THE UNIVERSITY OF TULSA

THE GRADUATE SCHOOL

HISTORY MATCHING, PREDICTION AND PRODUCTION OPTIMIZATION

WITH A PHYSICS-BASED DATA-DRIVEN MODEL

W
by
IE
Zhenyu Guo
EV

A dissertation submitted in partial fulfillment of

PR

the requirements for the degree of Doctor of Philosophy

in the Discipline of Petroleum Engineering

The Graduate School

The University of Tulsa

2018




ProQuest Number: 10789097




All rights reserved

INFORMATION TO ALL USERS
The quality of this reproduction is dependent upon the quality of the copy submitted.

In the unlikely event that the author did not send a complete manuscript
and there are missing pages, these will be noted. Also, if material had to be removed,
a note will indicate the deletion.



W

IE


EV
ProQuest 10789097

Published by ProQuest LLC (2018 ). Copyright of the Dissertation is held by the Author.


All rights reserved.
PR

This work is protected against unauthorized copying under Title 17, United States Code
Microform Edition © ProQuest LLC.


ProQuest LLC.
789 East Eisenhower Parkway
P.O. Box 1346
Ann Arbor, MI 48106 - 1346
 THE UNIVERSITY OF TULSA
THE GRADUATE SCHOOL

HISTORY MATCHING, PREDICTION AND PRODUCTION OPTIMIZATION

WITH A PHYSICS-BASED DATA-DRIVEN MODEL

W
by
Zhenyu Guo
IE
A DISSERTATION
EV
APPROVED FOR THE DISCIPLINE OF
PETROLEUM ENGINEERING
PR

By Dissertation Committee

Albert C. Reynolds, Chair

Rami Younis
Mustafa Onur
Richard Redner

ii
 ABSTRACT

Zhenyu Guo (Doctor of Philosophy in Petroleum Engineering)

History Matching, Prediction and Production Optimization with a Physics-Based Data-
Driven Model
Directed by Albert C. Reynolds
186 pp., Chapter 6: Conclusions

W
(681 words)

Assisted history matching and life-cycle production optimization are the two key
IE
components of closed-loop reservoir management, which are traditionally performed based
on a multitude of runs of full-scale reservoir simulation models which incur high computa-
EV

tional costs for large-scale problems. To reduce the computational cost spent on full-scale
simulation runs when performing history matching and production optimization, we focus
PR

on developing a new data-driven model, Interwell Numerical Simulation Model with Front
Tracking (INSIM-FT). INSIM-FT can be built without any prior knowledge of geological
information of the target reservoir. Although the INSIM-FT model is developed from pro-
duction data and requires no prior knowledge of rock property fields, it incorporates far
more fundamental physics than that of the popular Capacitance-Resistance Model (CRM).
INSIM-FT also represents a substantial improvement on an interwell numerical simulation
model (INSIM) developed by Zhao et al. (2016). Specifically, we introduce a theoretically
correct procedure to compute water saturation in INSIM-FT that generally gives more ro-
bust and accurate solutions than are obtained with INSIM where saturations are computed
with an ad hoc method. In addition, unlike INSIM, INSIM-FT incorporates the parameters
defining power-law relative permeability curves as additional history-matching parameters

iii
so that prior knowledge of relative permeabilities is no longer required as is the case with IN-
SIM. Also, we introduce imaginary wells and their associated interwell connections (stream
tubes) to enable more potential flow paths in the INSIM-FT model than are used in INSIM.
These additional flow paths enable INSIM-FT to honor more correctly the physics than is
done with the original INSIM model. With these modifications, one expects that INSIM-FT
will be more robust than INSIM, and via computational examples, we show that this is the
case.
After developing INSIM-FT for a two-dimensional reservoir model, we extend INSIM-
FT to full three-dimensional multi-layered reservoirs, where it is necessary to consider grav-
itational effects. The extended model, which is referred to as INSIM-FT-3D, can be used for

W
history matching, prediction and production optimization for a three-dimensional reservoir
under waterflooding. Compared to the original INSIM-FT model, INSIM-FT-3D replaces
IE
the original Riemann solver in INSIM-FT by a new Riemann solver based on a convex-hull
method that enables the solution of the Buckley-Leverett problem with gravity, where a frac-
EV
tional flow function may have more than one inflection point. Secondly, unlike the original
INSIM-FT model, which assumes all wells are vertical, the INSIM-FT-3D model allows for
the inclusion of wells with arbitrary trajectories with multiple perforations. Third, INSIM-
PR

FT-3D applies Mitchell’s best-candidate algorithm to automatically generate the imaginary

wells that are evenly distributed in the reservoir given a set of prefixed actual well nodes
and fourth INSIM-FT-3D utilizes our own modification of Delaunay triangulation to build
the 3D connection map necessary to use the general INSIM-FT-3D formulation.
The ensemble-smoother with multiple data assimilation (ES-MDA) is used for history
matching with INSIM-FT or INSIM-FT-3D. The history matching parameters for INSIM-
FT and INSIM-FT-3D are similar and include the connection-based parameters and the
parameters that define power-law permeabilities. In addition to the common parameters
included with the two methods, the parameters that define the well indices for the wells with
multiple perforations are included in the INSIM-FT-3D model. For production optimization
with INSIM-FT and INSIM-FT-3D, ensemble-based optimization (EnOpt) is used. Because

iv
initially the developed data-driven models only allow rate controls, pressure controls cannot
be used for production optimization. We provide a procedure to estimate the values of well
indices via the history matching process if bottom-hole pressure (BHP) data are available.
Then, BHP can be specified at each control step in the INSIM-FT(-3D) model to allow
optimization of the life-cycle net present value (NPV) of production, where the producer’s
BHPs at control steps are included in the optimization variables.
Computational results show that, history matching and production optimization per-
formed with INSIM-FT two- and three-dimensional models are far more computationally
efficient than are those performed with full-scale reservoir simulation models but still give
a characterization of a reservoir under waterflooding, future predictions and estimates of

W
the optimal NPV of production, that are similar to those obtained using computationally
expensive full-scale reservoir simulation models.
IE
EV
PR

v
 ACKNOWLEDGEMENTS

I would like to express my sincerest gratitude to my advisor, Dr. Albert C. Reynolds,

for his inspiration, encouragement, and insightful inputs for my research. His earnest manner
and responsive attitude towards scientific research has been a perfect model for me to learn.
I would also like to thank my dissertation committee members, Dr. Mustafa Onur,
Dr. Rami Younis and Dr. Richard Redner, for taking time to review my dissertation and

W
providing me with valuable feedback and comments.
Finally, I would like to show my greatest appreciation to my parents Minxin Guo and
IE
Xiaolan Zhang, for their unconditional support and love. I extend special gratitude to my
wife, Di Zhu, for her love, understanding and encouragement throughout my life.
EV
PR

vi
 TABLE OF CONTENTS

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

ACKNOWLEDGEMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

TABLE OF CONTENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii

W
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xx

CHAPTER 1: INTRODUCTION IE 1
1.1 Literature Review on History Matching . . . . . . . . . . . . . . . . . 2
1.2 Literature Review on Production Optimization . . . . . . . . . . . . . 5
1.3 Literature Review on Surrogate Model . . . . . . . . . . . . . . . . . . 7
EV
1.4 Research Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.5 Dissertation Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

CHAPTER 2: INSIM-FT MODEL FOR HISTORY-MATCHING, PREDIC-

TION AND CHARACTERIZATION OF WATERFLOOD-
PR

ING PERFORMANCE 15
2.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1.1 Relative Permeability Functions . . . . . . . . . . . . . . . . . . . . . 22
2.1.2 Front Tracking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.1.3 Water Saturation and Oil Production Rate at Well Nodes . . . . . . . 24
2.1.4 Demonstration Case . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.2 INSIM-FT versus INSIM . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2.1 Imaginary Wells . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.2.2 Total Rate Constraint on a Well’s Control Volume . . . . . . . . . . . 28
2.2.3 Interwell Connectivity . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.3 History Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.3.1 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.4 Application of INSIM-FT . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.4.1 Example 1: Homogeneous Reservoir with a Sealing Fault . . . . . . . 36
INSIM-FT without Adding Imaginary Wells: . . . . . . . . . . . . . 39
INSIM-FT with Imaginary Wells: . . . . . . . . . . . . . . . . . . . 41
Interwell Connectivity: . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.4.2 Example 2: Channelized Reservoir . . . . . . . . . . . . . . . . . . . . 48

vii
 2.4.3 Example 3: Field Example . . . . . . . . . . . . . . . . . . . . . . . . 54

CHAPTER 3: WATERFLOODING OPTIMIZATION WITH THE INSIM-

FT DATA-DRIVEN MODEL 62
3.1 Waterflooding Optimization Problem . . . . . . . . . . . . . . . . . . . 63
3.2 Average History-Matched INSIM-FT Model . . . . . . . . . . . . . . 65
3.3 Estimation of Optimal Well Controls . . . . . . . . . . . . . . . . . . . 66
3.4 Optimization Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.4.1 Ensemble-Based Optimization (EnOpt). . . . . . . . . . . . . . . . . . 68
3.4.2 Validation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.4.3 Complete Workflow. . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.5.1 A Homogeneous Reservoir with a Sealing Fault . . . . . . . . . . . . . 72
3.5.2 Channelized Reservoir . . . . . . . . . . . . . . . . . . . . . . . . . . 76
3.5.3 Field Example with Aquifer . . . . . . . . . . . . . . . . . . . . . . . 81

CHAPTER 4: INSIM-FT IN THREE-DIMENSIONS WITH GRAVITY 87

W
4.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.1.1 Pressure Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.1.2 Wells with Multiple Perforated Segments . . . . . . . . . . . . . . . .
IE 91
4.1.3 Saturation Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.1.4 Convex-Hull Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
4.1.5 Generate Imaginary Wells Using Mitchell’s Best-Candidate Algorithm 100
4.1.6 Create a Connection Map Using Delaunay Triangulation with A Mod-
EV
ification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
4.2 Example 1: Toy Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4.3 Example 2: Multilayer Channelized Reservoir . . . . . . . . . . . . . 106
4.4 Example 3: Field Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
PR

4.5 Example 4: Brugge Reservoir . . . . . . . . . . . . . . . . . . . . . . . . 113

CHAPTER 5: PRODUCTION OPTIMIZATION WITH INSIM-FT-3D 125

5.1 Formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
5.2 Optimization Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
5.3 Example 1: Multilayer Channelized Reservoir. . . . . . . . . . . . . . 135
5.4 Example 2: Brugge Reservoir . . . . . . . . . . . . . . . . . . . . . . . . 136

CHAPTER 6: DISCUSSION AND CONCLUSIONS 143

6.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

NOMENCLATURE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

APPENDIX A: DERIVATION OF INSIM-FT . . . . . . . . . . . . . . . . . . 162

APPENDIX B: DERIVATION OF FRONT TRACKING METHOD . . . . 167
B.1 Front Tracking Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 171
APPENDIX C: REVIEW OF CRM . . . . . . . . . . . . . . . . . . . . . . . . . 177

viii
 C.1 Estimation of Total Production Rate . . . . . . . . . . . . . . . . . . . 177
C.2 Estimation of Oil-Cut and Oil Production Rate with CRM . . . . . 181
APPENDIX D: GRAHAM’S SCAN ALGORITHM . . . . . . . . . . . . . . . 183
APPENDIX E: MITCHELL’S BEST-CANDIDATE ALGORITHM . . . . . 185

W
IE
EV
PR

ix
 LIST OF TABLES

2.1 Overall comparison among CRM, INSIM and INSIM-FT. . . . . . . . . . . . 17

2.2 Property of the three-connection reservoir. . . . . . . . . . . . . . . . . . . . 26
2.3 Property of the fault-segmented reservoir . . . . . . . . . . . . . . . . . . . . 38
2.4 Comparison of data mismatch for historical period (2250 days) and prediction
period for CRM, INSIM, INSIM-FT and INSIM (ES-MDA) history-matched

W
models; Example 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2.5 Comparison of data mismatch for historical period (440 days) and prediction
IE
period for CRM, INSIM, INSIM-FT and INSIM (ES-MDA) history-matched
models; Example 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
EV
2.6 Property of the channelized reservoir . . . . . . . . . . . . . . . . . . . . . . . 50
2.7 Well operating schedule for the Eclipse simulation model. . . . . . . . . . . . 53
2.8 Comparison of data mismatch for historical period (800 days) and prediction
PR

period among CRM, INSIM, INSIM-FT and INSIM (ES-MDA); Example 2,

channelized reservoir. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.9 Comparison of data mismatch for historical period (400 days) and prediction
period among CRM, INSIM, INSIM-FT and INSIM (ES-MDA) ; Example 2,
channelized reservoir. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.10 The comparison of data mismatch for historical period and prediction period
between INSIM (ES-MDA) and INSIM-FT; field example. . . . . . . . . . . . 60
3.1 Mean values and standard deviations of well indices for Example 1. . . . . . . 67

x
3.2 NPV comparison between INSIM-FT and Eclipse validations. INSIM-FT ini-
tial and optimal, respectively, represent the NPVs generated with INSIM-FT
using the initial and optimal INSIM-FT well controls. Eclipse V1 inital and
optimal, respectively, represent the results obtained by running Eclipse with
the same initial and optimal well controls from INSIM-FT; Eclipse V2 initial
denotes the NPV generated using the initial well controls used for INSIM-FT
in the true Eclipse simulation model. Eclipse V2 optimal denotes the results
by doing optimization directly on the reservoir simulation model (the truth). 77
3.3 Optimal NPVs obtained with INSIM-FT and Eclipse with two different initial
conditions of well controls. INSIM-FT represents the optimal NPV generated

W
with INSIM-FT; Eclipse V1 represents the NPV obtained by running Eclipse
with the optimal well controls from INSIM-FT; Eclipse V2 denotes the NPV
IE
by optimizing directly on the reservoir simulation model (the truth), faulted
reservoir. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
EV
3.4 The well operating conditions for historical period, channelized reservoir . . . 79
3.5 NPV comparison between INSIM-FT and Eclipse validations. INSIM-FT ini-
tial and optimal, respectively, represent the NPVs generated with INSIM-FT
PR

using the initial and optimal INSIM-FT well controls. Eclipse V1 inital and
optimal, respectively, represent the results obtained by running Eclipse with
the same initial and optimal well controls from INSIM-FT; Eclipse V2 initial
denotes the NPV generated using the initial well controls used for INSIM-FT
in the true Eclipse simulation model. Eclipse V2 optimal denotes the results
by doing optimization directly on the reservoir simulation model (the truth). 82

xi
3.6 Optimal NPVs obtained with INSIM-FT and Eclipse with two different initial
conditions of well controls. INSIM-FT represents the optimal NPV generated
with INSIM-FT; Eclipse V1 represent the NPV obtained by running Eclipse
with the optimal well controls from INSIM-FT; Eclipse V2 denotes the NPV by
optimizing directly on the reservoir simulation model (the truth), channelized
reservoir. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.1 Property of the channelized reservoir. . . . . . . . . . . . . . . . . . . . . . . 112
5.1 Comparison of NPV obtained with different scenarios. INSIM-FT-3D/CMG
refers to the NPV values obtained by inputting the INSIM-FT-3D generated
initial and optimal controls into the CMG true model; Simplex Opt refers to

W
the results obtained by optimizing well controls directly with CMG. . . . . . 137
5.2 Initial guesses of optimal controls for producer BHP. . . . . . . . . . . . . . 139
5.3
IE
Comparison of NPV obtained with different scenarios. INSIM-FT-3D/Eclipse
refers to the NPV values obtained by inputting the INSIM-FT-3D generated
EV
initial and optimal controls into the Eclipse true model; Simplex Opt refers to
the results obtained by optimizing well controls directly with Eclipse. . . . . 140
PR

xii
 LIST OF FIGURES

2.1 Connective units between wells of INSIM. . . . . . . . . . . . . . . . . . . . 18

2.2 The geometry for a three-connection reservoir. . . . . . . . . . . . . . . . . . 26
2.3 Water cut in W4 . The black line denotes the results obtained from INSIM, the
blue line denotes the results obtained from INSIM-FT and the red dashed line
denotes the results obtained from ECLIPSE. . . . . . . . . . . . . . . . . . . 26

W
2.4 To compute the flow rate from I2 passing through intermediate imaginary wells
to P1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
IE 31
2.5 Fault geology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.6 The total production rates obtained from the history matched CRM model.
EV
Red circles: observed total rates; red lines: true total production rates; gray
lines: estimated total rates with the history-matched CRM model. . . . . . . 38
2.7 The oil production rates calculated from the history matched CRM model.
PR

Red circles: observed oil rates; red lines: true values of the oil rates; gray lines:
estimated oil production rates by the history-matched CRM model. . . . . . . 39
2.8 Connection map generated without adding imaginary wells. . . . . . . . . . . 41
2.9 The estimated oil production rates obtained from the prior INSIM-FT models.
Red circles: observed oil production rates; red lines: true values of the oil
production rates; gray lines: estimated oil rates obtained by running the prior
INSIM-FT models; Example 1 without imaginary wells. . . . . . . . . . . . . 41

xiii
2.10 The oil production rates obtained with the history-matched INSIM-FT models.
Red circles: observed oil rates; red lines: true oil production rates; gray lines:
estimated oil rates with the history-matched INSIM-FT models; Example 1
without imaginary wells. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.11 Prior oil-water relative permeability curves, fault. The red solid lines represent
the true relative permeability curves and blue lines represent the prior models
of relative permeabilities; Example 1 without imaginary wells. . . . . . . . . . 42
2.12 Posterior oil-water relative permeability curves, fault. The red solid lines rep-
resent the true relative permeability curves and blue lines represent the history-
matched models of relative permeabilities; Example 1 without imaginary wells. 43

W
2.13 INSIM-FT well placement with adding imaginary wells; Example 1. . . . . . 44
2.14 INSIM-FT connection map with imaginary wells and connections added; Ex-
IE
ample 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.15 The oil production rates obtained from the prior INSIM-FT models. Red
EV
circles: observed oil rates; red lines: true oil rates; gray lines: estimated oil
rates with the prior INSIM-FT models; Example 1 with imaginary wells. . . 45
2.16 Comparison of oil production rates calculated with the CRM, INSIM, INSIM
PR

(ES-MDA) and INSIM-FT history-matched models where only INSIM-FT uses

imaginary wells; Example 1. Red circles: observed data; red lines: true data
of oil production rate; gray curve: posterior oil production rate. . . . . . . . . 46
2.17 Interwell connectivity obtained from FrontSim, CRM, INSIM, INSIM (ES-
MDA) and INSIM-FT for the fault case. The length of a narrow red triangular
denotes the magnitude of total interwell flow rate between an injector-producer
pair and the direction of the triangular indicates which producer belongs to
this injector-producer well pair. . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.18 Log permeability field, channelized reservoir. . . . . . . . . . . . . . . . . . . 51
2.19 Connection Map, channelized reservoir. . . . . . . . . . . . . . . . . . . . . . 51

xiv
2.20 Oil-water relative permeability curves obtained with INSIM-FT, channelized
reservoir. The red solid lines represent the true relative permeability curves
and blue lines represent the estimated relative permeabilities. . . . . . . . . . 53
2.21 Prior oil production rate; Example 2, channelized reservoir. Red circles: ob-
served data; red lines: true data of oil production rate; gray lines: prior re-
sponses; vertical black dashed lines: a separator for matching and prediction. 54
2.22 Comparison of oil production rates calculated with the CRM, INSIM, INSIM(ES-
MDA) and INSIM-FT history-matched models for four wells; Example 2, chan-
nelized reservoir. Red circles: observed data; red lines: true data of oil pro-
duction rate; gray curve: posterior oil production rate. . . . . . . . . . . . . . 55

W
2.23 Interwell connectivity obtained from FrontSim, CRM, INSIM and INSIM-FT
for the channelized reservoir. The length of a narrow red triangular denotes
IE
the magnitude of total interwell flow rate between a corresponding injector-
producer pair and the direction of the triangular indicates which producer
EV
belongs to this injector-producer well pair. . . . . . . . . . . . . . . . . . . . 56
2.24 Connection map of the field example. . . . . . . . . . . . . . . . . . . . . . . 57
2.25 Prior oil production rates of the field example. Red circles: observed oil pro-
PR

duction rates; gray lines: prior responses of the oil production rates obtained
from the prior INSIM-FT models; vertical black dashed lines: separator for
matching and predictions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
2.26 History-matched oil production rates of the field example obtained with INSIM(ES-
MDA) and INSIM-FT. Red circles: observed oil rates; gray lines: the oil rates
estimated with the history-matched INSIM-FT or INSIM(ES-MDA) models;
vertical black dashed lines: separator for matching and predictions. . . . . . . 60
2.27 Estimated relative permeabilities obtained with INSIM-FT, field example. The
red lines denote the true relative permeability curves and the blue lines are
the estimates of the relative permeability curves. . . . . . . . . . . . . . . . . 61

xv
3.1 NPV versus simulation runs, fault case. Red stars denote the independent
optimization with Eclipse; blue + denote the optimization with INSIM-FT. . 76
3.2 Estimated optimal water injection rate for injectors at different control steps. 76
3.3 Estimated optimal BHP for producers at different control steps. . . . . . . . . 77
3.4 Cumulative oil production versus time, Example 1. Green curve represents cu-
mulative oil production obtained from the initial guess of optimal well controls
using INSIM-FT as the forward model; red curve represents the cumulative oil
production from Eclipse with the initial guess of optimal well controls; black
pluses denote the results calculated with the optimized INSIM-FT control us-
ing the INSIM-FT forward model; blue circles denote the results from Eclipse

W
true model using the optimal controls estimated with INSIM-FT; pink circles
denote the results from independent optimization by Eclipse. . . . . . . . . . 78
3.5
IE
Oil saturation distributions obtained at the end of production life by applying
the optimal well controls from INSIM-FT and Eclipse into the true faulted
EV
reservoir model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.6 NPV versus simulation runs, channelized reservoir. Red stars denote the in-
dependent optimization with Eclipse; blue + denote the optimization with
PR

INSIM-FT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
3.7 Estimated optimal water injection rate for injectors. . . . . . . . . . . . . . . 82
3.8 Estimated optimal BHP for producers at different control steps. . . . . . . . . 82
3.9 Cumulative oil production versus time, Example 2. Green curve represents cu-
mulative oil production obtained from the initial guess of optimal well controls
using INSIM-FT as the forward model; red curve represents the cumulative oil
production from Eclipse with the initial guess of optimal well controls; black
pluses denote the results calculated with the optimized INSIM-FT control us-
ing the INSIM-FT forward model; blue circles denote the results from Eclipse
true model using the optimal controls estimated with INSIM-FT; pink circles
denote the results from independent optimization by Eclipse. . . . . . . . . . 83

xvi
3.10 Oil saturation distributions obtained at the end of production life by applying
the optimal well controls of INSIM-FT and Eclipse, channelized reservoir. . . 83
3.11 Initial guess of optimal total liquid production rate. . . . . . . . . . . . . . . 85
3.12 Optimal liquid production rate obtained with INSIM-FT. . . . . . . . . . . . 85
3.13 NPV versus INSIM-FT simulation runs. . . . . . . . . . . . . . . . . . . . . 86
4.1 Connective units between wells of INSIM. . . . . . . . . . . . . . . . . . . . 89
4.2 The fractional flow function with gravity included. The top dashed curve
represents the fractional flow function for a vertical downwards flow; the solid
curve represents the fractional flow function for horizontal flow when gravity
has no effect; the bottom dashed curve represents the fractional flow function

W
for vertical upwards flow. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.3 The convex hull of the set of points shown. . . . . . . . . . . . . . . . . . . . 96
4.4
IE
A piecewise linear approximation for a fractional flow function. The red curve
and blue curve comprise the convex hull of the fractional flow function relative
EV
to a given saturation interval between Sw1 and Sw2 . The red curve is the upper
part of the convex hull and the blue curve is the lower part of the convex hull. 100
4.5 A 2D reservoir with four horizontal wells. The black dots represent the differ-
PR

ent perforated segments of wells and the red line segments represent the well
trajectories. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.6 The imaginary well nodes generated with Mitchell’s best candidate. The open
circles represent the imaginary well nodes; red lines represent wells and the
solid circles represent actual well nodes. . . . . . . . . . . . . . . . . . . . . . 102
4.7 Delaunay triangulation for the node set of Fig. 4.6. . . . . . . . . . . . . . . . 105
4.8 The connection map by applying a modification after Delaunay triangulation. 105

xvii
4.9 Water cut results from INSIM-FT compared with those from Eclipse; black
solid curve represents the downward-flow results from Eclipse; dark green solid
curve represents the horizontal flow results from Eclipse; red solid curve rep-
resents the upward flow results from Eclipse; INSIM-FT-3D results are shown
open circles of same color as corresponding Eclipse results. . . . . . . . . . . 106
4.10 Horizontal absolute permeability map for a six-layer channelized reservoir. The
dark blue areas indicate the shale zones; the light blue zones represent the
levee facies and the yellow zones represent the channel facies. The color bar
represents the scale of absolute permeability. . . . . . . . . . . . . . . . . . . 110
4.11 Well locations for the channelized reservoir. . . . . . . . . . . . . . . . . . . . 111

W
4.12 The well trajectories and actual and imaginary well nodes, channelized example.111
4.13 The connection map generated by Delaunay triangulation with modification,
IE
channelized example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
4.14 The estimated oil production rates obtained from the prior INSIM-FT-3D
EV
models. Red circles denote the observed oil production rates; red curves denote
the true values of the oil production rate generated with CMG; gray curves
represent the estimated oil rates obtained by running the 200 prior INSIM-
PR

FT-3D models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

4.15 The estimated oil production rates obtained from the history-matched INSIM-
FT-3D models. Red circles denote the observed oil production rates; red curves
denote the true values of the oil production rate generated with CMG; gray
lines represent the estimated oil rates obtained by running the 200 history-
matched INSIM-FT-3D models. . . . . . . . . . . . . . . . . . . . . . . . . . 114
4.16 The prior and posterior relative permeabilities obtained with INSIM-FT-3D
based on an ensemble size of 200. The blue curves represent the estimated
relative permeabilities and the red curves represent the true relative perme-
abilities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

xviii
4.17 Prior oil production rates of the field example. Red circles are the observed oil
production rates; gray curves represent the prior responses of the oil production
rates obtained from the prior INSIM-FT models. . . . . . . . . . . . . . . . . 115
4.18 History-matched oil production rates of the field example. Red circles are the
observed oil rates; gray curves are the oil rates estimated with the history-
matched INSIM-FT or INSIM-FT-3D models. The top two sub-figures are
results obtained with INSIM-FT and the bottom two are obtained with INSIM-
FT-3D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
4.19 History-matched oil production rates of the field example. Red circles are the
observed oil rates; gray curves are the oil rates estimated with the history-

W
matched INSIM-FT or INSIM-FT-3D models. The top two sub figures are
results obtained with INSIM-FT and the bottom two are obtained with INSIM-
IE
FT-3D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
4.20 Top structure of Brugge field. . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
EV
4.21 Well trajectories and actual and imaginary well nodes, Brugge example. . . . 119
4.22 Connection map generated with Delaunay triangulation, Brugge example. . . 120
4.23 Prior oil production rates of the field example. Red circles are observed oil
PR

rates; gray curves are the oil rates estimated with INSIM-FT-3D models. . . 121
4.24 Posterior oil production rates of the Brugge example. Red circles are the
observed oil rates; gray curves are the oil rates estimated with INSIM-FT-3D
models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
4.25 Posterior field oil production rates of the Brugge example. Red curve represents
the true field oil production rates generated from the true Eclipse model; blue
curves represent the field oil production rates calculated with the INSIM-FT-
3D posterior realizations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

xix
4.26 Comparison of history match and predictions for two production wells from the
first and second history-match runs. The top two figures show the results for
the first round of history-match and the bottom two figures show the results
for the second round of history-match. . . . . . . . . . . . . . . . . . . . . . 124
5.1 NPV versus forward runs, red stars represent CMG results; blue pluses repre-
sent INSIM-FT-3D results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
5.2 Comparison of optimal injection rates obtained with INSIM-FT-3D and CMG. 137
5.3 Comparison of optimal production BHP obtained with INSIM-FT-3D and CMG.138
5.4 Comparison of optimal NPV values versus forward runs obtained with INSIM-
FT-3D and Eclipse, Brugge example. . . . . . . . . . . . . . . . . . . . . . . 140

W
5.5 Comparison of optimal injection rates obtained with INSIM-FT-3D and Eclipse,
Brugge example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
5.6
IE
Comparison of optimal production BHP obtained with INSIM-FT-3D and
Eclipse, Brugge example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
EV
5.7 Optimized oil saturation field. . . . . . . . . . . . . . . . . . . . . . . . . . . 142
B-1 Approximation of the initial condition. Here the initial condition of Sw is a
continuous function of x. We approximate it by a piecewise constant func-
PR

tion with four constant states, which produce three local Riemann problems
between each two neighboring constant states. . . . . . . . . . . . . . . . . . 172
B-2 Solution waves of three local Riemman problems shown in Fig. B-1. Each
of the dash lines represents a characteristic in the corresponding rarefaction.
Each of the solid lines represents a shock. The solutions of all these three
Riemann problems are the composite rarefaction-shock waves. These three
Riemann fans are valid until the shock in the second Riemann fan intersects
with the first characteristic in the third Riemann fan. A new Riemann problem
generates at the intersection point (x1 , t1 ). . . . . . . . . . . . . . . . . . . . . 172
B-3 A case of three shock clusters. . . . . . . . . . . . . . . . . . . . . . . . . . . 175
B-4 Insert the new shock cluster . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

xx
 CHAPTER 1

INTRODUCTION

Closed-loop reservoir management is a general decision-making framework which com-

bines the procedures of assisted history matching and production optimization. Its ultimate
objective is to reduce the reservoir uncertainty, predict the reservoir future performance and
maximize the economic return. The entire procedure of closed-loop management involves

W
alternating data assimilation (history matching) steps with life-cycle production optimiza-
tion based on the most recently updated reservoir model(s), to plan the optimal production
IE
strategy for the future. Since closed-loop reservoir management usually requires a multitude
number of forward runs, keeping the computational cost of the forward model as low as
EV
possible is desirable.
As one essential part of closed-loop reservoir management, assisted history matching
is an inverse problem which aims to find the model parameters (reservoir variables) that can
PR

honor historical dynamic data while maintaining geological plausibility. Due to uncertainties
in the model parameters and noise in the observations of dynamic data, it has become
popular to generate an ensemble of history-matched models for the purpose of assessing the
uncertainty in reservoir properties and future reservoir performance predictions. Assisted
history matching with uncertainty quantification is usually performed by using conventional
full-scale simulators. Depending on the scale of the reservoir simulation model, a single
forward simulation run may take the order of one hour to one or more days to complete.
Assisted history matching with uncertainty quantification can take several hundred forward
simulation runs, which is computationally expensive.
Life-cycle production optimization is also a vital part of closed-loop reservoir man-
agement. In life-cycle production optimization, given a reservoir model (or models) obtained

1
from history matching, one applies an optimization algorithm to estimate the well controls
at all control steps (time intervals) which maximize some cost or objective function such
as net-present-value (NPV) or total oil production. Traditionally, production optimization
is performed by using full-scale grid-based simulators to predict the new value of the cost
function at each time the optimization variables (well controls in this work) are updated dur-
ing the iterative optimization process. Depending on the optimization method utilized, the
number of forward simulation runs required to achieve the optimal well controls may range
from the order of one hundred to a few thousand. Similar to assisted history matching, this
process is computationally expensive for large-scale reservoir simulation models.
Due to the high computational cost for closed-loop reservoir management that involves

W
the procedures of assisted history matching and life-cycle production optimization with full-
scale reservoir simulation models, other methods to undertake the same tasks but with far
IE
less computational costs are badly needed. One possibility is to use a data-driven model,
which can be treated as a computationally efficient surrogate model to replace the traditional
EV
full-scale reservoir simulation model.
In this research, we develop and utilize a physics-based data-driven model as a com-
putationally efficient surrogate model to perform history matching and production optimiza-
PR

tion.

1.1 Literature Review on History Matching

The first focus of this research is to reduce the computational cost of assisted history
matching by using a data-driven model. Assisted history matching is a complicated ill-
conditioned inverse problem, to which an infinite number of solutions may exist that all
match the observations, when the number of observed data is far less than the number of
uncertain history-matched parameters to be tuned. Based on the Bayesian point of view,
these solutions are represented by a set of reservoir models that follow a posterior distribution
conditioned to the observed data. Many papers (Oliver et al., 1996; Reynolds et al., 1999;
Nævdal et al., 2002; Gu and Oliver, 2007; Li and Reynolds, 2009; Chen and Oliver, 2012;

2
Emerick and Reynolds, 2012, 2013a,b; Le et al., 2016) have focused on obtaining at least an
approximation of the correct sampling from the posterior distribution, which can properly
reflect the uncertainty in the model space and the uncertainty in future predicted reservoir
performance. To achieve this goal, a set of history matched models instead of a single model
should be obtained to approximate the posterior sampling. This process requires a large
number of full-scale forward simulation runs and incurs a heavy computational cost if the
simulation model is large scale.
In general, the history matching problem can be considered as an optimization prob-
lem where the objective function usually consists of model mismatch and data mismatch
terms. One or multiple history-matched models can be found by applying different types

W
of optimization algorithms such as gradient-based optimization algorithms (Li et al., 2003;
Reynolds et al., 2004; Gao and Reynolds, 2006; Kahrobaei et al., 2013), model-based derivative-
IE
free optimization algorithms (Zhao et al., 2013), direct pattern search derivative-free opti-
mization algorithms (Gao et al., 2016), stochastic derivative-free optimization algorithms
EV
(Gao et al., 2004), and their hybrid counterparts. Starting from an initial guess of the reser-
voir model, the optimization procedure iteratively updates the model until a local or a global
minimum of objective function is found. Multiple history-matched models can be found by
PR

starting from different initial guesses.

When an adjoint-based gradient is available, the gradient-based optimization algo-
rithms (Li et al., 2003; Reynolds et al., 2004; Gao and Reynolds, 2006; Kahrobaei et al.,
2013) perform better than other optimization algorithms that do not use the adjoint gradi-
ent. One of the popular gradient-based optimization methods used for history matching is the
Gauss-Newton method, where the Hessian matrix of the objective function can be evaluated
analytically using the first-order derivatives of data (or the sensitivity matrix). Alterna-
tively, one may apply a quasi-Newton method such as limited-memory Broyden-Fletcher-
Goldfarb-Shanno (LBFGS) (Liu and Nocedal, 1989). As benchmarked by Gao et al. (2016),
Gauss-Newton methods perform better than quasi-Newton methods for history matching or
least-square problems.