Influence of chemical reactions on convective dissolution:

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UNIVERSITÉ LIBRE DE BRUXELLES Faculté des Sciences

Service de Chimie Physique et Biologie Théorique

Influence of chemical reactions on convective dissolution:

a theoretical study

Vanessa Loodts

Promotrice : Prof. A. De Wit

Co-promotrice : Prof. L. Rongy

Thèse présentée en vue de l’obtention du grade de Docteur en Sciences

Année académique 2016 - 2017

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À mes parents et ma tantine Noëline.

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Acknowledgements — Remerciements

“ Good friends, good books, and a sleepy conscience: this is the ideal life.”

Mark Twain.

Ces quatre années dans le domaine de la recherche furent très enrichissantes.

J’ai beaucoup appris et j’ai progressé, tant du point de vue professionnel que du point de vue personnel. Avant d’entrer dans le vif du sujet, je souhaite exprimer ma reconnaissance envers toute les personnes qui ont rendu ce beau projet de thèse possible.

Tout d’abord, je tiens à remercier tout spécialement ma promotrice Anne De Wit pour sa direction, ses conseils et son soutien durant la réalisation de cette thèse. J’ai non seulement beaucoup appris auprès d’elle mais j’ai aussi pris beaucoup de plaisir à travailler à ses cotés. Je tiens aussi à remercier ma co-promotrice Laurence Rongy, qui a activement participé à l’encadrement de mon travail depuis le mémoire, avant même de devenir officiellement co- promotrice. Ses conseils avisés ont été précieux pour l’aboutissement de ce travail. Un autre membre de cette équipe de “CO

₂

girls” que je souhaite remercier est Carelle Thomas, avec qui j’ai eu le plaisir d’écrire deux articles publiés avec succès.

Further, I would like to thank Philip Trevelyan for helping me with mathe- matical and computational issues and writing an article with me.

C’est également un plaisir de remercier Bernard Knaepen pour avoir fait partie de mon comité d’accompagnement et pour m’avoir fourni le code YALES2, maintenu au CORIA par Vincent Moureau et Ghislain Lartigue.

Merci à vous trois de m’avoir permis d’utiliser YALES2 et de m’avoir aidée à l’utiliser. Mes pensées spéciales vont aussi à Mathieu Caby pour ses coups de main concernant YALES2 et le Fortran.

This study has also benefited from discussions with many other scientists and from comments sent by referees during submissions to peer-reviewed journals. I am grateful to all those people who helped me to improve the quality of my work.

Je tiens également à remercier les membres de mon jury de thèse de prendre le temps de lire et de commenter mon travail, ainsi que de se déplacer pour le jour de la défense.

De plus, je remercie le support financier que j’ai obtenu du Fonds de la Re- cherche Scientifique (F.R.S.)-FNRS pour ma bourse d’aspirante doctorante

i

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et mon crédit de fonctionnement, ainsi que le Département de Chimie pour la bouse de voyage de Brouckère-Solvay.

Je souhaiterais remercier nos secrétaires Fabian Trillet et Delphine Vanti- ghem pour leur excellent soutien logistique. Plusieurs fois ils ont été proactifs et m’ont fourni de l’aide sans que je ne les sollicite : ce sont de vrais anges ! De manière similaire, j’ai bénéficié de l’aide du support IT, notamment de Gérald Houart de la Faculté des Sciences mais aussi de l’équipe support IT pour le cluster HYDRA.

Next, I would like to thank all my current and former colleagues researchers who make the Service de Chimie Physique et Biologie Théorique a lively and scientifically exciting place: Marcello, Luis, Domenico, Florence, Laurane, Lorena, Priyanka, Valérie, Fabian B., Igal, Rachel, Shyam, Jorge, Gabor, Alexis, Reda, Alexia, Geneviève, Didier, Jean-Christophe, Hugo, Alen, Ben- jamin, Nathalie, Hind... I am indebted to them for their scientific advice but also for their kindness and friendly support. I would like in particular to thank Chinar for reading and commenting parts of the draft for this thesis.

Additional special thanks to my sport mates Ceri, Aurore, Joëlle and Virat, and to JF for interesting discussions and gossips at tea time.

Enfin, je suis reconnaissante envers mes proches, mes amis et ma famille pour leur affection et leur soutien sans faille durant cette période de ma vie.

Misaotra betsaka Andriamanitra sy ny fianakaviana eto Madagasikara : ny renibeko Mamasoa, ny dadatoako sy ankizy, ary Dadatoa René noho ny fanampianao.

Mes pensées vont vers tous ceux qui en croisant mon chemin ont illuminé mon quotidien. Plus particulièrement, je remercie ici Émile, Laurence P. et Nabil. Même si je ne vous vois pas aussi souvent que je le souhaiterais, vous gardez une place spéciale dans mon coeur.

Je suis immensément reconnaissante envers mes parents et ma tantine - mar- raine Noëline pour leur amour, leur confiance et leurs bons petits plats :).

Je souhaite également remercier la famille de Kevin qui m’a accueillie avec chaleur et en particulier mes beaux-parents Jean et Marie-Paule.

Le mot de la fin est consacré à Kevin, ma plus grande source d’inspiration.

Cela fait déjà neuf ans que tu es entré dans ma vie. Tu m’as fourni de bons conseils d’informaticien et physicien amateur. Dans les moments de doute tu m’as écoutée et encouragée à aller au bout de mes projets, quels qu’ils soient. C’est une grande joie de savoir que malgré le chaos de la vie, nous pouvons compter l’un sur l’autre. Merci pour tout.

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Abstract Studying the coupling between buoyancy-driven instabilities and chemical reactions is not only relevant to fundamental research, but has also recently gained increased interest because of its relevance to CO

₂

sequestra- tion in subsurface geological zones. This technique aims to limit the emis- sions of CO

₂

to the atmosphere, with a view to mitigating climate change.

When injected in e.g. a saline aquifer, CO

₂

dissolves into the brine occupying the geological formation, thereby increasing the density of the aqueous phase.

This increase of density upon dissolution leads to a denser fluid boundary layer rich in CO

₂

on top of less dense fluid in the gravity field, which drives dissolution-driven convection. This process, also called convective dissolu- tion, accelerates the transport of dissolved CO

₂

to the host phase and thus improves the safety of CO

₂

sequestration. The same kind of instability can develop in other contexts involving the dissolution of a phase A into a host phase, such as solid dissolution or transfer between partially miscible liq- uids. In this context, the goal of our thesis is to understand how chemical reactions coupled to dissolution-driven convection affect the dynamics of the dissolving species A in the host solution. To do so, we introduce a general reaction of the type A + B → C where A, B and C affect the density of the aqueous solution. We theoretically analyze the influence of the relative physical properties of A, B and C on the convective dynamics. Our theo- retical analysis uses a reaction-diffusion-convection model for the evolution of solute concentration in a host fluid solvent occupying a porous medium.

First, we quantify the characteristic growth rate of the perturbations by us- ing a linear stability analysis. Thereby we show that a chemical reaction can either accelerate or slow down the development of convection, depending on how it modifies the density profile that develops in the reactive solution. In addition, new dynamics are made possible by differential diffusion effects.

Then, by analyzing the full nonlinear dynamics with the help of direct nu- merical simulations, we calculate the dissolution flux into the host phase. In particular, the dissolution flux can be amplified when convection develops earlier, as CO

₂

is then transported faster away from the interface. Finally, we compare these theoretical and numerical predictions with results of lab- oratory experiments and discuss the possible implications of this study for CO

₂

sequestration.

Keywords: Chemical reactions, Fluid dynamics, Convection, Linear stabil- ity analyses, Numerical simulations.

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Résumé L’étude du couplage entre réactions chimiques et instabilités liées à des différences de densité est non seulement intéressante pour la recherche fondamentale, mais a aussi attiré plus d’attention récemment grâce à son intérêt pour la séquestration du CO

₂

dans les sous-sols géologiques. Cette technique a pour objectif de limiter les émissions de CO

₂

dans l’atmosphère, dans le but de limiter les changements climatiques. Durant son injection dans un aquifère salin par exemple, le CO

₂

se dissout dans de la saumure, augmentant ainsi la densité de la phase aqueuse. Cette augmentation de densité lors de la dissolution mène à une zone plus dense au-dessus d’une zone moins dense dans le champ de gravité, ce qui entraîne de la convec- tion liée à la dissolution. Ce processus, aussi appelé dissolution convective, accélère le transport du CO

₂

dissous et donc améliore la sécurité de la sé- questration. Le même type d’instabilité peut se développer dans d’autres contextes impliquant la dissolution d’une phase A dans une phase hôte, tels que la dissolution d’un solide ou d’un liquide partiellement miscible dans un autre. Dans ce contexte, l’objectif de notre thèse est de comprendre com- ment les réactions chimiques couplées aux instabilités convectives affectent la dynamique d’une espèce A se dissolvant dans une phase hôte. Pour ce faire, nous introduisons une réaction générale de type A + B → C où A, B et C affectent la densité de la phase hôte. Nous analysons théoriquement l’influence des propriétés physiques relatives des espèces A, B et C sur la dynamique convective. Notre analyse théorique utilise un modèle réaction- diffusion-convection pour l’évolution de la concentration des solutés dans un fluide hôte occupant un milieux poreux. Tout d’abord, nous quantifions le taux de croissance caractéristique des perturbations à l’aide d’une analyse de stabilité linéaire. Nous montrons ainsi qu’une réaction chimique peut soit accélérer soit ralentir le développement de la convection, en fonction du profil de densité qui se développe dans la solution réactive. De plus, les différences entre coefficients de diffusion sont à l’origine de nouvelles dynamiques. En analysant la dynamique non-linéaire à l’aide de simulations numériques di- rectes, nous calculons le flux de A qui se dissout dans la phase hôte. En particulier, ce flux peut être amplifié quand la convection se développe plus tôt, vu que le CO

₂

est alors transporté plus rapidement loin de l’interface.

Enfin, nous comparons ces prédictions théoriques et numériques avec des ré- sultats d’expériences en laboratoire et discutons les possibles implications de cette étude pour la séquestration du CO

₂

.

Mots-clés : Réactions chimiques, Dynamique des fluides, Convection, Ana- lyse de stabilité linéaire, Simulations numériques.

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Symbols and Abbreviations

Superscripts

f ¯ Horizontally- or vertically-averaged f . h f i Volume-averaged f.

f

^∗

Characteristic f.

f

⁰

Value of f at the onset of instability.

f

^s

Base state f .

f ˜ , f ˆ Dimensional f (depending on the unit).

f Dimensionless f.

Subscripts

f

_c

f scale.

f

m

f corresponding to the most unstable mode.

f

_U

f computed for all modes from the volume-averaged squared velocity.

f

x

Horizontal component of vector f (along the x-direction).

f

_z

Vertical component of vector f (along the z-direction).

ix

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Nondimensional numbers

β Ratio B

0

/A

0

between the initial concentration B

0

of reactant B and the solubility A

0

of the dissolving species A in the host phase. [—]

δ

_i

Ratio D

_i

/D

_A

between the diffusion coefficient D

_i

of species i and D

_A

of the dissolving species A. [—]

D Damköhler number constructed with the height of the system. [—]

φ Porosity of the porous medium. [—]

R Rayleigh number constructed with the height of the system. [—]

R

i

Solutal Rayleigh number of species i. [—]

T Temperature of the system. [—]

Greek Symbols

α

_i

Solutal expansion coefficient of species i. [L/mol]

κ Permeability of the porous medium. [m

²

]

θ Orientation of the z axis with regard to gravity. [

^◦

]

λ ˜ Wavelength of the instability. [m

⁻¹

]

µ Dynamic viscosity of the fluid. [Pa s]

˜

ν Spectral wavenumber of the instability. [m

⁻¹

]

ρ

₀

Density of the solvent. [kg/L]

ρ

_B

Density of the host brine. [kg/L]

˜

ρ Density of the fluid. [kg/L]

˜

σ Growth rate of the instability. [s

⁻¹

]

x

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Roman Symbols

A ˜ Concentration of the dissolving species A. [mol/L]

A

₀

Solubility of A in the host phase [mol/L]

B ˜ Concentration of the dissolved reactant B. [mol/L]

B

0

Initial concentration of reactant B (everywhere except in Section 5.1) or inert solute B (Section 5.1) in the host phase. [mol/L]

C ˜ Concentration of the product C of the reaction. [mol/L]

d Gap width of the Hele-Shaw cell. [m]

D

i

Diffusion coefficient of species i. [m

²

/s]

g Gravity field of norm g. [m/s

²

]

H Height of the system (z-direction). [m]

˜ h

_CO₂

Henry’s constant for CO

₂

. [atm kg mol

⁻¹

]

˜ k Wavenumber of the instability. [m

⁻¹

]

K

i

Equilibrium constant for reaction i. [depends]

k

_i1

Kinetic constant of forward reaction i. [depends]

k

_i2

Kinetic constant of backward reaction i. [depends]

L Width of the system (x-direction). [m]

˜

m

i

Molality of species i. [mol kg

⁻¹

]

M

_i

Molar mass of species i. [g mol

⁻¹

]

˜

p Pressure in the fluid host phase. [atm]

p

_CO₂

Partial pressure of gaseous CO

₂

above the host solution. [atm]

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q Kinetic constant of the A+B → C reaction. [L/mol/s]

r Reaction rate. [—]

T ˆ Temperature of the system. [

^◦

C]

T ˜ Temperature of the system. [K]

˜ t Time. [s]

u ˜ Flow velocity at the macroscopic scale. [m/s]

v ˜ Flow velocity at the pore scale. [m/s]

˜

x Horizontal space coordinate. [m]

˜

z Vertical space coordinate along g. [m]

Frequently used abbreviations

LSA Linear stability analysis NR Non-reactive

RD Reaction-diffusion

RDC Reaction-diffusion-convection

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Introduction

The interplay between chemistry and fluid flows can produce a wide range of various complex dynamics. Indeed, chemicals can sometimes be more than simple passive tracers carried out by fluid flows when changes in solute concentrations actively change the physical properties of the fluid such as density (as illustrated in Fig. 1), but also viscosity, or surface tension. By modifying the local quantities of solutes, reactions consuming or producing chemicals can thus create dynamic gradients (of density, for example) vary- ing in space and time, which in turn alter the flow dynamics. Conversely, amplifying fluid flow increases the reaction rate by accelerating transport and thus mixing in the fluid, thereby bringing reactants more quickly into contact.

Understanding this interplay between chemistry and fluid dynamics is there- fore crucial in numerous fields of study. In the field of chemical engineering, predicting the output of a chemical process is important for a large number of applications, for example to improve the efficiency of chemicals produc- tion or, on the contrary, slow down the chemical degradation in the reser- voir where the process takes place. Studying that interplay is in addition relevant to understand or even control natural processes occurring in the environment, such as the dispersion of pollutants in the atmosphere, in the oceans, or in geological media for instance. In particular, analyzing reactive fluid flows in porous or fractured geological media, so-called “Geological fluid dynamics” [133]

¹

not only provides insight on the origin of geological forma- tions shaped by fluid flows coupled with dissolution and precipitation, but also helps tackle challenges in geoengineering such as sustainably managing groundwater resources, performing Enhanced Oil Recovery (EOR) or carbon dioxide (CO

₂

) sequestration, to cite but a few.

Among those challenges, CO

₂

sequestration in geological formations has attracted much attention because of its potential role to mitigate climate change. CO

₂

sequestration consists in isolating this greenhouse gas from

1

References are numbered according to alphabetical order of authors.

1

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2 Figure 1 – Chemical reactions and fluid dynamics are coupled via local changes in solute concentrations due to reactions, which can create density gradients varying in space and time. In turn, these density gradients can trigger and alter the flows, which influences the spatio-temporal distribution of the chemicals and thus the global reaction rate.

the atmosphere to reduce its contribution to global warming. When CO

₂

is injected into geological formations such as saline aquifers or oil fields, convec- tion can occur because the dissolution of CO

₂

in the host phase (brine or oil) located below it, coupled to diffusion, creates a buoyantly unstable density stratification: a denser boundary layer of CO

₂

-rich solution lies on top of less dense CO

₂

-poor bulk solution in the gravity field. After some time, the initially flat miscible contact line between the denser fluid layer and the less dense one deforms and so-called “fingers” of denser fluid sink into the lower part of the solution. That convective dissolution of CO

₂

into the host phase accelerates the mixing of both CO

₂

and host phases, thereby enhancing the safety of sequestration. The impact of reactions on the convective dynamics remains, however, poorly understood. Convective dissolution during carbon dioxide (CO

₂

) sequestration in subsurface formations [42, 75] is very likely to be affected by the interplay between chemistry and fluid flows. Given that geochemical reactions can occur inside the storage site, understanding their impact on CO

₂

transport could help select storage sites with geochemistry optimal to accelerate the dissolution of CO

₂

into the host phase.

In this context, we theoretically examine the effects of chemical reactions on

the density-driven instability that develops upon the dissolution of a reservoir

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3 phase into a fluid host phase. We analyze the impact of chemistry on the fingering dynamics and on the resulting dissolution rate into the host phase.

Our aim is to understand whether chemical reactions can accelerate or slow down the development of convection. With a view to applying our results for CO

₂

sequestration in geological formations, we quantify the development of that instability in a partially miscible stratification occupying a porous medium and discuss the impact of chemistry on the time scales needed for complete CO

₂

dissolution and thereby on the safety of the whole process.

Our results are organized as follows.

In Chapter 1, we introduce the concepts of CO

₂

sequestration and fluid

flows in porous media, important to understand the context of this thesis,

and we review the literature on convective dissolution in partially miscible

stratifications. Next, we classify in Chapter 2 the effects of reactions on

convective dissolution according to whether they accelerate or slow down

the early-time development of the instability. In Chapter 3, we expand

the conclusions drawn in Chapter 2 to later-time dynamics when non-linear

effects are dominant, and more specifically we quantify the effects of reactions

on the dissolution rate into the host solution. Afterwards, we present the new

convective dynamics arising from changing the relative diffusion coefficients

of the solutes in Chapter 4. In Chapter 5 we apply our general classification

to the specific case of CO

₂

dissolving into aqueous solutions, in order to

compare our theoretical predictions to the results of laboratory experiments

in Hele-Shaw cells. Finally, we draw conclusions and present the prospects

that this work has opened.

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Chapter 1

Convective dissolution in the context of carbon dioxide

(CO ₂ ) sequestration

The interplay between chemical reactions and fluid flows has an important impact on the transport dynamics in a wide range of systems. More specifi- cally, we here consider a partially miscible stratification of a reservoir phase dissolving with a finite solubility into a host fluid. As this dissolution typ- ically modifies the density of the fluid, a complex density profile builds in time in the host phase. This profile can be modified by chemical reactions which can alter or even trigger buoyancy-driven (or density-driven) convec- tive flows, i.e. flows driven by differences of density in the gravity field.

These flows accelerate the transport processes in the host phase, thereby improving the global reaction rate and the mixing of both phases. Such buoyancy-driven flows take place in different types of partially miscible strat- ifications: a solid dissolving into a liquid [92, 93, 158], a gas dissolving into a liquid [16, 100, 115, 130, 136–139, 165, 166, 182–186], a liquid dissolving into another one [19, 106], solutions separated by a semi-permeable membrane [11, 25], to cite but a few. Analyzing convective dissolution is of tantamount importance for a wide range of applications including assessing the safety of nuclear energy generation [92, 93], or predicting the fate of carbon dioxide (CO

₂

) during its capture [182–186] or sequestration in geological formations [42, 75]. As CO

₂

sequestration plays a major role in the attempts to miti- gate climate change, we more specifically analyze convective dissolution in the context of CO

₂

sequestration, and therefore consider that the partially miscible stratification occupies a porous medium.

5

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6 Chapter 1 Convective dissolution and CO

₂

sequestration

Let us now discuss with more detail the context of this study. In Section 1.1, we present Carbon Capture and Sequestration (CCS), which is a major ap- plication of the research on convective dissolution in partially miscible strat- ifications. Next, in Section 1.2 we discuss fluid flow and solute transport in porous media, like the geological formations used to store CO

₂

. This flow coupled to diffusive processes and chemical reactions contributes to the spatio-temporal changes in solute concentrations, possibly creating density gradients at the origin of hydrodynamic instabilities. These notions are use- ful to understand how physicochemical processes impact the storage during CO

₂

sequestration. In Section 1.3, we briefly present buoyancy-driven flows in the more general context of partially miscible stratifications and contrast them with their counterparts in miscible stratifications. Finally, we present the aims and organization of this thesis in Section 1.4.

1.1 Geological sequestration of CO 2

Geological sequestration of carbon dioxide (CO

₂

) is one of the most promis- ing options to reduce CO

₂

emissions, with a view to mitigating climate change and ocean acidification [12–14, 77, 134]. CO

₂

sequestration is part of Carbon Capture and Sequestration (CCS) techniques, consisting in cap- turing CO

₂

from large emitters of exhaust gas like power plants or chemical industries, and transporting it to a storage site where it will be isolated from the atmosphere for long periods of time or even indefinitely

¹

. To explain the importance of CO

₂

sequestration in geological formations, we first discuss the contributions of this greenhouse gas to climate change and ocean acid- ification. Then, we discuss the role of CCS to help reduce CO

₂

emissions and precise the importance of geological sequestration among other CCS techniques.

1.1.1 Why should we limit CO

₂

emissions?

The quantity of CO

₂

in the atmosphere has risen since the industrial era due to the increasing use of fossil fuels. The pre-industrial level of CO

₂

in the atmosphere amounted

²

to 280 ppm [13]. This atmospheric concentration of CO

₂

has increased by 40 % since 1950, rising from less than 320 ppm in 1950 up to 391 ppm in 2011 (see Fig. 1.1). In 2015, the atmospheric

1

More precisely, the terms

storage

and

sequestration

refer to isolation from the atmo- sphere for long periods of time and indefinitely, respectively [13]. However in this study we will use one or the other interchangeably.

2

280 ppm CO

2

means 280 molecules of CO

2

per million molecules constituting dry

air.

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Geological sequestration of CO

₂

7 concentration of CO

₂

amounted to 399 ppm, with an average growth of 2 ppm/year in the last ten years [76]. The large anthropogenic CO

₂

emissions (over 32 GtCO

₂

/year in 2014 [76]) explain most of that increase [78].

Figure 1.1 – The increase of the atmospheric concentration of CO

₂

, a green- house gas, very likely explains most of the increase of the average annual temperature shown in Fig. 1.2. The letters represent different predictive models of IPCC: FAR is First Assessment Report, SAR Second Assessment Report, TAR Third Assessment Report, AR4 Fourth Assessment Report where A1B, A2 and B1 are predictive models based on different hypotheses.

Reproduced from IPCC (Intergovernmental Panel on Climate Change) [78].

The increase of the atmospheric concentration of CO

₂

, along with that of other greenhouse gases (GHGs)

³

, has very likely caused global warming

⁴

. CO

₂

is a major contributor to global warming as CO

₂

emissions due to energy generation represent 60% of the global anthropogenic emissions of greenhouse gases, and most of these emissions result from the oxidation of carbon in fuels during combustion [76]. Because of GHGs, the average annual temperature has never been so high in the last ten thousand years. It has risen by 0.85

^◦

C over the period 1880 – 2012 and more recently, by about 0.72

^◦

C over the period 1951 – 2012 (see Fig. 1.2) [78]. Hot days become more frequent and warmer, while cold days are less frequent and less cold

3

Other GHGs include water (H

2

O), methane (CH

4

), dinitrogen monoxide (N

2

O), and chlorofluorocarbons (CFCs), among others [78].

4

GHGs absorb the heat radiated by the earth’s surface, which maintains the tempera-

ture above 0

^◦

C. However, an increase in the atmospheric concentration of GHG perturbs

this equilibrium and leads to a global increase of the average temperature [168].

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8 Chapter 1 Convective dissolution and CO

₂

sequestration

[78]. As a consequence, the amounts of snow, sea ice and ice sheets on Earth decrease each year, while the average sea level rises [78]. Ocean acidification is another problem linked to the increase of CO

₂

atmospheric concentration [52, 63, 149]. When dissolving into the ocean, CO

₂

forms carbonic acid (H

₂

CO

₃

) in water, which releases protons

⁵

and thus increases the acidity of the ocean.

Figure 1.2 – The increase of the temperature anomaly is one of the indicators of global climate change. This anomaly represents the difference of global an- nual temperature relative to the 1961–1990 period. For further details on the caption, please refer to Fig. 1.1. Reproduced from IPCC (Intergovernmental Panel on Climate Change) [78].

Climate change and ocean acidification can have catastrophic consequences for the ecosystems and the species depending on them, including humans [52]. Through changes in evaporation, precipitation and runoff patterns, climate change might reinforce the problem of water scarcity, already present in vulnerable regions due to population growth: this lack of water might dramatically impact sanitation, public health, as well as food availability via decreases of crop yields [150]. In addition, extreme events such as drought conditions, heavy precipitations, flood hazards or extreme sea level rises are very likely to occur more frequently in some parts of the world, which could further affect the living conditions of billions of people [78, 150]. Ocean acidification and warming, shifting patterns of currents, and sea ice loss

5

See Chapter 5.

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Geological sequestration of CO

₂

9 are other stresses that could endanger marine ecosystems, from fisheries to coral reefs [149]. The loss of sea ice directly affects sea ice algae and sub ice phytoplankton through a loss of habitat, and indirectly impacts other species, such as arctic foxes, through effects on movement, population mixing, and pathogen transmission [140].

1.1.2 The role of Carbon Capture and Sequestration (CCS) to limit CO

₂

emissions

In this context, acting quickly to reduce CO

₂

emissions is mandatory. This reduction can be achieved by acting on the five factors impacting the mag- nitude of CO

₂

emissions [13, 71, 77]: the population size, the economic standard of living, the energy intensity of the economy, the carbon intensity of the energy system, and the uptake of atmospheric CO

₂

. Here below we briefly discuss all possible mitigation measures, based on these five factors.

First, we can already rule out the reduction of the population size, or of the economic standard of living, typically represented by GDP

⁶

per capita. The world population is unlikely to stop growing this century: an increase up to between 9.6 and 12.3 billion inhabitants worldwide in 2100 is expected [57].

Although an economy of “degrowth” is sometimes advocated [22], a policy aiming the decline of the economic output will probably not be accepted [13, 134]. Further, historical evidence shows that population size and economic decline are geographically localized and temporally limited [13].

Another possibility to reduce CO

₂

emissions is to decrease the energy inten- sity of the economy, i.e. the amount of energy needed to achieve a given standard of living. It is difficult to further improve the efficiency of the con- version of primary energy into useful energy (like electricity) [70]. Therefore, the only way of reducing the energy intensity of the economy is to improve the efficiency of energy end-use and energy management: improving isolation of houses, modifying behaviors in particular concerning family cars driving in cities, etc. [13, 70, 168]

Reducing the carbon intensity of the economy is also an option to reduce CO

₂

emissions. This means switching from carbon-rich fossil fuels (oil, coal) to hydrogen-rich ones such as natural gas, to renewable energy sources, or to nuclear energy generation [13]. Reducing the use of carbon-rich fossil fuels is difficult, especially in developing countries, because oil and coal are cheap and abundant [52, 121, 168]. Switching from oil or coal to natural gas can re- duce CO

₂

emissions by more than 40 % [168]. However, it can be short-lived

6

GDP stands for Gross Domestic Product.

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10 Chapter 1 Convective dissolution and CO

₂

sequestration

because of the limited supply of natural gas, and CO

₂

remains a byproduct of the energy generation [13]. Transitioning to renewable energy sources has several drawbacks like the difficulty in storing the energy generated, pub- lic resistance to wind energy, conflicts between expansion of land devoted to biomass production and other land uses, while nuclear energy raises the problems of nuclear waste disposal and security concerns [13].

The last option is the increase of the uptake of CO

₂

. This increase can be achieved either by stimulating natural carbon sinks (e.g. reforestation at the condition that forest resources are exploited rather than burned [168]), which have a relatively small capacity [13] or either through Carbon Capture and Sequestration (CCS) techniques.

1.1.3 The importance of geological sequestration in CCS Various technologies are available for CO

₂

capture, separation, transport, and storage, as shown in Fig. 1.3. These different options can be combined together depending on the case under consideration to form a full CCS sys- tem. Let us first briefly describe the technologies used for CO

₂

capture and transport before moving to CO

₂

storage.

a CO

₂

capture and transport

The first step is to capture CO

₂

from exhaust gas and to separate it from other components and impurities. This can be achieved by post-combustion, pre-combustion or through oxy-fuel combustion [52, 63, 134, 142, 168, 169].

In post-combustion systems, the exhaust gas is flown through a solvent that preferably absorbs CO

₂

. Examples of solvent include aqueous solutions of alkanolamines such as monoethanolamine (MEA) [134, 182], ammonia or amino acid salt solutions [134]. These solutions can be recycled, after heating to release the captured CO

₂

. In pre-combustion technologies, carbonless fuels like hydrogen (H

₂

) are produced from carbon fuels, with CO

₂

as by-product.

The output stream then contains CO

₂

at a higher concentration and pressure than in post-combustion technology [134, 142]. Oxy-fuel combustion consists in performing the combustion with pure oxygen (O

₂

) separated from the nitrogen (N

₂

) contained in air. This separation step allows to recover as combustion products essentially CO

₂

and water (H

₂

O), easily separated.

Similarly to pre-combustion technology, the gas stream then contains a high

CO

₂

concentration which facilitates further treatment [77, 134, 142, 169].

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Geological sequestration of CO

₂

11 Figure 1.3 – Schematic representation of possible CCS systems. CO

₂

is

emitted from combustion of coal, biomass, gas or oil, and captured in power

plants, petrochemical plants, or cement/steel/refineries, etc. H

₂

possibly

produced during the capture process can be exploited further as a carbon-

less fuel. Next, CO

₂

is transported to storage sites. The storage options

for CO

₂

are mineral carbonation, industrial uses, ocean storage, or geolog-

ical storage possibly coupled to EGR (Enhanced Gas Recovery) or EOR

(Enhanced Oil Recovery — not represented here). Reproduced from IPCC

(Intergovernmental Panel on Climate Change) [77].

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12 Chapter 1 Convective dissolution and CO

₂

sequestration

Once isolated, CO

₂

transport to storage sites is required in the very likely case that power plants and storage sites are not located at the same place.

Possible options are transport by pipeline, ships or for smaller quantities trains or tanker trucks [77, 134].

b CO

₂

storage

After its capture and transport, CO

₂

can be stored in several possible pro- cesses such as industrial uses, mineral carbonation, ocean storage or geolog- ical storage [77, 142].

Current industrial uses of CO

₂

include production of urea, refrigeration systems, inert agent for food packaging, beverages, fire extinguishers, etc.

[77, 161]. In the future, CO

₂

could also be used as a carbon source for the production of organic chemicals such as polymers (polyurethanes, polycar- bonates), liquid carbon-based fuels (e.g. gasoline and methanol) or biomass, for example starch that can be converted afterwards into industrial fuels like methane, methanol, hydrogen or biodiesel [77, 161]. This could however store only a small fraction of all CO

₂

emissions, and only for a short period of time as these products have a limited life cycle [77].

Mineral carbonation consists in accelerating the natural weathering to fix CO

₂

into minerals [77, 134, 161]. This safe but expensive process allows to store CO

₂

for a long period of time [134]. Although the carbonation reaction is favorable from a thermodynamic viewpoint, the kinetics of the reaction is so slow that energy supply is required to accelerate the reaction [77]. In addition, this technique requires mining and transporting large quantities of reactant mineral, as well as transporting the products towards their disposal site (geological formations) or further use site (e.g. construction) [77].

Ocean storage is an option still in development as the consequences are still not clearly understood. The idea is to inject liquid CO

₂

, denser than sea water (see density of CO

₂

phase as a function of temperature and pressure in Fig. 1.4), at great depths where it would sink at the bottom of the sea and dissolve as bicarbonate [74, 77]. In fact, the injection would accelerate the natural absorption of atmospheric CO

₂

by the ocean. Ocean storage raises concern because it could impact marine ecosystems through an increase in ocean acidity, as explained in Section 1.1.1. Furthermore, this option is rather unpopular as ocean currents coupled to local supersaturation could cause a quick release of the stored CO

₂

in the atmosphere [74].

Geological sequestration, consisting in injecting CO

₂

into the subsurface (see

Fig. 1.5), is one of the most promising options to reduce the emissions of

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Geological sequestration of CO

₂

13 Figure 1.4 – The density of CO

₂

depends on the type of phase — gas, liquid,

or supercritical — and typically increases when temperature decreases or

when pressure increases. Reprinted from Prog. Energ. Comb., 34, Bachu ,

CO

₂

storage in geological media: role, status and barriers to deployment,

254–273, Copyright (2008), with permission from Elsevier [13].

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14 Chapter 1 Convective dissolution and CO

₂

sequestration

CO

₂

to the atmosphere [52, 77, 154, 160]. This technology is mature as the transport and injection of CO

₂

have already been performed in the con- text of EOR (Enhanced Oil Recovery) and EGR (Enhanced Gas Recovery) [7, 13, 52, 77, 134, 154]. In addition, underground sequestration has a large potential capacity (1000–1800 Gt CO

₂

worldwide) and environmental im- pacts are more acceptable for underground environments than for ocean, due to less biocomplexity [154]. Saline aquifers are promising candidates for CO

₂

sequestration because, unlike hydrocarbon reservoirs, they are evenly distributed in various parts of the world [13, 52]. For example in Belgium, several saline aquifers are present in the north as well as in the south of the country, which compensates the absence of oil or gas reservoirs [181]. An- other advantage of saline aquifers is that the brine they contain typically has a long residence time, ensuring that the dissolved CO

₂

will actually remain in the reservoir [14, 119]. That brine is not drinkable and cannot be used for agriculture either [52], so that CO

₂

sequestration in saline aquifers does not compete with other uses. Finally, saline aquifers allow a higher injectivity than the unmined coal sites and abandoned coal mines present in Belgium [181].

Figure 1.5 – Schematic illustration of CO

₂

geological storage. Injected CO

₂

is typically less dense than the surrounding host phase in the geological

formation (for example oil, brine) and thus rises up to a caprock, i.e. an

impermeable boundary rock layer.

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Transport dynamics in geological formations 15

1.2 Transport dynamics in geological formations

As CO

₂

sequestration in subsurface sites is a promising option, it is crucial to analyze the CO

₂

transport dynamics in geological formations. For instance, the temporal evolution of the mixing zone between the CO

₂

phase and the brine of a saline aquifer can provide information on the times scales needed for complete sequestration. The quantity of CO

₂

that can be stored and its ultimate fate in the storage site, i.e. under what form it will be stored, are also important questions not only to assess the efficiency of the sequestration but also to address possible safety issues, such as contamination of nearby groundwater or leakage to the atmosphere, which could harm biodiversity and human health. As CO

₂

is denser than air, it can accumulate in depres- sions or buildings and cause harmful effects for the health at a concentration

⁷

above 0.5 %. Above 20 % concentration, death occurs in 20 – 30 min [13]. In this context, quantifying the safety, the rate and the duration of CO

₂

storage is of tantamount importance. These characteristics depend on the transport dynamics of CO

₂

in underground geological formations, composed of porous materials. Therefore, we first explicit the definition of porous media and present classical equations to model fluid flow and solute transport therein (Section 1.2.1). These notions are needed to understand how the physico- chemical processes occurring underground affect the transport dynamics of CO

₂

(Section 1.2.2).

1.2.1 Fluid flow and solute transport in porous media

A porous medium consists of a solid matrix with interconnected voids — small holes called pores — where fluids can flow through [129, 133]. Typical porous media (see e.g. Table 1.1) include biological tissues (sponge, wood, hair strands of mammals,...), industrial materials (membranes used for fil- tration, wire crimps, silica powder,...), or rocks forming the basis of geolog- ical formations (granite, sandstone, limestone, dolomite, etc.). The porous medium can be occupied by a fluid filling the pores: for instance brine in the case of saline aquifers, oil or gas for oil and gas reservoirs respectively, etc. The dynamics of that fluid are important for environmental sciences and geological engineering issues like avoiding contamination of aquifers by chemical waste, improving Enhanced Oil Recovery Techniques, understand- ing the history and morphology of geological formations, to cite but a few [133]. We therefore discuss how to model fluid flows in porous media.

7

The current usual atmospheric concentration of CO

2

of 391 ppm [76], mentioned in

Section 1.1, corresponds to 0.0391 % (in mole fraction of dry air).

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16 Chapter 1 Convective dissolution and CO

₂

sequestration

a Modeling fluid flows in porous media

In geological formations, there exist two different length scales: the system dimensions (for instance, the height of a typical acid gas injection site is of the order of 10 m – 100 m [62]), and the pore scale (generally 1 µm - 1 mm [133]). Due to the important number of pores and the inherent complexity of the network of pores with different sizes and geometries, modeling fluid flows at the pore scale requires large computing resources. To deal with this issue,

“mean field” properties characterizing the porous medium are defined on an intermediate mesoscopic length scale, much larger than the pore scale but still much smaller than the dimensions of the formation or the characteristic length of the flow pattern. Defining a mesoscopic scale is indeed possible in the context of geological formations given the large difference between the system dimensions and the pore scale. Phenomenological equations depend- ing on these mesoscopic properties are then used to model fluid flow and solute transport [28, 129, 133].

At such a mesoscopic scale, the flow velocity in a porous medium can be modeled with Darcy’s equation [85, 129, 133]:

∇ ˜ p ˜ = − µ

κ u ˜ + ˜ ρ g, (1.1)

which links the pressure p, the viscosity ˜ µ, the permeability κ, the fluid velocity u ˜ and the buoyancy term ρ ˜ g, where g is the gravity field and ρ ˜ is the density of the fluid. This equation means that in absence of buoyancy effects, the fluid flows in the opposite direction of the pressure gradient, i.e.

from large to low values of pressure. The intensity of this fluid flow is also proportional to the mobility ratio κ/µ between the the permeability of the porous medium and the viscosity of the fluid.

The dynamic viscosity µ (with units Pa s) quantifies the resistance of a fluid to shear stress. This property depends on the nature of the fluids, and for a given fluid depends on temperature and solute concentrations. A more viscous fluid such as glycerine flows more slowly than a less viscous one like water. This is also true in porous media as shown by Eq. (1.1).

The permeability κ (with units m

²

) measures the ability of the porous medium to let fluid flow through it. As shown by Eq. (1.1) for a fixed gradient of pressure, the flow velocity is larger in a medium with a larger permeability. In this study, we assume that the porous medium is isotropic and homogeneous, and thus κ is a constant scalar

⁸

[129].

8

In reality, the situation can be more complex: geological formations are often het-

erogeneous (i.e.

κ

is not constant but depend on the spatial coordinates) and anisotropic

(i.e.

κ

is not a scalar but a second-order tensor as it depends on the fluid flow direction)

[18, 23, 129].

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Transport dynamics in geological formations 17

The fluid velocity u ˜ is a “phenomenological” velocity, related to the velocity v ˜ of the interstitial fluid in the pores as

u ˜ = φ˜ v, (1.2)

where φ is the porosity of the porous medium, defined as the fraction of interconnected pore volume available to the fluid

⁹

, and thus in the range 0 to 1 [129, 133, 160]. Equation (1.2) means that, as the fluid only occupies a small fraction of the total volume, the observed fluid velocity u ˜ is smaller than the interstitial one ˜ v. We assume that the porosity φ is constant in the homogeneous porous medium at hand.

Table 1.1 gives ranges of typical permeabilities for some common porous media, varying between 10

⁻²¹

m

²

for granitic rocks to 10

⁻⁸

m

²

for wire crimps, and typical porosities, varying between 0.01 for granitic rocks to 0.8 for wire crimps. This table illustrates that the permeability in geological formations is typically lower than that in other types of porous media. Note that between different types of rocks, the permeability can vary by several orders of magnitude. As in some geological formations the porosity is much smaller than 1, the difference between the phenomenological velocity u ˜ and the velocity v ˜ in the pores can be large. Quantifying the porosity φ and permeability κ of a porous medium is important to select a site for CO

₂

sequestration as φ and κ affect the capacity (potential quantity of CO

₂

that can be stored) and injectivity (potential CO

₂

injection rate) of the geological formation [12].

Material Porosity φ Permeability κ (m

²

) Geological

Limestone (dolomite) (0.4 – 1) × 10

⁻¹

10

⁻¹¹

– 10

⁻¹⁰

Metamorphic/Granitic rocks 0.1 × 10

⁻¹

10

⁻²¹

– 10

⁻¹⁶

Sand 5 × 10

⁻¹

10

⁻¹¹

– 10

⁻¹⁰

Sandstone (0.5 – 4) × 10

⁻¹

10

⁻¹⁶

– 10

⁻¹²

Soil 5 × 10

⁻¹

10

⁻¹³

– 10

⁻¹¹

Industrial

Brick (1 – 3) × 10

⁻¹

10

⁻¹⁵

– 10

⁻¹³

Silica powder (4 – 5) × 10

⁻¹

10

⁻¹⁴

Wire crimps (7 – 8) × 10

⁻¹

10

⁻⁹

– 10

⁻⁸

Natural

Leather 6 × 10

⁻¹

10

⁻⁹

Table 1.1 – Properties of some common porous materials. Adapted from Nield and Bejan [129], Philips [133].

9

By consequence of this definition, if all the pores are connected, the fraction of solid

volume is calculated as

1−φ.

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18 Chapter 1 Convective dissolution and CO

₂

sequestration

The density ρ ˜ of the fluid appears in the buoyancy term ρg ˜ and thus impacts the flow dynamics. Without pressure gradient, the fluid flows in the direction of gravity g (Eq. (1.1)). The density ρ ˜ depends on the temperature T ˜ and solute concentrations c ˜

i

via an equation of state. In a dilute solution, non- ideal effects

¹⁰

can be neglected. With this assumption, the equation of state for ρ( ˜ ˜ T , { c ˜

_i

} ) is linear and writes [23]:

˜

ρ( ˜ T , { c ˜

_i

} ) = ρ

₀

( ˜ T

₀

)

"

1 + α

_T

( ˜ T − T ˜

₀

) + X

i

α

_i

( ˜ T ) ˜ c

_i

#

, (1.3)

where ρ

₀

( ˜ T

₀

) is the density of the solvent at the reference temperature T ˜

₀

, α

T

( ˜ T) =

_ρ¹

0

∂˜ρ

∂T˜

is the thermal expansion coefficient of the solvent, and α

i

( ˜ T ) =

_ρ¹

0

∂˜ρ

∂c˜i

is the solutal expansion coefficient of the solution with re- gard to the solute i. If the system is isothermal with a constant temperature T

0

, the only density differences inside the system are due to variations in solute concentrations and Eq. (1.3) is simplified to

˜

ρ( ˜ T , { c ˜

i

} ) = ρ

0

( ˜ T

0

)

"

1 + X

i

α

i

( ˜ T ) ˜ c

i

#

, (1.4)

Besides Darcy’s equation, the following mass balance equation must be ver- ified in the system:

∂

∂ ˜ t (φ˜ ρ) + ∇ ˜ · ( ˜ ρ˜ u) = 0, (1.5) If the porosity of the medium does not vary in time

¹¹

and the density differ- ences in the fluid are small, we can assume ∂(φ˜ ρ)/∂ ˜ t ≈ 0 and ∇ ˜ · ρ ˜ ≈ 0, so that Eq. (1.5) can be reduced to the condition of flow incompressibility:

∇ ˜ · u ˜ = 0. (1.6)

The small density variations in the fluid are then taken into account only in the buoyancy term ρg ˜ of Eq. (1.1): this corresponds to the Boussinesq approximation, valid for many common fluids [60].

b Modeling solute transport in porous media

Analyzing solute transport in porous media is crucial for a large number of applications. Solutes are dissolved in the interstitial fluid occupying the porous medium. Their concentration varies dynamically in space and time ˜ t

10

Non-ideal effects are linked to intermolecular interactions [10].

11

The porosity of the medium could vary in time due to mechanical effects such as an

increase of pressure for example [75].

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Transport dynamics in geological formations 19

because of fluid flows arising from either forced convection (= advection) or either natural (= free) convection, diffusion due to a concentration gradient, or chemical reactions

¹²

. This is expressed in the reaction-diffusion-convection (RDC) equation for the concentration c ˜

i

of a solute i in a porous medium [129]:

φ ∂ c ˜

i

∂ ˜ t + ∇ ˜ · ( u ˜ c ˜

i

) = φ ∇ ˜ (D

i

∇ ˜ c ˜

i

) + φ r ˜

i

( { c ˜

j

} ). (1.7) The reaction term r ˜

i

( { c ˜

j

} ) represents the variation of solute concentration

¹³

caused by chemical reactions. The term φ ∇ ˜ (D

_i

∇ ˜ c ˜

_i

) corresponds to the variation of solute concentration due to the diffusive flux − D

i

∇ ˜ c ˜

i

with D

i

the diffusion coefficient of species i. The advection-convection term ∇· ˜ ( u ˜ c ˜

i

) corresponds to the variation of solute concentration due to transport by the fluid flow with speed u. ˜

Assuming that the fluid is incompressible (Eq. (1.6)) and the diffusion coef- ficients D

_i

are constant, Eq. (1.7) is rearranged to read:

φ ∂ c ˜

i

∂ ˜ t + u ˜ · ∇ ˜ c ˜

i

= φD

i

∇ ˜

²

c ˜

i

+ φ r ˜

i

( { c ˜

j

} ). (1.8) The equation of state (1.4) for the density ρ ˜ depending on the solute con- centrations couples Eqs. (1.8) and (1.1), where ρ ˜ appears in the buoyancy term. Chemistry affects hydrodynamic instabilities because the properties of the fluid, such as surface tension, viscosity or in this study more specifically density, depend on concentrations and temperature, which are affected by chemical reactions [3, 29, 31, 36, 108].

1.2.2 Physicochemical processes in geological formations On the basis of the equations for fluid flow (Eqs. (1.1) and (1.6)) and solute transport in porous media (Eq. (1.8)), we now discuss the large variety of physicochemical processes taking place in subsurface formations. In the con- text of CO

₂

sequestration, it is important to understand these processes be- cause they impact CO

₂

transport dynamics underground [6, 13, 14, 42, 75].

For example, upon injection of CO

₂

in the storage site, the stress on the porous rock matrix due to the increase of pressure can cause a variation of the properties of the medium (porosity, permeability) [75], which can af- fect fluid flow (see Eq. (1.1)). Let us discuss other physical and chemical processes impacting transport dynamics during CO

₂

sequestration.

12

To simplify the problem, we neglect the effect of dispersion on solute transport [66].

13

More specifically, the reaction term

r˜i({c˜j})

represents the variation of solute quantity

per unit volume of

interstitial fluid.

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20 Chapter 1 Convective dissolution and CO

₂

sequestration

a Geochemistry

Different types of geochemical reactions are possible in subsurface formations [119, 133]. Dissolution/precipitation reactions of the type S −− ) −− * D

₁

+ D

₂

involve a solid S in equilibrium with two dissolved ions D

₁

and D

₂

. For example, the dissolution of solid calcium carbonate (CaCO

₃

in the form of calcite or aragonite) releases calcium cations Ca

²⁺

and carbonate anions CO

^2–₃

in solution [119]:

CaCO

_3(s)

−− ) −− * Ca

²⁺_(aq)

+ CO

²⁻_3(aq)

. (1.9) In replacement reactions of the type S

₁

+ D

₁

−− ) −− * S

₂

+ D

₂

, a dissolved ion D

₁

replaces another one in the solid S

₁

, which produces another solid S

₂

and a dissolved ion D

₂

. For instance, a magnesium cation Mg

²⁺

can replace a calcium cation in calcite, which results in dolomite CaMg(CO

₃

)

₂

and releases a calcium cation (dolomite replacement reaction) [133]:

2 CaCO

_3(s)

+ Mg

²⁺_(aq)

−− ) −− * CaMg(CO

₃

)

_2(s)

+ Ca

²⁺_(aq)

. (1.10) A second example is the so-called albitization of potassium-rich feldspar KAlSi

₃

O

₈

, where a potassium cation K

⁺

is replaced with a sodium cation Na

⁺

[133]:

KAlSi

₃

O

_8(s)

+ Na

⁺_(aq)

−− ) −− * NaAlSi

₃

O

_8(s)

+ K

⁺_(aq)

. (1.11) Fixation/combination reactions of the type S

₁

+ D −− ) −− * S

₂

+ S

₃

consist in the reaction of a dissolved ion D with a solid S

₁

, producing two solids S

₂

and S

₃

. An typical example of such reaction is dissolved CO

₂

reacting with wollastonite CaSiO

₃

, forming calcite and silica SiO

₂

[21, 133]:

CO

_2(aq)

+ CaSiO

_3(s)

−− ) −− * CaCO

_3(s)

+ SiO

_2(s)

. (1.12) Fixation reactions are the type of reactions most often considered in studies of reactive convective dissolution, where dissolved CO

₂

reacts with a solid, either in excess [8, 9, 21, 44, 58, 178, 179] or not [58, 180], to produce another solid. The reverse reaction consists in the combination of two solids S

₂

and S

₃

, generating a third solid S

₁

and a dissolved ion D.

Geochemical reactions that fix CO

₂

in a mineral can also be more complex reactions resulting from the combination of the elementary types presented here above. For example, CO

₂

can react with potassium-rich feldspar, pro- ducing dawsonite NaAlCO

₃

(OH)

₂

and silica (in the form of quartz) [58]:

CO_2(aq)+ H₂O_(l)+ KAlSi₃O_8(s)+ Na⁺_(aq)−−)−−*NaAlCO₃(OH)_2(s)+ K⁺_(aq)+ 3 SiO_2(s).

(1.13)

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Transport dynamics in geological formations 21

This reaction involves the replacement reaction (1.11) and a fixation reac- tion. More generally, geochemical reactions that fix CO

₂

in a mineral (for example as calcite, dolomite or dawsonite) improve the safety of the seques- tration because the solid mineral is immobilized. CO

₂

is thus less likely to escape to the atmosphere or to nearby aquifers. This immobilization is referred to as mineral trapping. For instance, tholeiitic flood basalts are par- ticularly suitable for CO

₂

sequestration because they contain relatively large quantities of magnesium and calcium oxides (MgO and CaO) which can fix CO

₂

in the solid matrix through chemical reactions [119]. In contrast to the common view that the mineral trapping of CO

₂

takes several hundreds to thousands of years, recent measurements in a specific storage site have shown that 95 % of the injected CO

₂

was mineralized to carbonate in less than 2 years [120].

b Fluid dynamics

Physical processes related to fluid dynamics and occurring during CO

₂

se- questration are schematized in Fig. 1.6: multiphase viscous fingering, capil- lary trapping, gravity currents, and convective dissolution.

Multiphase viscous fingering is a hydrodynamic instability that develops when a highly mobile phase displaces a less mobile one: the interface between both phases becomes unstable and deforms into fingers [17, 32–34, 72, 162].

The mobility of a phase can be quantified as the ratio (κ/µ

i

), with µ

i

the viscosity of phase i. Equation (1.1) indeed shows that a phase with a larger mobility has a larger velocity for the same pressure gradient. During CO

₂

se- questration, viscous fingering can occur because the injected free CO

₂

phase is typically less viscous than the host phase occupying the porous medium at hand [17, 42, 75, 122], for example brine (at a depth of 1000 m, µ

_CO₂

≈ 0.023 – 0.061 mPa s < µ

brine

≈ 0.195 – 1.58 mPa s [75]). As the mobility also depends on the permeability, geochemical reactions (precipitation for instance) at the origin of local variations of permeability can also cause a similar type of instability [127, 152]. We expect that viscous fingering im- pacts the spatio-temporal distribution of CO

₂

Influence of chemical reactions on convective dissolution:

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