, is specified equal to nt. During each control step, the BHP of each


x

y

1 2 3

4 5 6

7 8 9 10

20 40 60 80 100

20

40

60

80

100

2

4

6

8

(a) Realization 1

x

y

1 2 3

4 5 6

7 8 9 10

20 40 60 80 100

20

40

60

80

100

2

4

6

8

(b) Realization 2

x

y

1 2 3

4 5 6

7 8 9 10

20 40 60 80 100

20

40

60

80

100

2

4

6

8

(c) Realization 3

Figure 3.3: Three conditional realizations of log-permeability field for bimodal chan-nelized model. Fixed well configuration is also shown – circles denote producers andtriangles indicate injectors (Examples 3.1 and 3.3).

well is set to a constant value. In all cases, the producer BHP range is 1000–4100 psi

and the injector BHP range is 4600–7000 psi. With 10 wells and ncs = 5 control steps,

the number of control variables (and the length of the well-parameter vector) is 50.

We performed an out-of-sample validation study to determine the appropriate

size (NR) of the full set of realizations. With this approach, for a range of NR

values (starting with NR = 100), we generated a reference set of realizations and a

different set of equal size, referred to as the out-of-sample set. We then evaluated

the flow-responses for both the reference and out-of-sample sets and computed the

dissimilarity d (Eq. 3.4) between the two sets for each value of NR. We continued

increasing NR until the dissimilarity between the two sets plateaued. This occurred

at about NR = 200, which is the value used in this work.

Representative realizations for random well controls

Of the three selection methods considered, clustering with α = 1 does not require

any flow simulation (recall that α = 1 corresponds to a permeability-based feature

matrix). Any other α value entails flow information, which means we must simu-

late the full set of NR = 200 realizations. The base control strategy (x0) used in

these simulations corresponds to all injection wells operating at their maximum BHP

(7000 psi) and all production wells operating at their minimum BHP (1000 psi). The

number of flow-based features used in the clustering is 80, which is equal to the di-

mension of the flow-response vector (recall that there are two quantities per producer,


0 1000 2000 30004500

5000

5500

6000

6500

7000

Time (days)

BH

P(p

si)

Well 7Well 8Well 9Well 10

Figure 3.4: Injector BHP profiles corresponding to a random well-control vector xnew

(Example 3.1).

and one quantity per injector, for each of the nt time intervals). For computing the

permeability-based features, each realization is represented by l = 99 PCA parameters

(which corresponds to 65% of the total variation), except where otherwise indicated.

A total of 100 random well-control vectors (xnew) are generated. Each of these

vectors contains 50 elements (10 wells with five control steps each), which specify

the well-by-well time-varying BHP schedules. Each of the 50 entries in xnew is a

random sample from a uniform distribution between the lower and upper bounds for

the relevant well type (producer or injector). BHPs for the injectors, from one of

the xnew vectors, are shown in Fig. 3.4. The flow-response vectors for all NR = 200

realizations are evaluated and saved for each xnew. This involves a total of 20,000

simulation runs, which are performed using distributed computing with access to 200

compute nodes. The elapsed (wall-clock) time is thus equivalent to about the time

required for 100 serial simulation runs.

We now apply our methodology to select representative subsets of nr = 3 realiza-

tions from the full set. For each representative subset, the mean flow-response vector

for each of the 100 random control vectors is computed. Following this, the dissim-

ilarity values for mean flow responses, {d(rkrep, rkfull), k = 1, . . . , 100}, are computed.

There is one set of d values for each selection method. We let Dα=0, Dα=0.5 and

Dα=1 denote the sets of (100) dissimilarity values obtained for each of the selection

methods. Box plots for these three sets are shown in Fig. 3.5(a). A box plot displays

a summary of the variation in the sample, as described in the figure caption. From


Table 3.1: Median (mD) values of Dα=0, Dα=0.5, Dα=1 and Drand, for 100 randomwell control vectors, for different nr. Average mD values and average ranking are alsoprovided (Example 3.1).

nr α = 0 α = 0.5 α = 1 random

3 0.39 0.39 0.53 0.576 0.28 0.28 0.31 0.419 0.20 0.21 0.32 0.3912 0.16 0.23 0.24 0.2615 0.13 0.17 0.16 0.25avg (3-15) 0.21 0.26 0.33 0.37avg ranking (3-15) 1.08 2.15 3.15 3.62

Fig. 3.5(a), it is evident that dissimilarity values corresponding to clustering with

α = 0 (flow-based features) and α = 0.5 (both flow-based and permeability features)

are smaller than those from clustering with α = 1 (permeability-based features).

The above procedure is repeated for 3 ≤ nr ≤ 15. Box plots for nr = 6, shown

in Fig. 3.5(b), show that dissimilarity values from clustering with α = 0 are typically

smaller than those for the other two approaches. In general we expect the dissimilarity

to decrease with increasing nr, but since there is randomness in the problem this

decrease is not always monotonic. Comparing Figs. 3.5(b) and 3.5(a), we see that the

median d values decrease for all approaches.

We use the median d value, designated mD, as the basis of comparison between

methods. Recall that this quantifies the median dissimilarity in flow response over

the 100 xnew well-control vectors, evaluated using a particular selection method (for

a specified nr). Results for mD for five values of nr are presented in Table 3.1. For

comparison purposes, we also include results using a random selection of realizations.

With this approach, for each nr, we repeated the random selection nine times. The

results reported in Table 3.1 correspond to the median value of mD over these nine

selections. For each method, the average mD value over the range 3 ≤ nr ≤ 15,

along with the average ranking, are also shown. It is evident that the use of α = 0

(flow-based selection) is the overall best selection method for this problem. The use

of α = 1 provides better results than random selection.


0.35

0.4

0.45

0.5

0.55

α=0 α=0.5 α=1

D

(a) nr = 3

0.25

0.3

0.35

α=0 α=0.5 α=1

D

(b) nr = 6

Figure 3.5: Box plots of Dα=0, Dα=0.5 and Dα=1 for 100 random well control vectors.The red line within each box corresponds to the median, and the bottom and top ofeach box correspond to the 25th and 75th percentiles. The lines above and below theboxes correspond to the 2nd and 98th percentiles (Example 3.1).

As discussed in Section 3.2.1, flow-based features for clustering can also be gen-

erated from reduced-physics proxy simulations, and here we apply a tracer-type

constant-mobility proxy model (with krw = Sw, kro = 1− Sw, µw = µo). We perform

these proxy simulations for all realizations with the base control strategy to obtain

flow-based features for clustering. We then compute the flow-based distance ma-

trix (corresponding to α = 0), which is obtained by computing Euclidean distances

between feature vectors. With NR = 200, there are 19,900 distinct values in this

distance matrix. The correlation coefficient between these flow-based distances and

those computed from full-physics simulation was 0.8, which suggests the proxy flow

results should provide useful flow-based information in our procedure.

We let Dα,prx denote the set of dissimilarity (d) values obtained from realization

selection using proxy model flow features (here ‘prx’ designates proxy). Results for

median Dα,prx are presented in Table 3.2. Comparing these values to those for mD

using full-physics flow simulations (Table 3.1), we see that, on average, clustering

using proxy flow information performs nearly as well as clustering with full-physics

flow information for α = 0 (average mD of 0.23 versus 0.21), and equally well for

α = 0.5 (0.26 in both cases). Results are generally close for a particular nr value,

though some differences are apparent.

The average mD results for both α = 0 (prx) and α = 0.5 (prx) are clearly below


Table 3.2: Median values (mD) of Dα=0,prx and Dα=0.5,prx for 100 random well controlvectors, for different nr. Average mD values are also provided (Example 3.1).

nr α = 0 (prx) α = 0.5 (prx)

3 0.46 0.386 0.21 0.329 0.24 0.2112 0.18 0.1915 0.20 0.18avg (3-15) 0.23 0.26

those for α = 1 in Table 3.1 (which uses only permeability information and is therefore

not affected by the flow treatment). This is encouraging, as it indicates that, for this

example, much of the benefit in realization selection that derives from the use of flow

information can be achieved without performing expensive flow simulations.

We now assess two of the algorithmic treatments applied in this work relative

to alternate approaches. Specifically, we consider the use of PCA for permeability

representation, and the use of the two-stage k-means/k-medoids clustering procedure.

As discussed in Section 3.2, a PCA representation of each realization is incorpo-

rated in the permeability-based feature matrix. We now consider clustering based

on (1) the full permeability field, and (2) the ‘full’ PCA representation. In cluster-

ing based on the full permeability field, a distance matrix containing the Euclidean

distances between each pair of realizations is generated, and Algorithm 3.1 is then

applied to find representative realizations. In this case, the feature matrix is not

normalized by dividing each row by its corresponding standard deviation before com-

puting distances, since the standard deviation of permeability values at well grid

blocks is zero (due to conditioning to hard data). In the full PCA treatment, we use

the maximum allowable value for l, lmax = L− 1 = 999 (with each row in the feature

matrix normalized by its standard deviation). Recall that l is the number of columns

in the basis matrix UlΛl, and in the results up to this point we have used l = 99.

Results using the three treatments are shown in Table 3.3. These results all

correspond to the use of permeability-based clustering (α = 1), and the values in the

first column (l = 99, which corresponds to 65% of the total variation) are the same

as in Table 3.1. The results for the average mD values are identical (0.33), though


Table 3.3: Results for various permeability-based clustering treatments (α = 1 in allcases). Median (mD) values of Dl=99, Dfull PCA and Dfull perm, for 100 random wellcontrol vectors, for different nr. Average mD values are also provided (Example 3.1).

nr PCA (l = 99) full PCA (l = 999) full perm

3 0.53 0.77 0.466 0.31 0.30 0.429 0.32 0.29 0.3312 0.24 0.22 0.2215 0.16 0.26 0.30avg (3-15) 0.33 0.33 0.33

there are differences in mD for a particular value of nr. Truncation of the PCA

representation (i.e., the use of a value of l that is somewhat less then lmax) is known

to provide a useful ‘denoising’ effect, though this is not evident from the results in

Table 3.3. In subsequent results, we continue to use PCA with l = 99 to represent

permeability features since this is a concise and efficient representation, and there are

no obvious advantages associated with the other treatments considered.

We next compare our two-stage selection method (Algorithm 3.1) to a one-stage

k-medoids selection approach. Results using k-medoids clustering, with flow-based

features from full-physics simulations, are presented in Table 3.4. Comparing these

results to those in Table 3.1, it is evident that the two methods provide very similar

results, though the use of Algorithm 3.1 performs slightly better for α = 1. The last

column in Table 3.4 shows the computational time. The average time for k-medoids

clustering (using the implementation of Hornik [63]; other implementations may be

faster) is about 16 minutes, while the average computational time for the results

presented in Table 3.1 (using Algorithm 3.1) is about 9 seconds. This difference in

timing is consistent with the discussion of Hastie et al. [59, Ch. 14], who stated that

k-medoids clustering is ‘far more computationally intensive’ than k-means clustering.

In any event, since there is no apparent advantage to using a single-stage k-medoids

approach, we will continue to apply Algorithm 3.1 for subsequent examples.


Table 3.4: Results using k-medoids clustering. Median (mD) values of Dα=0, Dα=0.5

and Dα=1, for 100 random well control vectors, for different nr. Last column showscomputational time for selection with α = 0. Average values are also provided (Ex-ample 3.1).

nr α = 0 α = 0.5 α = 1 CPU time (minutes)

3 0.42 0.47 0.50 186 0.26 0.32 0.43 229 0.19 0.22 0.38 1312 0.14 0.21 0.28 1415 0.16 0.18 0.24 14avg (3-15) 0.21 0.26 0.37 16.2

Representative realizations for small changes in well controls

We now compare the three approaches for selecting representative subsets of realiza-

tions for cases where new well-parameter vectors xnew correspond to small changes

relative to base-case operations. As noted earlier, this corresponds to local pattern

moves, or search along a gradient, in pattern search (e.g., MADS) or gradient-based

optimization procedures. For the base operating condition, the BHP of each well is

specified to be the average of the upper and lower bounds. These settings are used

in the flow simulation of each of the NR = 200 realizations, from which the flow-

response vectors are constructed. New well-parameter vectors are generated in this

case using a pattern search (PS) type of procedure [13]. In the basic PS approach, a

conceptual mesh (in parameter space) around the current decision-parameter vector

is formed. The mesh points are obtained by incrementing or decrementing one or

more of the optimization variables by a specified amount. Here we construct xnew by

modifying two coordinates of x0 (at a time) by ±20%. The number of mesh points

is twice the dimension of x, which in this case is 100. Thus this assessment entails

200× 100 = 20, 000 flow simulations.

Box plots for D for nr = 3 and 6 are shown in Fig. 3.6. As was the case with

random controls, we again see that clustering with α = 0 leads to the smallest dissim-

ilarity values. The mD values and the average rankings are shown in Table 3.5. These

results are quite consistent with those in Table 3.1 and again indicate that clustering

with flow-based features is preferable. It is interesting to note that the box plots of


0.35

0.4

0.45

0.5

0.55

α=0 α=0.5 α=1

D

(a) nr = 3

0.26

0.27

0.28

0.29

0.3

0.31

0.32

α=0 α=0.5 α=1

D

(b) nr = 6

Figure 3.6: Box plots of Dα=0, Dα=0.5 and Dα=1 for 100 well control vectors corre-sponding to pattern search mesh points. The red line within each box correspondsto the median, and the bottom and top of each box correspond to the 25th and 75thpercentiles. The lines above and below the boxes correspond to the 2nd and 98thpercentiles (Example 3.1).

D in Fig. 3.6 span a narrower range than those in Fig. 3.5. This is because of the

smaller variation in the flow responses here than for the case with random controls.

Results using flow-based features obtained from proxy simulations are presented

in Table 3.6. These results are, on average, again comparable to those obtained from

selection using full-physics simulations (Table 3.5), indicating that proxy modeling

could be very useful in this setting.

3.3.2 Example 3.2: new well configurations

We now consider cases involving new well locations rather than new well controls.

This case is based on the same reservoir model as in Example 3.1 (Section 3.3.1).

However, since new well configurations are considered here, the realizations are not

conditioned to any hard data. In this example, well BHPs are held constant over the

run, with injector BHP equal to 7000 psi and producer BHP equal to 1000 psi. Three

realizations of the permeability field, along with three base well configurations, are

shown in Fig. 3.7. The simulation time frame is 10 years, which is divided into nt = 5

intervals for computing flow responses.


Table 3.5: Median (mD) values of Dα=0, Dα=0.5 and Dα=1, for 100 well control vectorscorresponding to pattern search mesh points, for different nr. Average mD values andaverage ranking are also provided (Example 3.1).

nr α = 0 α = 0.5 α = 1

3 0.36 0.41 0.536 0.26 0.28 0.329 0.22 0.16 0.2612 0.20 0.19 0.2415 0.13 0.16 0.24avg (3-15) 0.22 0.24 0.31avg ranking (3-15) 1.46 1.54 3.00

Table 3.6: Median values (mD) of Dα=0,prx and Dα=0.5,prx for 100 well control vectorscorresponding to pattern search mesh points, for different nr. Average mD values arealso provided (Example 3.1).

nr α = 0 (prx) α = 0.5 (prx)

3 0.33 0.276 0.30 0.269 0.32 0.2512 0.21 0.2515 0.13 0.15avg (3-15) 0.25 0.24

x

y

1 2 3

4 5 6

7 8 9 10

20 40 60 80 100

20

40

60

80

100

2

4

6

8

(a) x10

x

y

1

2

3

4

5

6

7

8

9

1020 40 60 80 100

20

40

60

80

100

2

4

6

8

(b) x20

x

y

1 2 3 4 5 6

7 8

9 10

20 40 60 80 100

20

40

60

80

100

2

4

6

8

(c) x30

Figure 3.7: Three realizations of log-permeability field and three base well configura-tions for computing flow-based features used in clustering. Circles denote producersand triangles indicate injectors (Example 3.2).


x

y

20 40 60 80 100

20

40

60

80

100

(a) x1new

x

y20 40 60 80 100

20

40

60

80

100

(b) x2new

x

y

20 40 60 80 100

20

40

60

80

100

(c) x3new

Figure 3.8: Three (out of 100) random well configurations, xnew, for computing flowresponses. Circles indicate producers and triangles denote injectors (Example 3.2).

Representative realizations for random well configurations

The well-parameter vectors xnew in this problem define the locations of the 10 wells.

This involves 20 integer variables, since each well location is prescribed by its (i, j)

location on the grid. We generate 100 random xnew vectors, with each constrained to

satisfy a minimum well-to-well distance of 5 grid blocks. Three of these configurations

are shown in Fig. 3.8. As noted earlier, although some of the random configurations

considered may appear to be unrealistic, they are fully consistent with the types of

configurations proposed and evaluated by global stochastic search algorithms. See, for

example, the (converged) PSO and GA solutions presented in Fig. 8 in Onwunalu and

Durlofsky [100]. In our assessment here, we evaluate 100 different well configurations,

each of which is simulated for the full set of NR = 200 realizations (for a total of

20,000 reservoir simulation runs). We then proceed with the selection of representative

realizations.

In this case we simulate three different flow problems to provide the flow infor-

mation used in the clustering. The well configurations specified in these simulations

are shown in Fig. 3.7. Multiple configurations are used with the intent of capturing

the impact of geological connectivity on the flow response. The specific configura-

tions were selected in order to generate large-scale flow in both coordinate directions

as well as diagonally. The number of flow-based features in this case is 240, while

the number of PCA parameters (l) is again 99. The clustering approaches are then

applied to select representative subsets of size nr = 3, . . . , 15.


Table 3.7: Median (mD) values of Dα=0, Dα=0.5, Dα=1 and Drand, for 100 randomwell configurations, for different nr. Average mD values and average ranking are alsoprovided (Example 3.2).

nr α = 0 α = 0.5 α = 1 random

3 0.50 0.50 0.48 0.546 0.37 0.36 0.34 0.389 0.30 0.29 0.29 0.3212 0.27 0.25 0.25 0.2715 0.24 0.23 0.24 0.24avg (3-15) 0.33 0.32 0.31 0.34avg ranking (3-15) 3.15 1.85 1.48 3.46

Box plots of D for nr = 6 and 12 are shown in Fig. 3.9 (we present results for

larger nr here since the impact of uncertainty is greater in well location problems).

We see in Fig. 3.9 that, for both nr values, the box plots corresponding to the different

selection methods are quite similar, though the d values corresponding to clustering

with α = 0 (flow-based information only) are slightly greater than those for α = 1.

This is in contrast to the results for random well controls, where we found that the

use of α = 0 was preferred.

Table 3.7 presents results for mD for a range of nr. Results for random selection,

computed as described earlier, are also shown. The mD values for all four methods

are quite similar in this case, and no one method clearly outperforms the others. The

use of α = 1 does lead to the best average ranking (1.48) and the smallest average

mD (0.31) over the range 3 ≤ nr ≤ 15, though again its advantage over the other

cluster-based methods is slight. It is also noteworthy that none of the cluster-based

selection methods greatly outperforms random selection. We note finally that all of

the methods in Table 3.7 display decreasing mD with increasing nr. This general

trend is expected.

Results using flow-based features (generated using the same three flow configura-

tions) from proxy simulations are shown in Table 3.8. These results are identical, in

terms of average mD values, to those in Table 3.7. This is perhaps not surprising,

since flow-based information is not very informative in this case.


0.2

0.3

0.4

0.5

0.6

α=0 α=0.5 α=1

D

(a) nr = 6

0.2

0.25

0.3

0.35

0.4

α=0 α=0.5 α=1D

(b) nr = 12

Figure 3.9: Box plots of Dα=0, Dα=0.5 and Dα=1 for 100 random well configurations.The red line within each box corresponds to the median, and the bottom and top ofeach box correspond to the 25th and 75th percentiles. The lines above and below theboxes correspond to the 2nd and 98th percentiles (Example 3.2).

Table 3.8: Median values (mD) of Dα=0,prx and Dα=0.5,prx for 100 random well config-urations, for different nr. Average mD values are also provided (Example 3.2).

nr α = 0 (prx) α = 0.5 (prx)

3 0.49 0.486 0.37 0.379 0.31 0.2912 0.27 0.2515 0.25 0.23avg (3-15) 0.33 0.32


x

y

20 40 60 80 100

20

40

60

80

100

(a) x0

x

y

20 40 60 80 100

20

40

60

80

100

(b) x1new

x

y

20 40 60 80 100

20

40

60

80

100

(c) x2new

Figure 3.10: Base-case well configuration, and two (out of 40) new well configurationscorresponding to pattern search mesh points. Circles denote producers and trianglesindicate injectors (Example 3.2).

Representative realizations for small changes in well locations

We now consider cases where new well locations correspond to local perturbations

around a base-case configuration. Again, this corresponds to the types of ‘moves’

that are performed in pattern search and gradient-based optimization. The base case

(x0) used in this example is shown in Fig. 3.10(a). As in the example in Section 3.3.1,

new well-parameter vectors correspond to pattern search mesh points around x0.

Since there are 20 decision parameters in this example, the number of mesh points

is 40. New well configurations correspond to shifts, in two coordinates of the well-

parameter vector, by ±9 grid blocks. Two of the xnew configurations (of the 40) are

shown in Fig. 3.10(b) and (c). Flow-based features for clustering are obtained from

a single flow simulation using the base configuration (Fig. 3.10(a)).

Box plots of D are shown in Fig. 3.11, and results for a range of nr are presented in

Table 3.9. In this case, clustering with α = 0.5 appears to be the preferred approach,

as it provides, on average, the smallest dissimilarity value (0.22) and the highest

ranking (1.46). Clustering with flow-based features (α = 0) provides comparable

results to those from α = 0.5. The use of permeability-based features (α = 1)

typically gives the largest dissimilarity values. It is interesting to note that our

findings for this case differ from those for random well configurations. This is likely

because flow-response quantities are more informative here since new configurations

correspond to relatively small, and systematic, changes in a base-case configuration.


0.1

0.2

0.3

0.4

0.5

α=0 α=0.5 α=1

D

(a) nr = 6

0.05

0.1

0.15

0.2

0.25

0.3

α=0 α=0.5 α=1

D

(b) nr = 12

Figure 3.11: Box plots of Dα=0, Dα=0.5 and Dα=1 for 40 well configurations corre-sponding to pattern search mesh points. The red line within each box correspondsto the median, and the bottom and top of each box correspond to the 25th and 75thpercentiles. The lines above and below the boxes correspond to the 2nd and 98thpercentiles (Example 3.2).

Results for selection using flow-based features from proxy simulations are pre-

sented in Table 3.10. The average mD values are again comparable to (but slightly

greater than) those computed based on selection using full-physics simulations (Ta-

ble 3.9). The results in Table 3.10 are consistent with those in Table 3.9, as both

show that the use of α = 0.5 is preferred in this case.

3.3.3 Summary of realization-selection results

In the previous subsections, we assessed the performance of various realization-selection

methods for a series of problems involving different types of decision parameters (well

controls and well locations). We considered xnew that correspond to both random

well parameters (consistent with solutions proposed in global stochastic search pro-

cedures), and to well parameters that represent local shifts relative to a base case

(consistent with solutions proposed in pattern search methods). The selection meth-

ods were compared in terms of the flow response computed for a subset of realizations

compared to that for the full set of 200 realizations. A distribution of flow responses

was generated by considering many possible (new) sets of well controls or locations,

and a median dissimilarity mD was reported in each case.

Table 3.11 provides a summary of the results, in terms of the average mD values,


Table 3.9: Median (mD) values of Dα=0, Dα=0.5 and Dα=1, for 40 well configurationscorresponding to pattern search mesh points, for different nr. Average mD values andaverage ranking are also provided (Example 3.2).

nr α = 0 α = 0.5 α = 1

3 0.43 0.40 0.676 0.27 0.27 0.329 0.19 0.18 0.2712 0.15 0.18 0.1815 0.15 0.13 0.19avg (3-15) 0.23 0.22 0.32avg ranking (3-15) 1.69 1.46 2.85

Table 3.10: Median values (mD) of Dα=0,prx and Dα=0.5,prx for 40 well configurationscorresponding to pattern search mesh points, for different nr. Average mD values arealso provided (Example 3.2).

nr α = 0 (prx) α = 0.5 (prx)

3 0.47 0.366 0.40 0.359 0.27 0.2412 0.20 0.1915 0.16 0.17avg (3-15) 0.28 0.24


for the four cases considered in this chapter. Results for α = 0 and α = 0.5 are

provided for selection using both full-physics and proxy simulations. For the cases

involving new well controls (either random or pattern search shifts relative to the base

case), the best performing method for selecting a representative subset was clustering

with flow-based features (α = 0). For randomly-generated new well configurations,

the three selection methods performed similarly, though selection with permeability-

based features (α = 1) provided slightly better results than the other approaches.

Finally, for small well-location changes relative to a base well configuration, realiza-

tions selected using α = 0.5 provided the best results. The results for binary systems

presented in Appendix B are generally consistent with the results in Table 3.11.

The use of tracer-type constant-mobility (proxy) simulations to obtain flow-based

features for clustering appears to be quite suitable for the cases considered here, as

it provides results of nearly the quality of those using full-physics simulations. It

would be expected, however, that this particular proxy model will be less effective for

cases with more challenging flow physics; i.e., systems with strong compressibility or

three-phase flow effects, or those involving complex recovery processes such as steam

injection. For such cases, alternative proxies could presumably be developed, though

this would require some amount of numerical experimentation.

Taken in total, these results demonstrate the benefit of using base-case flow infor-

mation for the selection of representative realizations in many situations. Flow data

are quite informative when well controls are varied, which seems reasonable since one

would expect that, with fixed well locations, the impact of geology for a particular

realization can be estimated from the base-case simulation. Flow information was

also found to be useful for selecting realizations in cases when the new well configu-

rations correspond to systematic (local) perturbations around the base case. This is

again reasonable as the base-case flow data in such situations are also expected to be

relevant to the new simulations. In general, however, for a particular system and set

of optimization parameters or decisions, some amount of numerical experimentation

may be required to determine the optimal value of α.

3.4. PRODUCTION OPTIMIZATION UNDER UNCERTAINTY 83

Table 3.11: Summary of results: average mD values for nr = 3, . . . , 15 for all cases.The smallest value for each case is indicated in bold.

Case α = 0 α = 0.5 α = 1

Random controls (full phys.) 0.21 0.26 0.33Random controls (proxy) 0.23 0.26 –PS mesh controls (full phys.) 0.23 0.24 0.31PS mesh controls (proxy) 0.25 0.24 –Random configs. (full phys.) 0.33 0.32 0.31Random configs. (proxy) 0.33 0.32 –PS mesh configs. (full phys.) 0.23 0.22 0.32PS mesh configs. (proxy) 0.28 0.24 –

3.4 Realization selection in production optimiza-

tion under uncertainty

In this section, we compare the performance of the various selection methods for

optimization under uncertainty. We consider production optimization, in which the

time-varying BHPs that maximize net present value (NPV) are determined. This

assessment is also relevant to the joint optimization of well locations and controls

(as was considered in Chapter 2) when the problem is treated in a nested fashion

[19], in which case the inner loop entails well control optimization. We first describe

the optimization problem and then present results for production optimization under

uncertainty.

3.4.1 Optimization of well controls with representative real-

izations

The objective of our optimization is to maximize the expected undiscounted NPV.

The undiscounted NPV for a particular realization mj, which depends on the decision-

parameter vector x, is given by

J(x,mj) =

NP∑k=1

(poQo,k − cwpQw,k)−NI∑k=1

cwiQwi,k −NP+NI∑i=1

cwell, (3.9)


where NP and NI denote the number of producers and injectors, po, cwp and cwi are

the oil price and the cost of handling produced and injected water (all in $/STB),

Qo,k and Qw,k are the cumulative oil and water production for producer k, and Qwi,k

designates the cumulative water injection for injector k. These quantities, in units of

STB, all correspond to production or injection over the full simulation time frame.

Finally, cwell defines the cost of drilling a well. Note that Eq. 3.9 is similar to Eq. 2.1

(used within CLFD), as a zero discount rate is specified in both. The only difference

is that in Eq. 3.9 the decision vector x contains a subset of variables of that in Eq. 2.1.

For the production optimization problem, the decision vector x includes the BHP

of each well at each control step. The expected NPV for the full set of realizations,

J , is computed as the average over all NR realizations:

J(x,Mfull) =1

NR

NR∑j=1

J(x,mj). (3.10)

Rather than optimizing over all NR realizations, which is of course time consuming

since NR simulations must be performed for each function evaluation in the opti-

mization, we optimize instead over a representative subset, Mrep. The optimization

problem can then be expressed as

maximize J(x,Mrep) =1

nr

nr∑j=1

J(x,mrj), subject to xl ≤ x ≤ xu, (3.11)

where xu and xl define the vectors of upper and lower bounds. In our optimizations,

nonlinear output constraints, such as maximum well flow rates, are handled in the

forward simulator. The optimization problem in Eq. 3.11 is similar to Eq. 2.4 (used

within CLFD), with a difference that nonlinear constraints are not included here in

Eq. 3.11.

Our approach is to select a representative subset Mrep, of size nr, and to perform

optimization to find the xopt that maximizes J(x,Mrep). We then simulate the full

set of NR realizations using this xopt, which allows us to compute J(x,Mfull) from

Eq. 3.10. If we had a ‘perfect’ set of representative realizations, the improvement

achieved from optimizing over Mrep would also be observed for Mfull. In general


Table 3.12: Economic parameters and bounds for Example 3.3Parameter Value

cwell $107

po $70/STBcwp $7/STBcwi $7/STBProd. BHP range 1000–4100 psiInj. BHP range 4600–7000 psi

this degree of improvement in Mfull will not be attained, but the improvement we do

observe will depend on the representativeness of subset Mrep; i.e., on the performance

of our realization-selection method.

3.4.2 Example 3.3: production optimization under uncer-

tainty

In these optimizations we use the bimodal channelized models introduced in Exam-

ple 3.1 (three realizations are shown in Fig. 3.3). In the optimizations, the simulation

time frame is 10 years. This is divided into ncs = 10 control steps, each of length

1 year. As there are 10 wells in the reservoir, the number of decision parameters is

100. Economic parameters and optimization bounds are provided in Table 3.12. As

simulation results are sensitive to the time stepping, we specify a maximum time step

length of 30 days for all simulations.

We apply the PSO–MADS hybrid algorithm [70] for these optimizations. This

is the same procedure used in the CLFD optimizations in Chapter 2. Each PSO–

MADS iteration requires the simulation of all selected realizations. The initial guess

corresponds to wells operating at their bounds (7000 psi for injectors and 1000 psi

for producers). The PSO–MADS algorithm is applied with 50 PSO particles, while

the initial MADS mesh size is specified as 0.2 of each variable range. The maximum

liquid rate for producers is 10,000 STB/day, while the maximum injection rate for

each injector is 20,000 STB/day. The PSO–MADS optimizations are terminated

when a minimum MADS mesh size (1% of each variable range) is reached. Due

to the stochastic nature of the PSO algorithm, and the complexity of the problem


(i.e., multiple local optima typically exist), each optimization case is run three times,

using a different initial random seed. Our comparisons will be based on average

performance over the three runs. When performing simulations for the full set of

NR = 200 realizations (using xopt from optimization over the representative subset),

if necessary we apply a ‘reactive-control’ strategy in which a producer is shut-in

(closed) if the cost of handling produced water exceeds the oil revenue for the well.

In addition to the three cluster-based methods, we also consider an approach based

on the cumulative distribution function (CDF) of the NPV. This method is essentially

that described in Chapter 2, though here we use a variant of this procedure, as applied

by Aliyev [5]. With this approach, NPV for all NR realizations is evaluated using the

initial-guess BHPs, and the realizations are then ranked in terms of NPV. Realizations

corresponding to even increments in NPV percentile are then selected (e.g., for nr = 3,

we use the realizations corresponding to P10, P50 and P90).

We first perform optimizations for selected subsets of nr = 3 realizations. The

realizations are selected at the start of the optimization and they are used through-

out the optimization (i.e., the selected subset is not updated, though this could be

considered). Results for the various realization-selection strategies are presented in

Table 3.13. The quantities in the table correspond to improvement in expected NPV

relative to the initial-guess expected NPV. All quantities are for improvement in

J(xopt,Mfull), with xopt computed over the corresponding representative subset. The

initial-guess expected NPV for this case is J(x0,Mfull) = $360.6 × 106, so the im-

provements are quite significant. The highest average improvement in expected NPV

for the full set is obtained using realizations selected from clustering with flow-based

features (α = 0). This result is 12.5% higher than that from the CDF approach

and 15% higher than clustering with permeability-based features. Although the CDF

approach also uses initial flow information, it does not use it as effectively as our

flow-based clustering procedure.

The PSO–MADS optimizations are performed using distributed computing. For

these runs, we had access to a maximum of 50 cores. Each of these optimization runs

requires about 142 iterations. For iterations that use PSO, a total of 150 simulations

(nr = 3 runs for each of 50 PSO particles) are required at each iteration. Each MADS

iteration entails the evaluation of expected NPV at 200 stencil ‘points,’ which means


Table 3.13: Improvement in expected objective (in $106) for the full set of 200 realiza-tions, J(xopt,Mfull) − J(x0,Mfull), evaluated using xopt obtained from optimizationruns with nr = 3 (Example 3.3).

Case CDF α = 0 α = 0.5 α = 1

Best 278.9 297.9 274.7 274.0Intermediate 253.2 281.2 267.7 271.1Worst 231.0 278.9 220.2 200.8Average 254.3 286.0 254.2 248.6

that 600 simulations are performed. The total number of simulations performed in

these optimizations is around 30,000. Because we used 50 cores, this corresponds

to about the elapsed (wall-clock) time required for 600 simulations. Given a fixed

number of cores, the computational time scales essentially linearly with the number

of realizations. Thus, optimization over the full set of 200 realizations would require

around 2×106 flow simulations and would increase the computational time by a factor

of about 67.

We repeated this assessment for optimizations over representative subsets of nr =

6 realizations. The objective function improvement for the full set of realizations is

reported in Table 3.14. The highest average improvement for the full set is obtained

in this case from selection with α = 0.5, though average results using α = 0 are within

0.3% of this result. The use of clustering with permeability-based features is again

the least effective on average (8.7% below the α = 0.5 result). The CDF approach is

also suboptimal compared to clustering with either α = 0.5 or α = 0.

The optimization results in Tables 3.13 and 3.14 are generally consistent with

our findings in Section 3.3. Specifically, we observe that the use of flow-based fea-

tures in the clustering applied for realization selection provides more representative

realizations than does the use of permeability-based features (for problems involving

changing well controls). Use of these more representative realizations in optimizations

provides better results for the target objective function (maximization of J(x,Mfull))

since they better capture the overall behavior of the full set of NR realizations.


Table 3.14: Improvement in expected objective (in $106) for the full set of 200 realiza-tions, J(xopt,Mfull) − J(x0,Mfull), evaluated using xopt obtained from optimizationruns with nr = 6 (Example 3.3).

Case CDF α = 0 α = 0.5 α = 1

Best 309.1 321.1 310.1 302.0Intermediate 290.9 296.1 308.7 280.1Worst 283.5 292.1 293.2 250.5Average 294.5 303.1 304.0 277.5

3.4.3 Additional observations

We also performed optimizations for a different example, involving binary channelized

models, with representative subsets of nr = 3 and nr = 6. We observed similar results

to those reported above; i.e., optimal solutions corresponding to clustering with α = 0

provided the highest average objective function values for the full set of realizations,

while the CDF approach, and clustering with α = 1, did not perform as well. From

these observations and the results in Tables 3.13 and 3.14, we thus recommend the

use of α = 0 for selecting realizations for production optimization under uncertainty.

Some preliminary numerical experimentation should be performed, however, since the

optimal α value may be somewhat problem dependent.

We additionally performed well placement optimization under uncertainty, again

using the PSO–MADS hybrid algorithm [70, 71], to determine the optimal locations

of 10 wells. The objective function was again expected NPV evaluated over all NR

realizations. These optimizations used the models described in Example 3.2, and

subsets of various sizes were considered. In contrast to the results for well control

optimization, we did not observe any of the methods to consistently provide superior

performance for the well location problem. This observation is in agreement with our

results for random configurations in Example 3.2 (Section 3.3.2), where the different

approaches provided similar results.

In the PSO–MADS algorithm (applied for well placement optimization or CLFD),

when the search is performed by the local MADS component, we might expect the

optimization to benefit from the use of some amount of flow information. However,

it may be that the random PSO component is dominant, in terms of sensitivity to


selected realizations, and the overall results reflect this. It is thus possible that an

adaptive-selection procedure, where the subset of realizations is updated during the

course of the optimization, might be beneficial. In such an approach, at early iter-

ations (when the search is dominated by PSO), representative realizations would be

selected using α = 1. At later (MADS-dominated) iterations, however, realizations

would be reselected using flow-based features (α = 0 or α = 0.5). Such an ap-

proach, which should be developed in future work, will require the determination of

appropriate ‘switching’ criteria and α values. This will require detailed investigation

since these quantities may depend on nr, the progress of the optimization, and other

parameters.

3.4.4 Summary

In this chapter, we developed a general method, based on clustering, to select a

representative subset of geological realizations from a large set. Prior to clustering,

each geological realization is represented by a feature vector composed of flow-based

and permeability-based quantities, weighted by a specified α factor. The use of both

full-physics and proxy-type tracer flow information was investigated in this setting.

We introduced and applied a statistical approach for comparing various selection

methods for a range of problems involving new well controls and new well locations,

generated either randomly or by small changes around a base case.

Based on our overall findings in this chapter, we recommend the use of flow-based

clustering (α = 0, using full-physics simulations if feasible) for optimizations involv-

ing well controls, and α = 0.5 for (local) pattern-search-based optimization of well

configurations. For optimization of well locations using a stochastic search procedure

(e.g., PSO or GA), the realization-selection methods are expected to perform compa-

rably. In this case permeability-based clustering is preferable since it does not require

any additional flow simulations. When practical, however, numerical experimentation

along the lines of the assessments presented in this chapter should be performed to

determine the appropriate value of α for the particular problem of interest. The com-

putational cost of such experimentation will often be reasonable, especially if proxy

flow information is used, compared to the cost associated with optimization under

geological uncertainty.

Chapter 4

Optimization of Economic Project

Life for Reservoir Operations

In the optimizations presented in Chapters 2 and 3, we specified the project life

and then optimized NPV. Project life here refers to the timeframe in which the

reservoir operates with the existing wells – actual reservoir life may be extended by

drilling new wells. This type of specification is the usual approach taken in reservoir

optimization problems. In this chapter we consider a much more detailed treatment

of reservoir economics that entails the optimization of economic project life along

with the optimization of well controls.

In particular, we present a new formulation for the joint optimization of economic

project life (EPL) and well controls. This formulation includes the specification of a

a minimum attractive rate of return (MARR or hurdle rate) for the project. After

formulating the joint optimization problem in a nested fashion, we present an iterative

procedure that involves driving the rate of return (which varies with project life) to the

specified MARR, while maximizing NPV. The problem specification and the results

we present should be useful in practical settings because they enable the operator

to plan for infill drilling or redevelopment operations to avoid situations where NPV

increases slowly in time, but the benefit relative to the capital employed is extremely

low.

In the next section, we present our approach for the computation of the rate of

91

92 CHAPTER 4. OPTIMIZATION OF ECONOMIC PROJECT LIFE

return and for the joint determination of EPL and optimal well controls. Computa-

tional results are then presented for two- and three-dimensional reservoir models. We

restrict ourselves here to the production optimization problem, as our formulation

is based on an existing set of wells. Computational results in this chapter are pre-

sented for deterministic problems (with a single realization), though we could treat

geological uncertainty by optimizing over multiple realizations.

4.1 Economic measures and production optimiza-

tion

4.1.1 Net present value computation

In this work we define the NPV objective function for production optimization as

J(x) = −Ccap +

Nl∑l=1

f l

(1 + rd)(tl+t0)/365, (4.1)

where x is the vector of operational settings, Ccap is the capital investment, Nl is the

number of simulation time steps, t0 designates the time lag between capital investment

and the start of production, tl is the simulation time (in days), and rd is the annual

discount rate. The variable f l is the cash flow at simulation time step l, given by

f l =

[NP∑k=1

(poqlo,k − cwpq

lw,k)−

NI∑k=1

cwiqlwi,k −

cf365

]∆tl. (4.2)

In the above equation, NP and NI denote the number of producers and injectors,

respectively, and po, cwp and cwi indicate the oil price and the cost of produced and

injected water (all in $/STB). Variables qlo,k and qlw,k denote oil and water production

rates for producer k at simulation time step l, qlwi,k is the water injection rate of

injector k (all in STBD), cf denotes the fixed costs (accounting for power costs and

human resources) in $/year, and ∆tl is the size (in days) of time step l. Note that

in Chapter 3 we used simplified forms of Eqs. 4.1 and 4.2, as presented in Eq. 3.9.

There we specified a zero discount rate (which allowed us to use cumulative quantities

4.1. ECONOMIC MEASURES AND PRODUCTION OPTIMIZATION 93

instead of production/injection rates at each time step) and zero fixed cost.

The variable x is the vector of decision parameters that specify the operational

settings (here we control BHP) of all wells. In the problem formulation, the project

life is divided into ncs equal intervals, where each interval defines a control step.

4.1.2 Modified internal rate of return and economic project

life

We now describe the computation of the rate of return. As discussed in Chapter 1,

there are various ways of computing this quantity. These include the internal rate

of return (IRR), modified internal rate of return (MIRR) [88], and average internal

rate of return (AIRR) [91]. Among these measures, we have found MIRR to be the

most suitable for determining economic project lifetime in the context of production

optimization. This is because MIRR accounts for capital value and it is sensitive

to project timeframe. IRR, by contrast, is not sensitive to project lifetime, while

computing AIRR requires the specification of the capital value of the project at each

control step, which involves the introduction of additional assumptions.

MIRR is defined as the discount rate at which the future value of return is equal to

the present value of investment. Computation of MIRR requires the specification of

a reinvestment rate, which is the interest rate applied when reinvesting intermediate

positive cash flows. The positive cash flows are then compounded to the end of the

project life, while the negative cash flows are discounted to time zero. This reduces

the sequence of time-varying cash flows to two key values, an equivalent initial cost

or “present value of investment” (designated Cp) and a final income or “future value

of return” (denoted by Rf). In this work, we assume that the reinvestment rate is

equal to the discount rate, i.e., funds are borrowed and reinvested at the same rate.

This assumption is consistent with the NPV computation, which applies the same

discount rate to positive and negative cash flows (see, e.g., Shull [116], Balyeat et al.

[14]). Depending on the specific project circumstances, however, different values could

be specified for the discount and reinvestment rates. We now describe the detailed

computation of MIRR.

The cash flow values in Eq. 4.1 are computed at every simulation time step.

Because time steps vary, it is more convenient to compute MIRR based on control


steps, which are of equal duration in our formulation. We let F ics designate the cash

flow for control step i, computed as

F ics =

∑l, ti−1<tl≤ti

f l. (4.3)

Here ti (= i∆tcs) is the end time for control step i, where control step i = 1, . . . , ncs

corresponds to the period (ti−1, ti), and ∆tcs is the length of each control step (for

i = 1, the control period is (0, t1)). After computing the cash flow for each control

step, we proceed with the computation of Cp and Rf. The present value of investment

(Cp) accounts for the capital investment and negative cash flows, i.e.,

Cp = Ccap +ncs∑

i,F ics<0

F ics(1 + rd)

−(ti+t0)/365, (4.4)

where the summation only includes negative cash flows (if any exist). The future

value of return, Rf, is expressed as

Rf =ncs∑

i,F ics>0

F ics(1 + rd)

(T−ti)/365, (4.5)

which only includes positive cash flows. Here T is the final simulation time (in days).

An example of a cash flow stream in a reservoir optimization problem is shown in

Fig. 4.1. This case corresponds to Example 4.1 below with rd = 0.05, t0 = 180 days,

T = 3240 days, and an initial investment of $360 MM. Cash flows are computed

every 90 days after the start of production. There are no negative cash flows so we

have Cp = $360 MM. Computing Rf, however, involves compounding all positive cash

flows to the end of project timeframe.

As stated earlier, MIRR, designated im, is the discount rate for which the future

value of return, Rf, becomes equal to the present value of investment, Cp. After

computing Rf and Cp, im is obtained from solving

Cp =Rf

(1 + im)(T+t0)/365. (4.6)


0 1000 2000 3000−400

−300

−200

−100

0

100

200

time (days)

cash

flow

($

MM

)

Figure 4.1: Example cash flow stream for a production optimization problem. Cashflows are computed for each 90-day control step.

MIRR can be computed over any interval (0, t). The MIRR at time t, denoted by

im,t, then reflects the rate of return of the project if it is ended at t. Fig. 4.2 shows

the MIRR trajectory for the cash flow stream in Fig. 4.1. Note that, in this and

subsequent figures of this type, when im is negative (at very early time), we set it to

zero on the plot.

We define the economic project life (EPL) as the time for which the MIRR cor-

responding to optimal well settings is equal to a desired value, namely the minimum

attractive rate of return (MARR, designated rmin) or hurdle rate. This EPL is des-

ignated T ∗. The underlying idea here is that when the rate of return of the project

becomes smaller than rmin, the capital value of the project does not economically jus-

tify continuing the existing operations. This is because there are (in concept) more

attractive investment opportunities with rate of return of at least rmin. Before reach-

ing the EPL, the operator should plan for infill drilling, well reconfiguration, or some

other field development option (that satisfies im ≥ rmin).

In this work, we specify rmin = rd+ 0.1, i.e., MARR is the discount rate plus 10%.

This ensures that the rate of return of the project is 10% greater than the market

interest rate. This corresponds to MARR of 0.15 for the example in Figs. 4.1 and 4.2.


0 1000 2000 30000

0.05

0.1

0.15

0.2

0.25

time (days)

MIR

R

Figure 4.2: MIRR trajectory corresponding to cash flow stream in Fig. 4.1. Thedashed vertical line shows the time where the rate of return becomes smaller thanthe specified MARR of 0.15.

4.1.3 Optimization problem statement

We now define the formal optimization problem. For simplicity, we assume that the

control step length is fixed and equal to ∆tcs, and the project life is always a multiple

of ∆tcs, i.e., T = ncs∆tcs. Our goal is to determine the optimal project life along with

the optimal controls that maximize NPV, subject to the constraint that im ≥ rmin.

The decision parameters include the well controls x and the project life T , though

the dimension of x depends on T (through ncs).

We define a nested formulation for this optimization problem:

maxT{maxx(T )

J(x,m)},

s.t. |im,T − rmin| ≤ ε, xl ≤ x ≤ xu,(4.7)

where we write x(T ) to emphasize that the number of controls and the values of the

controls change with T . Variables xu and xl define the vectors of upper and lower

bounds, and im,T designates the value of MIRR at T . In order to limit the number

of outer iterations, we require |im,T − rmin| ≤ ε, rather than im,T = rmin. Here we set


ε = 0.0025. We let x∗ and T ∗ designate the optimal solution of Eq. 4.7.

The inner maximization in Eq. 4.7 is performed by use of SNOPT [53], which is

a gradient-based optimization algorithm, though other optimization methods could

also be used. While we could also apply a formal optimization algorithm for the

outer loop (determination of T ∗), we have found that a simple graphical/interpolation

approach performs well and is computationally efficient. Our procedure is outlined

in Algorithm 4.1.

Algorithm 4.1 Joint optimization of economic project life and well controls

1: Specify an initial guess for project life, T .2: Specify the initial guess for x such that it spans the current estimate for project

life, T .3: Perform the inner optimization (maximize the NPV objective in Eq. 4.1) for the

timeframe (0, T ). If the maximum NPV is obtained at t < T , reduce the projectlife to correspond to the maximum NPV and repeat this step.

4: Compute im,T .5: if |im,T − rmin| ≤ ε then6: accept x and T as x∗ and T ∗.7: else if im,T < rmin then8: set T to the time corresponding to the end of the control step during which

im,t = rmin. Then go to Step 3.9: else if im,T > rmin then

10: increase T by a specified multiple, or by extrapolating im versus t out to im =rmin. Then go to Step 3.

11: end if

Step 8 of Algorithm 4.1 is illustrated in Fig. 4.2. In the plot, for T = 3240 days,

im,T = 0.135, which is smaller than rmin (= 0.15). The new project life is then set

to T = 2700 days. This is because, in the figure, the control step spanning from

t = 2610 days to t = 2700 days is the control step during which im,t = 0.15 (im,t is

also equal to 0.15 at an earlier time, but this corresponds to a lower NPV). We then

return to Step 3 to optimize the controls with this new T . In the examples considered

here, we typically require 2-3 outer iterations. Each of these entails about 60 inner

iterations.

Algorithm 4.1 defines our procedure for the case where NPV is still increasing at

the time when im,t = rmin. If NPV is decreasing when im,t = rmin, then T ∗ would


correspond to the time when NPV is a maximum. We have not observed this behavior

in any of our runs, but it could occur if capital costs are very low. There may be

other situations, such as those exhibiting one or more local maximums in NPV, for

which additional modification of Algorithm 4.1 is required.

4.2 Computational results

In this section, we present computational results for two different examples where

we apply our procedure to determine EPL and optimal BHPs. Both examples in-

volve two-phase oil-water flow. Oil and water viscosities are specified as 3 cp and

1 cp, respectively, and both fluids are considered to be incompressible. The rock

compressibility is specified to be 10−3 bar−1. Relative permeability curves are the

same as those used in Chapter 3 (Fig. 3.2). The reservoir initially contains oil and

connate water, with irreducible water saturation of Swc = 0.1. Deterministic opti-

mization (with a single realization) is considered in both cases. Flow simulations are

performed with Stanford’s AD-GPRS [157]. We apply the adjoint method within

AD-GPRS to compute the gradient of NPV with respect to the controls.

4.2.1 Example 4.1: 2D bimodal reservoir

The reservoir model in this example is two-dimensional and contains 100× 100 grid

blocks. The grid block dimensions are ∆x = ∆y = 100 ft, ∆z = 15 ft. The isotropic

log-permeability field, shown in Fig. 4.3, displays a bimodal distribution. The two

modes are of mean (in log-permeability) of 8 for the sand facies and 3 for shale facies,

with corresponding variances of 0.4 and 0.8. Porosity of the sand facies is equal to

0.2 and for the shale facies it is 0.1. There are 12 producers and six water injection

wells in this reservoir. All wells are operated under BHP control. Initial reservoir

pressure is 310 bar. Optimization parameters are shown in Table 4.1. The maximum

injection and total liquid production rates for a well are specified to 10,000 STBD.


x

y

1 2 3 4

5 6 7 8

9 10 11 12

13 14 15

16 17 18

20 40 60 80 100

20

40

60

80

100

2

4

6

8

Figure 4.3: Log-permeability field with well locations. Circles denote producers andtriangles denote injectors (Example 4.1).

Determination of optimal solution: x∗ and T ∗

We follow the procedure presented in Section 4.1.3 to determine EPL as the project

life for which MIRR is essentially equal to MARR, with well settings such that NPV

is maximized. For the discount rate of rd = 0.1, rmin is specified as 0.2.

We first specify the project life to T = 3960 days (corresponding to ncs = 44

with ∆tcs = 90). The number of control parameters n is ncsnw = 792, where nw

is the number of wells. The initial guess for the well BHPs (x) corresponds to the

average of the lower and upper bounds. The NPV for the initial guess is $-287.2 MM.

The SNOPT optimization with T = 3960 days converges to an optimal NPV value

of $315.0 MM. The NPV versus time trajectory for the optimal solution, shown in

Fig. 4.4(a), achieves its maximum value of $323.2 MM at t = 3240 days. Based on this

solution, it is clear that the reservoir operations should not proceed beyond t = 3240

days.

Following Algorithm 4.1, we specify the project life to T = 3240 days (ncs =

36) and repeat the optimization. We use the same initial guess for x as was used


Table 4.1: Economic parameters and BHP ranges for all examplesParameter Value

Ccap $360 MMcf $1 MM/yearpo $70 STBcwp $7 STBcwi $7 STBrd 10%rmin 20%t0 180 days∆tcs 90 daysProd BHP range 70− 310 barInj BHP range 310− 483 bar

earlier. Another choice for this initial guess would be the first 36 settings from

the inner optimization for T = 3960 days. We use the former approach because

limited numerical experimentation showed that this approach typically leads to a

better solution. The effect of the initial guess on SNOPT convergence to different

(locally optimal) solutions will be investigated later.

The NPV trajectory from the inner optimization with T = 3240 days appears to

increase monotonically (Fig. 4.4(b)), and it corresponds to a final NPV of $323.3 MM.

This value is essentially the same as the NPV value of $323.2 MM at t = 3240 days

from the first optimization. From Fig. 4.4(b), it is evident that the NPV versus time

curve is very flat at late time. This observation can be quantified by computing the

yearly cash flow percentage and the rate of return of the project, which are discussed

next.

The cash flow stream from the inner optimization with T = 3240 days, computed

for each control step, is shown in Fig. 4.5. We also present the percentage of cash

flow from each year (Fig. 4.6). Fig. 4.6(a) shows that more than 50% of positive cash

flow is obtained in the first year of production. From Fig. 4.6(b), it is evident that

the total contribution from the last three years of production is less than 4.5% of the

total cash flow.

This observation is closely related to the rate of return (im) of the project as a

function of time. We present this quantity for each production interval (0, t), where


0 1000 2000 3000 4000

−200

0

200

400

time (days)

NP

V (

$ M

M)

(a) Solution for T = 3960 days

0 1000 2000 3000

−200

0

200

400

time (days)

NP

V (

$ M

M)

(b) Solution for T = 3240 days

Figure 4.4: NPV trajectory from inner optimization with (a) initial guess T , and (b)T for which NPV in (a) is the maximum. The dashed vertical line in (a) shows thetime where the maximum NPV is obtained (Example 4.1).


0 1000 2000 3000−400

−300

−200

−100

0

100

200

time (days)

cash

flow

($

MM

)

Figure 4.5: Cash flow stream for the optimal controls with T = 3240 days (Exam-ple 4.1).

t = i∆tcs and 1 ≤ i ≤ ncs, in Fig. 4.7. As the cash flow is positive at all control steps

(Fig. 4.5), the present value of investment, Cp, only includes the capital investment,

which is $360 MM. It is evident in Fig. 4.7 that MIRR reaches a maximum of 0.257 at

900 days. The value of MIRR for the entire project (until 3240 days) is 0.176, which is

smaller than the specified MARR (0.2). Therefore, in accordance with Algorithm 4.1,

we reduce T and repeat the optimization. We determine the new project life as the

end time of the control step during which im(t) reaches 0.2 (Step 8 in Algorithm 4.1).

This corresponds to t = 2340 days in Fig. 4.7.

We thus specify T = 2340 days and repeat the optimization. The NPV trajectory

from optimization with T = 2340 days is shown in Fig. 4.8(a). In Fig. 4.8(b), the

NPV trajectories based on the two solutions with T = 2340 and T = 3240 days are

shown for the period after 2000 days (note that this plot ends at 2400 days). The

optimal NPV from optimization with T = 2340 days is $311.6 MM, which is greater

than $308.6 MM (this latter value is the NPV at t = 2340 days from optimization

with T = 3240 days). This reiterates the fact that the optimization must be repeated

when the project life changes.

The MIRR trajectory computed over the (optimal) EPL using the optimized well


1000 2000 30000

20

40

60

time (days)

% o

f tot

al p

ositi

ve c

ash

flow

(a) T = 3240 days

2200 2400 2600 2800 3000 32000

1

2

3

4

5

time (days)

% o

f tot

al p

ositi

ve c

ash

flow

(b) t ≥ 2160 days

Figure 4.6: (a) Cash flow percentage, computed yearly, versus time for optimal con-trols with T = 3240 days, and (b) magnification for the last three years (Example 4.1).


0 1000 2000 30000

0.05

0.1

0.15

0.2

0.25

0.3

time (days)

MIR

R

Figure 4.7: MIRR trajectory for optimal controls with T = 3240 days, computed forthe period (0, t) (Example 4.1).

settings is shown in Fig. 4.9. The MIRR for the optimal solution at T = 2340 days

is 0.202, which is sufficiently close to our specification for MARR (0.2). We thus

accept this as the optimal solution, which means T ∗ = 2340 days and the optimal

well controls from this run correspond to x∗. The optimal time-varying BHPs for

some of the wells are shown in Fig. 4.10.

The optimization results in terms of NPV and cumulative oil and water production

(Qo, Qw) and injection (Qwi) are shown in Table 4.2. Although the NPV could

be increased by 3.75% if we continued the operations beyond t = 2340 days (to

3240 days), the relative increase in NPV is small and it does not justify the capital

value invested in the project. This is reflected in the rate of return of the project

quantified by MIRR (im = 0.176 for T = 3240 days as opposed to 0.202 for T ∗ =

2340 days). In addition, for T = 3240 days, the amount of produced and injected

water are greater by 54% and 35%, respectively, than those for T ∗ = 2340 days.

The water saturation maps from these two solutions, shown in Fig. 4.11, highlight

the increased use of water in the T = 3240 day case, consistent with the results in

Table 4.2.


0 500 1000 1500 2000

−200

0

200

400

time (days)

NP

V (

$ M

M)

(a) Solution for T ∗ = 2340 days

2000 2100 2200 2300 2400290

300

310

320

time (days)

NP

V (

$ M

M)

T=3240T=2340

(b) Solutions for T ∗ = 2340 days and T = 3240 days

Figure 4.8: (a) NPV trajectory for the optimal solution with T ∗ = 2340 days, and (b)magnification of NPV trajectory for the period of (2000, 2400) days from solutions forT ∗ = 2340 days (which is the optimal solution) and T = 3240 days (Example 4.1).

Table 4.2: NPV ($ MM), corresponding MIRR and fluid production/injection (MMSTB) for optimal controls with different project life. The solution with T ∗ =2340 days represents an optimum (Example 4.1).

T (days) MIRR NPV Qo Qw Qwi

2340 0.202 311.6 2.66 2.69 5.223240 0.176 323.3 3.05 4.14 7.053960 0.164 315.0 3.27 5.35 8.46


0 500 1000 1500 20000

0.05

0.1

0.15

0.2

0.25

0.3

time (days)

MIR

R

Figure 4.9: MIRR trajectory for the optimal solution (T ∗ = 2340 days) computed forthe period (0, t).

0 1000 2000

100

200

300

Time (Days)

BH

P(b

ar)

Well 2Well 6Well 8

(a) producers

0 1000 2000300

350

400

450

Time (Days)

BH

P(b

ar)

Well 13Well 15Well 16

(b) injectors

Figure 4.10: Optimal controls x∗ for three producers and three injectors correspondingto T ∗ = 2340 days (Example 4.1).


(a) T ∗ = 2340 days (optimal) (b) T = 3240 days (suboptimal)

Figure 4.11: Final Sw distribution from optimal controls with different T . The wellconfiguration is also shown, with red circles denoting producers and blue circles indi-cating injectors (Example 4.1).

Sensitivity of optimal solution to initial-guess BHPs

The production optimization problem may have many local optima, and the solution

found will in general depend on the initial guess used in the SNOPT algorithm. We

therefore perform three additional inner loop optimization runs, with T = 2340 days,

using different initial guesses. These three initial guesses for x are generated such that

the BHP control of each well is specified to a random value (selected from a uniform

distribution between the corresponding lower and upper bounds) for five consecutive

control steps or until the end of project life. Fig. 4.12 shows an example of one such

set of initial BHP profiles.

Table 4.3 provides a summary of the optimization results, with the first row cor-

responding to the case presented earlier (the initial guess in that case corresponds to

the average of the lower and upper bounds). These results indicate that the optimal

NPV varies by about 6% based on the initial set of controls, though the variation is

small for MIRR (a range of 0.006). The best results here correspond to Run 2, which

provides both the highest NPV and the highest MIRR.


0 500 1000 1500 2000

100

150

200

250

300

time (days)

BH

P (

bar)

Well 1 Well 2 Well 3

Figure 4.12: Initial-guess BHP profiles for three producer wells (T = 2340 days).

Table 4.3: Optimal NPV, corresponding MIRR and fluid production/injection (MMSTB) from optimal controls with different initial-guess BHPs with T = 2340 days(Example 4.1).

Run # MIRR NPV ($ MM) Qo Qw Qwi

1 0.202 311.6 2.66 2.69 5.222 0.205 315.3 2.68 2.85 5.223 0.200 300.4 2.66 2.96 5.354 0.199 296.7 2.74 3.35 5.95


Table 4.4: Optimal NPV and the corresponding MIRR for different (specified) projectlife (Example 4.1).

T (days) NPV ($ MM) MIRR

1800 270.0 0.2181980 297.5 0.2162160 304.3 0.2082340 311.6 0.2022520 315.4 0.1962700 318.6 0.1912880 321.5 0.1853060 322.6 0.1813240 323.3 0.176

Relationship between NPV and rate of return for optimal controls

In order to investigate further the relationship between optimal NPV and MIRR, we

repeat the inner loop (well control) optimizations for a range of specified T values,

from T = 1800 days to T = 3240 days, in 180-day intervals. These optimizations are

performed using the Run 1 initial guess. After each optimization, the corresponding

MIRR value is computed. Note that only the solution with T = T ∗ = 2340 days is

optimal in the sense of Eq. 4.7. The other solutions correspond to the use of optimized

controls for the specified value of T .

The resulting plot of MIRR versus optimal NPV (Fig. 4.13) displays a conflicting

relationship between the two quantities. Numerical values, reported in Table 4.4,

additionally demonstrate that MIRR decreases with project life. This is because the

cash flows at late times are small compared to the capital value of the project, and

this acts to reduce MIRR.

Sensitivity of optimal solution to discount rate

We now investigate the sensitivity of EPL and optimal NPV to the discount rate,

rd. Following the procedure presented in Algorithm 4.1, we determine T ∗ and x∗ for

three additional values of rd. For each rd, we specify MARR as rmin = rd + 0.1. We

reiterate that MIRR is computed based on the assumption that the reinvestment rate

is equal to rd.


280 300 320

0.17

0.18

0.19

0.2

0.21

0.22

0.23

NPV ($ MM)

MIR

R

Figure 4.13: Relationship between MIRR and optimal NPV for optimizations withdifferent specified project life. Only the solution corresponding to MIRR= 0.202,NPV=$311.6 MM, is optimal in the sense of Eq. 4.7 (Example 4.1).

Results are presented in Table 4.5. There we see that T ∗ and optimal NPV

decrease with increasing rd. This is because by increasing the discount rate, the

relative impact of small cash flows at late project life (which appear in Rf, the future

value of return) is reduced. As a consequence, EPL decreases. Note, however, that

the larger T ∗ associated with small values of rd lead to greater optimized NPVs.

Table 4.5: Optimal NPV and the corresponding MIRR and EPL (T ∗) from optimiza-tions with different discount rates (Example 4.1).

rd MARR MIRR T ∗ (days) NPV ($ MM)

1% 0.11 0.113 2970 478.62.5% 0.125 0.124 2880 429.85% 0.15 0.151 2700 390.310% 0.20 0.202 2340 311.6


4.2.2 Example 4.2: 3D binary reservoir

This example involves a three-dimensional binary channelized reservoir model, defined

on a grid of dimensions 50 × 50 × 6 (see Fig. 4.14). The grid block dimensions are

∆x = ∆y = 100 ft, ∆z = 15 ft. The isotropic horizonal permeabilities for sand and

shale are, respectively, 500 mD and 10 mD. The ratio of vertical permeability to

horizontal permeability is specified to be constant and equal to 0.2. The sand and

shale porosity are 0.2 and 0.1, respectively. There are five horizontal producers (in

Layers 1 and 2) and six vertical injectors (in Layers 4-6), as shown in Fig. 4.14. The

initial reservoir pressure is 310 bar.

The capital investment of the project is specified to be Ccap = $700 MM. The

discount rate is rd = 0.1, while the time lag between investment and the start of

production is t0 = 180 days. MARR is specified to be 0.2. Other optimization and

simulation parameters are identical to those in Example 4.1 (e.g., ∆tcs = 90 days).

We follow the procedure presented in Algorithm 4.1 to determine the optimal EPL

and optimal controls. We first specify T = 4950 days and optimize the NPV objective.

The NPV and MIRR trajectories are shown in Figs. 4.15 and 4.16. The maximum

NPV ($850 MM) is obtained at 4050 days. The value of MIRR at T = 4950 days is

0.164, which is smaller than 0.2. The control step where MIRR becomes less than

0.2 extends from 2970 days to 3060 days. Therefore, we specify the project life to

T = 3060 days and repeat the well control optimization.

The optimal NPV (at convergence of SNOPT) is $822 MM for T = 3060 days. As

Fig. 4.17 shows, the value of MIRR at T = 3060 days is equal to 0.200, which is the

exact value of MARR. Thus we have reached the optimum, meaning T ∗ = 3060 days

and the well controls associated with the result in Fig. 4.17 correspond to x∗. The

final water saturation maps for the optimal controls and optimal project life are

shown in Fig. 4.18. As none of the injectors is perforated in the top three layers,

Layer 1 displays large regions of unswept oil. Most of the other layers, however, show

a reasonable degree of sweep.


x

y

1 2

3 4

Layer 1

10 20 30 40 50

10

20

30

40

50 0

0.2

0.4

0.6

0.8

1

(a) Layer 1

x

y

5

Layer 2

10 20 30 40 50

10

20

30

40

50 0

0.2

0.4

0.6

0.8

1

(b) Layer 2

x

y

Layer 3

10 20 30 40 50

10

20

30

40

50 0

0.2

0.4

0.6

0.8

1

(c) Layer 3

x

y

67 8

Layer 4

10 20 30 40 50

10

20

30

40

50 0

0.2

0.4

0.6

0.8

1

(d) Layer 4

x

y

67 8

910

11

Layer 5

10 20 30 40 50

10

20

30

40

50 0

0.2

0.4

0.6

0.8

1

(e) Layer 5

x

y

67 8

910

11

Layer 6

10 20 30 40 50

10

20

30

40

50 0

0.2

0.4

0.6

0.8

1

(f) Layer 6

Figure 4.14: Binary permeability field, with red indicating sand facies (permeabilityof 500 mD), and blue indicating shale facies (permeability of 10 mD). The well con-figuration, which includes five horizontal producers, denoted by circles and lines, andsix vertical injectors, denoted by triangles, is also shown (Example 4.2).


0 1000 2000 3000 4000 5000

−500

0

500

1000

time (days)

NP

V (

$ M

M)

Figure 4.15: NPV trajectory for optimal controls with T = 4950 days (Example 4.2).

0 1000 2000 3000 4000 50000

0.1

0.2

0.3

time (days)

MIR

R

Figure 4.16: MIRR trajectory for optimal controls (T = 4950 days) computed for theperiod (0, t). Dashed horizontal line shows the value of MARR (Example 4.2).


0 1000 2000 30000

0.1

0.2

0.3

time (days)

MIR

R

Figure 4.17: MIRR trajectory for optimal solution (T ∗ = 3060 days) computed forthe period (0, t) (Example 4.2).

Layer 1, 3060 Days

1 2

3 4

10 20 30 40 50

10

20

30

40

50 0.2

0.3

0.4

0.5

0.6

0.7

0.8

(a) Layer 1

Layer 2, 3060 Days

5

10 20 30 40 50

10

20

30

40

50 0.2

0.3

0.4

0.5

0.6

0.7

0.8

(b) Layer 2

Layer 3, 3060 Days

10 20 30 40 50

10

20

30

40

50 0.2

0.3

0.4

0.5

0.6

0.7

0.8

(c) Layer 3Layer 4, 3060 Days

67 8

10 20 30 40 50

10

20

30

40

50 0.2

0.3

0.4

0.5

0.6

0.7

0.8

(d) Layer 4

Layer 5, 3060 Days

67 8

910

11

10 20 30 40 50

10

20

30

40

50 0.2

0.3

0.4

0.5

0.6

0.7

0.8

(e) Layer 5

Layer 6, 3060 Days

67 8

910

11

10 20 30 40 50

10

20

30

40

50 0.2

0.3

0.4

0.5

0.6

0.7

0.8

(f) Layer 6

Figure 4.18: Final water saturation maps from optimal solution (x∗ with T ∗ =3060 days). (Example 4.2).

4.3. SUMMARY 115

4.3 Summary

In this chapter, we introduced a procedure for the simultaneous determination of

optimal project life (referred to as economic project life, EPL) and optimal well

controls. Because the optimal controls depend on the specified project life T , the

optimal controls must be recomputed if T is varied. We formulated this problem as

a nested optimization in which we maximize NPV while requiring that the modified

internal rate of return of the project is (essentially) equal to the minimum attractive

rate of return (MARR or hurdle rate). The outer loop of the optimization entails

optimization of T , while the inner loop involves optimization of the controls given the

current T . We successfully applied the methodology for determining optimal EPL

and optimal controls to two examples involving a two-dimensional bimodal reservoir

model and a three-dimensional binary model with horizontal production wells.

In this work we only considered deterministic optimization (with a single real-

ization). The methodology, however, can be extended for optimization under un-

certainty. For multiple realizations, the optimization with sample validation (OSV)

framework (presented in Chapter 2) can be applied to accelerate the computations.

The realization selection method, presented in Chapter 3, can be used within the

OSV framework. Computation of rate of return for multiple realizations, however,

may require additional treatments, such as those presented by Hazen [61].

Chapter 5

Summary, Conclusions and Future

Work

In this work, we introduced and applied advanced techniques relevant for reservoir op-

timization under uncertainty. We first developed a general framework for closed-loop

field development optimization under uncertainty that incorporates data from new

wells as they are drilled. We then investigated two particular aspects of closed-loop

reservoir optimization as standalone topics. These include selecting a representa-

tive subset from a large set of geological realizations and the joint optimization of

economic project life (EPL) and well settings.

5.1 Conclusions

The key contributions from our work on closed-loop field development (CLFD) are

as follows:

• We introduced a general methodology for closed-loop field development (CLFD).

The framework includes optimization, data collection and history matching per-

formed in a repeated sequence. Uncertainty is taken into account by history

matching, and optimizing over, multiple geological realizations. The optimiza-

tion step, which uses the recently developed PSO-MADS hybrid algorithm,

allows the determination of well type, locations and controls for new wells, and

117

118 CHAPTER 5. SUMMARY, CONCLUSIONS AND FUTURE WORK

(future) well controls for existing wells. The history matching step entails the

use of the RML method, which is applied within an adjoint-gradient setting

using AD-GPRS.

• A key feature of CLFD is that, at each optimization step, the full development

plan is optimized, i.e., the location of each well is determined based on the

fact that it is one well in a sequence. The future field development plan (type,

locations and controls for planned wells) does, however, change as new data are

collected.

• We showed that a greedy approach for field development optimization, in which

the location, type and controls of each well are optimized independently (one

at a time), may lead to a suboptimal solution. This further motivates the need

to optimize the parameters associated with all wells simultaneously.

• We introduced an optimization with sample validation (OSV) procedure as

a computationally efficient means for optimizing under geological uncertainty.

In OSV, the optimization at each CLFD step is first performed on a (small)

subset of representative realizations. A validation step is then applied to assess

(quantitatively) whether the selected subset is sufficiently representative of the

entire set of realizations. If not, a larger subset of realizations is selected and the

CLFD optimization step is repeated. At each OSV step, a representative subset

of realizations was reselected based on the NPVs computed using the current

best field development plan, though other flow or permeability information could

also be used for this selection.

Recently, Aliyev and Durlofsky [4] incorporated the OSV procedure in their

multilevel optimization approach to accelerate the joint optimization of well

locations and controls under geological uncertainty.

• We presented appropriate CLFD history matching treatments for (Gaussian)

models described by two-point geostatistics and channelized models described

by multipoint geostatistics. The CLFD procedure for channelized models, pre-

sented in Appendix A, involves a two-stage conditioning where hard data are

5.1. CONCLUSIONS 119

first incorporated by use of geostatistical simulation, and then production data

are integrated through O-PCA-based history matching.

Our main findings regarding the selection of representative models are:

• We developed and tested a new framework for selecting a representative subset

of realizations from a large set, for the purpose of optimization or decision mak-

ing under uncertainty. A low-dimensional flow-response vector was defined to

efficiently quantify base-case flow simulation results for a particular realization,

and principal component analysis was applied to concisely represent the het-

erogeneous permeability field. We introduced a clustering algorithm that can

use flow-based or geology-based information, or a combination of the two (with

arbitrary weightings), for realization selection.

• Three different realization-selection approaches, which can all be defined within

our overall framework, were applied for four different cases. These cases in-

volved the (separate) consideration of new well settings and new well locations,

and well-parameter vectors (which define new well settings or locations) that

were generated either randomly (as in PSO or GA), or through systematic per-

turbations around a base case (as in pattern search methods). The selection

methods were assessed in terms of the difference in flow response between the

selected subset and the full set of realizations. Many new well-parameter vec-

tors were considered, and results were expressed concisely in terms of median

performance. We found that, for all cases except for random configurations of

new wells, the use of clustering with flow-based information was clearly benefi-

cial. For random well configurations, the various selection approaches provided

similar performance, though all approaches outperformed random selection.

• We considered the use of both full-physics simulations as well as proxy (tracer-

type) flow information for obtaining flow-based features in clustering. Our

results showed that simplified (proxy) flow simulations provided a viable alter-

native for obtaining flow-based features for clustering for the cases considered.


• We investigated the use of representative subsets selected by the various ap-

proaches in production optimization under uncertainty. The optimization re-

sults were consistent with our findings for random or perturbation-based well-

parameter vectors, and demonstrated the benefit of using flow-based informa-

tion for realization selection.

Our work on the simultaneous determination of optimal operational settings and

optimal project life (EPL) leads us to the following observations:

• We introduced a nested formulation for the joint optimization of well controls

and economic project life (EPL), with EPL optimized in the outer loop, and

the associated well controls in the inner loop.

• Our methodology determines the optimal project life and well settings such that

the maximum NPV is obtained at the end of project life, and the rate of return

of the project is (essentially) equal to the hurdle rate or the minimum attractive

rate of return (MARR). This avoids situations with negligible increase in NPV

at late times.

• Our computational results indicated that increasing reservoir life may increase

the optimal NPV, but decrease the rate of return (as reflected by modified

internal rate of return). This highlights a fundamental trade off between these

two quantities.

5.2 Future work

There are many directions that could be pursued in areas relevant to CLFD. Our

suggestions for future research are as follows:

• Because the CLFD procedure is computationally demanding, it will be useful to

develop surrogate models to accelerate the optimization runs, or to implement

more efficient optimization methods.

• Value of information should also be included in CLFD. Relevant work in this

area has been presented in [15, 38].

5.2. FUTURE WORK 121

• In order to protect against downside risk due to geological uncertainty, bi-

objective optimization [69] could be incorporated into CLFD. We could then,

for example, minimize a risk objective while maximizing expected performance.

• In our current CLFD implementation, we assume the geological ‘scenario’ is

known and then consider uncertainty in the permeability distribution. It will

be of interest to consider situations where the geological scenario (i.e., training

image) is also uncertain and must be determined as part of the CLFD history

matching step. This could be accomplished using, e.g., the approach presented

by [107, 108].

• Treatments for accommodating 4D seismic data, in addition to production data,

should be introduced into the general CLFD framework.

• It will be useful to develop an adaptive realization-selection procedure that is

appropriate for use in challenging well placement (or joint well placement and

control) optimizations. Within the context of a hybrid stochastic – local search

algorithm, such as PSO–MADS, this would entail the use of different weightings

for flow versus permeability features as the optimization proceeds. Treatments

could also be devised for the nested optimization approach presented in [19]. In

this case, realizations could be selected based on the well locations prescribed

in the outer loop of the optimization.

• It will also be of interest to investigate the application of our selection ap-

proaches to realistic and more complex recovery processes involving, e.g., three-

phase flow or steam injection. Uncertainty in other simulation quantities, such

as relative permeability parameters, should also be considered.

• The procedures described here for selection of representative realizations should

be implemented within the CLFD framework. They could also be applied for

related subsurface flow problems such as CO2 sequestration or groundwater

management.

• In this work, we captured the flow response of a realization by dividing the

reservoir life into a few time intervals. It may be worth investigating the ap-

plication of functional data analysis (see, e.g., [105]) to define an alternative


flow-response vector that concisely captures flow simulation results.

• Our methodology for the joint optimization of project life and well controls

was presented for deterministic problems. It will be of interest to extend this

methodology to include multiple realizations. Treatments should be developed

to compute an appropriate rate of return when multiple realizations are consid-

ered.

• An approach for the joint optimization of well locations and control, together

with project life, could be developed. Along these lines, our method for the

joint optimization of project life and well controls can be incorporated in the

nested formulation of [19]. The inner optimization in this case will determine

the optimal well controls and project life, while the outer optimization provides

the well locations.

Nomenclature

Abbreviations

AD-GPRS Automatic Differentiation-based General Purpose Research Simulator

BHP bottom-hole pressure

CDF cumulative distribution function

CLFD closed-loop field development

CLRM closed-loop reservoir management

EPL economic project life

GA genetic algorithm

IRR internal rate of return

MADS mesh adaptive direct search

MAP maximum a posteriori

MARR minimum attractive rate of return

MIRR modified internal rate of return

NPV net present value

OSV optimization with sample validation

O-PCA optimization-based PCA

PCA principal component analysis

PDF probability density function

PSO particle swarm optimization

PS pattern search

RML randomized maximum likelihood

SCP sequential convex programming

SNOPT sparse nonlinear optimizer

STB stock tank barrels

123

124 NOMENCLATURE

STBD stock tank barrels per day

TSVD truncated singular value decomposition

Variables

cf fixed costs

cwi cost of injected water

cwp cost of produced water

C covariance matrix

Ccap capital investment

Cp present value of investment

Cw drilling cost per well

d dissimilarity measure between mean flow response vectors

D set of dissimilarity values between full set and a representative subset

d observed data

f cash flow at every simulation time step

Fcs cash flow at every control step

g simulated data

im modified internal rate of return (MIRR)

J net present value

J expected net present value

kro relative permeability for oil phase

krw relative permeability for water phase

l number of PCA parameters

L number of realizations for obtaining PCA parameters

mD median of set D

m geological realization

m mean of permeability realizations

M , M matrix of geological realizations

n number of decision parameters

nr number of representative realizations

ns maximum number of OSV subproblems

nw maximum number of wells

NOMENCLATURE 125

NI number of injector wells

NP number of producer wells

NR number of realizations in the full set

po oil price

q well flow response vector

Q total liquid production/injection

r flow-response vector

r mean flow-response vector

rd discount rate

rmin minimum attractive rate of return (MARR) or hurdle rate

Rf future value of return

RI relative improvement

S mismatch in history matching

SN normalized mismatch in history matching

Sw water saturation

t time

t0 time lag

T project life

T ∗ optimal economic project life (EPL)

U matrix of left singular vectors

V matrix of right singular vectors

x vector of decision variables

x∗ optimal well controls

Z feature matrix for clustering

Zf flow-based feature matrix for clustering

Zp permeability-based feature matrix for clustering

Z normalized feature matrix for clustering

Greek Symbols

α weighting factor for permeability-based features in clustering

Λ diagonal matrix of singular values

µ viscosity

126 NOMENCLATURE

θ validation criterion

ξ vector of PCA parameters

Subscripts/Superscripts

b number of base flow problems for obtaining flow-based features in clustering

cs control step

d data

full full set of geological realizations

h hard data

l lower bound

m median

m model

o oil

p production data

rep representative set of geological realizations

u upper bound

w water

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Appendices

145

Appendix A

Closed-Loop Field Development

for Channelized Models

In the work presented in Chapter 2, we represented reservoir properties in terms of

two-point spatial statistics. A more recent approach to geological modeling is the use

of multipoint spatial statistics (MPS). In this case, a training image that represents

key geological patterns, such as sinuous channels with a specified orientation and

thickness, or deltaic fans of a particular geometry is introduced [139]. Realizations

are then generated that honor available hard data and are consistent with the training

image. Several specific approaches for generating complex channelized models have

been presented [131, 130].

In this Appendix, we describe the application of closed-loop field development to

channelized systems where MPS modeling is used. The workflow is similar to that

described in Chapter 2, though the history matching, described in the next section, is

different. The modified CLFD procedure will then be applied to a binary channelized

system.

A.1 History matching for channelized models

The CLFD history matching for a channelized reservoir described by MPS is different

from that described for Gaussian models in Chapter 2. For history matching channel-

ized models, we apply the optimization-based principal component analysis (O-PCA)

147

148 APPENDIX A. CLFD FOR CHANNELIZED MODELS

parameterization [139, 140]. The CLFD history matching for channelized reservoirs

consists of two steps. In the first step, new conditional realizations are generated

using a geostatistical approach with hard data from all wells (including the most

recent well). Here we use the multiscale cross-correlation simulation (MS-CCSIM)

geostatistical algorithm [131] for generating these realizations. In the second step,

the O-PCA-based sampling method [140] is applied for history matching NR realiza-

tions to production data.

We now briefly describe these procedures. The O-PCA method requires generating

L realizations of the permeability field conditioned to hard data at well locations. We

use L = 1000 in this work. New realizations must be generated at each CLFD

step since new conditioning data become available as we proceed in time. Then the

centered matrix of realizations, Xc, is computed,

Xc =[m1 − m . . . mL − m

], (A.1)

where m is the mean of the L realizations. A truncated SVD of Xc is then computed

as UlΛlVTl , where l < L. Given a random l-dimensional vector ξ ∼ N (0, 1), a new

realization can be generated by solving the following optimization problem:

m = argminz{‖UlΛlξ + m− z‖22 + γR}, (A.2)

where R is a regularization term that is specified such that the realization is generated

consistent with the training image and γ is the weighting factor (see [139] for details).

In the O-PCA sampling method, a history matched RML realization is generated

by minimizing the following objective function

S(ξ) =1

2(ξ − ξuc)

T (ξ − ξuc) +1

2(gp(ξ)− dpuc)

TC−1d,p(gp(ξ)− dpuc), (A.3)

where ξuc corresponds to a projected MPS realization, i.e., ξuc = Λ−1l UT

l (muc − m).

Here muc is a realization that is unconditioned to production data, but conditioned

to hard data. Note that the hard data mismatch term, (gh(m)−dhobs)TC−1

d,h(gh(m)−dhobs), in Eq. 2.7, does not appear since hard data are already honored in the realiza-

tions and thus in the O-PCA representation. Minimization of Eq. A.3 is performed

A.2. EXAMPLE A1: CLFD FOR A CHANNELIZED MODEL 149

by SNOPT [53]. Note that for each trial ξ, the vector m is obtained from solving

Eq. A.2. Predicted data, gp, are then generated by performing a reservoir simulation

run using this m. The mismatch objective function in Eq. A.3 is then computed. The

gradient of S with respect to m is constructed through an adjoint solution, which is

then projected using the chain rule to obtain derivatives with respect to ξ. For more

detail on O-PCA-based history matching, please refer to [138].

A.2 Example A1: CLFD for a channelized model

This example involves a binary channelized reservoir model described on a two-

dimensional uniform grid of dimensions 60× 60 with ∆x = ∆y = 100 ft, ∆z = 15 ft.

A binary channelized training image from [138] is taken as the prior geological de-

scription, from which a set of unconditional realizations are generated using the MS-

CCSIM geostatistical algorithm [131]. The realizations are not constrained to honor

the sand/shale ratio observed in the training image. The sand permeability is 500 mD,

while the shale permeability is 10 mD. The true permeability field, along with an ini-

tial guess for the well locations (x0), is shown in Fig. A.1. Three prior realizations of

the permeability field are shown in Fig. A.2.

Reservoir life is 3000 days, which is divided into seven control steps, with the first

five control steps of length 180 days, and the last two control steps of length 1050

days. In this example, we assume that two rigs are available and therefore a maximum

of two wells can be drilled at each CLFD step. The optimal number of wells, however,

is determined from the optimization. The drilling (and completion) time is specified

as 180 days. Therefore, the ti values are given by {0, 180, 360, 540, 720}. The last

optimization is performed at 540 days, which determines the decision parameters

corresponding to well type and location of the last two wells and operational settings

of existing wells. The cost of drilling each well is specified to be $10 MM. The

maximum injection and total liquid production rates for a well are specified to 12,500

and 25,000 STBD, respectively. Other simulation and optimization parameters are

identical to those in Example 2 of Chapter 2.

The number of decision parameters is 80. These correspond to 10 categorical

variables for well types, 20 integer variables for well locations, and 50 continuous


x

y

1

2

3

4

5

6

7

8

9 10

20 40 60

10

20

30

40

50

60

Figure A.1: True permeability field, with red indicating sand facies (permeability of500 mD), and blue indicating shale facies (permeability of 10 mD). The initial wellconfiguration is also shown, with circles denoting producers and triangles denotinginjectors (Example A1).

X

Y

20 40 60

10

20

30

40

50

60

(a) Realization 1

X

Y

20 40 60

10

20

30

40

50

60

(b) Realization 2

X

Y

20 40 60

10

20

30

40

50

60

(c) Realization 3

Figure A.2: Three prior realizations of the permeability field, with red indicatingsand facies (permeability of 500 mD), and blue indicating shale facies (permeabilityof 10 mD).


3000 Days

2

4

5 67

8

1 3

9

10

20 40 60

10

20

30

40

50

60 0.2

0.3

0.4

0.5

0.6

0.7

0.8

Figure A.3: Well configuration from deterministic optimization (using mtrue), withred denoting producer, blue denoting injector, and the well numbers indicating thedrilling sequence. Background shows final oil saturation. Note that two wells aredrilled at a time (Example A1).

parameters for well settings. We first apply deterministic optimization using the

true model. The PSO-MADS algorithm is applied with 50 PSO particles, and the

minimum mesh size for MADS is specified to be 1% of the variable range. The

optimal development plan and the final water saturation map are shown in Fig. A.3.

The NPV from the initial guess is $441 MM while the optimal NPV is $783 MM. We

next apply CLFD using OSV for this case.

Observed data for CLFD history matching include production data measured at

30-day intervals and hard data from all existing wells. History matching is performed

every 180 days by first generating new geological realizations conditioned to hard data

using the MS-CCSIM algorithm, and then applying O-PCA-based history matching

using all production data from time zero. At each history matching step NR =

50 RML realizations together with MAP estimate are generated. Optimization is

then performed on a set of representative models through the OSV framework. The

progress of the true NPV with CLFD step is shown in Fig. A.4. The final truth-case

NPV from CLFD is $646 MM, which is 36.6% higher than the NPV from optimization


0 180 360 540400

500

600

700

800

Time (Days)

NP

V (

$ M

M)

J(xi,M irep)

J(xi,mtrue)

DeterministicFinal NPV

$ 646.2 MM

Figure A.4: Optimal expected NPV, and the corresponding NPV for the true model,versus CLFD step. The number of realizations at each CLFD step is determinedusing OSV. The star shows the final true NPV from CLFD (Example A1).

over prior realizations (by use of OSV).

Fig. A.5 presents the P10–P50–P90 results for NPV, determined by simulating all

50 realizations and then constructing the cdf, at each CLFD step. The expected NPV

based on the current representative subset (which satisfies the validation criterion of

RI ≥ 0.5) is also displayed. It is evident that the optimal expected NPV for the

representative subset falls within the P10–P90 range.

Fig. A.6 shows the evolution of the well configuration and the geological model

for two realizations. Note that realizations at t1 are generated by MS-CCSIM and

conditioned to hard data at Wells 1 and 2. Updated realizations at t3 and t4 are

conditioned to both hard data and production data (all history matched models are

generated using O-PCA). It is evident that the well scenario involves three injectors

at t1, but four injectors at later times. Each realization continues to show differences

through the CLFD steps due to conditioning to new hard and production data.

As discussed earlier, the CLFD history matching step involves integrating produc-

tion data from all previous wells (except the most recent well) together with hard data


0 180 360 540100

200

300

400

500

600

700

Time (Days)

NP

V (

$ M

M)

M1

x1

M2

x2

M3

x3

M4

x4

J(xi,M irep)

P10-P50-P90

Figure A.5: P10, P50, P90 NPVs evaluated for the entire set of 50 realizations,along with the expected NPV for the representative subset, versus CLFD step (Ex-ample A1).

from all wells including the most recent well. Integrating both hard data and produc-

tion data is required to achieve optimal CLFD performance. At each CLFD history

matching step, new realizations conditioned to hard data are generated. We let mprior

designate the mean of these NR = 50 realizations (conditioned to all available hard

data) at each update step. Figs. A.7(a)-A.7(d) show the evolution of the prior mean

with CLFD step. We also compute the mean of NR = 50 posterior realizations (con-

ditioned to both production and hard data). These are shown in Figs. A.7(e)-A.7(h).

The optimal development plan at each CLFD step is also shown. It is evident that

the mean of the posterior realizations (conditioned to production data) more closely

resembles the true model (Fig. A.1). This indicates the importance of integrating

production data to reduce the uncertainty in the geological description.

We repeated the CLFD procedure using four additional mtrue. Table A.1 summa-

rizes the results (the first row corresponds to the true model discussed earlier). For

true models 2 to 5, the true NPV obtained from optimization with sample validation

using prior realizations (third column of Table A.1) is greater than that obtained

from optimization with nr = 5 prior representative realizations (for true model 1,

the two values are very close). The CLFD (using OSV) procedure improves the true

NPV by an average of 17.9% over the true NPV obtained from OSV applied to prior


x

y

1

2

34

5

6

7

8

9

20 40 60

10

20

30

40

50

60

(a) Realization 1 at t1 with x1

x

y1

2

34

5

6

7

8

9

20 40 60

10

20

30

40

50

60

(b) Realization 1 at t3 with x3

x

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6

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8

920 40 60

10

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50

60

(c) Realization 1 at t4 with x4

x

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9

20 40 60

10

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60

(d) Realization 2 at t1 with x1

x

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9

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60

(e) Realization 2 at t3 with x3

x

y

1

2

34

5

6

7

8

920 40 60

10

20

30

40

50

60

(f) Realization 2 at t4 with x4

Figure A.6: Evolution of two RML realizations for different CLFD steps, with redindicating sand facies (permeability of 500 mD), and blue indicating shale facies(permeability of 10 mD). Current optimal well configuration and drilling sequence isalso depicted. Solid white circles and triangles denote producers and injectors (drilledor in the process of being drilled), and yellow circles and triangles denote plannedproducers and injectors. Numbers indicate the drilling sequence (Example A1).


x

y

1

2

3

20 40 60

10

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30

40

50

60 0

0.2

0.4

0.6

0.8

1

(a) mprior, 180 days

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20 40 60

10

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10

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89

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1

(h) mpost, 720 days

Figure A.7: Evolution of mean of (NR = 50) prior realizations (conditioned to harddata) and mean of (NR = 50) posterior realizations of facies distribution, for differentCLFD steps. Current optimal well configuration and drilling sequence is also depicted.White circles and triangles denote producers and injectors, respectively. Wells withcolored (red or blue) numbers are drilled, while outlined red circles and blue trianglesdenote planned producers and injectors. For the prior model (a-d) only the drilledwells are shown. Numbers indicate the drilling sequence (Example A1).


Table A.1: NPV values ($ MM) from optimization over prior realizations with nr = 5and by use of OSV (where nr is increased to satisfy RI > 0.5) and from CLFD byuse of OSV, for five different true models (Example A1)

True model prior (nr = 5) prior (OSV) CLFD (OSV)

1 474 473 6462 409 539 6173 354 608 6584 290 462 5525 390 519 573

Table A.2: NPV values ($ MM) from optimization over prior realizations by use ofOSV (where nr is increased to satisfy RI > 0.5) and from CLFD by use of OSV, fortrue model 4 (Example A1)

Run # prior (OSV) CLFD (OSV)

1 462 5522 509 5683 452 542

realizations.

We repeated these experiments two more times for one of the true models (true

model 4). The results are summarized in Table A.2, where the first row corresponds

to the fourth row of Table A.1. These results show that there is less variation in true

NPV using CLFD (this range is $26 MM) than in true NPV from optimization over

the prior realizations (range of $57 MM). Furthermore, CLFD improves the NPV

from prior optimization by at least 11.6% for this true model.

Appendix B

Representative Realizations for a

Binary System

In this Appendix, the various procedures for selecting representative realizations, pre-

sented in Chapter 3, are applied to a binary channelized reservoir model (recall that,

in Chapter 3, bimodal realizations were considered). In the results presented here,

flow-based features for clustering are computed based on full-physics flow simulations

only; proxy-type tracer flow is not considered.

B.1 Example B1: new well settings

We repeat the numerical experiments presented in Section 3.3 for a set of binary

channelized realizations. The model is two-dimensional and is defined on a 60 ×60 grid. The (isotropic) permeability field, consisting of 3600 grid-block values, is

uncertain. Porosity is specified to be uniform and equal to 0.2. A binary channelized

training image, taken from Vo [138], is considered to provide the facies description. A

set of realizations is then generated using the MS-CCSIM geostatistical method [131].

These realizations are not conditioned to any hard data. The permeability values for

sand and non-sand (shale/mud) facies are 500 mD and 10 mD, respectively. Three

permeability realizations are shown in Fig. B.1. There are six producers and three

injectors in the reservoir (as shown in Fig. B.1). Other simulation parameters are

identical to those in Section 3.3.

157

158 APPENDIX B. REPRESENTATIVE MODELS FOR A BINARY SYSTEM

x

y

1 2 3

4 5 6

7 8 9

20 40 60

10

20

30

40

50

60

(a) Realization 1

xy

1 2 3

4 5 6

7 8 9

20 40 60

10

20

30

40

50

60

(b) Realization 2

x

y

1 2 3

4 5 6

7 8 9

20 40 60

10

20

30

40

50

60

(c) Realization 3

Figure B.1: Three unconditional realizations of binary channelized model. Red indi-cates sand facies (permeability of 500 mD) while blue shows non-sand facies (perme-ability of 10 mD). Fixed well configuration is also shown – circles denote producersand triangles indicate injectors (Example B1).

The reservoir life is 3500 days, which is divided into nt = 7 equal time intervals

(of length 500 days) in the construction of the flow-response vectors. With nine wells

in the model and ncs = 7 control steps, the number of control variables (and the

dimension of the well-parameter vector) is 63. The producer BHP range is 1000–

4100 psi and the injector BHP range is 4600–7000 psi.

B.1.1 Representative realizations for random well controls

The base control strategy for computing flow-based features corresponds to producer

BHPs at their lower bound (1000 psi) and injector BHPs at their upper bound

(7000 psi). The number of flow-based features used in the clustering is 105, which

is equal to the dimension of flow-response vector. For computing the permeability-

based features, each binary realization is represented by l = 70 PCA parameters

(which corresponds to 65% of the total variation).

A total of 300 random well-control vectors (xnew) are generated. BHPs for the

three injectors, from one of the xnew vectors, are shown in Fig. B.2. The flow-response

vectors for all NR = 200 realizations are evaluated and saved for each xnew. This

involves a total of 60,000 simulation runs, which are performed using distributed

computing with access to 200 compute nodes.

Box plots of D for nr = 3 and nr = 6 are shown in Fig. B.3. These results indicate

B.1. EXAMPLE B1: NEW WELL SETTINGS 159

0 500 1000 1500 2000 2500 3000 35004500

5000

5500

6000

6500

7000

Time (Days)B

HP

(psi

)

Well 7Well 8Well 9

Figure B.2: Injector BHPs corresponding to a random well-control vector xnew (Ex-ample B1).

that dissimilarity values from clustering with α = 0 are typically smaller than those

for the other two approaches. Results for mD for five values of nr are presented in

Table B.1. For comparison purposes, we also include results using a random selection

(as described in Section 3.3.1). For each method, the average mD value over the

range 3 ≤ nr ≤ 15, along with the average ranking, are also shown. It is evident that

the use of α = 0 (flow-based selection) is the overall best selection method for this

problem. The use of α = 1 and random selection are roughly comparable.

These results are quite consistent with those for Example 1 in Section 3.3.1

(cf. Fig. 3.5 and Table 3.1), and again indicate that the best-performing selection

method corresponds to clustering using only flow-based information (α = 0). This

consistency is noteworthy, since these two examples involve different permeability

fields, different conditioning (recall that models in Section 3.3.1 were conditioned to

hard data), and different well locations.

B.1.2 Representative realizations for small changes in well

controls

We now compare the three approaches for cases where new well-parameter vectors

xnew correspond to small changes relative to base-case operations. For the base oper-

ating condition, the BHP of each well is specified to be the average of the upper and

lower bounds. Here we construct xnew by modifying two coordinates of x0 (at a time)

by ±20%. The number of mesh points is twice the dimension of x, which in this case


0.3

0.4

0.5

0.6

0.7

α=0 α=0.5 α=1

D

(a) nr = 3

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

α=0 α=0.5 α=1

D

(b) nr = 6

Figure B.3: Box plots of Dα=0, Dα=0.5 and Dα=1 for 300 random well control vectors.The red line within each box corresponds to the median, and the bottom and top ofeach box correspond to the 25th and 75th percentiles. The lines above and below theboxes correspond to the 2nd and 98th percentiles (Example B1).

Table B.1: Median values (mD) of Dα=0, Dα=0.5, Dα=1 and Drand, for 300 randomwell control vectors, for different nr. Average mD values and average ranking are alsoprovided (Example B1).

nr α = 0 α = 0.5 α = 1 random

3 0.39 0.51 0.42 0.556 0.27 0.36 0.46 0.399 0.26 0.28 0.21 0.3112 0.23 0.23 0.25 0.2515 0.20 0.22 0.27 0.26Avg (3-15) 0.31 0.34 0.40 0.40Avg ranking (3-15) 1.47 2.07 3.07 3.53

B.2. EXAMPLE B2: NEW WELL CONFIGURATIONS 161

0.32

0.34

0.36

0.38

0.4

0.42

0.44

α=0 α=0.5 α=1

D

(a) nr = 3

0.25

0.3

0.35

0.4

α=0 α=0.5 α=1

D

(b) nr = 6

Figure B.4: Box plots of Dα=0, Dα=0.5 and Dα=1 for 126 well control vectors corre-sponding to pattern search mesh points. The red line within each box correspondsto the median, and the bottom and top of each box correspond to the 25th and 75thpercentiles. The lines above and below the boxes correspond to the 2nd and 98thpercentiles (Example B1).

is 126. Thus this assessment entails 200× 126 = 25, 200 flow simulations.

Box plots for D for nr = 3 and 6 are shown in Fig. B.4. As was the case with

random controls, we again see that clustering with α = 0 leads to the smallest dis-

similarity values. The mD values and the average rankings are shown in Table B.2.

These results are quite consistent with those in Table B.1 and again indicate that

clustering with flow-based features is preferable.

The box plots of D in Fig. B.4 span a narrower range than those in Fig. B.3. This

is because of the smaller variation in the flow responses here than for the case with

random controls. This effect was also apparent in Example 1 in Section 3.3.1 (see

Figs. 3.6 and 3.5).

B.2 Example B2: new well configurations

We now consider cases involving new well locations. We use the same reservoir model

and NR = 200 realizations as in Example B1 (Section B.1). Now, however, there

are 10 wells in the reservoir (six producers and four injectors). The reservoir life is

specified to be 3000 days, which is divided into nt = 3 time intervals for computing


Table B.2: Median (mD) values of Dα=0, Dα=0.5 and Dα=1, for 126 well control vectorscorresponding to pattern search mesh points, for different nr. Average mD values andaverage ranking are also provided (Example B1).

nr α = 0 α = 0.5 α = 1

3 0.33 0.38 0.426 0.25 0.27 0.419 0.21 0.22 0.1912 0.21 0.20 0.2415 0.19 0.23 0.30Avg (3-15) 0.27 0.28 0.39Avg ranking (3-15) 1.27 1.73 2.87

x

y

1 2 3 4 5 6

7 8 9 10

20 40 60

10

20

30

40

50

60

(a) x10

x

y 1 2 3 4 5 6

7 8

9 10

20 40 60

10

20

30

40

50

60

(b) x20

x

y

1

2

3

4

5

6

7

8

9

10

20 40 60

10

20

30

40

50

60

(c) x30

Figure B.5: Three realizations and three base well configurations for computing flow-based features used in clustering. Circles denote producers and triangles indicateinjectors (Example B2).

flow responses. In this example, well BHPs are held constant over the run, with

injector BHP equal to 7000 psi and producer BHP equal to 1000 psi. Three reservoir

models along with three base-case well configurations are shown in Fig. B.5.

B.2.1 Representative realizations for random well configura-

tions

The well-parameter vectors xnew in this problem define the locations of the 10 wells.

This involves 20 integer variables. We generate 300 random xnew vectors, with each

constrained to satisfy a minimum well-to-well distance of 5 grid blocks. Three of these


x

y

20 40 60

10

20

30

40

50

60

(a) x1new

x

y

20 40 60

10

20

30

40

50

60

(b) x2new

x

y

20 40 60

10

20

30

40

50

60

(c) x3new

Figure B.6: Three (out of 300) random well configurations, xnew, for computing flowresponses. Circles indicate producers and triangles denote injectors (Example B2).

configurations are shown in Fig. B.6. We simulate the 300 different well configurations

for the full set of NR = 200 realizations (for a total of 60,000 reservoir simulation

runs).

In this case, consistent with our approach in Section 3.3.2, we simulate three

different flow problems to provide the flow information used in the clustering. The

well configurations specified in these simulations are shown in Fig. B.5. The number

of flow-based features in this case is 144, while the number of PCA parameters is 70

(as in Example B1).

Box plots of D are displayed in Fig. B.7, and additional results are compiled

in Table B.3. Results for random selection, computed as described earlier, are also

shown. The mD values for all four methods are quite similar in this case. The use

of α = 0.5 leads to the best average ranking (1.87) over the range 3 ≤ nr ≤ 15, but

the smallest average mD (0.37) corresponds to the use of α = 1. These results are

analogous to those for Example 2 in Section 3.3.2 (cf. Fig. 3.9 and Table 3.7), in that

no selection method substantially outperforms the other approaches.

B.2.2 Representative realizations for small changes in well

locations

We now consider cases where new well locations correspond to local perturbations

around a base-case configuration. The base case (x0) used in this example is shown


0.2

0.3

0.4

0.5

0.6

α=0 α=0.5 α=1

D

(a) nr = 6

0.15

0.2

0.25

0.3

0.35

0.4

α=0 α=0.5 α=1D

(b) nr = 12

Figure B.7: Box plots of Dα=0, Dα=0.5 and Dα=1 for 300 random well configurations.The red line within each box corresponds to the median, and the bottom and top ofeach box correspond to the 25th and 75th percentiles. The lines above and below theboxes correspond to the 2nd and 98th percentiles (Example B2).

Table B.3: Median (mD) values of Dα=0, Dα=0.5, Dα=1 and Drand, for 300 randomwell configurations, for different nr. Average mD values and average ranking are alsoprovided (Example B2).

nr α = 0 α = 0.5 α = 1 random

3 0.57 0.49 0.50 0.536 0.36 0.34 0.34 0.399 0.28 0.28 0.29 0.3012 0.26 0.26 0.24 0.2715 0.21 0.22 0.20 0.23Avg (3-15) 0.39 0.38 0.37 0.40Avg ranking (3-15) 2.53 1.87 2.00 3.53


x

y

20 40 60

10

20

30

40

50

60

(a) x0

x

y

20 40 60

10

20

30

40

50

60

(b) x1new

x

y

20 40 60

10

20

30

40

50

60

(c) x2new

Figure B.8: Base-case well configuration, and two (out of 40) new well configurationscorresponding to pattern search mesh points. Circles denote producers and trianglesindicate injectors (Example B2).

Table B.4: Median (mD) values of Dα=0, Dα=0.5 and Dα=1, for 40 well configurationscorresponding to pattern search mesh points, for different nr. Average mD values andaverage ranking are also provided (Example B2).

nr α = 0 α = 0.5 α = 1

3 0.40 0.38 0.426 0.35 0.41 0.439 0.22 0.31 0.2712 0.19 0.19 0.2415 0.12 0.20 0.30Avg (3-15) 0.26 0.31 0.34Avg ranking (3-15) 1.27 2.13 2.53

in Fig. B.8(a). New well configurations correspond to shifts, in one of the coordinates

of the well-parameter vector, by ±5 grid blocks. Two of the xnew configurations are

shown in Fig. B.8(b) and (c).

Results for the three selection methods are presented in Fig. B.9 and Table B.4.

In this case, clustering with flow-based features (α = 0) provides, on average, the

smallest dissimilarity value (0.26) and the highest ranking (1.27). These results differ

from those for Example 2 in Section 3.3.2 (Fig. 3.11 and Table 3.9), in that the use

of α = 0.5 outperformed the other selection methods in Example 2. Both cases,

however, demonstrate the advantage of using flow information.


0.3

0.35

0.4

0.45

0.5

0.55

α=0 α=0.5 α=1

D

(a) nr = 6

0.15

0.2

0.25

0.3

α=0 α=0.5 α=1

D

(b) nr = 12

Figure B.9: Box plots of Dα=0, Dα=0.5 and Dα=1 for 40 well configurations corre-sponding to pattern search mesh points. The red line within each box correspondsto the median, and the bottom and top of each box correspond to the 25th and 75thpercentiles. The lines above and below the boxes correspond to the 2nd and 98thpercentiles (Example B2).

Table B.5: Summary of results: average mD values for nr = 3, . . . , 15 for all cases.The smallest value for each case is indicated in bold.

Case α = 0 α = 0.5 α = 1

Random controls (Ex. B1) 0.31 0.34 0.40PS mesh controls (Ex. B1) 0.27 0.28 0.39Random configs. (Ex. B2) 0.39 0.38 0.37PS mesh configs. (Ex. B2) 0.26 0.31 0.34

B.2.3 Summary of realization-selection results

Table B.5 provides a summary of the results, in terms of the average mD values,

for the four cases considered in this Appendix. For both cases involving new well

controls (either random or pattern-search shifts relative to the base case), the best

performing method for selecting a representative subset was clustering with flow-based

features (α = 0). For randomly-generated new well configurations, the three selection

methods performed similarly, though selection with permeability-based features (α =

1) provided slightly better results than the other approaches. Finally, for small well-

location changes relative to a base well configuration, realizations selected using α = 0


provided the best results. These observations are consistent with those for the bimodal

systems considered in Chapter 3, with the exception that α = 0.5 provided the best

results for small changes in well locations in Chapter 3, while the best results here

are achieved using α = 0.

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