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OPTIMIZATION OF CO2 STORAGE SYSTEMS WITH
CONSTRAINED BOTTOM-HOLE PRESSURE INJECTION
A THESIS
SUBMITTED TO THE DEPARTMENT OF
ENERGY RESOURCES ENGINEERING
OF STANFORD UNIVERSITY
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF
MASTER OF SCIENCE
Srinikaeth Thirugnana Sambandam
August 2018
© Copyright by Srinikaeth Thirugnana Sambandam 2018
All Rights Reserved
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Abstract
Of concern with CO2 storage in underground aquifers is the potential leakage of
mobile CO2 into the regions above or surrounding the storage site. In this study,
we perform an optimization study to minimize the mass of mobile CO2 at the top
of the formation at the end of a typical CO2 storage project. We define and solve a
comprehensive optimization problem involving well placement and control, along with
appropriate constraints. The computations are performed within Stanford’s Unified
Optimization Framework.
We consider two different storage aquifer models – a channelized aquifer and an
aquifer characterized by multi-Gaussian statistics. For both models, we perform a
heuristic sensitivity study to estimate the required pore volume of the region sur-
rounding the storage aquifer such that the bottom-hole pressure constraint is not
violated. We consider different injection scenarios by varying the number of CO2
injection wells. Optimizations are performed using a combination of particle swarm
optimization (PSO) and mesh adaptive direct search (MADS). Multiple runs are con-
ducted to account for the stochastic nature of the optimizations. The sensitivity of
the optimized objective to the number of injection wells is assessed for both models.
We observe a 45% decrease in the objective function value as the number of wells is
increased from one to four for the channelized aquifer model. For the Gaussian model,
the corresponding decrease in the objective function value is 33%. We also perform
a study to determine the minimum pore volume of the surrounding region that al-
lows for a feasible injection strategy (in terms of maximum bottom-hole pressure), for
each of the injection scenarios in both of the aquifer models. We observe that with
four injection wells, a surrounding region of pore volume 17 times that of the storage
aquifer is required for the channelized model, and 10 times that of the storage aquifer
is required for the Gaussian model.
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Acknowledgments
Firstly, I would like to thank my adviser, Prof. Louis J. Durlofsky for his support
and encouragement during my time at Stanford. His expertise, insights and writing
skills were valuable in guiding this project to completion.
I would also like to thank Dr. Nikhil Padhye for providing me assistance with the
Unified Optimization Framework and for useful suggestions to improve my research
skills. I would like to acknowledge Dr. Oleg Volkov for his help with the reservoir
simulations in AD-GPRS and ECLIPSE. I am grateful to Larry Jin, Wenyue Sun and
David Cameron for providing me with the simulation models used in this study. I am
extremely grateful to the Smart Fields Consortium and Stanford Center for Carbon
Storage (SCCS) for providing financial support during my studies.
I am fortunate to have met some wonderful people during my time at Stanford,
especially Payal Bajaj, Jason Hu, EJ Baik, Greg Von Wald and Dante Orta. Their
company has made the past two years one of the best times of my life. I am deeply
indebted to my mother, Kavitha Thiru and my sister, Sivastuti Thiru for their uncon-
ditional love and support. I’d like to make a special acknowledgment to my father,
Thirugnanasambandam and my grandfather, Venkatachalam for the values that they
passed on to me. I believe that they are watching me from the heavens and are proud
of who I am today.
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Contents
Abstract v
Acknowledgments vii
1 Introduction 1
1.1 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.1 CO2 Storage and Relevant Trapping Mechanisms . . . . . . . 2
1.1.2 Optimization Procedures . . . . . . . . . . . . . . . . . . . . . 3
1.1.3 Constraints in CO2 Sequestration Optimization . . . . . . . . 5
1.1.4 Pressure Constraints for CO2 Injection . . . . . . . . . . . . . 5
1.2 Scope of Work and Thesis Outline . . . . . . . . . . . . . . . . . . . . 6
2 Optimization Procedures 8
2.1 Optimization Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Unified Optimization Framework . . . . . . . . . . . . . . . . . . . . 10
2.3 Particle Swarm Optimization . . . . . . . . . . . . . . . . . . . . . . 11
2.4 Mesh Adaptive Direct Search . . . . . . . . . . . . . . . . . . . . . . 13
3 Simulation Model Description 16
3.1 Channelized Aquifer Model . . . . . . . . . . . . . . . . . . . . . . . 16
3.2 Sensitivity Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.3 Constraint Handling . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.4 Comparisons Between ECLIPSE and AD-GPRS . . . . . . . . . . . . 23
3.4.1 Aquifer Model with kz = 0.218kx . . . . . . . . . . . . . . . . 23
viii
3.4.2 Aquifer Models with Lower Vertical Permeability . . . . . . . 27
4 Optimization Results 35
4.1 Channelized Model Results . . . . . . . . . . . . . . . . . . . . . . . . 35
4.2 Aquifer Characterized by Multi-Gaussian Statistics . . . . . . . . . . 43
4.2.1 Simulation Model . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.2.2 Sensitivity Study . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.2.3 Optimization Results . . . . . . . . . . . . . . . . . . . . . . . 46
4.3 Support Volume Study . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5 Conclusions and Future Work 55
Bibliography 57
ix
List of Tables
2.1 Summary of optimization variables. . . . . . . . . . . . . . . . . . . . 9
4.1 Results of the support volume study for the Gaussian aquifer. . . . . 54
4.2 Results of the support volume study for the channelized aquifer. . . . 54
ix
List of Figures
2.1 Representation of the Unified Optimization Framework (adapted from
Kim [24]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Velocity components for a PSO particle (from Onwunalu and Durlofsky
[26]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3 Depiction of a MADS iteration in a 2D search space (adapted from
Isebor et al. [21]). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.1 Schematic of the complete simulation model. . . . . . . . . . . . . . . 17
3.2 Permeability field of the storage aquifer model. . . . . . . . . . . . . . 18
3.3 Results of the sensitivity study for different injection scenarios. . . . . 21
3.4 Comparison between the ECLIPSE and AD-GPRS model. . . . . . . 24
3.5 Gas saturation in x-z cross sections at 100 years for the Wells W1 and
W2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.6 Gas saturation in x-z cross sections at 100 years for Well W3 and W4. 26
3.7 Comparison of the average gas saturation at the top layer between the
two models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.8 Comparison between the ECLIPSE and AD-GPRS model for kz =
0.05kx. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.9 Gas saturation in x-z cross sections at 100 years for the Wells W1 and
W2 (kz = 0.05kx). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.10 Gas saturation in x-z cross sections at 100 years for Well W3 and W4
(kz = 0.05kx). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.11 Comparison of the average gas saturation at the top layer between the
two models (kz = 0.05kx). . . . . . . . . . . . . . . . . . . . . . . . . 31
x
3.12 Comparison between the ECLIPSE and AD-GPRS model for kz =
0.01kx. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.13 Gas saturation in x-z cross sections at 100 years for the Wells W1 and
W2 (kz = 0.01kx). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.14 Gas saturation in x-z cross sections at 100 years for Well W3 and W4
(kz = 0.01kx). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.15 Comparison of the average gas saturation at the top layer between the
two models (kz = 0.01kx). . . . . . . . . . . . . . . . . . . . . . . . . 34
4.1 Summary of optimization runs for channelized aquifer. . . . . . . . . 36
4.2 Progression of best optimization run for each injection scenario. . . . 37
4.3 Single injection well scenario. . . . . . . . . . . . . . . . . . . . . . . 38
4.4 Two injection wells scenario. . . . . . . . . . . . . . . . . . . . . . . . 38
4.5 Three injection wells scenario. . . . . . . . . . . . . . . . . . . . . . . 39
4.6 Four injection wells scenario. . . . . . . . . . . . . . . . . . . . . . . . 39
4.7 CO2 gas saturation at 100 years in the top layer for optimized four-well
scenario. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.8 Optimized well behavior for single injection well scenario. . . . . . . . 41
4.9 Optimized well behavior for two injection wells scenario. . . . . . . . 41
4.10 Optimized well behavior for three injection wells scenario. . . . . . . 42
4.11 Optimized well behavior for four injection wells scenario. . . . . . . . 42
4.12 log kx for the Gaussian aquifer. . . . . . . . . . . . . . . . . . . . . . 43
4.13 Results of sensitivity study for the Gaussian aquifer. . . . . . . . . . . 46
4.14 Summary of optimization runs for the Gaussian model. . . . . . . . . 47
4.15 Progression of best optimization run for each injection scenario for the
Gaussian model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.16 Single injection well scenario (Gaussian model). . . . . . . . . . . . . 49
4.17 Two injection wells scenario (Gaussian model). . . . . . . . . . . . . . 49
4.18 Three injection wells scenario (Gaussian model). . . . . . . . . . . . . 50
4.19 Four injection wells scenario (Gaussian model). . . . . . . . . . . . . 50
xi
4.20 Optimized well behavior for single injection well scenario (Gaussian
model). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.21 Optimized well behavior for two injection wells scenario (Gaussian
model). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.22 Optimized well behavior for three injection wells scenario (Gaussian
model). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.23 Optimized well behavior for four injection wells scenario (Gaussian
model). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
xii
Chapter 1
Introduction
The sequestration of CO2 in subsurface formations represents a potential means for
reducing greenhouse gas emissions. Several projects, such as the Sleipner project
offshore Norway [4] and the Quest CCS project in Canada [36], have demonstrated
the viability of carbon capture and storage (CCS) at large scale. Other projects are
also being developed or considered in the U.S. and Europe [31]. A main concern with
any CCS project is to ensure the safe storage of CO2, with minimum risk of leakage.
Leakage out of a storage aquifer can result from the flow of CO2 in the gas phase
(note that supercritical CO2 is often referred to as gas). Thus the risk can be miti-
gated by ensuring that a minimal quantity of CO2 is mobile, either during, or at the
end of, the CCS project. In a heterogeneous reservoir, one way this can be achieved is
by placing and operating the CO2 injection wells in an advantageous manner. Along
these lines, previous investigators have applied computational optimization techniques
to find optimal well placement locations and injection strategies for CO2 storage in
reservoirs. In this work, we will draw upon existing contributions to CO2 seques-
tration modeling and optimization, formulate optimization problems with realistic
constraints, and solve them using established optimization techniques.
1
2 CHAPTER 1. INTRODUCTION
1.1 Literature Review
In this section we will summarize some of the existing literature on CO2 sequestra-
tion optimization that is relevant to this study. First, we will briefly outline the
trapping mechanisms relevant to CO2 storage. Next, details regarding optimization
procedures, including the algorithms and the types of objective functions considered,
are presented. We will then describe the constraints imposed in previous studies and
explain the need for a pressure constraint during CO2 injection. Appropriate values
for this pressure constraint are then discussed.
1.1.1 CO2 Storage and Relevant Trapping Mechanisms
CO2 sequestration studies in the literature usually involve 100 – 1000 year projects
that typically have an initial period of CO2 injection that lasts for around 20 or
30 years. Typical injection quantities are 1 – 5 MT/year of CO2 (1 MT = 109 kg)
during this injection period. The total volume of CO2 injected usually corresponds
to 1 – 4% of the total pore volume of the aquifer (depending on the reservoir temper-
ature), as suggested by the Intergovernmental Panel on Climate Change (IPCC) [34].
As explained by Cameron [6] and many other authors, CO2 is stored in the sub-
surface via four major trapping mechanisms. These are dissolution trapping, where
CO2 is dissolved in the brine present in the aquifer; residual trapping, where CO2 is
immobilized due to relative permeability and relative permeability hysteresis effects;
structural trapping, where buoyant CO2 is trapped beneath an impermeable geolog-
ical feature such as the cap rock; and mineral trapping, where the CO2 chemically
reacts with the rock to form stable carbonates and other minerals. Structural trap-
ping is often considered to be the least secure of the four mechanisms since the CO2
is present in the gas phase (and is thus highly mobile) and could potentially leak
through any existing or induced fractures. Thus, many optimization studies involve
minimizing the amount of CO2 stored in this form. Further discussion regarding the
trapping mechanisms and their associated time frames can be found in Cameron [6]
and Benson and Cole [5]. Simulations involving mineralization are computationally
intensive, and mineral trapping often represents only a relatively small fraction of the
1.1. LITERATURE REVIEW 3
CO2 stored in a typical sandstone aquifer system (as explained in [15]). For these rea-
sons, most (if not all) previous optimization studies do not include chemical reactions
or mineralization.
1.1.2 Optimization Procedures
Optimization techniques have been developed and used extensively for a variety of
subsurface flow problems such as oil and gas production optimization and history
matching. Yeten et al. [38] applied a genetic algorithm (GA) for the optimiza-
tion of nonconventional well placement, while Onwunalu et al. [26] used particle
swarm optimization (PSO). Hybrid algorithms using a global search procedure in
conjunction with a local pattern search method have also been employed, as reported
by Isebor et al. [21], where PSO was used along with mesh adaptive direct search
(MADS). In that work, both well locations and time-varying bottom-hole pressures
were optimized. In [21], the PSO-MADS hybrid algorithm switched back and forth
between PSO and MADS based on algorithmic parameters.
The application of computational optimization for CO2 storage problems has been
explored since 2007. In an early study [25], structurally trapped CO2 in heteroge-
neous 2D models was minimized using a conjugate gradient procedure. Significant
reduction in the quantity of structurally trapped CO2 was observed (up to 43%) with
an optimal injection strategy. Cameron and Durlofsky [7] considered a more compre-
hensive optimization problem involving four horizontal wells in a 3D model. They
optimized well placement and injection strategy (time-varying rates) to minimize the
fraction of mobile CO2 throughout the aquifer at the end of the equilibration period.
An implementation of the Hooke-Jeeves direct search (HJDS) algorithm [19] was used
for the optimization, and the ECLIPSE [32] simulator was used to model the CO2
sequestration process. HJDS is a pattern search method, in which the search space is
traversed using a stencil-based approach. A stochastic global search using PSO was
also considered in [7], but it did not provide significant improvement over the HJDS
results. This study also considered examples that included brine cycling, which when
optimized provided significant additional improvement in the objective function.
4 CHAPTER 1. INTRODUCTION
Other studies, such as [17], have also considered minimizing the fraction of mobile
CO2. Here, the well placement in a 2D reservoir model was optimized using an
adaptive evolutionary Monte Carlo (AEMC) algorithm. This procedure combines
a heuristic-based sampling method (evolutionary Monte Carlo) with a metamodel-
based method (ordinary kriging). In a more recent study, Goda and Sato [18] used a
population-based global search algorithm, called iterative Latin hypercube sampling
(ILHS), for minimizing a similar objective function (fraction of mobile CO2 in the
aquifer at the end of the simulation). Both of these studies considered a simulation
time frame of 100 years, with an injection period of 20 years and an equilibration
period of 80 years. They used the TOUGH2 simulator [29] with the ECO2N module
[30] for flow simulations.
Other investigators have considered maximizing the fraction of CO2 stored due to
residual and/or dissolution trapping. For example, Shamshiri and Jafarpour [33] con-
sidered a well control problem in which they maximized the total volume of residually
trapped and dissolved CO2 over the entire simulation period (300 years in their case).
A gradient-based optimization technique was used, and simulations were performed
with ECLIPSE. Pan et al. [27] and Babaei et al. [2] also considered maximizing the
volume fraction of immobile CO2 at the end of the simulation for well control and
well placement problems, respectively. In a more recent study, Babaei et al. [3] con-
sidered a multi-objective optimization problem. One objective involved maximizing
residually trapped and dissolved CO2, and the other objective entailed minimizing
the fraction of CO2 gas outside the storage region.
Various mobility-based objective functions have also been explored in this context.
In [8], Cameron and Durlofsky considered the time-averaged CO2 mobility at the top
of the aquifer as the objective to be minimized. The use of this cost function should
lead to a more uniform distribution of CO2 at the top of the aquifer, in addition to
a decrease in the overall mobile CO2. Petvipusit et al. [28] proposed a cost function
associated with the volume fraction of mobile CO2, and minimized the time average
of this quantity.
1.1. LITERATURE REVIEW 5
1.1.3 Constraints in CO2 Sequestration Optimization
In many of the studies involving multiple CO2 wells, the optimization involved both
well placement and well control. For well control, the injection period is usually
divided into a number of control periods, with a target volumetric or mass flow rate of
CO2 to be injected. The optimization variables are then the (constant) injection rate
of each well during a control period, with a constraint on the total CO2 injection rate
(such as in [3, 27]). The constraint can also be implicitly enforced by specifying the
optimization variables as fractions of the target injection rate in each control period
(such as in [7, 18]). In this case, there would be Nw − 1 well control variables (where
Nw is the number of injection wells) for each control period. Additional variables and
constraints are required for cases with brine cycling, as discussed in [7].
In other studies, such as [10, 28], an additional constraint was placed on the well
bottom-hole pressure (BHP) during CO2 injection. This is one way of ensuring that
the injection pressure for each well is not unrealistic. This approach also minimizes
the risk of induced fracturing during CO2 injection. Cameron and Durlofsky [7]
treated this implicitly (and approximately) by applying upper bounds on the injection
fraction for each well. This meant, for example, that a single well could not inject
all of the CO2 (at a very high BHP) in a given control period. In this study, we will
apply an explicit constraint on the BHP for each injection well using a penalty-based
procedure. We next discuss previous work that will be used to guide our treatment
of this aspect of the problem.
1.1.4 Pressure Constraints for CO2 Injection
It is necessary to limit the pressure build-up in CO2 storage processes since too much
over-pressure could lead to fracturing of the surrounding formation, or more impor-
tantly, the cap rock. According to regulatory documents (such as [14]), the maximum
injection pressure should be less than the measured fracture pressure of the forma-
tion. The fracture pressure can be determined on a case-by-case basis using direct or
indirect measurement techniques. These are dependent on the local stress field and
other geomechanical properties that are specific to the storage site. For example, in
6 CHAPTER 1. INTRODUCTION
[28], an upper limit on the well BHP was enforced in the optimization problem. The
value of this upper bound (60 MPa) in [28] was based on prior information about the
aquifer and the geologic setting.
In other studies, the maximum injection pressure is calculated using a combination
of heuristics applied to available site-specific data. In [35], the maximum allowable
pressure build-up was calculated as 90% of the minimum horizontal stress in the
Alberta basin for the Basal Cambrian sandstone. Using this information, an upper
bound for the injection pressure was estimated to be a 39% pressure increase from the
initial pressure at the well. In another study, Zhou et al. [39] considered a maximum
pressure build-up of 6 MPa, which corresponds to 50% of the initial pressure, in their
numerical simulations of several closed aquifer systems. Wang et al. [37] employed
a similar upper bound. They specified a maximum pressure build-up of 50% of
the initial pressure in their simulations involving a stratified saline aquifer. Typical
heuristics-based estimates for the maximum pressure increase are thus in the range
of 35% – 50% of the initial formation pressure. Consistent with this, a BHP limit
corresponding to a 40% increase over the initial hydrostatic pressure at the top of the
model will be used in our study.
1.2 Scope of Work and Thesis Outline
To the best of our knowledge, a comprehensive optimization study involving injection
well placement and time-varying control, with variable well lengths and an explicit
pressure constraint, has not been reported. Formulating this optimization problem
and studying the sensitivity of the solution to the number of CO2 injection wells will
be the main topic of this study.
The specific goals of this research are to:
• Assess the impact of the size of the region surrounding the storage aquifer model
on pressure build-up and the BHP constraint.
• Perform computational optimization for the CO2 storage problem with an ex-
plicit pressure constraint.
1.2. SCOPE OF WORK AND THESIS OUTLINE 7
• Consider the effect of the number of CO2 injection wells on the optimized ob-
jective function.
A PSO-based algorithm (based on [8]) will be used as the first-stage optimizer
for the optimizations performed in this study. MADS will then be applied after a
specified number of PSO iterations (though we will not go back and forth between
PSO and MADS as in [21]).
This thesis is organized as follows. In Chapter 2 we describe the optimization
problem, including the form of the objective function and the type and number of
optimization variables. This is followed by an overview of the Stanford Unified Op-
timization Framework (UOF), which is used in this work. Brief descriptions of the
PSO and MADS algorithms are then presented.
In Chapter 3, one of the simulation models used in this study is considered in
detail. We perform a sensitivity study to estimate the impact of the size of the
region surrounding the storage aquifer. A comparison between results from ECLIPSE
and AD-GPRS is then provided. We also discuss the manner in which the required
constraints are handled in the optimizer.
In Chapter 4, we present optimization results for two different aquifer models.
These optimizations attempt to minimize the total mass of gas-phase CO2 in the top
layer of the model at the end of a 100-year CO2 sequestration project. Multiple runs
of the PSO-MADS algorithm are considered. The sensitivity of the optimum to the
number of CO2 injection wells is also assessed. The conclusions drawn from this study
and ideas for future work are provided in Chapter 5.
Chapter 2
Optimization Procedures
As explained in Chapter 1, we aim to solve a comprehensive optimization problem
for CO2 storage systems. The details of this optimization problem will be presented
in this chapter. The optimizations are performed using algorithms within the Stan-
ford Unified Optimization Framework (UOF). The two primary algorithms used in
this study are mesh adaptive direct search (MADS) and particle swarm optimization
(PSO). Some specifics regarding the UOF and these algorithms are also discussed in
this chapter.
2.1 Optimization Problem
Following Isebor et al. [21], the optimization problem considered in this study can be
expressed as:
minx∈X,uc∈U
J(x,uc), (2.1)
where J is the objective function that we will minimize, x ∈ X ⊂ Z3Nw are the integer
variables pertaining to the problem, X is the subspace of Z3Nw that represents the
feasible region for x, uc ∈ U ⊂ R(Nw−1)Nc are the real (continuous) variables in the
optimization, and U is the subspace of R(Nw−1)Nc that represents the feasible region
for uc. Here, Nw denotes the number of CO2 injection wells and Nc represents the
number of control periods during CO2 injection.
8
2.1. OPTIMIZATION PROBLEM 9
In this study, the objective function (J) is defined as,
J =∑k∈Dtl
Sgask × Vk × φk × ρmolar, (2.2)
where Dtl corresponds to the grid blocks in the top layer of the storage aquifer model,
Sgask is the CO2 gas saturation in grid block k at 100 years, Vk is the bulk volume of
grid block k, φk is the porosity of grid block k, and ρmolar is the molar density of CO2
evaluated in grid block k.
Thus, J corresponds to the total mass of mobile CO2 (in kmol) in the top layer
of the storage aquifer model after 100 years. The optimization variables consist of
CO2 injection well locations, well lengths, and injection rates. These optimization
variables are summarized in Table 2.1 and are explained in detail below. Note that
the well location and length variables are integers, but in PSO we treat them as real
and round to the nearest integer.
Table 2.1: Summary of optimization variables.
Opt. variable # vars. Type LimitsHeel location (x) Nw integer [1 25]Heel location (y) Nw integer [1 25]Well length Nw integer [1 4]Injection fraction (Nw − 1)Nc real [0 0.9]
In order to limit the size of the search space during optimization, the injection
wells are specified to be horizontal wells oriented along the x direction in a particular
layer. Since the wells are placed through the entire grid block, the well lengths are
represented by the number of grid blocks in which the well is completed. This means
that we need three variables to represent a single well – the (x, y) location of the heel
of the well and the corresponding well length. This leads to 3Nw variables for well
location in the optimization problem.
For optimization problems involving more than one well, the corresponding well
injection rates are also optimization variables. The injection rates are expressed
as fractions of the total injection rate assigned to each injection well. This gives
(Nw − 1)Nc real variables for the injection rate specification. The well fraction of
10 CHAPTER 2. OPTIMIZATION PROCEDURES
well Nw is not a separate optimization variable since it can be calculated using the
injection fraction information of the other Nw − 1 injection wells (i.e., the fractions
must sum to one). Therefore, the total number of control variables in the optimization
problem, Nv, is Nv = 3Nw+(Nw−1)Nc. In this study, we take Nc = 4 and Nw = 1, 2, 3
or 4. Thus Nv varies from a minimum of 3 to a maximum of 24.
As explained in Chapter 1, a constraint is applied on the maximum BHP during
injection to restrict excessive pressure build-up. This is enforced through use of a
penalty that is added to the objective function value when the constraint is violated
by any of the injection wells. A second constraint to limit the movement of CO2 out of
the target region is also applied. As will be explained in Chapter 3, the aquifer model
used in the flow simulations has a surrounding region of grid blocks that provides
pressure support to the system. The second constraint ensures that the injected
CO2 stays within the storage region and does not migrate to the surrounding region.
This is also enforced through use of a penalty. The detailed penalty treatments are
described in Chapter 3.
2.2 Unified Optimization Framework
The Stanford Unified Optimization Framework (UOF) used in this work includes sev-
eral local and global search methods, such as PSO, MADS and differential evolution
(DE). These algorithms can be applied to optimization problems involving continu-
ous, integer and categorical variables. The UOF can be customized to employ any
of these algorithms separately or in combination using appropriate switching criteria.
In this study, PSO followed by MADS is applied.
A high-level representation of the structure of the UOF is presented in Fig. 2.1.
The user-provided input files are processed in the ‘INPUT’ block and fed into the
‘MAIN’ block, which calls the ‘OPTIMIZER’ (with the optimization algorithm and
parameters specified). The ‘OPTIMIZER’ interfaces with the ‘SIMULATOR’ to cal-
culate the value of the objective function. In this study, Stanford’s AD-GPRS is
used as the simulator. For supported algorithms (such as PSO and MADS), each call
to the simulator can be parallelized. Once the optimization is complete, a report is
2.3. PARTICLE SWARM OPTIMIZATION 11
generated by the ‘OUTPUT’ block. This report contains the best solution at each
iteration, along with the algorithm used at that step and the corresponding objective
function value.
Figure 2.1: Representation of the Unified Optimization Framework (adapted fromKim [24]).
Input to the UOF includes specifications for both the simulator and the optimizer.
The simulator input consists of the model setup files required for performing subsur-
face flow simulation in AD-GPRS. The optimizer input specifies the optimization
algorithms (and required parameters), switching criteria (if relevant), termination
conditions, and information about the type and number of optimization variables.
An initial guess can be provided to the optimizer if desired.
2.3 Particle Swarm Optimization
Particle swarm optimization (PSO) is a stochastic global search method that was
introduced by Eberhart and Kennedy [13]. Being a global search algorithm, it reduces
the possibility of the optimizer finding a poor local minimum. PSO has been applied
in the context of CO2 sequestration optimization by Cameron and Durlofsky [8],
among others.
PSO is a population-based algorithm that uses a swarm of potential solutions
(called particles) to explore the search space with the goal of objective function im-
provement. During each iteration, the movement of each of these particles is governed
12 CHAPTER 2. OPTIMIZATION PROCEDURES
by the following velocity expression:
vi(k + 1) = wvi(k) + c1r1(uPbesti (k)− ui(k)) + c2r2(u
Gbesti (k)− ui(k)). (2.3)
Here, vi(k) and vi(k + 1) are the velocity of particle i at iteration k and k + 1
respectively, and ui(k) represents the current position of particle i in the search space.
Note that u here corresponds to u = [xT,uTc ]T, with x and uc as defined previously.
The personal best (uPbesti ) is the best location that particle i has visited so far during
the optimization, and the global best (uGbesti ) is the best location that any particle
in the neighborhood of particle i has visited so far. The coefficients w, c1 and c2 are
pre-defined weight factors that are taken to be 0.729, 1.494 and 1.494 respectively.
These values are based on suggestions by Clerc [11], and they have been shown to be
effective in other optimization studies such as [20] and [24]. The coefficients r1 and r2
are random numbers between 0 and 1 that are sampled from a uniform distribution
at each iteration. This provides a stochastic component to the search. PSO proceeds
until a termination criterion is reached. Here, we specify a maximum number of
iterations as the termination criterion.
In Equation (2.3), the first term represents the inertial component, which main-
tains the motion of the particle from the previous iteration. The second term is
the cognitive component, which represents the attraction of the particle to the best
location it has reached so far, and the third term is the social component which rep-
resents its attraction to the best location that has been reached by any particle in its
neighborhood. In this study, we consider a random neighborhood as described in [12]
and [20]. A representation of the velocity components is shown in Fig. 2.2. Further
details regarding the algorithm can be found in [26] and [21].
The swarm size (number of particles in the swarm) is generally determined
heuristically. In our study, the number of particles (Np) is given by the heuristic
Np = 10 + 2√Nv, as suggested by Fernandez Martinez et al. [16]. As noted above,
the optimization problems considered in this study have Nv varying from 3 (for a
2.4. MESH ADAPTIVE DIRECT SEARCH 13
single injection well) to 24 (for four injection wells), but the swarm size for the four-
injection-well case (Np = 20) is used in all of the optimization runs.
Figure 2.2: Velocity components for a PSO particle (from Onwunalu and Durlofsky[26]).
PSO can be readily parallelized since each function evaluation (flow simulation)
can be performed independently. As noted above, well location and well length vari-
ables are handled by rounding the corresponding continuous value to the nearest
integer before the function evaluation. Since the method is stochastic in nature, mul-
tiple optimization runs should be performed. In this study, three separate runs are
performed for each optimization case.
2.4 Mesh Adaptive Direct Search
Mesh adaptive direct search (MADS) is a pattern search method introduced by Audet
and Dennis [1]. In contrast to PSO, in many cases MADS is guaranteed to converge to
a local optimum based on local convergence theory. When used appropriately with a
14 CHAPTER 2. OPTIMIZATION PROCEDURES
global search method such as PSO, MADS can potentially provide local convergence
in promising regions of the search space.
MADS explores the search space using a stencil of possible solutions that are
centered at the best known solution up to the current iteration. In each iteration,
these potential solutions are evaluated and compared with the current best objective
function value (value at the stencil center). The stencil is then re-centered at the
location with the best improvement in the objective function, and the process is
repeated in the next iteration.
This process is depicted in Fig. 2.3, where the red star represents the local op-
timum. The blue point at the stencil center is the best location up to the current
iteration, and the red stencil point provides the most improvement in the objective
function over the current solution. Thus, the stencil would be centered around the
red point at the next iteration. If none of the locations provides an improvement,
the stencil size is reduced and the process is continued. A lower limit on the size of
the stencil can be specified, and the algorithm terminated if that limit is reached. A
termination criterion based on the total number of function evaluations can also be
specified, and this treatment is used in this study.
Figure 2.3: Depiction of a MADS iteration in a 2D search space (adapted from Iseboret al. [21]).
Similar to PSO, MADS can be readily parallelized since the evaluations are all
2.4. MESH ADAPTIVE DIRECT SEARCH 15
performed independently. Integer variables in MADS are handled by defining the
search stencil in a discrete mesh. The number of evaluation points in the stencil
(called polling points) is assigned to be 2Nv (as explained in [21]), meaning that
MADS requires 2Nv function evaluations (flow simulations) for each iteration.
In this study, the optimization problem is solved sequentially by applying PSO
followed by MADS. We use PSO for the first 1000 function evaluations, and MADS for
the next 500 function evaluations. As mentioned earlier, in this work, for simplicity,
we do not switch back and forth between PSO and MADS as in [21]. It is possible
that improved solutions could be identified by iterating between the two algorithms.
Chapter 3
Simulation Model Description
One of the aquifer models used for flow simulation is described in this chapter. We will
also present results for the sensitivity studies performed to estimate the required size
of the region surrounding the storage aquifer (called the support region). Comparisons
between results from an industry standard simulator, ECLIPSE, and the simulator
that was used for this study, AD-GPRS, are also included in this chapter. The other
aquifer model used in this work is described and assessed (in less detail) in Chapter 4.
3.1 Channelized Aquifer Model
The simulation model discussed in this chapter is based on the model used by Jin
and Durlofsky [22] in their study on reduced-order modeling for CO2 sequestra-
tion. The storage aquifer is represented on a 25 × 25 × 10 grid, with each grid
block having dimensions 436 m × 436 m × 10 m, resulting in an aquifer of size
10.9 km × 10.9 km × 100 m. A total of 1.47 MT (1 MT = 109 kg) of CO2 is injected
into the aquifer every year for 20 years, resulting in a total injection of about 8000 m3
of CO2 per day. The total CO2 injected corresponds to 3.52% of the pore volume of
the storage aquifer, which is within the 1% – 4% range specified by the Intergovern-
mental Panel on Climate Change [34]. The full reservoir model used in the simulations
consists of the central storage aquifer region surrounded by additional grid blocks to
provide pressure support. A schematic of this setup is shown in Fig. 3.1. The size of
16
3.1. CHANNELIZED AQUIFER MODEL 17
the surrounding region used in the optimizations is determined by a sensitivity study,
presented in Section 3.2.
Figure 3.1: Schematic of the complete simulation model. The CO2 does not enterthe surrounding region. The additional grid blocks provide pressure support.
As explained in [22], the porosity and permeability values are obtained from the
Stanford VI model [9]. This synthetic geological model is a highly heterogeneous
channelized reservoir system with significant vertical variation. The log-permeability
field and the variation in permeability across the layers is shown in Fig. 3.2. The
permeability is in the range of 1 − 1000 mD and the porosity is in the range of
0.05 − 0.25. The total pore volume of the aquifer is 1.657 × 109 m3. For this model
ky = 0.8kx and kz = 0.218kx, where kx, ky and kz are the permeability values in the
x, y and z directions.
As discussed in Chapter 2, the flow simulations are performed using Stanford’s
Automatic Differentiation General Purpose Research Simulator (AD-GPRS). We use
a compositional simulation model with two components (CO2 and water) and two
phases (gas and water). The relative permeabilities for both phases are defined using
the Brooks-Corey relation with residual gas saturation Sgr = 0.1, the irreducible
water saturation Swi = 0, and exponents of 2 for both phases. Capillary pressure
between the phases is neglected.
18 CHAPTER 3. SIMULATION MODEL DESCRIPTION
(a) log kx for the full model (b) log kx for three layers
Figure 3.2: Permeability field of the storage aquifer model.
The initial reservoir pressure at the top layer is 17 MPa and the reservoir temper-
ature is set to 372 K. The system is isothermal. The initial mole fraction of the in-situ
fluid is 0.999 water and 0.001 CO2. The injected fluid is 0.999 CO2 and 0.001 water.
All of the injection wells are horizontal wells oriented along the x direction and located
in layer 8 of the storage aquifer model (third layer from the bottom). The model is
run for a total of 100 years, with the injection period being the first 20 years. The
simulations are run fully implicitly. Each run takes about 3-4 minutes on average,
and an upper limit of 7 minutes is enforced during the optimization procedure.
3.2 Sensitivity Study
For this system, we have observed that modeling only the storage aquifer (with no-flow
conditions at the boundaries of the storage aquifer) leads to a maximum injection BHP
of around 160 MPa (1600 bar) for the target CO2 storage quantity considered. This
is significantly higher than the allowable maximum BHP of 24 MPa (240 bar) for this
geological system. Therefore, it is necessary to have a simulation model that includes
a region of grid blocks surrounding the storage aquifer to provide pressure support.
This is also in line with how a realistic CCS project would be structured, where the
3.2. SENSITIVITY STUDY 19
storage region is embedded within a larger reservoir volume. In this section, we will
detail the sensitivity study applied to estimate the required size of this surrounding
region.
For the sensitivity study, we investigate the variation of the maximum BHP ob-
served (during CO2 injection) as a function of the size of the supporting region around
the aquifer. This is done by progressively adding a layer of support blocks around
the aquifer and identifying the highest injection BHP observed during the entire sim-
ulation. The grid spacing for each additional layer of grid blocks is higher than the
previous layer. Considering the total pore volume of the (channelized) storage aquifer
as PV caq, the porosity for each additional layer is assigned such that its pore volume
is 1×PV caq more than the previous layer.
The variation of the maximum BHP with the pore volume of the supporting region
is also studied under different injection scenarios. The parameters considered are the
number of injection wells, well lengths, and the placement of the wells. We consider
1 – 4 injection wells that are either one or three grid blocks in length (436 m or
1308 m). These are placed either in a high-permeability region (∼ 1000 mD) or a
low-permeability region (∼ 1 mD). In all scenarios, 8000 m3/day of CO2 is injected,
which corresponds to the target quantity of 1.47 MT/year of CO2, for 20 years. For
cases involving multiple wells, each of the wells injects an equal amount of CO2, and
the wells are of the same length (436 m or 1308 m).
The results of the sensitivity study are presented in Fig. 3.3 for the different cases
considered. The BHP limit of 24 MPa (240 bar), which corresponds to a 40% increase
over the initial pressure (as explained in Chapter 1), is also indicated in these figures.
It can be observed that for each type of well configuration, the maximum BHP during
injection decreases as the pore volume of the surrounding region increases. This is
expected because adding more surrounding pore volume provides additional pressure
support to the aquifer, thus enabling CO2 injection at a lower well pressure.
Note that the maximum injection BHP is dependent not only on the size of the
surrounding region, but also on the locations of the injection wells. For example, in
Fig. 3.3(a), when the single injection well is placed in a low-permeability location
(dashed-blue curve), we require a surrounding region of at least 25×PV caq for CO2
20 CHAPTER 3. SIMULATION MODEL DESCRIPTION
injection within the BHP limit. If the well is placed in a high-permeability region
(solid-red curve), around 17×PV caq is sufficient.
We also observe that the maximum injection BHP can be affected by the well
length. This effect is evident in Fig. 3.3(b), for two injection wells placed in a low-
permeability region. When they are long wells, a surrounding region of at least
22×PV caq is required. However, for short wells, we need more than 35×PV c
aq to avoid
violating the BHP limit. For one or two wells in high-permeability regions, the results
in Figs. 3.3(a) and 3.3(b) show that maximum BHP behavior is very similar for short
and long wells (the solid-blue and solid-red curves in these figures essentially overlap).
The maximum injection BHP is additionally dependent on the number of CO2
injection wells being considered. For example, when we have a single short well in
a low-permeability region (Fig. 3.3(a)), we require around 34×PV caq to inject CO2
without reaching the BHP limit. However, when we have four short wells in a low-
permeability region (Fig. 3.3(d)), it is possible to inject with a surrounding region of
around 26×PV caq.
3.2. SENSITIVITY STUDY 21
(a) One injection well (b) Two injection wells
(c) Three injection wells (d) Four injection wells
Figure 3.3: Results of the sensitivity study for different injection scenarios.
Based on the results in Fig. 3.3, it is evident that if we specify the pore volume
of the surrounding region to be 22×PV caq, feasible solutions exist for 1 – 4 injection
well cases. If we set the pore volume of the surrounding region to be 15×PV caq, by
contrast, many solutions will violate the constraint. We also note that a 22×PV caq
surrounding region is much smaller than that used in some previous studies. For
example, in [7], the surrounding region corresponds to about 165×PVaq (where PVaq
is the pore volume of the aquifer in [7]).
22 CHAPTER 3. SIMULATION MODEL DESCRIPTION
3.3 Constraint Handling
As mentioned in Chapter 2, penalty-based treatments are used to enforce the two
constraints (maximum BHP and no CO2 leakage out of the storage aquifer) during
optimization. We consider two types of penalty treatments. In the first approach,
fixed values are added to the original objective function (J , defined in Equation (2.2))
if the constraints are violated, i.e.,
J1 = J + λ+ γ, (3.1)
where J1 is the new objective function, λ is the penalty that is added if the BHP
constraint is violated by any of the wells at any time, and γ is the penalty that is added
if any CO2 has migrated outside of the storage region at the end of the simulation
period. The parameters λ and γ are both set to 1010 (kmol) based on numerical
experiments (note that typical values for J during optimization are O(108) kmol).
To describe the second type of penalty treatment, we first explain the manner
in which CO2 injection is specified in the simulator. Injection wells are specified to
operate on a rate-controlled basis until the maximum BHP is reached (240 bar). If
any well exceeds this value, it is switched to BHP control with the BHP set to 240 bar.
Thus, when the BHP constraint is violated, the actual mass of CO2 injected over the
20-year injection period (qact) is less than the target mass (qtrgt).
Hence, with the second type of penalty treatment, the objective function J2 is
now given by,
J2 = J + α(qtrgt − qact) + βMleak. (3.2)
Here, the α(qtrgt−qact) term addresses the BHP constraint (α is an appropriate scaling
factor), and the βMleak term is a CO2 leakage penalty. In this term Mleak is the total
mass of CO2 that has migrated outside the storage region at the end of the simulation
period and β is a scaling factor. We considered both fixed and variable scaling factors
(α, β). In the fixed case, we set α = 103 and β = 106. In the variable case, the values
of α and β were increased every 200 function evaluations.
3.4. COMPARISONS BETWEEN ECLIPSE AND AD-GPRS 23
In both cases, we did not observe any appreciable improvement in optimization
performance over the penalty treatment described above (Equation (3.1)). Because of
its simplicity, we therefore use J1, defined in Equation (3.1), in all of the optimizations
presented in Chapter 4.
3.4 Comparisons Between ECLIPSE and AD-GPRS
As discussed in Chapter 1, there are several mechanisms that trap CO2, namely
residual trapping, dissolution trapping, mineral trapping and structural trapping. In
this study, we model CO2 storage using only dissolution and structural trapping due
to simulator limitations. In this section, we will compare our AD-GPRS results to
ECLIPSE results that also include residual trapping to assess the limitations of our
simulations.
3.4.1 Aquifer Model with kz = 0.218kx
The ECLIPSE model used for this study is very similar to the reservoir model in AD-
GPRS. The model geometry and nearly all fluid and rock-fluid properties are specified
to be the same in both models. The key difference is the relative permeability input,
which in the ECLIPSE model includes hysteresis. This capability is not currently
available in the AD-GPRS version used in this study. Similar to [6], hysteresis is
represented using Killough’s method [23], with a Lands trapping coefficient of 1. As
mentioned earlier, in this model kz = 0.218kx. All of the simulations are compositional
runs, with the same equation of state (Peng-Robinson) as in the AD-GPRS runs. In
both the AD-GPRS and ECLIPSE models we have four injection wells, with each
injecting 2000 m3/day (corresponding to a total of 1.47 MT/year) of CO2 for 20 years.
As noted earlier, the wells are located in the third layer from the bottom. The
projected well locations are indicated in Fig. 3.4.
The objective function used in our optimizations, as explained earlier, involves
the mass of gaseous CO2 in the top layer of the model. A comparison between the
top-layer gas saturation at 100 years, in the AD-GPRS and ECLIPSE models, is
24 CHAPTER 3. SIMULATION MODEL DESCRIPTION
presented in Fig. 3.4. There is a difference of 7.5% in the objective function values
(J , as defined in Equation (3.1)) between the two simulation models. This difference
arises from the residually trapped CO2 in the ECLIPSE model.
In Figs. 3.5 and 3.6 we show x-z cross sections at each of the well locations. The
ECLIPSE model shows clear trapping of CO2 within the cross section. This effect is
absent in the AD-GPRS results. In Fig. 3.7, we show the variation in the average
CO2 gas saturation at the top layer of the model over time. There, we see that the
differences between the two models are more significant well after the injection stage
(i.e., after around 60 years).
(a) ECLIPSE model (J = 2.33× 108) (b) AD-GPRS model (J = 2.52× 108)
Figure 3.4: Gas saturation at the top layer at 100 years for the two models. Projectedwell locations are also shown. Wells are located in layer 8. J is in kmol.
3.4. COMPARISONS BETWEEN ECLIPSE AND AD-GPRS 25
(a) Well W1: ECLIPSE (b) Well W1: AD-GPRS
(c) Well W2: ECLIPSE (d) Well W2: AD-GPRS
Figure 3.5: Gas saturation in x-z cross sections at 100 years for Wells W1 and W2.
26 CHAPTER 3. SIMULATION MODEL DESCRIPTION
(a) Well W3: ECLIPSE (b) Well W3: AD-GPRS
(c) Well W4: ECLIPSE (d) Well W4: AD-GPRS
Figure 3.6: Gas saturation in x-z cross sections at 100 years for Wells W3 and W4.
3.4. COMPARISONS BETWEEN ECLIPSE AND AD-GPRS 27
Figure 3.7: Comparison of the average gas saturation at the top layer between thetwo models.
3.4.2 Aquifer Models with Lower Vertical Permeability
To further quantify the importance of residual trapping, two other reservoir models
with lower vertical permeability, corresponding to kz = 0.05kx and kz = 0.01kx, are
now considered. For both of these models, we simulate the four-well case described
in Section 3.4.1. Comparisons of the ECLIPSE and AD-GPRS results for these cases
are presented in Figs. 3.8 – 3.15. It can be observed that the contribution of residual
trapping to CO2 storage increases significantly as the vertical permeability decreases.
This is especially visible in Fig. 3.14(c), where the region around Well W4 in the
ECLIPSE model contains a significant quantity of residually trapped CO2, which is
not seen in the AD-GPRS model (Fig. 3.14(d)).
In Figs. 3.11 and 3.15, we present a comparison of the average gas saturation at
the top layer of the two simulation models. For the kz = 0.05kx case (Fig. 3.11), we
can see that the differences between the ECLIPSE and AD-GPRS results during the
injection period (20 years) are small, similar to the kz = 0.218kx model (Fig. 3.7).
28 CHAPTER 3. SIMULATION MODEL DESCRIPTION
However, for the kz = 0.01kx model (Fig. 3.15), the difference in CO2 in the top
layer is apparent from the early stages of the simulation, indicating the importance
of residual trapping for simulation models with very low vertical permeability.
(a) ECLIPSE model (J = 1.95× 108) (b) AD-GPRS model (J = 2.22× 108)
Figure 3.8: Gas saturation at the top layer at 100 years for the two sets of modelswith kz = 0.05kx. Projected well locations are also shown. Wells are in layer 8. J isin kmol.
3.4. COMPARISONS BETWEEN ECLIPSE AND AD-GPRS 29
(a) Well W1: ECLIPSE (b) Well W1: AD-GPRS
(c) Well W2: ECLIPSE (d) Well W2: AD-GPRS
Figure 3.9: Gas saturation in x-z cross sections at 100 years for Wells W1 and W2(kz = 0.05kx).
30 CHAPTER 3. SIMULATION MODEL DESCRIPTION
(a) Well W3: ECLIPSE (b) Well W3: AD-GPRS
(c) Well W4: ECLIPSE (d) Well W4: AD-GPRS
Figure 3.10: Gas saturation in x-z cross sections at 100 years for Wells W3 and W4(kz = 0.05kx).
3.4. COMPARISONS BETWEEN ECLIPSE AND AD-GPRS 31
Figure 3.11: Comparison of the average gas saturation at the top layer between thetwo models (kz = 0.05kx).
(a) ECLIPSE model (J = 0.619× 108) (b) AD-GPRS model (J = 1.99× 108)
Figure 3.12: Gas saturation at the top layer at 100 years for the two sets of modelswith kz = 0.01kx. Projected well locations are also shown. Wells are in layer 8. J isin kmol.
32 CHAPTER 3. SIMULATION MODEL DESCRIPTION
(a) Well W1: ECLIPSE (b) Well W1: AD-GPRS
(c) Well W2: ECLIPSE (d) Well W2: AD-GPRS
Figure 3.13: Gas saturation in x-z cross sections at 100 years for Wells W1 and W2(kz = 0.01kx).
3.4. COMPARISONS BETWEEN ECLIPSE AND AD-GPRS 33
(a) Well W3: ECLIPSE (b) Well W3: AD-GPRS
(c) Well W4: ECLIPSE (d) Well W4: AD-GPRS
Figure 3.14: Gas saturation in x-z cross sections at 100 years for Wells W3 and W4(kz = 0.01kx).
34 CHAPTER 3. SIMULATION MODEL DESCRIPTION
Figure 3.15: Comparison of the average gas saturation at the top layer between thetwo models (kz = 0.01kx).
In the results in Fig. 3.4, we saw that there was only a 7.5% difference in the
objective function for the vertical permeability considered in the channelized aquifer
optimizations. Since this difference is relatively small, the use of AD-GPRS for CO2
sequestration optimization is reasonable for this case.
Chapter 4
Optimization Results
In this chapter we first present optimization results for the channelized model de-
scribed in Chapter 3. We then introduce a second aquifer model for which analogous
assessments and optimizations are performed. Finally, we present the results of a
study to determine the smallest support volume possible for the different injection
scenarios considered.
4.1 Channelized Model Results
The aquifer model we now consider was discussed in detail in Chapter 3. The opti-
mization is performed with the number of wells varying from one to four. Since PSO
is a stochastic search algorithm, we perform three separate optimization runs (with
different initial-guess solutions) for each injection scenario. In all of the optimization
runs (as explained in Chapter 3), we inject 1.47 MT/yr of CO2 for 20 years, with a
BHP limit of 24 MPa (240 bar) on the injection wells. The variation of the objective
function value (J) with the number of wells is presented in Fig. 4.1. We see that there
is some variation in the optimized J among the three runs for each of the different
injection scenarios.
The values of the initial-guess solutions for each of the optimization runs are also
indicated in Fig. 4.1. We see a clear improvement (from the initial-guess solution) in
the objective function due to optimization. For example, for the four-well case, the
35
36 CHAPTER 4. OPTIMIZATION RESULTS
Figure 4.1: Summary of optimization runs for channelized aquifer.
best initial-guess solution has an objective function value of 1.40× 108 kmol (Run 3),
whereas the best optimized solution has a value of 6.99× 107 kmol (Run 2).
We also observe a decreasing trend in optimized J with an increase in the number
of injection wells, as would be expected. The best optimized solution for the single-
well scenario is J = 1.28× 108 kmol (Run 1), whereas the best optimized solution for
the four-well scenario is J = 6.99× 107 kmol (Run 2). There is thus a 45% decrease
in optimized J between these two cases. The increased number of wells enables the
CO2 to be spread over a larger volume in the aquifer, increasing the quantity of CO2
that can be dissolved. This in turn reduces the quantity of mobile CO2 at the top of
the model.
In Fig. 4.2, we present the progression of the best optimization runs for each of
the four injection scenarios. As mentioned in Chapter 2, we use PSO for the first
1000 function evaluations, followed by MADS for the next 500 function evaluations.
We can observe that for the single-well scenario, the optimization does not provide
any appreciable improvement after around 300 function evaluations. However, with
three or four injection wells, we see continued improvement up to the end of the
4.1. CHANNELIZED MODEL RESULTS 37
optimization runs. In the two-well scenario, we can see that although the PSO run
does not show much improvement after around 500 function evaluations, there is some
improvement in J during the MADS run.
Figure 4.2: Progression of best optimization run for each injection scenario.
We will now further investigate the characteristics of the best optimized solutions
for the different injection scenarios. Figs. 4.3 – 4.6 show the CO2 gas saturation at the
top layer of the aquifer model after 100 years (along with the projected well locations)
for the initial-guess solution and the optimized solution. The J values indicated in
the figures quantify the decrease in the CO2 saturation due to optimization. In the
optimized four-well scenario (Fig. 4.6), the wells are placed sufficiently far apart such
that the CO2 plumes from each of the injectors do not interact, and this leads to a
lower CO2 saturation at the top layer of the aquifer.
Note that in some cases, it might appear as if there is some flow of CO2 outside
the storage aquifer. For example, in Fig. 4.6(b), it appears as though the CO2 plume
around well W1 might extend outside the aquifer. However, the penalty constraint
discussed in Chapter 3 limits the movement of CO2 outside the borders of the storage
38 CHAPTER 4. OPTIMIZATION RESULTS
aquifer. To demonstrate this, we show the CO2 gas saturation after 100 years at
the top layer for the entire simulation model (including the region surrounding the
storage aquifer) in Fig. 4.7. There we see that the CO2 does not reach the surrounding
region. Note that the grid blocks in the surrounding region in Fig. 4.7 are not shown
to scale, since the grid block dimensions in those sections are around ten times those
in the aquifer, and representing them to scale would obscure the CO2 plumes inside
the aquifer.
(a) Initial guess (J = 1.76× 108) (b) Optimized solution (J = 1.28× 108)
Figure 4.3: Single injection well scenario. J is in kmol. In this and subsequent figures,projections of wells onto the top layer are also shown (in all cases, wells are locatedin layer 8).
(a) Initial guess (J = 1.65× 108) (b) Optimized solution (J = 1.06× 108)
Figure 4.4: Two injection wells scenario. J is in kmol.
4.1. CHANNELIZED MODEL RESULTS 39
(a) Initial guess (J = 2.05× 108) (b) Optimized solution (J = 0.743× 108)
Figure 4.5: Three injection wells scenario. J is in kmol.
(a) Initial guess (J = 1.48× 108) (b) Optimized solution (J = 0.699× 108)
Figure 4.6: Four injection wells scenario. J is in kmol.
40 CHAPTER 4. OPTIMIZATION RESULTS
Figure 4.7: CO2 gas saturation at 100 years in the top layer for optimized four-wellscenario. Region within the white box corresponds to the storage aquifer. Note thatthe region outside the aquifer is not represented to scale.
In Figs. 4.8 – 4.11, we show the well behavior for the best optimized solution for
each of the four scenarios. For the single-well case, well control is not possible and
the injection rate must stay constant. However, for cases with multiple wells, the
well behavior is quite variable during the injection period. For example, in the four-
well scenario (Fig. 4.11(a)), Well 2 injects very little CO2 over the entire injection
period. Well 4, by contrast, injects much more than the other wells, particularly from
10 – 20 years.
The injection BHPs of the wells are also shown in Figs. 4.8 – 4.11. Note that BHP
corresponds to the local reservoir pressure when there is no injection during a specific
period, such as in Fig. 4.10, where Well 3 does not inject any CO2 after 10 years.
The general trend is that the BHP continuously increases when the injection rate
is constant (as it is over a control period). This is expected since continuous CO2
injection increases the pressure in the aquifer, and to maintain a constant rate the
CO2 must be injected at increasing pressure. Note that the upper BHP constraint of
240 bar is not violated in any of the wells in Figs. 4.8 – 4.11. However, several of the
4.1. CHANNELIZED MODEL RESULTS 41
well BHPs do approach this limit.
(a) Well injection profile (b) Well BHP profile
Figure 4.8: Optimized well behavior for single injection well scenario.
(a) Well injection profiles (b) Well BHP profiles
Figure 4.9: Optimized well behavior for two injection wells scenario.
42 CHAPTER 4. OPTIMIZATION RESULTS
(a) Well injection profiles (b) Well BHP profiles
Figure 4.10: Optimized well behavior for three injection wells scenario.
(a) Well injection profiles (b) Well BHP profiles
Figure 4.11: Optimized well behavior for four injection wells scenario.
4.2. AQUIFER CHARACTERIZED BY MULTI-GAUSSIAN STATISTICS 43
4.2 Aquifer Characterized by Multi-Gaussian
Statistics
In this section we will first describe the second aquifer model. We will then discuss the
results of the sensitivity study (similar to that in Section 3.2) performed to estimate
the required size of the surrounding region. We will then present optimization results
for this case.
4.2.1 Simulation Model
The second simulation model used in this study is based on the model used by
Cameron and Durlofsky [7]. The storage aquifer is represented on a 25 × 25 × 8
grid, with each grid block of dimensions 436 m × 436 m × 13 m, resulting in an
aquifer of size 10.9 km × 10.9 km × 104 m. The log-permeability field for this model
is shown in Fig. 4.12. In this model, we have ky = kx and kz = 0.1kx. The porosity in
the model ranges from 0.15 – 0.25, and the total pore volume of the storage aquifer is
1.93× 109 m3. The target CO2 injection quantity is the same as for the channelized
model (1.47 MT/year for 20 years), indicating that the total injected CO2 occupies
about 3.03% of the pore volume. The initial reservoir conditions and other simulation
parameters are the same as for the channelized aquifer model. Wells are located in
layer 6.
Figure 4.12: log kx for the Gaussian aquifer.
44 CHAPTER 4. OPTIMIZATION RESULTS
4.2.2 Sensitivity Study
Similar to the assessment performed for the channelized aquifer model, a sensitivity
study is now conducted for the Gaussian aquifer model. We progressively add pore
volume around the storage aquifer and record the highest injection BHP observed
during the simulation. The dimensions and pore volume of the additional layers are
specified in a similar manner to that for the channelized model (in Section 3.2). We
once again consider 1 – 4 injection wells, that are either one or three grid blocks in
length (436 m or 1308 m) and are placed either in a high-permeability (∼ 1000 mD)
or a low-permeability (∼ 1 mD) region. As noted earlier, the total injection rate
is specified to be about 8000 m3/day (corresponding to 1.47 MT/year) of CO2 for
20 years. For cases with more than one injection well, each of the wells injects an
equal amount of CO2 and they are all of the same length (436 m or 1308 m). Note
that the quantity PV gaq used here corresponds to the total pore volume of the Gaussian
aquifer model (PV gaq = 1.93 × 109 m3), which is larger than the pore volume of the
channelized aquifer model (PV caq = 1.66 × 109 m3).
Results of the sensitivity study are presented in Fig. 4.13. As expected, we observe
a similar overall trend, where the maximum BHP decreases with an increase in the
size of the surrounding region. However, the difference in the maximum BHP between
the short and long well cases in low-permeability regions is significantly higher here
than in the results for the channelized aquifer. For example, the highest difference
between the two cases in the Gaussian model is about 180 bar (single injection well,
Fig. 4.13(a)), whereas it is only about 50 bar in the channelized model (two injection
wells, Fig. 3.3(b)).
Another difference between the results for the two models is in the required size
of the supporting region for the different injection scenarios considered. We can
see that for the Gaussian model, with wells in high-permeability locations (long or
short), the required size of the surrounding region is in the 12×PV gaq – 16×PV g
aq range
(corresponding to the solid-blue or solid-red curves in Figs. 4.13(a) – 4.13(d)). For
the channelized aquifer, the corresponding range is 16×PV caq – 21×PV c
aq for analogous
wells.
4.2. AQUIFER CHARACTERIZED BY MULTI-GAUSSIAN STATISTICS 45
In the case of long wells placed in low-permeability regions, we require a sur-
rounding region of at least 16×PV gaq (for the Gaussian model) to inject CO2 without
violating the BHP constraint (corresponding to the dashed-blue curve in Fig 4.13(d)).
The corresponding size for the channelized model is 21×PV caq for a comparable injec-
tion scenario. We also observe that, for short wells placed in low-permeability regions,
we require a surrounding region of size greater than 25×PV gaq for the Gaussian model
(corresponding to the dashed-red curves in Figs. 4.13(a) – 4.13(d)). Based on the
results in Fig. 4.13, the surrounding region is taken to be 16×PV gaq for optimizations
using the Gaussian model.
46 CHAPTER 4. OPTIMIZATION RESULTS
(a) One injection well (b) Two injection wells
(c) Three injection wells (d) Four injection wells
Figure 4.13: Results of the sensitivity study for the Gaussian model.
4.2.3 Optimization Results
We now present optimization results for this case. In Fig. 4.14, the variation of
the optimized solution with the number of wells is shown. We can observe that the
optimizations performed for each injection scenario lead to a significant reduction in
J from the corresponding initial-guess solution. On average, the improvement from
the initial-guess solution is in the 40% – 45% range for the Gaussian model (for
different injection scenarios), whereas in the channelized aquifer, this improvement
4.2. AQUIFER CHARACTERIZED BY MULTI-GAUSSIAN STATISTICS 47
was on average in the 30% – 35% range. However, when comparing the best optimum
across the four different injection scenarios, we find that there is a 33% decrease in
the objective between one and four injection wells in this case, which is lower than
the 45% decrease in the channelized aquifer case.
Figure 4.14: Summary of optimization runs for the Gaussian model.
In Fig 4.15, we present the progression of the best objective function value during
the course of the optimization. Of interest here is the rapid decrease in J at early
iterations. We again observe improvement from MADS for the two, three and four-
well cases.
48 CHAPTER 4. OPTIMIZATION RESULTS
Figure 4.15: Progression of best optimization run for each injection scenario for theGaussian model.
Comparisons of the CO2 gas saturation in the top layer after 100 years, for the
initial-guess solution and the best optimized solution in each injection scenario, are
shown in Figs. 4.16 – 4.19. As discussed above, improvement in the optimized solu-
tions over the initial-guess solutions is evident in all cases. Even in the single-well sce-
nario, we achieve an improvement of about 43% in the objective function value, which
is higher than the corresponding improvement for the channelized aquifer (27%). As
noted earlier, although it may appear that some CO2 has left the storage region, this
is not the case.
4.2. AQUIFER CHARACTERIZED BY MULTI-GAUSSIAN STATISTICS 49
(a) Initial guess (J = 3.70× 108) (b) Optimized solution (J = 2.11× 108)
Figure 4.16: Single injection well scenario for the Gaussian model. J is in kmol. Inthis and subsequent figures, projections of wells onto the top layer are also shown (inall cases, wells are located in layer 6).
(a) Initial guess (J = 3.32× 108) (b) Optimized solution (J = 1.89× 108)
Figure 4.17: Two injection wells scenario for the Gaussian model. J is in kmol.
50 CHAPTER 4. OPTIMIZATION RESULTS
(a) Initial guess (J = 3.27× 108) (b) Optimized solution (J = 1.76× 108)
Figure 4.18: Three injection wells scenario for the Gaussian model. J is in kmol.
(a) Initial guess (J = 2.79× 108) (b) Optimized solution (J = 1.42× 108)
Figure 4.19: Four injection wells scenario for the Gaussian model. J is in kmol.
We present the optimized well behavior for each of the four injection scenarios
in Figs. 4.20 – 4.23. Similar to the cases in the channelized aquifer, injection rates
vary considerably in time for scenarios involving multiple wells. For example, in
Fig. 4.23(a), we see that Well 1 injects relatively little, and that Well 4 has no injection
after 10 years. The BHP behavior also shares similarities with the previous case.
We again observe that the maximum BHP constraint is not violated in any of the
scenarios.
4.2. AQUIFER CHARACTERIZED BY MULTI-GAUSSIAN STATISTICS 51
(a) Well injection profile (b) Well BHP profile
Figure 4.20: Optimized well behavior for single injection well scenario (Gaussianmodel).
(a) Well injection profiles (b) Well BHP profiles
Figure 4.21: Optimized well behavior for two injection wells scenario (Gaussianmodel).
52 CHAPTER 4. OPTIMIZATION RESULTS
(a) Well injection profiles (b) Well BHP profiles
Figure 4.22: Optimized well behavior for three injection wells scenario (Gaussianmodel).
(a) Well injection profiles (b) Well BHP profiles
Figure 4.23: Optimized well behavior for four injection wells scenario (Gaussianmodel).
4.3. SUPPORT VOLUME STUDY 53
4.3 Support Volume Study
The sensitivity study results presented earlier were based on well behavior after man-
ual placement of wells. Although the well placement was chosen with reference to the
local permeability field, no optimizations were performed for either well locations or
rates. Our goal now is to determine the smallest surrounding region that allows us
to find a feasible injection strategy for each of the injection scenarios. We proceed
by continuously decreasing the pore volume (PV) of the surrounding region, and at
each pore volume we optimize well locations and rates. The minimum pore volume at
which the optimization yields a feasible solution is taken to be the minimum required
pore volume. This study is performed for both of the aquifer models. In this assess-
ment we again perform three optimization runs for each case, and the best result is
presented.
Results of this assessment for the Gaussian model are presented in Table 4.1.
The last column in this table shows the corresponding optimized J values for the
16×PV gaq case discussed earlier. As expected, scenarios with more injection wells can
have smaller surrounding regions than the single-well scenario. For example, we can
inject CO2 without violating the BHP constraint with a surrounding region of size
10×PV gaq if there are four wells, but we need at least 12×PV g
aq for one or two wells.
In addition, the smallest size determined by the optimization is smaller than that
estimated by the sensitivity study. For example, from Fig. 4.13(d), we can observe
that the smallest surrounding region where the maximum BHP does not violate the
BHP constraint is about 12×PV gaq, whereas with the optimization, we see that a
surrounding region of 10×PV gaq has a feasible solution.
We also note that in all of the reduced PV cases, the optimized J is higher than
the optimized J from the 16×PV gaq cases. For example, when we have a support
volume of 12×PV gaq, the optimized J with four wells is 1.67×108 kmol, whereas in
the 16×PV gaq case, the optimized J is 1.42×108 kmol. This is because a reduced PV
leads to higher BHPs during injection. As a result, previous (optimum) solutions can
become infeasible in the reduced volume cases. Hence, the optimization problem is
more constrained in the reduced volume cases than in the 16×PV gaq case.
54 CHAPTER 4. OPTIMIZATION RESULTS
Table 4.1: Results of the support volume study for the Gaussian aquifer. The sym-bol ‘×’ means no feasible solutions exist. Last column corresponds to results inSection 4.2.3. All values are in 108 kmol.
# wells J(9×PV gaq) J(10×PV g
aq) J(11×PV gaq) J(12×PV g
aq) J(16×PV gaq)
1 × × × 2.31 2.112 × × × 2.02 1.893 × × 1.93 1.83 1.764 × 1.82 1.75 1.67 1.42
Results of an analogous study performed for the channelized aquifer model are
presented in Table 4.2. Similar to the results of the Gaussian aquifer model, the four-
well scenario has a feasible injection strategy at a smaller support volume than the
other injection scenarios. The smallest support volume for feasible CO2 injection in
the four-well scenario is 17×PV caq, whereas for the one, two and three-well scenarios,
it is 18×PV caq. Once again, in all of the reduced PV cases, the optimized J is higher
than the optimized J in the 22×PV caq cases (shown in the last column of Table 4.2).
Table 4.2: Results of the support volume study for the channelized aquifer. Thesymbol ‘×’ means no feasible solutions exist. Last column corresponds to results inSection 4.1. All values are in 108 kmol.
# wells J(16×PV caq) J(17×PV c
aq) J(18×PV caq) J(22×PV c
aq)
1 × × 1.47 1.282 × × 1.46 1.063 × × 1.21 0.7434 × 1.63 1.04 0.699
Chapter 5
Conclusions and Future Work
Minimizing the risk of leakage is of key importance in the design of large-scale CCS
projects. In this thesis, we performed a comprehensive optimization study with the
objective of minimizing the mass of gas-phase CO2 that is structurally trapped at
the end of a 100-year CO2 storage project. CO2 in this form is susceptible to leakage
through (potentially unknown) fractures in the cap rock. Our optimizations involved
injection well placement, length, and time-varying rate control. BHP constraints were
satisfied, and CO2 was kept from flowing out of the storage aquifer, through use of
penalty treatments. Stanford’s Unified Optimization Framework (UOF) was used for
all of the optimizations in this study.
A heuristic sensitivity study was first performed to assess the impact of the size of
the region surrounding the storage aquifer on the pressure build-up during injection.
Using the results of this study, an appropriate support volume for the storage aquifer
was chosen for optimization. This support volume enabled (feasible) solutions in
which the maximum BHP during injection corresponded to at most a 40% increase
over the initial pressure in the aquifer. In this work, the simulation model used
for optimization included dissolution and structural trapping mechanisms – residual
trapping and mineralization were not modeled. To assess the impact of neglecting
residual trapping, we compared the results of our AD-GPRS flow simulations to
the results from equivalent ECLIPSE simulations that included relative permeability
hysteresis. For the channelized model used in our optimizations, we observed that
55
56 CHAPTER 5. CONCLUSIONS AND FUTURE WORK
the difference in the objective function (to be minimized) between the two models
was acceptable for our purposes.
We then performed optimizations using PSO followed by MADS. Two different ge-
ological models – a channelized aquifer and an aquifer characterized by multi-Gaussian
statistics – were considered. Clear improvement in the objective function was con-
sistently observed. The sensitivity of the optimized solution to the number of CO2
injection wells was also studied. We observed a 45% decrease (channelized model) and
a 33% decrease (Gaussian model) in the optimized objective function as the number
of CO2 injection wells was increased from one to four.
We also performed a detailed optimization study to determine the smallest sur-
rounding region that allows for a feasible injection strategy, for each of the injection
scenarios that were considered. We found that it is possible to have feasible injec-
tion strategies (with four injection wells) with a surrounding-region pore volume of
10×PV gaq for the Gaussian model and 17×PV c
aq for the channelized model (where PV gaq
and PV caq correspond to the total pore volume of the respective storage aquifer mod-
els). From these results, we observed that the optimized objective value increased as
the size of the region surrounding the storage aquifer was decreased. This assessment
can be extended to larger pore volumes of the surrounding region (for potential further
reduction in the objective). A Pareto front could then be constructed to represent
the trade-off between top-layer CO2 mass and the pore volume of the surrounding
region.
In future work, it will be of interest to explore the performance of other opti-
mization procedures, including hybrid approaches (such as PSO-MADS) that switch
back and forth between algorithms based on user-specified parameters. It would also
be useful to incorporate other types of objective functions, such as those based on
the economics of the CO2 sequestration project. From a modeling standpoint, the
use of properly upscaled aquifer models that capture fine-scale flow behavior in the
optimization would be of interest. Modeling brine removal for pressure management,
or brine cycling as in [7], could also lead to improved solutions. Finally, as observed
in Chapter 3, residual trapping can be a dominant storage mode under certain aquifer
conditions, so it is important to include this mechanism for general cases.
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