Optimization of CO2 Storage Systems with Constrained Bottom … · 2019. 6. 21. · optimization of...

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

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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

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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

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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


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


two models (kz = 0.05kx). . . . . . . . . . . . . . . . . . . . . . . . . 31

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3.12 Comparison between the ECLIPSE and AD-GPRS model for kz =

0.01kx. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31


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


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

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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

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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.

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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.


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


[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


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


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


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]).


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


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.


(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.




Figure 3.6: Gas saturation in x-z cross sections at 100 years for Wells W3 and W4.


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).


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.


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.




Figure 3.9: Gas saturation in x-z cross sections at 100 years for Wells W1 and W2(kz = 0.05kx).






Figure 3.11: Comparison of the average gas saturation at the top layer between thetwo models (kz = 0.05kx).


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.










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


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).


Figure 4.4: Two injection wells scenario. J is in kmol.



Figure 4.5: Three injection wells scenario. J is in kmol.


Figure 4.6: Four injection wells scenario. J is in kmol.


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


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.



Figure 4.10: Optimized well behavior for three injection wells scenario.


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.


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.


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.


(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


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.


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.



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).


Figure 4.17: Two injection wells scenario for the Gaussian model. J is in kmol.



Figure 4.18: Three injection wells scenario for the Gaussian model. J is in kmol.


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.


(a) Well injection profile (b) Well BHP profile

Figure 4.20: Optimized well behavior for single injection well scenario (Gaussianmodel).


Figure 4.21: Optimized well behavior for two injection wells scenario (Gaussianmodel).



Figure 4.22: Optimized well behavior for three injection wells scenario (Gaussianmodel).


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.


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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