Keywords: function vs, τ, 5, 17 Image differences during reduced-gravity phase

The understanding and prediction of the processes involved in the convective dissolution of carbon dioxide (CO 2 ) in brine are of major concern for storage applications and to reach the 2050 carbon neutral target. Indeed, the efficiency of the CO 2 storage process in deep saline aquifers is related to the dissolution rate of CO 2 in brine. The latter increases when the mass transfer switches from pure diffusion to convection. This thesis presents an experimental study of the convective dissolution of CO 2 at high pressure, and in three-dimensions (3D), i.e. in the Rayleigh-Taylor regime. The experiments were carried out in pure water and in salted water for multiple salt compositions and concentrations. In this study, we develop a non-destructive experimental approach based on the dynamic shadowgraphy method, and a high-pressure cell adapted for the characterisation of convection with or without a temperature gradient. The results obtained with homogeneous temperature show that the injection pressure of CO 2 plays a dominant role on the dissolution rate, and influences the shape of the 3D convective patterns. The analysis of the refractive index variations over time shows that the onset time of convection is a function of the solutal Rayleigh number following a power-law, which is independent of the brine composition and its height. To allow the up-scaling to the saline aquifers scale, the results are rationalised with dimensionless equations. At these scales, convection induced by the Soret effect in the liquid mixture could be significant. Thus, the influence of a temperature gradient was checked. These latest studies have been carried out on fluids mimicking the water-CO 2 system on Earth, and under reduced-gravity conditions taking part in a parabolic flight campaign. i 5.1 p -T phase diagram of the system 1-hexanol-CO 2 . . . . . . . . . . . 5.2 p -x CO 2 phase diagram of the system 1-hexanol-CO 2 . . . . . . . . . . 5.3 Scheme of the second set-up . . . . . . . .

Résumé

Comprendre et prédire les mécanismes impliqués dans la dissolution convective du dioxyde de carbone (CO 2 ) dans la saumure est un enjeu majeur pour atteindre la neutralité carbone en 2050. En effet, l'efficacité du procédé de stockage du CO 2 dans les aquifères salins profonds est associée au taux de dissolution du CO 2 dans la saumure. Ce dernier augmente lorsque le mode de transfert de masse devient rapidement convectif. Ce mémoire de thèse présente une étude expérimentale de la dissolution convective du CO 2 à haute pression et en trois dimensions (3D), c'est-à-dire en régime de Rayleigh-Taylor. Les expériences ont été réalisées dans de l'eau pure, puis de l'eau salée à différentes compositions et concentrations de sel. L'approche expérimentale non-destructive développée dans cette étude repose sur l'ombroscopie dynamique et une cellule haute pression adaptée pour la caractérisation de la convection en présence ou non de gradients de température. Les résultats obtenus à température homogène montrent que la pression d'injection du CO 2 joue un rôle prépondérant sur le taux de dissolution et influence les motifs 3D de convection. L'analyse des variations moyennes d'indice de réfraction au cours du temps montre que le temps de mise en place de la convection en fonction du nombre de Rayleigh solutal suit une loi puissance indépendante de la composition de la saumure et de sa hauteur. Les résultats sont rationalisés avec des équations adimensionnelles permettant leur projection aux échelles des aquifères salins. A ces échelles, la convection induite par effet Soret dans le mélange liquide, due à la présence du gradient géothermique, devient non négligeable et a été étudié dans le dernier chapitre de ce mémoire. Ces dernières études ont été menées sur des fluides mimant le système eau + CO 2 sur Terre ainsi qu'en impesanteur.

Aknowledgements

The present doctoral work has been a long journey within the chair CO2ES, supported by E2S-UPPAS, TotalEnergies, the BRGM, and the CNES. During more than three years in the campuses of Anglet of the Université de Pau et des Pays de l'Adour, I met many people in the campuses of Anglet and Pau, in international conferences, and in several meetings. They all participate in their own way to this work. I want to have a word for them in the following paragraphs.

First, I would like to thank Fabrizio Croccolo, my supervisor, that hired me and trusted me to carry out this research project. I express also many gratitude to Cédric Giraudet, that co-supervised me during the last two years. Their expertise on many subjects were very helpful. The three of us had many instructive talks and debates. This docotral work would not have been possible without them. I would like to thank as well Henri Bataller. His disponibility and help were much appreciated.

I express my profound gratitude to the jury members. Abdelkader Mojtabi that accepted to be the president of the jury. Alberto Vailati and Patrice Meunier that reviewed this dissertation. Their feedback was essential for the quality of this work. Alberto was already following and contributed to this work: his expertise on convection was essential. Patrice, who supervised my master thesis, gave me the taste of experimental research. I also thank the other reviewers Alfredo Soldati, Hannelore Derluyn, and Mahmoudreza Jazayeri Noushabadi for the interest they had in my work and their commentaries during meetings we had together.

To all the people I met both in Anglet and Pau, with who I shared my time. The members of the CO2ES Chair, the Geomécanique group, the LFCR. A big thank to the other PhD students, the team of the 'Pavillon', and the students we hosted for internship. The daily moments, meal, drinks we had together made me cheerful.

Finally, I want to have a word for my family and closest friends, that always supported and pushed me to do my best. This thesis work would not have been possible iii without any of you.

List of Figures

Introduction

Carbon is one of the most important atom for life development. We find it in any living being body, coal mines, or in the atmosphere in its well-known gaseous form: the carbon dioxide (CO 2 ). Besides its involvement in the photosynthesis phenomenon, the CO 2 largely contributes to the greenhouse effect that was essential for the emergence of life and its evolution on Earth.

Since the industrial revolution, the concentration of CO 2 in the atmosphere has increased at an unprecedented pace due to the anthropogenic activities. In 1967, it has been shown by Wanabe et al. [START_REF] Manabe | Thermal equilibrium of the atmosphere with a given distribution of a relative humidity[END_REF], the 2021 Nobel price for physics, that the rise in the concentration of CO 2 in the atmosphere induces its warming. Despite this, it is only in the latest decades that a consensus has risen from scientific community and policy makers to admit that the increase in CO 2 emissions causes global warming and climate change. Since 1992, 197 states are meeting every year in the context of the united nations framework convention on climate change (UNFCCC) to find solutions against global warming. One big lever is the cut in CO 2 emissions, which is achievable with several solutions, one of them being the storage of carbon in saline aquifers.

In this part, we present the ecological and socio-economical context of climate change in which CO 2 has a central role. After briefly presenting the portfolio of solutions available to reduce CO 2 emissions, we will introduce the carbon capture and storage (CCS) and the physical processes occurring in the sequestration of CO 2 . Finally, we will present the objectives of this thesis and the research project in which it operates.

Context 1.Ecological and socio-economical context

A consensus in the scientific community states that the global warming has direct consequences on climate and is already a threat to humankind. Here, before introducing some solutions that can be adopted to mitigate the climate change, we present the current situation, its causes, and the future consequences expected.

Climate crisis

According to the 2020 state of the global climate of the world meteorological organization [START_REF]State of the global climate 2020[END_REF], the earth average surface temperature has increased by 1.2 ± 0.1 • C over the period 1850 to 2020; see figure 1.1. Consequently, the evaporation and precipitation processes become stronger, the ice melting is accelerated, and the water cycle is more intensive. Thus, extreme meteorological events are more powerful and more frequent. We already noticed a rise in the strength of tropical cyclones. We also experience heat wave or heavy rain episodes more frequently that lead to droughts and floods. It causes huge damages on field crops, public buildings, and private houses. The sea level has already risen by about 0.2 m this last century, see figure 1.2, and entire cities could be erased from the map if this continues.

The climate change rises awareness and forces policy makers to take strong and quick measures otherwise the climate crisis could lead to a huge economical crisis as well. The fact that places across the globe become no longer habitable would induce migration waves and probably civil wars.

In order to tackle this dramatic scenario, actions must be taken right now. Especially because the climate change is a slow process with a large inertia. Current emissions of CO 2 , or other greenhouse gases (GHGs) such as methane or nitrous oxide, will have an impact in decades. 2020 is the best evidence of this fact. Even though GHG emissions have been reduced due to extreme measures taken for the sanitary situation, it was one of the three warmest years recorded ever. This illustrates the fact that actions taken against global warming have no short-term impact. With a similar mind-thinking, the greenhouse gases emitted decades ago impact our current and future lives. It is essential to act as of now to limit the ecological and economical crisis as much as possible.

Causes of global warming: the responsibility of GHGs

The Earth temperature depends on several thermal fluxes received or emitted by our planet. Carbon dioxide, water vapour, methane and nitrous oxide are the gases responsible for the most part of greenhouse effect. The infra-red energy emitted by Earth is absorbed by the GHGs and partly re-emitted towards the Earth surface. Without this process, all this thermal energy would leave the planet and temperatures would not be high enough to get liquid water on Earth surface. The development of life as we know would not have been possible. Thus, the greenhouse effect is essential to our lives on Earth.

However, over the last couple of centuries, the atmospheric concentration in GHGs has increased due to anthropogenic activities and the massive use of fossil fuels; see figure 1.3. The greenhouse effect is, thus, enhanced and Earth gets warmer. That has direct consequences as the acidification of the ocean or the ice melting and has huge impacts on fauna and flora as well as on air pollution. Figure 1.3: Atmospheric greenhouse gases concentrations for carbon dioxide (CO 2 , green), methane (CH 4 , orange), and nitrous oxide (N 2 O, red). The dots correspond to data extracted from ice core measurements, the lines are direct atmospheric measurements. [START_REF] Meyer | Climate change 2014: Synthesis report[END_REF] According to the intergovernmental panel on climate change (IPCC) fourth assessment report [START_REF] Meyer | Climate change 2007: Synthesis report[END_REF], it is extremely likely that the rise in the global average surface temperature is due to anthropogenic increase in GHGs emissions, for more than a half. The latest report of IPCC, group I, published in August 2021 [START_REF] Masson-Delmotte | Summary for policymakers[END_REF], is even more unequivocal stating that "human influence has warmed the atmosphere, ocean and land. Widespread and rapid changes in the atmosphere, ocean and biosphere have occurred". This is evidenced by the simulations shown in the right part of figure 1.1. The first model takes into account the human and natural factors causing the global warming (brown line), while the second one considers the natural factors only (blue-green line). According to these results, it clearly appears that the raise in Earth surface temperature by 1.2 • C is only associated with human activities.

Shared socioeconomic pathways

In 2021, the IPCC introduced five shared socioeconomic pathways (SSPs) that provide the new emission scenarios [START_REF] Masson-Delmotte | Summary for policymakers[END_REF] providing the CO 2 atmospheric concentration according to different climate policies and social adaptations. They are called sustainability (taking the green road) (SSP1), middle of the road (SSP2), regional rivalry (a rocky road) (SSP3), inequality (a road divided) (SSP4), and fossil-fuel development (taking the highway) (SSP5). These scenarios are combined with representative concentration pathways (RCPs) that describe the future of the climate providing the radiative forcing values in the year 2100 depending on the evolution of GHGs concentration in the atmosphere. The common value of radiative forcing are 1.9, 2.6, 3.4, 4.5, 6, 7, and 8.5 W/m². In the figure 1.4 from the IPCC sixth assessment report, five scenarios are then under study: SSP1-1.9, SSP1-2.6, SSP2-4.5, SSP3-7.0, and SSP5-8.5. The future emissions of GHGs range from very low emissions, where the CO 2 emissions cut to net zero around 2050, to very high emissions, where the CO 2 emissions triple by 2075. They correspond to global warming between 1.0 and 5.7 • C over pre-industrial level in 2100.

Future consequences

The global warming is not the only challenge we must take up. For instance, air pollution, waste management, or the diminution of wildlife diversity are huge challenges. We already experienced a rise in the number of strong weather events and this could continue to increase with more power each time. The rise of CO 2 atmospheric concentration causes ocean acidification and, thus, impacts on the coral reef barrier. This has strong consequences on the entire food chain that can be completely disrupted. With a global warming of 4 • C the consequences would be irrevocable and would induce huge global crisis.

Policymakers responsability

In 1992, at the Earth summit organized in Rio de Janeiro, 154 states signed the united nations framework convention on climate change (UNFCCC). Since 1995, every year the committed parties, that are now 197, meet during the conference of the parties (COP) that led to the signature of several important treaties. In 1997, the Kyoto protocol obliged developed countries to reduce their GHGs emissions in the period 2008-2012 to maintain the global warming below 2 • C relative to pre-industrial level. In 2015, the Paris agreement was signed during the 21 st COP with similar objectives, and to limit the global warming to 1.5 • C if possible. However, even rich nations have difficulties to meet their objectives because of the continuously growing global energy demand.

The responsibility of the policy makers is huge. They have to build our future world taking into account ecological matters and economical goals. Luckily, investments in global warming limitation will have co-benefits in various fields as health and unemployment, as well as on the biodiversity. As each action of one actor is affecting other actors, an international cooperation is essential. This seems to be the only way to develop a sustainable, fair, and environmentally friendly world. Mitigation solutions would have the best efficiency if governments are working side by side.

Mitigation solutions

To overcome the challenges of cutting GHGs emissions while meeting the growing global energy demand, several solutions already exist or are under development. Many fields are concerned such as energy supply, transport, buildings, industry, agriculture, forestry and other land use. Besides the change in daily behaviour of Earth inhabitants, the main idea of mitigation solutions is to reduce the GHG emissions.

In 2020, the International Energy Agency (IEA) published the energy technology perspectives 2020 [START_REF]Energy technology perspectives[END_REF]. Therein, they proposed 2 scenarios by 2070 to reach carbon neutrality. The "Sustainable Development Scenario" uses a portfolio of technologies to reach net-zero emissions by 2070. The second one, the "Stated Policies Scenario", uses the existing policies and commitments and their likelihood to be fully met. In figure 1.5, it is shown the CO 2 emission reductions by sector in the "Sustainable Development Scenario" relative to the "Stated Policies Scenario" in the period 2019-2070.

The energy efficiency in industrial processes, the thermal insulation, heating and cooling of buildings, the fuel of vehicles, and the renewable energies such as wind turbine and solar photovoltaic are the lever that can be improved as of now. These Figure 1.5: Global CO 2 emissions reductions by energy sector in the "Sustainable Development Scenario" relative to the "Stated Policies Scenario". Both scenarios are introduced in the energy technology perspectives 2020 of the international energy agency [START_REF]Energy technology perspectives[END_REF]. This shows the reductions of emissions that could be made by adopting greener climate policies.

enhancement have already been implemented. However, this is a continuous work that could contribute to more than 70% in the GHGs emissions reductions over the next two decades.

In the pathway for net-zero emissions by 2070, four technologies that are emerging are essential: the electrification of end-use sectors, the low-carbon hydrogen and synthetic fuels, sustainable bioenergy, and carbon capture use and storage. These technologies are less developed than the one mentioned above but they start to be deployed and would take an active part in the emission reductions in longer terms.

Carbon Capture and Storage (CCS)

As mentioned in the previous part, the carbon capture utilization and storage (CCUS) is a very promising technology. It can contribute to reduce the CO 2 emissions by 15% by 2070, capturing around 10 Gt/year of CO 2 globally at this time. In the "Sustainable Development Scenario", CCUS deployment is separated in three phases. In the first decade, the captured CO 2 is related to fossil fuel and industries that produce cement, iron, and chemicals. Then, between 2030 and 2050, renewable energies and bioenergy carbon capture and storage (BECCS) are integrated in the mix. After 2050, during the final phase, the goal is recentred on BECCS or direct air capture to generate negative emissions.

In this part, we will present the general concept of the CCS, and the techniques to capture CO 2 , before focusing on the storage processes.

General concept

CCS combines a huge carbon dioxide emitters as a power supply or an industry that produces cement, chemicals or other, with a geographically close storage site. The idea is to prevent the CO 2 from being released into the atmosphere and to store it underground where it will remain for thousands of years. This process involves three steps. First, CO 2 is captured from huge emitters before being released in the atmosphere. Then, it is liquefied and transported to the injection site. Finally, the CO 2 is injected in a geological formation where it is stored by natural trapping mechanisms. The storage must be ensured for centuries to millenniums to help states reduce their greenhouse gases emissions and meet the objectives they are committed to.

Capture techniques

Generally, CO 2 is emitted during combustion reactions. The result of the combustion reaction is the release of energy and smokes containing multiple gases. The smokes contain between 5 to 15% of CO 2 . The CO 2 must be separated from other gases before its transportation. Three main technologies are used for that purpose: pre-combustion, oxyfuel combustion, and post-combustion technologies.

Pre-combustion

There are two main routes for the pre-combustion process. In a gasifier, steam is added to traditional carbonated fuels, the process is "steam reforming"; oxygen O 2 is added to the carbonated fuels, we talk about"partial oxidation"if the fuels are in gaseous or liquid phase, and about "gasification" if the fuel is in solid phase. These two processes produce a synthesis gas called syngas, that is a mixture of carbon monoxide and hydrogen [START_REF] Jansen | Pre-combustion CO 2 capture[END_REF].

The syngas is purified and then reacts with water vapour to increase the percentage of CO 2 [START_REF] Osman | Recent advances in carbon capture storage and utilisation technologies: a review[END_REF]. This reaction is called water gasshift reaction. Finally, the CO 2 is separated from other residual gases using chemical and physical absorption processes [START_REF] Jansen | Pre-combustion CO 2 capture[END_REF][START_REF] Li | Prediction of CO 2 absorption by physical solvents using a chemoinformatics-based machine learning model[END_REF].

Post combustion

The flue gas emitted goes first through denitrification and desulphurisation, plus dust removal. These processes allow to prevent the solvent degradation [START_REF] Osman | Recent advances in carbon capture storage and utilisation technologies: a review[END_REF]. After these preliminary operations, the gas flue contains mainly CO 2 , water (H 2 O), and nitrogen (N 2 ). This goes in a chamber containing the absorber, that is usually monoethanolamine, which allows the isolation of CO 2 .

Oxy-fuel combustion

The oxy-fuel combustion is considered as the most promising energy efficient process for the capture of CO 2 [START_REF] Wienchol | Waste-to-energy technology integrated with carbon capture -challenges and opportunities[END_REF]. First, an air-separator-unit isolates oxygen form air. Then, the captured carbonated fuel is combusted with the pure oxygen stream in a boiler. A particle removal unit is necessary to remove flying ashes from the mixture. This operation is repeated several times.

Thereafter, the mixture goes in a condenser unit to cool down and condense the remaining water vapour. Before repeating the condensing operation, the mixture goes through a sulphur removal unit.

Transportation

Once CO 2 is captured, it is compressed to liquid or supercritical state. Its density strongly increases, from 2 kg • m -3 to reach almost 800 kg • m -3 at pressures and temperatures of about 10 MPa and 350 K, respectively, i.e. thermodynamic conditions similar to those at the depths of the aquifers. Thus, larger quantities of CO 2 can be transported at relatively low costs. The CO 2 is then transported from emitters to injection site by pipelines, trucks, or ships.

Injection sites

Storage sites must be sufficiently deep to have the necessary thermodynamic conditions, i.e. high pressure and high temperature, in order to dissolve the maximum of CO 2 in brine. The sequestration in oceans has also been considered. Meanwhile, this would acidify the ocean and have huge impacts on submarine fauna and flora. Thus, this solution has been dropped. CO 2 can be stored in different kinds of geological formations. Former depleted natural reservoirs can easily be used. Indeed, they already proved their sealing efficiency while the injected techniques are well known by the oil and gas industry. Furthermore, the required installations for injection are already in place, limiting injection costs.

Unmineable coal seams are another site possibility. This kind of site can be used to store CO 2 while methane is retrieved. However, the global storage capacity is limited and this solution has been dropped because of its weak storage potential.

Saline aquifers

The most promising formations are the saline aquifers. These are porous geological formations filled of highly salted water (brine) and sandwiched between two waterproof layers, called caprock, that prevent CO 2 leakage. They are located at depths between 800 m and 3000 m. In fact, the complete dissolution takes many years. The aquifers have a thickness up to 200 m and can spread laterally over kilometres. Inside them, the water is undrinkable due to its high salinity and does not participate in the water cycle used by humankind. Thus, storing CO 2 inside saline aquifers will not affect the society. The global storage capacity of saline aquifers all over the world is up to 10,000 Gt of CO 2 . Moreover, they have a good geographic distribution that can reduce the transportation costs. Indeed, saline aquifers exist both under sea and ground.

After a deep risk-analysis, the evaluation of the storage capacity, and the measurements of physical parameters such as permeability, porosity, and salinity, a site is selected. Equipments are installed and the injection starts and will last for around 30 years.

Trapping processes

Four processes occur at different time and space scales: the stratigraphic trapping, the residual or capillary trapping, the solubility trapping, and the mineral trapping. These processes are responsible for different amount of CO 2 stored, at different time-scales, and their combination ensures a growing storage security as can be seen in figure 1.6.

Structural or statigraphic trapping

CO 2 is injected at supercritical state (s-CO 2 ). When it reaches the saline aquifer, its density varies between 266 and 766 kg • m -3 [START_REF] Emami-Meybodi | Convective dissolution of CO 2 in saline aquifers: Progress in modeling and experiments[END_REF] depending on the thermal and pressure conditions of the chosen aquifer. The water has a larger density in the range 945-1230 kg • m -3 [START_REF] Bachu | Sequestration of CO 2 in geological media in response to climate change: Capacity of deep saline aquifers to sequester CO 2 in solution[END_REF]. Therefore, s-CO 2 will rise until it reaches the caprock and will spread across the aquifer according to its shape. This is the stratigraphic trapping. At this step, the leakage risk is high and must be well-assessed. Indeed, seismic movements can create faults. The latter can provide leakage pathways for CO 2 which could reach shallower formations. Fortunately, the other storage mechanisms strongly mitigate this risk.

Residual trapping

During its motion through the porous rocks, s-CO 2 displaces the brine present in the pore space. When the brine returns in the pore spaces, liquid CO 2 can be stuck inside pores due to capillary forces. This is the capillary or residual trapping.

Dissolution trapping

After s-CO 2 reaches the top of the aquifer, a layer of s-CO 2 is formed above a layer of brine. s-CO 2 dissolves in the brine to reach an equilibrium between the two phases. This creates an intermediate layer composed of CO 2 and brine that diffuses and is destabilized by gravity, leading to convection. This results in the formation of downward plumes that are the illustration of the convective dissolution process and significantly accelerate the dissolution of CO 2 . This is the solubility trapping. After dissolution, the leakage risks are close to zero which strongly enhances the storage security.

Mineral trapping

At a much larger time-scale, the dissolved CO 2 will react with minerals in the surrounding rocks to form carbonate minerals. It ensures a safe and permanent storage for hundreds of millennia. This last mechanism is called mineral trapping.

Risks

As any project, CCS projects involve risks that must be managed. The risks are twofold: risks related to the monitoring and security of the CO 2 injection all over the CCS projects lifetime; and risks related to the acceptance and feasibility of the projects considering the socio-economic and political context and impacts of CCS projects. In this part, we present these risks and how they can be managed.

Risks related to the injection of CO 2

The geological risks after the injection of s-CO 2 in the aquifer are mostly related to leakage, ground movement and seismicity, as well as brine displacement [START_REF] Damen | Health, safety and environmental risks of underground CO 2 storage -overview of mechanisms and current knowledge[END_REF]. Leakages can either come from the injection well, from abandoned well, or even from faults and fractures. Indeed, after its injection into the well, the CO 2 rises up due to buoyancy effects. In theory, the CO 2 is blocked by the caprock. Nevertheless, it could leak from injecting or abandoned wells if their sealings are not perfect.

The injection of large amount of CO 2 in saline aquifers increases the pore pressure. Thus, the porous matrix encounters modification on its mechanical state that could lead to fractures or the creation of faults. It could cause damages in the infrastructures or lead to CO 2 leakage. We can mitigate these risks by controlling the injection pressure. For that, a study of the pore space of the aquifer is required to determine the maximum injection pressure.

The displacement of brine could contaminate shallower water reservoirs and increase their salinity. Therefore, it perturbs drinking water extraction and water used for irrigation.

These risks must be well assessed to get all the authorizations to start CO 2 injection in a selected site. Derman et al. (2006) [START_REF] Damen | Health, safety and environmental risks of underground CO 2 storage -overview of mechanisms and current knowledge[END_REF] explain that the leakages induce a loss of less than 0.2% of the total amount of CO 2 stored. Anyway, a study of [START_REF] Deng | Leakage risks of geologic CO 2 storage and the impacts on the global energy system and climate change mitigation[END_REF] [START_REF] Deng | Leakage risks of geologic CO 2 storage and the impacts on the global energy system and climate change mitigation[END_REF] concludes that the risks and their assessment are unlikely to block the deployment of CCS projects. Furthermore, considering the expecting costs of risk management, the CCS technology remains essential to fight global warming and achieve the 2 • C over pre-industrial levels goals.

Risks related to the socio-economic context

A huge risk that could hinder the deployment of CCS projects is the social rejection of this type of projects. CCS involves the building of huge infrastructures. Local inhabitants could be reluctant to see a change on their daily landscape, as it is with wind turbines, nuclear reactors, or hydropower plants.

Besides the social acceptance, CCUS needs the support of policy makers to be an economically viable option. Today, the storage of a ton of CO 2 is more expensive than the carbon taxes you must pay for the emission of the same. However, things are moving and need to move fast to allow the CCS technology to be largely and globally deployed.

Objectives

This thesis is enrolled in the CO 2 Enhanced Storage (CO2ES) industrial chair created to investigate the processes involved in CO 2 geological storage. After presenting the subjects studied in the CO2ES Chair framework, we will outline the goals of this thesis.

CO2ES Chair

The CO2ES Chair aims at enlarging the general understanding of the various CO 2 trapping and transport processes involved in CO 2 geological storage. It focuses on the mechanisms that occur during the solubility and mineral trappings presented above and in figure 1.6. It will help to develop monitoring techniques and injection procedures to improve the security and the efficiency all over the time scale of CO 2 storage projects.

The chair is divided in several work packages including experimental investigations conducted in both two-and three-dimensional samples and in both porous and free medium. This experimental work is done at the mesoscopic and macroscopic scales and involves coupled thermal and concentration gradients. This experimental part is centred on the CO 2 storage by dissolution and by mineral trapping.

Another important part is the up-scaling of CO 2 processes. This step involves numerical simulations including the new data and the new models developed within the experimental work packages. This will possibly lead to develop new pilots to demonstrate the efficiency of the numerical models previously developed.

Finally, the CO2ES chair involves also experiments performed in microgravity conditions. This is linked with the research activities of the group on transport properties on complex fluids and also with the investigations on the dissolution trapping. Indeed, this trapping involves convection that is driven by the non-equilibrium fluctuations (NEFs) and gravity. Working under microgravity conditions would help to better understand the motor of convection. Some experiments will be launched to the international space station (ISS) and other are performed during parabolic flight campaigns organized by the Centre National d'Etude Spatiale (CNES).

This thesis

The present work is part of the work package about the storage of carbon dioxide by dissolution and reactive transport in brine. It focuses on the experimental study of the convective dissolution. In the literature, this process has already been experimentally and numerically assessed. However, the instability induced by the dissolution of CO 2 in brine still need some efforts to be fully understood.

We have noticed a lack of experiments conducted in a three-dimension sample in the literature. Indeed, most of the experiments, with an optical access to the sample, are conducted in a Hele-Shaw cell on two-dimensional samples. There is an evident need to experimentally study the instability in a cell with a three-dimensional geometry.

The other main issue is that most of the experiments in the literature are conducted at low pressure and ambient temperature, i.e. far from the thermodynamic conditions of typical aquifers.

Therefore, the objective of this thesis is to develop a new experimental set-up and methodology to study the convective dissolution of CO 2 in brine under processrelevant conditions to characterize the convective dissolution of CO 2 in brine in a threedimensional sample.

Chapter 2 Fundamentals and state-of-the-art on convective dissolution

In this chapter, we present all the fundamentals necessary for a proper understanding of the convective dissolution process of CO 2 in brine. It involves thermodynamic and hydrodynamic concepts as well as the study of the diffusion of CO 2 in the liquid phase. Here, by diffusion, we refer to Fick's diffusion, i.e. the basis of the microscopic mixing process through which a solute is dissolved in a solvent.

The first section presents some general thermodynamic concepts describing the initial and final equilibria when a gas is placed on top of a liquid layer. First, we recall the thermodynamics fundamentals to describe phase equilibrium, explain the concept of dissolution, focusing on the dissolution of a gas in a liquid, and introduce the system CO 2 -brine.

The second section focuses on the convective dissolution. After explaining the operational procedure of this process, we will present the studies available in the literature on this topic. From this, we will define what needs to be investigated and, thus, the outlines of this doctoral work.

Fluid-phase equilibria

The convective dissolution involves the dissolution process, which is "the mixing of two phases with the formation of one new homogeneous phase (i.e. the solution)" [START_REF] Clarke | Definitions of terms relating to phase transitions of the solid state[END_REF]. At the end of the dissolution process, the system reaches a new macroscopic equilibrium. Starting from the definition of the chemical potential, it is possible to describe the equilibrium states depending on pressure and temperature to obtain the composition of both the gaseous and liquid phases.

Thermodynamic fundamentals

Here, we study the thermodynamics of fluid-phase equilibrium allowing the description of two or more homogeneous phases that can exchange energy and matter. The intensive properties such as density, pressure, temperature, and composition are of particular interest to describe quantitatively the equilibrium conditions [START_REF] Prausnitz | Molecular Thermodynamics of Fluid-Phase Equilibria[END_REF]. Thus, we would be able to predict unknown properties, e.g., the concentration of CO 2 in the aqueous phase, by providing some of the equilibrium properties of the system.

Chemical potential

In 1875, Josiah Willard Gibbs defined the chemical potential as the mathematical function allowing to define the fluid-phase equilibrium. It expresses the capacity of a system to give or exchange particles. In a homogeneous system, which is a single phase system including multiple chemical components, we define the chemical potential of the i-th component, µ i , at constant temperature and pressure

µ i = ∂G ∂n i p,T,n j =i , (2.1)
where G is the Gibbs free-energy of the system, n i is the number of moles of the i-th component, p is the total pressure of the system, and T the temperature [START_REF] Spycher | CO 2 -H 2 O mixtures in the geological sequestration of CO 2 . I. Assessment and calculation of mutual solubilities from 12 to 100 • C and up to 600 bar[END_REF].

Considering a heterogeneous system at macroscopic thermodynamic equilibrium, where two or more phases coexist, the free energy is at its minimum and, thus, the total chemical potential is constant in any phase

µ i,α = µ i,β . (2.2)
In equation (2.2), µ i,α and µ i,β are the chemical potential of the i-th component in the phase α and β.

Fugacity and activity coefficient

The chemical potential enables a mathematical definition of the phase equilibrium. However, to have a better physical vision of the problem, we need to introduce two further functions: the activity coefficient and the fugacity. These two functions allow us writing equations relating the chemical potential of each phase to temperature, pressure, and the composition of the system. Usually, the composition of the system is described by the molar fractions of every component i in all phases. Gilbert Newton Lewis [START_REF] Prausnitz | Molecular Thermodynamics of Fluid-Phase Equilibria[END_REF] used the equation

µ i (T, p) = µ 0 i - T T 0 s i dT + p p 0 v i dp, (2.3)
where µ 0 i is the chemical potential at the standard state, T 0 and p 0 are the standard temperature and pressure, wisely chosen to simplify the calculation, and the ideal gas equation pV = nRT , V being the total volume, n the total number of moles, s i the molar entropy of the i-th component, v i the molar volume of the i-th component, and R = 8.314 J • K -1 • mol -1 is the universal gas constant, to obtain

µ i -µ 0 i = RT ln p p 0 (2.4)
that is valid for pure ideal gases at constant temperature. To generalise the latter to any phase and non ideal or mixed systems, Lewis introduced the fugacity f , so that equation (2.4) becomes

µ i -µ 0 i = RT ln f i f 0 i (2.5)
with f i the fugacity of the i-th component of the system, and f 0 i the fugacity of the i-th component in its standard state.

The ratio f i /f 0 i is called as the activity coefficient a i of the i-th component and provides a measurement of how "active" is the system, or how far the system is from its standard state. As it will be discussed later, the activity coefficient is also of capital interest for predicting the Fick diffusion coefficient.

Using equation (2.2), we can say that equilibrium occurs at a given temperature and pressure, when the fugacity of a component is the same in any phase

f α i = f β i . (2.6)
This is equivalent to the equilibrium defined through the chemical potential.

Fugacities in liquid mixtures

In liquids, the fugacity is related to the mole fraction of the i-th component in the liquid phase x i by

f L i = a i x i f 0 i . (2.7)
In equation (2.7), a i is the activity coefficient the i-th component. We chose the standard state in a way that the activity coefficient is close to unity. Then, for an ideal solution, a i is equal to unity.

Several models exist to compute the fugacity depending on the thermodynamic properties of the system, whether it is in solid, liquid, or gaseous phase.

The solubility of gases in liquids

According to Le Chatelier's principle, a system responds to a stress, in pressure or temperature, and tries to minimize it. For instance, if the stress is an increase in the pressure of the gaseous phase, the system tends to reduce the pressure by dissolving gaseous solute in the liquid solvent. Thus, for a proper study of the process, we must quantify how the solubility is affected by changes in thermodynamic and physico-chemical parameters.

The solubility of a i-th component in the liquid phase, is the mole fraction of the component itself in this phase at equilibrium. It is denoted x L i , or simply x i as x already refers to the mole fraction in the liquid phase. It strongly depends on the species, and on thermodynamic parameters such as pressure and temperature [START_REF] Nikolay | Thermodynamic model of aqueous CO 2 -H 2 O-NaCl solutions from -22 to 100 • C and from 0.1 to 100 MPa[END_REF][START_REF] Spycher | CO 2 -H 2 O mixtures in the geological sequestration of CO 2 . I. Assessment and calculation of mutual solubilities from 12 to 100 • C and up to 600 bar[END_REF][START_REF] Valtz | Vapour-liquid equilibria in the carbon dioxide-water system, measurement and modelling from 278.2 to 318.2[END_REF].

Starting from equation (2.6), we can find the mutual solubilities of each species in the gaseous phase α and liquid phase β. The first way to simplify this equation on fugacities, is to use the assumption introduced by Raoult

p i = x i p s i , (2.8)
where p i is the partial pressure of the i-th component, p s i is the vapour pressure of the pure component i at the temperature of the system, and x i is the ideal gas solubility. However, this first approximation provides only a rough estimation of gas solubility.

Henry's law

The solubility of a gas is highly dependent on its partial pressure. In the beginning of the 19 th century, J. W. Henry proposed an empirical law, stating that the concentration of a gas solute in a liquid solvent at equilibrium is directly proportional to the partial pressure of the gas. This is known as Henry's law.

f i a i = f 0 i p = H i,solvent x i (2.9)
where H i,solvent is the Henry's constant of the i-th component in the solvent, and x i the molar fraction of the i-th component dissolved in the solvent. The Henry's constant, H i,solvent , depends on temperature and pressure. However, this linear law is valid only for pressures lower than 0.5-1 MPa and low solubility not exceeding 3 mol. %. This upper limit of validity depends, however, on the system under study. In our case, where a polar solute (CO 2 ) is dissolved in a polar solvent (brine), we observed a loss of linearity at about 1.5 mol. %. At higher pressure, the behaviour deviates from the linear one as we will see later in figure 2.3. Above this limit, the Henry's law is no longer valid and we need to use a convenient model for the calculation of fugacity and, then, the solubility. ) and the carbonate ion (CO 2- 3 ). We associate two acid dissociation constants to the acid base reactions:

K a1 = [HCO - 3 ][H + ]/[H 2 CO 3 ] and K a2 = [CO 2- 3 ][H + ]/[HCO - 3 ]
. However, the hydration equilibrium constant is relatively low. Thus, the carbon dioxide is mostly remaining in its liquid phase, and any chemical reactions can be neglected in this study.

In figure 2.1, we present the phase diagram of pure CO 2 extracted from data given by [START_REF] Span | A new equation of state for carbon dioxide covering the fluid region from the triple-point temperature to 1100 K at pressures up to 800 MPa[END_REF] [START_REF] Span | A new equation of state for carbon dioxide covering the fluid region from the triple-point temperature to 1100 K at pressures up to 800 MPa[END_REF]. The plait critical point is of particular interest in this thesis. The corresponding critical temperature T c is given around 304.15 K, and the critical pressure p c around 7.4 MPa. Beyond these values CO 2 is in its supercritical state. The CO 2 -H 2 O system Due to the bipolar and quadrupolar moment of both the carbon dioxide and water molecules, the resulting mixture is highly non-ideal. This prompts interesting properties to the mixture, especially about its density.

The phase diagram of the mixture CO 2 -H 2 O is classified as type V in the classification of van Konynenbourg and Scott (1980) [START_REF] Van Konynenburg | Critical lines and phase equilibria in binary van der Waals mixtures[END_REF]. In figure 2.2, the different stable phases are displayed at relatively low pressure and temperature. The phase diagram describes what are the phases at thermodynamic equilibrium of the system, depending on pressure and temperature. We notice a quadrupole point Q 1 where four phases coexist in equilibrium. We also notice the upper critical end point (UCEP) close to the plait critical point (CP) of pure CO 2 . In the proximity of these points, there is a risk to cross the triphasic line while we are changing the thermophysical properties of the system, especially when we will rise the pressure during the experiments. Thus, a CO 2 -rich liquid layer would appear between the brine-rich liquid one and the CO 2 -rich gaseous one, and the process under study would be completely different than the one occurring in saline aquifers.

Convective dissolution

The dissolution of a gas or a solid in a liquid induces the creation of a boundary layer composed of the mixture solute-solvent. When the density of the mixture is higher than that of the solvent below, the density gradient combined with the gravity field may lead to a convective motion inside the liquid phase. This highly enhances the dissolution rate of the solute in the solvent.

Convection is a combination of diffusion and advection. In this section, we will present the Fickian's diffusion and the equations that describe it. Then, we will explain how the advective motion is driven and the equations governing the instability, both in porous and free media.

Fickian's diffusion

Once a gaseous or solid solute is dissolved in a liquid solvent, it creates a region more concentrated than the bulk. The random walk of molecules, due to the combination of thermal agitation and collisions, known as Brownian motion, results in the mixing of the higher concentrated zone with the less concentrated one. This process induces mass transport without requiring any motion in the fluid, and is called diffusion.

In the middle of the 19 th century, Fick introduced an empirical model to describe diffusion in a binary mixture, which is based on analogies with the Fourier's work on heat transfer. The first Fick's law defines a total one-dimensional flux J 1 of the solute in the solvent through a surface S, that requires the definition of a binary diffusion coefficient D between the solute and the solvent

J 1 = Sj 1 = -SD∇c 1 , (2.10)
where j 1 is the diffusive flux per unit area of the solute in the solvent and c 1 the concentration of the solute in the solvent, where the subscript 1 denotes the solute and the subscripts 2 the solvent. This law has been justified and generalised to a multicomponant system with the equation of Maxwell-Stefan [START_REF] Bardow | Prediction of multicomponent mutual diffusion in liquids: Model discrimination using NMR data[END_REF]. From (2.10), Fick also developed a more general conservation equation:

∂c 1 ∂t = D ∂ 2 c 1 ∂z 2 + 1 S ∂S ∂z ∂c 1 ∂z (2.11)
which is known as the Fick's second law for a constant surface S, where z is the direction of diffusion. For a one-dimensional analysis of diffusion, we can derive an analytical solution under specific boundary conditions. In the following, as we are interested on what is happening in the liquid phase, we will denote c 1 as c.

Description of the convective dissolution process

The dissolution of the solute in the solvent creates a boundary layer growing diffusively with time. For specific systems, the density of the solute-solvent mixture is higher than the one of the solvent. Then, we are in a situation where a denser fluid layer is placed above a lighter one. This is prone to the apparition of a hydrodynamic instability known as the Rayleigh-Taylor instability. At the interface between the two fluid layers, small fluctuations of concentration are amplified by the gravity field and can lead to an instability with the shape of fingerings. This instability is also referred to as densitydriven convection. is a transient step where the pressure of the gaseous phase has been risen and CO 2 starts to dissolve in the liquid phase forming a boundary layer. c 0 and c eq are the equilibrium concentrations at p 0 and p eq , respectively. Here, the concentration in the boundary layer is given as a constant equal to the concentration at the gas-liquid interface, which is true in most of the boundary layer. g is the acceleration of gravity, h the height of the liquid layer, z the axis parallel to the gravity acceleration pointing downward, and d(t) the height of the boundary layer growing with time.

Geometry of the system

To describe quantitatively the instabilities, we must, at first, describe the geometry of the system under study. Let's consider a three dimensional sample in which there are two phases: a gaseous CO 2 -rich phase at the top and a liquid brine-rich phase at the bottom. In figure 2.4, a two-dimensional cross-section visualisation of the sample is provided at three different times. (a) is the initial equilibrium at pressure p 0 and (c) the final equilibrium at pressure p eq . (b) is the transient step that occurs after the pressure in the gaseous phase has been risen. c 0 and c eq are the equilibrium concentrations at p 0 and p eq , respectively. In the following equations describing the problem, c = c(x, y, z, t) is the concentration of the solute in the solvent, p(t) the pressure of the gaseous phase when the solute is a gas, h denotes the liquid layer thickness, and d(t) the boundary layer thickness.

The container cell can be cylindrical or rectangular, the medium could be porous or free, depending on the type of experiments studied.

Governing equations in free medium

In fluid mechanics, the flow is governed by the Navier-Stokes equations. Here, we use them in their incompressible form because we encounter small density differences during one experiment regarding to the density of the liquid phase. The equations are known as the continuity equation and the momentum equation and are completed by a mass transport equation for the solute concentration c

∇ • u = 0,
(2.12)

∂u ∂t + (u • ∇)u = - 1 ρ ∇p + ν∇ 2 u -g, (2.13
)

∂c ∂t + (u • ∇)c = D∇ 2 c (2.14)
where ρ = ρ(x, y, z, t) is the density of the liquid phase, u = u(x, y, z, t) the velocity in the liquid phase, ν the kinematic viscosity in the liquid phase, g the acceleration of gravity vector, c = c(x, y, z, t) the concentration of solute, and D the Fickian diffusion coefficient of the solute in the solvent. In the case of buoyancy-driven flows, we use the Boussinesq's approximation. It states that the density variations are small compared to the density of the fluids but that the gravity is strong enough to make them influence the flow. Then, the density variations are negligible except in the terms multiplied by g, corresponding to the buoyancy force. In this specific work, the temperature is kept constant and, thus, the variations of density are only due to the variations of concentration of carbon dioxide inside the liquid phase

ρ = ρ 0 [1 + β s (c -c 0 )], (2.15)
with ρ 0 the density at the concentration c 0 . We also assume that the pressure can be written as the pressure of a hydrostatic equilibrium p 0 and a perturbation δp, p = p 0 +δp. For the hydrostatic case, we have ∇p 0 = ρ 0 g. Therefore, the momentum equation becomes ∂u ∂t

+ (u • ∇)u = - 1 ρ 0 ∇δp + ν∇ 2 u + β s (c -c 0 )g. (2.16)
Then, we scale the previous equations by introducing scaling quantities and defining dimensionless parameters denoted why a * superscript. The scaling length corresponds to the height of the liquid layer h, and the scaling time is the diffusive time over this layer h 2 /D. Then, we define a scaling velocity U = D/h, a scaling pressure U 2 ρ 0 ν/D, and a scaling concentration ∆c = c eq -c 0 . We thus have

p = U 2 ρ 0 ν D p * , u = U u * , c = c 0 + ∆c • c * , r = hr * , t = h 2 D t * , (2.17)
and the momentum equation becomes

∂u * ∂t * + (u * • ∇ * )u * = - ν D ∇ * δp * + ν D ∇ * 2 u * + β s ∆ch 3 D 2 c * e z . (2.18)
We introduce two dimensionless numbers: the Schmidt number Sc = ν/D and the solutal Rayleigh number Ra s = β s ∆cgh 3 /νD. Then, we remove the * superscript to lighten the equation and, as the pressure will be kept constant during the experiment after the injection, we assume that δp=0. Then, we finally obtain this set of dimensionless equations

Du Dt = Sc∇ 2 u + ScRa s ce z (2.19) ∇ • u = 0, (2.20)
Dc Dt = ∇ 2 c (2.21)
with Du/Dt = ∂u/∂t + (u • ∇)u for any vector or scalar u.

Governing equations in porous medium

In a porous medium, at the mesoscopic scale, i.e. at a scale much larger than the pore scale but much smaller than the system scale, the fluid velocity is governed by the Darcy's law.

u Da = - k η (∇p -ρg) (2.22)
where u Da is the Darcy fluid velocity, k the permeability of the porous medium and g the acceleration of gravity. η, p, and ρ are respectively the dynamic viscosity, pressure and density of the liquid phase One important parameter appear here: the mobility ratio k/η. The real velocity of the interstitial fluid v, that occurs inside the pore spaces, is related to u Da by u Da = φv, where φ is the porosity of the porous medium. Using similar arguments than in free medium, the density is expressed as

ρ = ρ 0 [1 + β c (c -c 0 )]; (2.23)
with β c the volumetric solutal expansion coefficient. c 0 and ρ 0 are the reference concentration and density at the initial time. The system also verify a mass balance equation.

In the context of the Boussinesq's approximation and for a constant porosity in time we write it as

∇ • u = 0. (2.24)
The concentration varies in space and time due to convection. This is given by the convective-diffusion equation

φ ∂c ∂t + (∇ • u)c = φD∆c, (2.25)
with D the Fickian's diffusion coefficient. We define new scaling quantities as in the free medium: a scaling velocity U = g∆ρk/η, a scaling length h, a scaling time φh/U = φhη/g∆ρk, a scaling pressure ∆ρgh, and a scaling concentration ∆c = c eq -c 0 . We thus have

p = ∆ρghp * , u = U u * , c = c 0 + ∆c • c * , r = hr * , t = φhη g∆ρk t * . (2.26)
We introduce the dimensionless reduce pressure P * = p * -(ρ 0 /∆ρ)z * and, removing the * superscript of the dimensionless quantities to lighten the writing, the equations become ∇ • u = 0, (2.27)

u Da = -(∇P -ce z ), (2.28)
∂c ∂t + (∇ • u)c = 1 Ra Da ∆c, (2.29)
where we introduced a new dimensionless number, the Rayleigh-Darcy number

Ra Da = U h φD = ∆ρgkh φηD .
(2.30)

Boundary conditions

The boundary and initial conditions of the process are:

c(t = 0) = c 0 (2.31) c(z = 0) = c eq , for t > 0 (2.32) ∂c ∂z z=h = 0, ∀t (2.33)

Further dimensionless numbers

We already introduced the solutal Rayleigh number, or solutal Rayleigh-Darcy number, that characterise the predominance of convection over diffusion. Other adimensional numbers could be useful in describing the system behaviour. The Peclet number P e = hu/D is the ratio of the advection on the diffusion transport, with u the local flow velocity. In buoyancy-driven flow, the Peclet number is typically equal to the Rayleigh number, but it can be particularly interesting for the investigation of the influence of a background flow in an aquifer.

The Damkölher number is used in studies on the influence of reactive transport on the convective dissolution. We define Da = kC n-1 A γ, where A is a reactant, k its reaction constant, n the order of the reaction, and γ the mean residence time. It characterises the influence of reactive transport due to the geochemical response of the aquifer compared with the diffusion transport. In the definition of this number, we can also replace the typical diffusion time scale by the typical convective time scale.

The Lewis number Le is used to characterize flows inside a fluid that is subject to both thermal and mass transfer, comparing the thermal and solutal diffusivity. Le = D T /D, with D T the thermal diffusivity and D the mass Fick diffusivity. A larger Lewis number means that the characteristic time of mass diffusion would be larger than the characteristic time of thermal diffusion. In the saline aquifers, there is often a small geothermal gradient between the top and the bottom of the domain. Under these conditions, a double diffusive-convective mixing could appear. It would be important to consider this phenomenon during the modelling of the flows inside an aquifer.

Finally, the Sherwood number Sh = k m h/D, with k m the mass transfert coefficient, is of major interest in these studies. It characterises the mass transfer by taking the ratio of the mass transfer induced by convection to the one induced by diffusion. Sometimes, we will also make use of the Nusselt number, N u = k T h/λ, with k T the convective heat transfer coefficient and λ the thermal conductivity of the fluid. It is the ratio between the heat transfer due to convection and the one due to heat conduction. Even though it is appropriated for heat transfer experiments, its solutal equivalent N u s frequently used in studies about solutal convection instead of the Sherwood number but the two of them refer to the very same quantity.

Analysis of the convective dissolution

The solutal Rayleigh number allows a rapid quantitative comparison between the advective force provided by buoyancy and the dissipation given by diffusion and viscosity. Depending on the boundary conditions and on the kind of medium, a critical value Ra * s can be defined [START_REF] Messlinger | Transient diffusive boundary layers at high Rayleigh numbers in simple and double diffusive fluids: Latency time scaling for the convection onset[END_REF]. Once the solutal Rayleigh number exceeds the critical value associated with the geometry concerned, convection starts. That allows us to define a critical time corresponding to the time at which convection sets-in.

The onset time of convection is thus related to the behaviour of the system before the instability. It is also interesting to study the behaviour of the system at longer time by studying the shut-down time of convection, t sd corresponding to the moment at which convection ceases.

Additionally, the three-dimensional spatial structure of the system can provide useful information about the instability. [START_REF] Messlinger | Transient diffusive boundary layers at high Rayleigh numbers in simple and double diffusive fluids: Latency time scaling for the convection onset[END_REF] [START_REF] Messlinger | Transient diffusive boundary layers at high Rayleigh numbers in simple and double diffusive fluids: Latency time scaling for the convection onset[END_REF] provide equations for critical wavenumber or critical wavelength depending on the boundary conditions. Furthermore, spatial information on the plumes can be extracted from experiments. Their size, length, and speed can be measured to characterise the phenomenon.

Two other quantities are of major interest, especially for the industrial applications to CCS projects: the dissolution rate and the quantity of CO 2 dissolved during its injection in the aquifer, which are essential to determinate the efficiency of the injection procedure. Here, the solutal Nusselt or Sherwood number is useful to quantitatively describe the relative amount of CO 2 transferred by convection and, thus, the efficiency of dissolution.

State-of-the-art on convective dissolution of CO 2 in water and salted solutions

The convective dissolution has been widely investigated for the dissolution of a gas or solid solute in a liquid solvent. Here, we focus on the studies relevant to the convective dissolution of CO 2 in water or brine both in porous and free medium. This will include studies of a solid dissolving in a liquid solvent, and some liquid-liquid systems presenting similar density differences to the system CO 2 -brine.

In the literature, several studies describe the convective dissolution by introducing several regimes. From experimental and numerical investigations, [START_REF] Slim | Dissolution-driven convection in a Hele -Shaw cell[END_REF] [START_REF] Slim | Dissolution-driven convection in a Hele -Shaw cell[END_REF] highlighted six distinct regimes during the entire process: a diffusive regime, during which the concentration profile is solution of the diffusion equation and the flux decays as the square root of t; a linear-growth regime where the concentration profile diverges from its diffusive growth due to the apparition of convective fingers; a flux-growth regime in which the flux deviate significantly from its pure diffusive value; a merging regime where fingers are merging and the flux is decaying; a reinitiation regime where new primary fingers appear; and a saturation regime where the fingers reached the bottom of the bulk and the flux is decaying rapidly. These regimes are separated by five well-defined events: onset of instability, onset of convection, unsteady-flux peak, constant flux regime, and convective shutdown. In some studies [START_REF] De Paoli | Rayleigh-Taylor convective dissolution in confined porous media[END_REF], the process has been simplified in three regimes: diffusion, convection with the apparition of fingerings, and shutdown.

The onset time of convection is a critical parameter providing information on the efficiency of the process due to their impact on CCS applications. In the literature, the onset time of convection, as known as the critical time t c , is suggested to be described by the equation t c = a 1 Ra -b s , with the exponent b being equal to -2 for experiments performed in a porous medium and -2/3 in a free medium. It is possible to estimate the exponent by linear stability analysis (LSA) [START_REF] Bestehorn | Effect of fluctuations on the onset of density-driven convection in porous media[END_REF][START_REF] Chan | Linear stability analysis on the onset of buoyancy-driven convection in liquid-saturated porous medium[END_REF][START_REF] Cheng | Effect of permeability anisotropy on buoyancy-driven flow for CO 2 sequestration in saline aquifers[END_REF][START_REF] Teres | On the time scales of nonlinear instability in miscible displacement porous media flow[END_REF][START_REF] Emami | Stability analysis of two-phase buoyancy-driven flow in the presence of a capillary transition zone[END_REF][START_REF] Ennis | Role of convective mixing in the longterm storage of carbon dioxide in deep saline formations[END_REF][START_REF] Ennis-King | Onset of convection in anisotropic porous media subject to a rapid change in boundary conditions[END_REF][START_REF] Hassanzadeh | Modelling of convective mixing in CO 2 storage[END_REF][START_REF] Javaheri | Linear stability analysis of double-diffusive convection in porous media, with application to geological storage of CO 2[END_REF][START_REF] Chan | Onset of buoyancy-driven convection in a liquid-saturated cylindrical porous layer supported by a gas layer[END_REF][START_REF] Meulenbroek | The effect of interface movement and viscosity variation on the stability of a diffusive interface between aqueous and gaseous CO 2[END_REF][START_REF] Myint | Onset of convection with fluid compressibility and interface movement[END_REF][START_REF] Riaz | Onset of convection in a gravitationally unstable diffusive boundary layer in porous media[END_REF][START_REF] Slim | Onset and cessation of time-dependent, dissolution-driven convection in porous media[END_REF][START_REF] Tilton | Nonlinear stability of gravitationally unstable, transient, diffusive boundary layers in porous media[END_REF][START_REF] Wessel-Berg | On a linera stability problem related to underground CO 2 storage[END_REF][START_REF] Xu | Convective stability analysis of the long-term storage of carbon dioxide in deep saline aquifers[END_REF] and direct numerical simulation (DNS) [START_REF] Teres | On the time scales of nonlinear instability in miscible displacement porous media flow[END_REF][START_REF] Slim | Solutal-convection regimes in a two-dimensional porous medium[END_REF]. In numerical studies, the different values obtained for the onset time of the instability is due to differences in the method used to estimates its value, the definition of the boundary conditions, and the geometry of the simulations. Many numerical studies and simulations [START_REF] Azin | Measurement and modeling of CO 2 diffusion coefficient in saline aquifer at reservoir conditions[END_REF][START_REF] Cheng | Effect of permeability anisotropy on buoyancy-driven flow for CO 2 sequestration in saline aquifers[END_REF][START_REF] Teres | On the time scales of nonlinear instability in miscible displacement porous media flow[END_REF][START_REF] Emami | Stability analysis of two-phase buoyancy-driven flow in the presence of a capillary transition zone[END_REF][START_REF] Farajzadeh | Numerical simulation of density-driven natural convection in porous media with application for CO 2 injection projects[END_REF][START_REF] Hassazadeh | Scaling behavior of convective mixing, with application to geological storage of CO 2[END_REF][START_REF] Hidalgo | Scaling of convective mixing in porous media[END_REF][START_REF] George | High-resolution simulation and characterization of density-driven flow in CO 2 storage in saline aquifers[END_REF][START_REF] Riaz | Onset of convection in a gravitationally unstable diffusive boundary layer in porous media[END_REF][START_REF] Slim | Solutal-convection regimes in a two-dimensional porous medium[END_REF] computed the value of the prefactor for the onset time of convection according to equations of the previous paragraph. The reported results are in the range 845-5616. The huge difference can be explained by the differences in the numerical schemes and initial perturbations used in the simulations.

The onset of convection induces a rise in the dissolution flux that reaches a maximum. Some results present also a linear scaling law for the maximum flux with the Rayleigh number in the range 1 × 10 2 < Ra s < 3 × 10 3 . However, a sublinear scaling law has been suggested by Nazari-Moghadam et al. (2015) [START_REF] Nazari-Moghaddam | Scaling analysis of the convective mixing in porous media for geological storage of CO 2 : An experimental approach[END_REF] and [START_REF] Seyyedi | Experimental study of density-driven convection effects on CO 2 dissolution rate in formation water for geological storage[END_REF] [START_REF] Seyyedi | Experimental study of density-driven convection effects on CO 2 dissolution rate in formation water for geological storage[END_REF] in a wider range of solutal Rayleigh numbers. Some studies have shown that the permeability of the porous medium has an influence on the convective dissolution. [START_REF] Xu | Convective stability analysis of the long-term storage of carbon dioxide in deep saline aquifers[END_REF] [START_REF] Xu | Convective stability analysis of the long-term storage of carbon dioxide in deep saline aquifers[END_REF] showed that the system tends to be destabilized more easily with a higher permeability both in vertical and horizontal configuration. [START_REF] Cheng | Effect of permeability anisotropy on buoyancy-driven flow for CO 2 sequestration in saline aquifers[END_REF] [START_REF] Cheng | Effect of permeability anisotropy on buoyancy-driven flow for CO 2 sequestration in saline aquifers[END_REF] studied the effect of the anisotropy of the horizontal permeability on the onset time. They drew the conclusion that an increase in the anisotropy decreases the onset time of convection.

Experimental studies

In the literature, two main types of experiments are available. On one side, there are experiments performed in PVT (pressure volume temperature) cells consisting in recording the pressure decay induced by the dissolution of CO 2 in water or brine. On the other side, there are experiments performed in two-dimensional Hele-Shaw cells where the dissolution of CO 2 in brine is observed transversally to see how the fingerings are developing inside the bulk.

Experiments in PVT cells

A PVT cell is a cylindrical chamber in which we measure the evolution of pressure and temperature of the system. This type of cell is of major interest to investigate mass transfer between gas and liquid. These experiments are particularly used to study quantitatively the dissolution of CO 2 in brine in both porous [START_REF] Kneafsey | Laboratory flow experiments for vizualizing carbon dioxide-induced, density-driven brine convection[END_REF][START_REF] Nazari-Moghaddam | Scaling analysis of the convective mixing in porous media for geological storage of CO 2 : An experimental approach[END_REF][START_REF] Nazari Moghaddam | Quantification of density-driven natural convection for dissolution mechanism in CO 2 sequestration[END_REF][START_REF] Seyyedi | Experimental study of density-driven convection effects on CO 2 dissolution rate in formation water for geological storage[END_REF][START_REF] Shi | Measurement and modeling of CO 2 mass transfer in brine at reservoir conditions[END_REF] or free medium [START_REF] Farajzadeh | Mass transfer of CO 2 into water and surfactant solutions[END_REF][START_REF] Farajzadeh | Enhanced mass transfer of CO 2 into water: Experiment and modeling[END_REF][START_REF] Farajzadeh | Numerical simulation of density-driven natural convection in porous media with application for CO 2 injection projects[END_REF][START_REF] Gholami | Prediction of carbon dioxide dissolution in bulk water under isothermal pressure decay at different boundary conditions[END_REF][START_REF] Khosrokhavar | Visualization and investigation of natural convection flow of CO 2 in aqueous and oleic systems[END_REF][START_REF] Nazari Moghaddam | Quantification of density-driven natural convection for dissolution mechanism in CO 2 sequestration[END_REF][START_REF] Shi | Measurement and modeling of CO 2 mass transfer in brine at reservoir conditions[END_REF][START_REF] Tang | Experimental study on the density-driven carbon dioxide convective diffusion in formation water at reservoir conditions[END_REF][START_REF] Yang | Accelerated mass transfer of CO 2 in reservoir brine due to density-driven natural convection at high pressures and elevated temperatures[END_REF][START_REF] Zhao | Nonconstant diffusion behavior for CO 2 diffusion into brine: Influence of densitydriven convection[END_REF]. Some of these experiments also study the convective dissolution of CO 2 in oil [START_REF] Amarasinghe | CO 2 convective dissolution in oil-saturated unconsolidated porous media at reservoir conditions[END_REF][START_REF] Farajzadeh | Mass transfer of CO 2 into water and surfactant solutions[END_REF][START_REF] Farajzadeh | Numerical simulation of density-driven natural convection in porous media with application for CO 2 injection projects[END_REF][START_REF] Khosrokhavar | Visualization and investigation of natural convection flow of CO 2 in aqueous and oleic systems[END_REF], most for enhanced oil recovery applications.

The approach of PVT experiment is based on the "Pressure-decay method". Using the diffusion equation developed by Fick and Henry's laws, one can detect the deviation between the pressure decay and the one modelled by a diffusion process. The measured deviation stems from the contribution of convection. The mass transfer rate induced by the latter is much stronger than the one predicted by the diffusive dissolution. To quantify this difference, the same diffusive approach is used and a pseudo-diffusive coefficient D 1 is introduced. After the onset of convection, the pseudo-diffusive coefficient is one to three orders of magnitude larger than the molecular diffusive coefficient D 2 . The coefficient D 1 gradually decreases and eventually goes back to the diffusive value D 2 .

These studies are based on a method developed by Zhang et al. [START_REF] Zhang | Measurement of gas diffusivity in heavy oils[END_REF] to measure the diffusivity of a gas into oil by the analysis of the pressure decay in a cylindrical PVT cell. This kind of investigation needs to make use of few assumptions [START_REF] Gholami | Prediction of carbon dioxide dissolution in bulk water under isothermal pressure decay at different boundary conditions[END_REF] the swelling of the liquid phase is negligible and the boundary conditions remain constant, especially the height of the liquid level z 0 is constant; the concentration at the interface is the equilibrium concentration; the temperature remains constant during the experiment; the diffusion coefficient does not change significantly with concentration; the liquid-phase is non-volatile. In the dissolution of CO 2 into water the water evaporation will be considered negligible; the gas is pure (single component) gas;

the initial concentration of gas in the liquid phase is equal to zero.

The diffusion problem is thus reduced to the unsteady-state one-dimensional diffusion equation:

dc dt = D d 2 c dz 2 (2.34)
where c is the CO 2 concentration into the brine. Boundary conditions are set as follow:

z = 0, t > 0, c = c eq(p) (2.35) z = h, t ≥ 0, dc dz = 0 (2.36) 0 < z ≤ h, t = 0, c = 0 (2.37)
where c eq(p) is the equilibrium concentration at the interface which only varies with the pressure as the temperature is kept constant. John Crank [START_REF] Crank | The mathematics of diffusion[END_REF] gave the analytical solution of this problem:

c c eq = 1 - 4 π ∞ n=0 (-1) n 2n + 1 cos (2n + 1)πz 2z 0 exp - (2n + 1) 2 π 2 D 4z 2 0 t . (2.38)
For mass conservation, the moles of gas transferred from the interface are equal to the moles removed from the gas phase. Then,

V g Z g RT dp(t) dt = -DA dc dz z=0 , (2.39)
where V g is the volume of the gas phase, A is the surface of the interface, R, is the universal gas constant, T is the temperature, and Z g is the compressibility factor. V g , A, and Z g are considered constant. Thus, integrating the pressure over time from t to infinity, we obtain:

peq p(t) dp(t) = -BD ∞ t dc dz z=0 dt, (2.40)
where

B = Z g RT h = constant. (2.41)
p eq is the equilibrium pressure reached at infinite time, and h is the gas height. Combining the previous equations, we get a relationship between pressure and time:

ln [p(t) -p eq ] = ln 8Bz 0 c eq(p) π 2 - π 2 Dt 4z 2 0 , (2.42)
which is the analytical solution of the problem considering only the diffusion process. Then, by fitting the slope of the tangent of the curve given by equation (2.42), we are able to extract the pseudo-diffusive coefficient, as shown in figure 2.5.

From this analysis, it is also possible to derive other quantitative numbers. Some studies provide the mass flux of CO 2 through the gas-liquid interface [START_REF] Gholami | Prediction of carbon dioxide dissolution in bulk water under isothermal pressure decay at different boundary conditions[END_REF][START_REF] Nazari-Moghaddam | Scaling analysis of the convective mixing in porous media for geological storage of CO 2 : An experimental approach[END_REF][START_REF] Nazari Moghaddam | Quantification of density-driven natural convection for dissolution mechanism in CO 2 sequestration[END_REF][START_REF] Seyyedi | Experimental study of density-driven convection effects on CO 2 dissolution rate in formation water for geological storage[END_REF][START_REF] Yang | Accelerated mass transfer of CO 2 in reservoir brine due to density-driven natural convection at high pressures and elevated temperatures[END_REF].

Others introduced the Sherwood adimensional number [START_REF] Farajzadeh | Mass transfer of CO 2 into water and surfactant solutions[END_REF][START_REF] Gholami | Prediction of carbon dioxide dissolution in bulk water under isothermal pressure decay at different boundary conditions[END_REF][START_REF] Nazari-Moghaddam | Scaling analysis of the convective mixing in porous media for geological storage of CO 2 : An experimental approach[END_REF][START_REF] Seyyedi | Experimental study of density-driven convection effects on CO 2 dissolution rate in formation water for geological storage[END_REF]. This number increases at the beginning, reaches a maximum and then decreases. They are able to find a scaling law between the maximum Sherwood number Sh max and the Rayleigh number Ra: Sh max = αRa β , where α and β are the fitting parameters. In the literature, the value of β is close to 0.77.

From these experiments, one can study the convective dissolution of CO 2 in water, brine, or oil. Some quantitative values are obtained, like the dissolution rate, to characterise the effect of convection on dissolution. Nevertheless, the approach of the two pseudo-diffusive coefficients has no real physical sense. Indeed, this approach relies on Fick's way of diffusion, which cannot be valid anymore since advective motion appears inside the bulk due to the destabilisation of the system. The second limit of this type of experiment is that it does not provide any information on the different convective regimes.

Experiments in Hele-Shaw cell

The other kind of experiments in the literature is related to experiments performed in a two-dimensional Hele-Shaw cell. This cell consists in two parallel transparent plates, spaced by a thin distance e, across which we observe the flow in a two-dimensional plane. The particularity of this cell is that the flow is driven by the Darcy's law, and the fluid behaviour is comparable to the one in a porous medium. Here, the permeability k will be k = e 2 /12. In absence of porous medium, the porosity is usually φ = 1. However, it is possible to add blockers or beads inside the bulk to mimic the porous medium more realistically, and to get the porosity smaller than unit.

In the Hele-Shaw cells, the classical solutal Rayleigh number should be replaced by a Rayleigh-Darcy number, like in a porous medium:

Ra Da,s = ∆ρgkh φηD = ∆ρgh 3 12φηD . ( 2

.43)

The Hele-Shaw cells are widely used in the experimental investigations of convective dissolution of CO 2 in brine. Indeed, it provides an easy optical access to visualise the flow inside the cell. There are several optical methods to observe the flow: pH tracking [START_REF] Titly | Experimental and numerical studies of density driven natural convection in saturated porous media with application to CO 2 geological storage[END_REF][START_REF] Kneafsey | Laboratory flow experiments for visualizing carbon dioxide-induced, density-driven brine convection[END_REF][START_REF] Kneafsey | Laboratory flow experiments for vizualizing carbon dioxide-induced, density-driven brine convection[END_REF][START_REF] Mahmoodpour | Convective dissolution of carbon dioxide in deep saline aquifers: Insights from engineering a high-pressure porous Hele-Shaw cell[END_REF][START_REF] Seyyedi | Experimental study of density-driven convection effects on CO 2 dissolution rate in formation water for geological storage[END_REF]145,[START_REF] Thomas | Effect of precipitation mineralization reactions on convective dissolution of CO 2 : An experimental study[END_REF], color tracking [START_REF] Taheri | Experimental and simulation studies of density-driven-convection mixing in a Hele-Shaw geometry with application for CO 2 sequestration in brine aquifers[END_REF], light-scattering method based on a LED wall system [START_REF] Alipour | Concentration-based velocity reconstruction in convective Hele-Shaw flows[END_REF][START_REF] Backhaus | Convective instability and mass transport of diffusion layers in a Hele-Shaw geometry[END_REF][START_REF] De Paoli | How non-Darcy effects influence scaling laws in Hele-Shaw convection experiments[END_REF][START_REF] Jiang | Change in convection mixing properties with salinity and temperature: CO 2 storage application[END_REF][START_REF] Slim | Dissolution-driven convection in a Hele -Shaw cell[END_REF], shadowgraphy [START_REF] Peichun | Density-driven convection enhanced by an inclined boundary: Implications for geological CO 2 storage[END_REF], Schlieren method [START_REF] Loodts | Dissolutiondriven convection in reactive systems[END_REF][START_REF] Tang | Experimental study on the density-driven carbon dioxide convective diffusion in formation water at reservoir conditions[END_REF][START_REF] Thomas | Convective dissolution of CO 2 in water and salt solutions[END_REF][START_REF] Thomas | Effect of precipitation mineralization reactions on convective dissolution of CO 2 : An experimental study[END_REF], and spectrophotometric method [START_REF] Teng | New spectrophotometric method for quantitative characterization of density-driven convective instability[END_REF].

In the literature, we found studies of convective dissolution for multiple systems: CO 2 -water [START_REF] Titly | Experimental and numerical studies of density driven natural convection in saturated porous media with application to CO 2 geological storage[END_REF][START_REF] Kneafsey | Laboratory flow experiments for visualizing carbon dioxide-induced, density-driven brine convection[END_REF]145,[START_REF] Thomas | Convective dissolution of CO 2 in water and salt solutions[END_REF][START_REF] Zhang | Experimental and numerical investigation on interfacial mass transfer mechanism for rayleigh convection in Hele-Shaw cell[END_REF], CO 2 -brine [START_REF] Jiang | Change in convection mixing properties with salinity and temperature: CO 2 storage application[END_REF][START_REF] Kneafsey | Laboratory flow experiments for vizualizing carbon dioxide-induced, density-driven brine convection[END_REF][START_REF] Loodts | Dissolutiondriven convection in reactive systems[END_REF][START_REF] Mahmoodpour | Convective dissolution of carbon dioxide in deep saline aquifers: Insights from engineering a high-pressure porous Hele-Shaw cell[END_REF][START_REF] Seyyedi | Experimental study of density-driven convection effects on CO 2 dissolution rate in formation water for geological storage[END_REF][START_REF] Tang | Experimental study on the density-driven carbon dioxide convective diffusion in formation water at reservoir conditions[END_REF][START_REF] Teng | New spectrophotometric method for quantitative characterization of density-driven convective instability[END_REF][START_REF] Thomas | Convective dissolution of CO 2 in water and salt solutions[END_REF][START_REF] Thomas | Effect of precipitation mineralization reactions on convective dissolution of CO 2 : An experimental study[END_REF], KMnO4water [START_REF] Alipour | Concentration-based velocity reconstruction in convective Hele-Shaw flows[END_REF][START_REF] De Paoli | How non-Darcy effects influence scaling laws in Hele-Shaw convection experiments[END_REF][START_REF] Slim | Dissolution-driven convection in a Hele -Shaw cell[END_REF], water-propylen glycol (PPG) [START_REF] Backhaus | Convective instability and mass transport of diffusion layers in a Hele-Shaw geometry[END_REF][START_REF] Macminn | Buoyant currents arrested by convective dissolution[END_REF][START_REF] Peichun | Density-driven convection enhanced by an inclined boundary: Implications for geological CO 2 storage[END_REF], Glycerin and brine [START_REF] Ennis | Role of convective mixing in the longterm storage of carbon dioxide in deep saline formations[END_REF], and brine-water [START_REF] Taheri | Experimental and simulation studies of density-driven-convection mixing in a Hele-Shaw geometry with application for CO 2 sequestration in brine aquifers[END_REF]. Jafari-Raad et al. ( 2016) [START_REF] Mostafa Jafari-Raad | On the choice of analogue fluids in CO 2 convective dissolution experiments[END_REF] have shown that the choice of the analogue system is critical. Indeed, the impacts on the convective dissolution process are shown by a shift in the onset time of convection and a modification of the scaling laws [START_REF] Liyanage | Multidimensional observations of dissolution-driven convection in simple porous media using X-ray CT scanning[END_REF]. For instance, the system water-PPG is less unstable than the system CO 2water and has different dynamics [START_REF] Mostafa | Onset of dissolution-driven instabilities in fluids with nonmonotonic density profile[END_REF].

Most of the studies are performed at atmospheric temperature, especially those for systems that mimic the same density differences than CO 2 and water. However, only a few experiments are conducted at pressure close to reservoir conditions [START_REF] Kneafsey | Laboratory flow experiments for vizualizing carbon dioxide-induced, density-driven brine convection[END_REF][START_REF] Mahmoodpour | Convective dissolution of carbon dioxide in deep saline aquifers: Insights from engineering a high-pressure porous Hele-Shaw cell[END_REF][START_REF] Tang | Experimental study on the density-driven carbon dioxide convective diffusion in formation water at reservoir conditions[END_REF].

Qualitative observations are easily obtained with a Hele-Shaw apparatus. Quantitative results can be obtained as well. Some studies determined the length and wavelength of the fingerings [START_REF] Backhaus | Convective instability and mass transport of diffusion layers in a Hele-Shaw geometry[END_REF][START_REF] Kneafsey | Laboratory flow experiments for visualizing carbon dioxide-induced, density-driven brine convection[END_REF][START_REF] Macminn | Buoyant currents arrested by convective dissolution[END_REF][START_REF] Peichun | Density-driven convection enhanced by an inclined boundary: Implications for geological CO 2 storage[END_REF]. Others investigated the mass flux or convective flux by introducing the Sherwood number [START_REF] De Paoli | How non-Darcy effects influence scaling laws in Hele-Shaw convection experiments[END_REF][START_REF] Seyyedi | Experimental study of density-driven convection effects on CO 2 dissolution rate in formation water for geological storage[END_REF], the solutal Nusselt number [START_REF] Backhaus | Convective instability and mass transport of diffusion layers in a Hele-Shaw geometry[END_REF][START_REF] Peichun | Density-driven convection enhanced by an inclined boundary: Implications for geological CO 2 storage[END_REF], or by determining directly the mass flux [START_REF] Mahmoodpour | Convective dissolution of carbon dioxide in deep saline aquifers: Insights from engineering a high-pressure porous Hele-Shaw cell[END_REF] and the dissolution rate [START_REF] De Paoli | How non-Darcy effects influence scaling laws in Hele-Shaw convection experiments[END_REF]. They proposed scaling laws, for instance, N u s = αRa β Da,s with α = [0.0228-0.045] and β = [0.76-0.85]. We will see that in three-dimensional sample, these coefficients change.

A typical quantitative value derived from these investigations is the critical time or the onset time of convection [START_REF] Ennis | Role of convective mixing in the longterm storage of carbon dioxide in deep saline formations[END_REF][START_REF] Titly | Experimental and numerical studies of density driven natural convection in saturated porous media with application to CO 2 geological storage[END_REF][START_REF] Jiang | Change in convection mixing properties with salinity and temperature: CO 2 storage application[END_REF][START_REF] Kneafsey | Laboratory flow experiments for vizualizing carbon dioxide-induced, density-driven brine convection[END_REF][START_REF] Macminn | Buoyant currents arrested by convective dissolution[END_REF][START_REF] Mahmoodpour | Convective dissolution of carbon dioxide in deep saline aquifers: Insights from engineering a high-pressure porous Hele-Shaw cell[END_REF][START_REF] Seyyedi | Experimental study of density-driven convection effects on CO 2 dissolution rate in formation water for geological storage[END_REF][START_REF] Slim | Dissolution-driven convection in a Hele -Shaw cell[END_REF]145,[START_REF] Tang | Experimental study on the density-driven carbon dioxide convective diffusion in formation water at reservoir conditions[END_REF][START_REF] Thomas | Convective dissolution of CO 2 in water and salt solutions[END_REF][START_REF] Zhang | Experimental and numerical investigation on interfacial mass transfer mechanism for rayleigh convection in Hele-Shaw cell[END_REF]. Theoretically, the latter is given by the equation t c,Hele-Shaw = a 1 Ra -2 Da,s . However, the studies that validate the theoretical approach on the onset time of convection are limited or are done on a limited range of Ra Da,s .

We also find some studies investigating the effect of precipitation and reactive transport on the convective dissolution [START_REF] Thomas | Convective dissolution of CO 2 in water and salt solutions[END_REF][START_REF] Thomas | Effect of precipitation mineralization reactions on convective dissolution of CO 2 : An experimental study[END_REF]. It is found that, depending on the system, the precipitation intensifies (CaCO 3 ) or hinders (CaCl 2 ) the convective dissolution. A similar conclusion on the effect of reactive transport has been given by Loodts et al (2017) [START_REF] Loodts | Dissolutiondriven convection in reactive systems[END_REF]. Some studies deal with the influence of the inclination of an aquifer on the convective dissolution. The idea is to incline the sample cell and measure how it influences quantitative numbers as the onset time of convection or the dissolution rate. Tsai et al. (2013) [START_REF] Peichun | Density-driven convection enhanced by an inclined boundary: Implications for geological CO 2 storage[END_REF] find that the plume width is not altered by the inclination of the cell, while the propagation through the interface is enhanced with an inclined boundary. Mac Minn et al. (2013) [START_REF] Macminn | Buoyant currents arrested by convective dissolution[END_REF] obtained similar conclusions.

Alipour et al. (2020) [2] introduced a concentration-velocity reconstruction (CVR)

algorithm with which they were able to determine the concentration and velocity fields from light scattering images.

In Hele-Shaw cell experiments, it is important that the gap e between the confining plates remains small enough to limit three-dimensional effects due to Taylor dispersion. [START_REF] Fernandez | Density-driven unstable flows of miscible fluids in a hele-shaw cell[END_REF] [START_REF] Fernandez | Density-driven unstable flows of miscible fluids in a hele-shaw cell[END_REF] showed that diffusion dominates dispersion if e < 10D/u b where u b = ∆ρgk/η is the buoyant speed. This criterion has been refined by [START_REF] Slim | Dissolution-driven convection in a Hele -Shaw cell[END_REF] [START_REF] Slim | Dissolution-driven convection in a Hele -Shaw cell[END_REF] who stated that diffusion dominates dispersion if e < 2D/u b . More recently, De Paoli et al. (2020) [START_REF] De Paoli | How non-Darcy effects influence scaling laws in Hele-Shaw convection experiments[END_REF] showed that three regimes exist depending on the gap and the Rayleigh number. They introduced a new parameter, called the anisotropy ratio, 2 Ra, with = e/ √ 12h. The three regimes are: a Darcy regime ( 2 Ra → 0), a Hele-Shaw regime ( 2 Ra 1), and a three-dimensional regime 2 Ra ≥ 1).

Other experiments

In the literature, further experiments have been conducted in three dimensional medium. The convective dissolution is also studied for the dissolution of a solid in the liquid solvent. This is investigated by [START_REF] Philippi | Solutal convection induced by dissolution[END_REF] [START_REF] Philippi | Solutal convection induced by dissolution[END_REF]. A solid ionic-crystal salt NaCl is placed in a water bulk. The solid dissolves and convective dissolution is laterally observed by shadowgraphy. The same approach than that presented in the previous section is used here to estimate the onset-time of convection, based on the creation of a boundary layer growing diffusively at the beginning of the process. Some studies presented above also investigated the convective dissolution of a solid solute (KMnO 4 ) in a liquid solvent [START_REF] Alipour | Concentration-based velocity reconstruction in convective Hele-Shaw flows[END_REF][START_REF] De Paoli | How non-Darcy effects influence scaling laws in Hele-Shaw convection experiments[END_REF].

Few experiments are conducted in a porous three dimensional medium. To visualize the flow inside it, they make use of magnetic resonance imaging [START_REF] Shattuck | Onset and stability of convection in porous media: Visualization by magnetic resonance imaging[END_REF][START_REF] Wang | Enhanced mass transfer by density-driven convection during CO 2 geological storage[END_REF].

There are studies investigating the behaviour of other systems than CO 2 -water, like PPG-water. These systems mimic the dynamics of the CO 2 -water system because they involve similar density differences. However, there are limitations with these systems and it has been shown that the PPG-water system, for instance, is less unstable than the CO 2 -water system and, thus, does not accurately mimic the behaviour of the system CO 2 -water [START_REF] Mostafa | Onset of dissolution-driven instabilities in fluids with nonmonotonic density profile[END_REF].

Position of this thesis in the literature

As stated before, some issues can limit the applicability of the studies done with PVT cells. Even if they have a three-dimensional shape, there is no optical access to observe the flow motions in the bulk. Furthermore, the analysis done from pressure decay data has no real physical meaning. The pressure decay is disturbed by the gas relaxation induced after the injection of CO 2 , which makes it difficult to define the onset time of convection. In addition, the pressure used with these experiments are relatively far from the reservoir one's. Thus, we will not use this type of cell, even if we will see that the three dimensional geometry is an interesting point. We also keep in mind that it is easier to work at constant pressure to avoid issues caused by the relaxation of CO 2 after its injection. This approach is also more realistic to reservoir conditions where the pressure of the system is not influenced so much by the convective process at relatively short time scales.

The experiments performed in Hele-Shaw cells simulate the porous medium found in the aquifers being governed by Darcy's flow. They also provide a direct optical access, which gives qualitative information on the flow inside the cell. Thus, it is easy to define the onset time of convection as it corresponds to the moment when the interface starts to be destabilised. Furthermore, the spatial analysis of the plumes is also made possible by the optical access. Velocity, size, merging of the plumes, and spacing between them are precisely measured. The optical access also gives the possibility to retrieve a CO 2 concentration field from the images obtained by optical techniques such as shadowgraphy or by pH-tracking. Besides the quantitative data obtained with the Hele-Shaw cell, some limit can be identified. First, the cell is two-dimensional which is a major limit since the reservoirs are three-dimensional. We also noticed a lack in experiments performed under relevant thermodynamic conditions meaning high pressures, i.e. supercritical CO 2 . Most of these experiments are performed at low pressure or even atmospheric pressure for the systems that mimic the density difference of the CO 2 -brine system.

In conclusion, there is a clear lack of experiments performed in a three-dimensional cell, at high pressure, with an optical access to the interior of the cell. The goal of this thesis is, thus, to develop an experimental set-up allowing the observation of the flow inside the liquid phase under relevant thermodynamic conditions. Considering the expertise of the laboratory on the shadowgraph method, used to study transport phenomena, we will use this optical technique in combination with a three-dimensional cylindrical cell. Another specific competence of the hosting laboratory is the knowledge related to high-pressure cells which provides the possibility to work at thermodynamic conditions close to those of the aquifers.

Chapter 3 Experimental methods

In this chapter, we describe the experimental apparatus and methodology developed within the framework of this thesis. After the description of the materials used to build the set-up, we will explain the overall functioning of the shadowgraph method. Finally, we will present the experimental protocols as well as the experimental plan.

Materials

The experimental set-up is composed of four main parts: high-pressure filling and pressure control of the cell, temperature control, sample confinement and optical system. They are all described in the following sections and presented in the schematics in figure 3.1 and 3.3.

The experimental cell

The experimental cell is presented in the right part of figure 3.1 and can be described as follows. A polytetrafluoroethylene (PTFE) ring (yellow) is inserted inside a stainlesssteel block (purple) that is confined between two sapphire windows (blue-green). This allows containing a cylindrical sample of 5 mm height and 20 mm diameter. In the PTFE and stainless-steel rings, there are two entries. One is oriented upwards and the second downwards. They are connected to the high-pressure hydraulic circuit to inject CO 2 and water or brine inside the cell.

PTFE being highly hydrophobic, the contact angle between the solid and the liquid layer is typically larger than 90 • . It can induce a concave interface and deforming the image. To minimize this spurious effect, we added a thin hydrophilic plastic ring with

Hydraulic circuit and pressure control

As shown in figure 3.3, the hydraulic circuit is divided in two parts. One is related to the CO 2 injection and the other one to the brine injection inside the cell.

To inject CO 2 , we use a combination of two Teledyne ISCO pumps. The model is the 500D high-pressure syringe pump with a capacity of 507.38 mL, and a flow rate ranged from 0.001 to 204 mL • min -1 with an accuracy of 0.5%. The working pressure range is comprised between 0.07 and 25.8 MPa and the pump can operate between 278.15 and 313.15 K. With one pump, we can control precisely the pressure and perform an injection of CO 2 at constant pressure. The second pump is necessary to work at pressure larger than the inlet bottle. Indeed, the pressure delivered by the bottle of CO 2 is around 6 MPa. Thus, we need the second pump to do a pre-compression and obtain a pressure larger than the bottle pressure. More details are given in the experimental protocols section.

A pressure sensor is connected to the inlet of the cell in order to monitor the pressure. It is a pressure sensor from the series 33X of Keller working at temperatures between -10 to 80 • C and pressure up to 100 MPa. The transmitter integrates a thermocouple on the membrane to compensate pressure with temperature. It is connected to the converter K-114 that is plugged to the computer via USB connection. It has a maximum deviation of ± 0.05% in the temperature range 283-313 K and the full pressure range.

Two micrometric valves (V2.2 and V3.2.3 in figure 3.3) are added to the hydraulic circuit to control the injection of CO 2 in the cell and its discharge. This lets us adjust the aperture of the valve to have a smooth but rapid injection of CO 2 and have a good repeatability of CO 2 injection.

The injection of water or brine is performed with a syringe pump. We use the Fusion 4000 syringe pump from Chemyx. It works at temperature in the range from 4 to 40 • C and can deliver a flow rate between 0.0001 to 170.5 ml • min -1 in either infuse or withdraw mode. It has a good accuracy and reproducibility. It allows us to inject pure water or brine at a small flow rate. This is especially necessary for viscous brines.

Temperature control

The sample cell is thermally controlled at the top and bottom with the help of two independent Peltier elements controlled by two independent PID thermoelectric temperature controllers with autotune (LFI-3751 from Wavelength Electronics) providing a driving current of 5 A and 40 W. The Peltier elements have a precision of 0.5 mK. As shown in the right part of figure 3.1, two aluminium plates are inserted between the sapphire windows and the Peltier elements. Each aluminium plate hosts a thermocouple connected to the temperature controller to regulate the Peltier element. Both the Peltier elements and the aluminium plates have a 12.5 mm diameter circular hole in their centre to let the light beam pass. Finally, two additional thick aluminium plates, embedding a water circuit, clamp the cell. The blocks are connected to a ministat 240 thermal bath from Huber to remove the excess heat from the Peltier elements.

Beside these thermal control, we observed issues in the repeatability of the experiments. Indeed, we performed pressure decay experiments, closing the valve that connects the pump and the cell after the injection of CO 2 , and measuring the pressure decay. Looking at the figure 3.4, we observe two phenomenon hindering the repeatabil- ity. First, it is hard to have a leakage proof circuit considering the multiple connections and the intensive use of the apparatus. Second, there are thermal effects due to the daily variations of the temperature inside the room that are illustrated by the oscillations with a 24 hours period in the pressure evolution curve. The limited repeatability is also illustrated by the differences of pressure observed in figure 3.4 at a similar time. Indeed, for t = 2.0 × 10 5 s, the pressure reached varies from 2.3 to 2.9 MPa while the pressure of injection was similar. This can be explained partly because of the temperature of the room at the beginning of the experiment.

To minimize the influence of external perturbations, three thermal layers are added to the outside of the cell; see figure 3.5. The first layer is in two parts and is placed in contact with the sapphire windows, the thin aluminium plates and the Peltier elements, both at the top and bottom of the block. The second layer covers both the first layers and the stainless steel annulus. The third layer confines in a box all the cell, including the thick aluminium plates. To measure the efficiency of the insulation layers, a thermocouple is placed in contact with the external part of the stainless steel annulus and gives us the external temperature of the cell. As shown in figure 3.6, the pressure decay shows a plateau after about 500 s for a ∆p ≈ 2 MPa when the insulation has been improved. Furthermore, the pressure reached at the end of the experiment ranges between 1.7 and 1.8 MPa, giving an uncertainty of 0.1 MPa instead of the previous 0.6 MPa we obtained without the insulation layer.

Optical table

The core of the apparatus, including the cell and the shadowgraph optical set-up, is mounted on a column placed on a vibration isolated workstation from Newport connected to a pneumatic circuit. It considerably reduces the vibrations from the external environment, especially those in the range 10-50 Hz that are the common vibrations of multi-floor buildings. Our system is particularly sensitive to vibration due to the high sensitivity of the shadowgraph on one side and the presence of a gas-liquid interface in the system. The inclination of the table is regulated precisely and monitored by a 2-axis calibrated inclination sensor module (Digi-Pas DWL-5000XY) with a precision of 0.001°. The horizontal inclination of the optical table is permanently kept below 0.1°d uring all our measurements.

Optical arrangement

The shadowgraph set-up consists of a parallel quasi-monochromatic beam, at a wavelength λ = (675 ± 13) nm, delivered by the combination of a super luminescent diode (SLD) and an achromatic doublet lens with a focal distance f = 200 mm. The beam passes through the sample cell in which refractive index variations occur due to density variations. As it will be explained in more details in section 3.2, the beam is scattered by the local variations of the refractive index and interferes with the direct transmitted beam in the near field. A scientific-complementary-metal-oxide semiconductor (sCMOS) placed at a fixed distance from the cell records intensity maps. The camera ORCA Flash4.0 V3 from Hamamatsu displays recording frequencies between 1 and 100 Hz in full resolution mode. The sensor consists of 2048×2048 pixels of size 6.5×6.5 µm². Two polarisers are used to control the intensity of the light beam and prevent saturation issues. The acquired images are stored as a 16-bit grey level images.

Horizontal and vertical configurations

The set-up has two configurations: a horizontal and a vertical one. In figure 3.7, we show two pictures of the experimental set-up in its two positions. The optical apparatus and the high-pressure cell are mounted on a column that can be flipped in order to change the configuration. In the "horizontal configuration", the column is placed horizontally and this is refered to "Hele-Shaw-like configuration", even though the thickness of the cell is much larger than in a classical Hele-Shaw cell. In this configuration, the shadowgraph The second one is the "vertical configuration" in which the column is placed vertically. The observation is now made from above. We no longer observe the vertical spreading of the patterns, but their distribution in the two-dimensional horizontal plane. The combination of these two configurations provides full access to the threedimensional representation of the gravitational instability; see figure 3.8 (b).

Acquisition of housekeeping data and images

Images, pressure, temperature, and inclination data are recorded on a desktop computer. For the recording of images, we use the software HCImage Live. It has two main features: the "Timelapse" mode and the "High speed streaming" mode. The "Timelapse" mode is used to record images at a desired time delay that we can modify during the recording. Thus, we record at a short time delay of 0.05 s until the convective patterns have appeared and then we reduce the time delay to 0.5 s or even 2 s to have a recording all along the convection process. The "High Speed Streaming" mode is used to record images at the maximum frame rate of the camera that is 100 Hz for full-frame images.

The shadowgraph method

Light scattering techniques are used in many experimental investigations in the literature to study refractive index fluctuations in a transparent medium [START_REF] Brogioli | Giant fluctuations in diffusion processes[END_REF][START_REF] Croccolo | A light scattering study of non equilibrium fluctuations in liquid mixtures to measure the Soret and mass diffusion coefficient[END_REF][START_REF] Segre | Light-scattering measurement of nonequilibium fluctation in a liquid mixture[END_REF]. Considering a parallel quasi-monochromatic beam passing through a transparent medium, the local variations of the refractive index scatter a part of the beam while the main part of the beam is transmitted. In near field scattering experiments, transmitted and scattered beams interfere causing local variations of the intensity. In shadowgraph experiments, the latter are magnified. A simple explanation of the shadowgraph as an imaging technique can be provided by considering that the local maxima and minima of the refractive index modulations within the sample act as tiny lenses, making the light converge or diverge. This modifies the intensity of the light at the detector plane and can explain how density variations within the fluid are transformed into measurable light intensity variations on the camera.

To get a quantitative description of the working principle of the shadowgraph and get its transfer function T (q), i.e. the equation describing its sensitivity as a function of the wave vector q (inverse of the wavelength), one should take into account the diffraction of light generated by refractive index fluctuations in the plane perpendicular to the optical beam path. Spatial fluctuations δn(x, y) can be decomposed by 2D-Fourier transform into periodical modulations that act like gratings. Each grating coherently diffracts light that, at first approximation, propagates at an angle +θ and -θ, where θ is the angle of the scattered light. The further interference of these two beams with the much more intense unscattered beam provides intensity fluctuations at the sensor plane with a wavelength identical to that of the Fourier component. This is commonly referred to as Talbot effect [START_REF] Talbot | Facts relating to optical science[END_REF]. The interference depends on the propagating distance between the sample and the detector z, on the incident light wavelength, and on the wave vector of the Fourier component. Mathematical development based on physical optics provides the transfer function of the shadowgraph as T (q) = 4sin 2 ((q 2 z)/(2k 0 )) [START_REF] Croccolo | Quantitative fourier analysis of schlieren masks: the transition from shadowgraph to schlieren[END_REF][START_REF] Trainoff | Physical optics treatment of the shadowgraph[END_REF].

Experimental protocol

In this section, we describe the experimental protocol we implemented and used during all our experimental campaigns. The protocols covert the cleaning of the cell, its assembling, the injection of brine and CO 2 , the setting of experimental parameters, and the recording of the images.

Cleaning protocol

For purity reasons, the entire hydraulic circuit must be cleaned between each measurement series. First, the injection part of CO 2 is emptied and flushed several times with pure CO 2 before it is used. In a similar fashion, the injection system for brine is completely disassembled, cleaned with pure water and technical ethanol at 95 % purity, and, then, rinsed with the solution of interest.

In the same way, the cell is disassembled to clean each part separately. Before reassembling the cell, to remove all dust particles from the sapphire surfaces, we use a Typhoon blast bottle containing compressed tetrafluoropropene from Chemtronics.

Assembling protocol

After cleaning, the cell is re-assembled. To do so, we must assemble every elements together and ensure a homogeneous thermal conductivity between aluminium plates and sapphire windows. We use the Thermal Grizzly Hydronaut thermally conductive paste that is silicon-free and has a good thermal resistance of 7.6 mK/W. In detail, first, we spread a thin homogeneous layer of the thermal paste on the thick aluminium plate and stick the Peltier element on it. We repeat the same step spreading the thermal paste on the Peltier element and sticking the thin aluminium plate this time. Finally, we spread thermal paste on the thin aluminium plate and stick the sapphire windows on it. The procedure is repeated for both the top and bottom parts of the cell.

Then, we confine the stainless steel annulus containing the PTFE ring between the two sides of the cell, and tight all the elements with six screws. The screws are tighten with a wrench torque at a couple of 20 N•m. At this point, we can add the two first insulation layers, and fix the cell to the column and the rest of the hydraulic circuit. Then, we need to plug the Peltier elements, the thermocouples, and the primary and the secondary hydraulic circuits. Subsequently, we need to adjust the inclination of the cell and to measure its relative inclination with respect to that of the table. Indeed, during the experiment, we will measure the inclination of the table rather than that of the cell. To do so, we set the inclination of the cell to that of the optical table. Therefore, we adjust the horizontality of the table to be better than 0.1 • . Finally, we fix the third insulation layer and put the cell in contact with the isolated optical pathway to avoid external perturbation such as mean light intensity changes in the room or dust passing through the optical pathway.

Injection of fluids

For our experimental protocol, we chose to start each experiment from an equilibrium state between a CO 2 -rich gaseous phase and a brine-rich liquid phase. Here, the purpose is to reduce the influence of further gases contamination. CO 2 is then injected in the gaseous phase to get a pressure increase and let the experiment start. After few seconds, both the pressures of the ISCO pump and the cell equalize.

Before injecting brine, we do vacuum inside the circuit to remove the air and the potential remaining fluids. Then, we inject a small amount of CO 2 with an opened outlet to let it go to the atmospheric pressure and obtain a 0.1 MPa atmosphere inside the cell. Then, we inject brine with the syringe pump at a flow rate of 1 mL • min -1 . The small flow rate is necessary to avoid that the viscous brine creates bubbles larger than the size of the cell. Once we have injected enough brine, an almost flat interface is forming and we are able to observe the variations of the refractive index inside the sample.

From this moment, we wait the diffusive time to let the two phases go to equilibrium. Once the equilibrium is reached, we can start the experiment by injecting CO 2 and recording shadowgraph images.

Experimental plan

Our main goal is to study the influence of multiple physico-chemical and thermodynamic parameters on the convective dissolution of CO 2 in water or brine. We have first performed experiments in pure water and then in brine at multiple concentrations and multiple types of brine. Comparing the results between these experiments will highlight the effect of the presence of salts inside the liquid phase.

To vary the solutal Rayleigh number that characterizes the instability, we vary the density difference between the boundary layer and the bulk by varying the initial and final pressures of the system. In pure water, the initial pressure were 0.1 MPa and the final pressure varies from 0.35 to 6.1 MPa, giving pressure differences ranging between 0.25 and 6 MPa. For brine experiments, we performed successive injections at a constant pressure difference from 0.1 to 9.1 MPa. The pressure differences varied between 1 and 6 MPa.

Each experiment has been repeated 5 times for the experiments with pure water, and 3 times with brine in order to have a statistical average of the experimental results.

53

Chapter 4

Convective dissolution of CO 2 in brine

From primary quantitative remarks made on the raw images obtained with the shadowgraph apparatus, we build new analysing methods to investigate both temporally and spatially the convective dissolution of CO 2 in brine. In this chapter, after precisely describing these methods, we show the obtained results, first for the convective dissolution of CO 2 in pure water, and, then, in brine. Further theoretical calculations needed to properly discuss the results are also presented. Finally, the complete temporal and spatial behaviour of the convective dissolution of CO 2 in brine is described, and modelled using dimensionless equations for upscaling purposes.

Analysis methods

Primary qualitative remarks

Each experiment provides access to the evolution of the local variations of the refractive index in the sample cell. The images provide qualitative information on the convective dissolution. In fact, from the shadowgraph images we can see that, after the rapid pressure jump due to the turbulent injection, patterns appear in the gaseous phase. These patterns persist longer if the pressure jump is larger. For low pressure differences, these patterns vanish quickly and the shadowgraph images become stable, i.e. there is no variation from an image to the successive one, except for noise disturbances arising from, e.g., the non-equilibrium fluctuations.

Next, some localized darker regions appear within the images corresponding to denser fluid moving downward. At the beginning, an isolated line appears. Then, additional interconnected lines become visible together with round patterns all along these lines. These patterns are moving, sometimes combining, sometimes emerging from a pattern-free area, especially for the round patterns. After few minutes, the patterns become less and less visible until they completely fade away.

The analysis of the different series of images allows us to see the temporal and spatial evolution of the convective patterns. We can thus notice that the convection does not appear at the same time, depending on the pressure difference imposed at the beginning of the experiments. In a similar way, the size of the patterns depends on the imposed pressure difference. We also notice that the spatial density of the patterns is higher for a larger pressure difference, and the patterns are wider.

Although these qualitative remarks are interesting as preliminary results, we cannot extract quantitative information from the raw shadowgraph images. Therefore, we need to develop further analysis method, that we show in the following paragraphs, both from the temporal and the spatial points of view.

Temporal analysis

As mentioned in the previous paragraph, the patterns tend to vanish with time, which results in a decrease of the contrast of the images. Thus, we decided to compute the variance of image difference to see its evolution with time, and verify if we can extract quantitative information from it.

First, we normalize the intensity of each image by its intensity mean value i(x, y, t) = I(x, y, t)/I(t), where I(t) is the mean spatial value of the images at the time t, and I(x, y, t) is the intensity of the image. Then, we define a reference image which is taken as the mean value of the five first images of the series, before the injection of CO 2 . The latter provides a reference intensity I ref (x, y), and the normalized reference intensity is i ref (x, y). To highlight the convective patterns, we study the difference between each normalized intensity and the reference normalized intensity δi(x, y, t) = i(x, y, t) -i ref (x, y).

(4.1)

Thereafter, we compute the variance the intensity resulting from the difference of normalized images (< δi 2 >). In some of the experiments, we noticed the presence of particles, like dust, inside the sample cell. Due to the advective motion induced by the dissolution of CO 2 in brine, or by the injection of CO 2 in the bulk at early times, these particles are moving in the optical pathway and can slightly perturb the analysis of the variance. To avoid this problem, we implemented a filter before the calculation of the variance. First, we compute the Fourier transform of δi and keep only the dimensionless wave vectors in the range 7-57. This range was empirically defined and optimizes the filter efficiency. Then, we compute the inverse Fourier transform of the filtered Fourier transform, to obtain a new filtered intensity, δi filt . Finally, the variance is computed as

var(t) =< δi 2 filt > . (4.2)
A typical variance curve, obtained with the method described above, is plotted in figure 4.1. At the bottom, carefully chosen shadowgraph images are shown to illustrate the various stages of evolution of the variance curve. For all the temporal analyses, we assume that the injection takes place at t = 0. We observe first the turbulent injection (a), before a period of constant variance (b). Then, there is a rapid growth of the variance curve that corresponds to the apparition of the first convective pattern (c). The images become more and more contrasted and further patterns appear (d), until they have developed in the entire bulk (e), corresponding to a maximum of the variance. Next, the contrast of the patterns decreases and the variance reaches a plateau (f). Finally, they lose contrast again and the variance slowly decreases (g) until it reaches its initial baseline value (h).

From these observation, we are able to extract quantitative temporal values that provide important insights on the behaviour of the convective dissolution process. In the graph abscissa, we have highlighted four timescales that will help to characterise the instability: the onset time of convection t c ; the time when the variance is at its maximum level t m ; a latency time t r after which the variance reaches a plateau; and, finally, a shut-down time t sd that highlights the stop of convection. We also define two decay times τ 1 and τ 2 , corresponding to two exponential decays, one between t m and t r , and the second one between the end of the constant variance period and t sd .

Onset time of convection

At first, the injection is turbulent which can be seen in the image (a) of figure 4.1. We see turbulent patterns in the gaseous phase due to the rapid increase of pressure. This is mirrored by large disturbances in the variance curve. After the gaseous phase reaches a macroscopic equilibrium, the shadowgraph images are stable, and the variance is constant (b). This defines a baseline value of the variance. Then, the first convective Figure 4.1: Temporal evolution of the variance of shadowgraph images during the dissolution of CO 2 in pure water, for a pressure increase from 0.1 to 2.1 MPa. Here, several characteristic times are highlighted: t c is the onset time of convection, or critical time, corresponding to a huge rise in the variance curve; t m when the variance reaches its maximum value; it is followed by a first exponential decay that ends at t r ; the variance reaches a plateau ; a second exponential decay lasts until t sd after which the variance goes back to its baseline value.

pattern appears at the onset time of convection t = t c , corresponding to a rise in the variance curve. Indeed, in the middle of image (c), we see a dark line appearing that gets more contrasted in the image (d), and causes a huge rise in the variance curve. This is associated with the time at which the Rayleigh number exceeds its critical value. As we will explain in further details in the section 4.4.1, the dissolved CO 2 spreads diffusively inside the liquid phase in a boundary layer. As it grows, the solutal Rayleigh number increases until it reaches its critical value. After that, the boundary layer starts to destabilize and forms convective patterns.

Maximum of the image variance

After the appearance of convection in the bulk, the first convective patterns get more contrasted, which is observable when we compare the images (c) and (d) of figure 4.1. This is also combined with the emergence of new convective patterns all over the sample cell, and causes a rise of the variance up to its maximum (e) at t = t m . At this moment, the convective scheme is complete and fully developed over the entire bulk. The evolution of this time provides information on the dissolution rate efficiency. Indeed, as all the patterns have appeared, the dissolution rate is maximum. Furthermore, we observe the beginning of the formation of Voronoï-like structures around the patterns (e). This witnesses the fact that plumes spread around the downward fingers. At this moment, the CO 2 -rich plumes have reached the bottom of the cell and start to spread the dissolved CO 2 at the cell bottom. However, the Voronoï-like structure is not completely formed at this moment and the CO 2 has not mixed with the entire bulk. As a consequence, the concentration difference at the interface, ∆c = c eq -c 0 is still maximum, like the density difference ∆ρ. Thus, the dissolution rate is maximum around t m .

Characteristic decay times of the image variance

After t = t m , the variance starts to decrease exponentially with a characteristic decay time τ 1 . Comparing the images (e) and (f), we see that there are less tubular patterns. In both images, we do not see the Voronoï-like structure anymore, showing that the CO 2 has started to mix in the entire bulk. Thus, the density difference has decreased and the driving force of convection is less strong. This explains why the patterns become less contrasted between (e) and (f) and leads to a decrease in the variance curve until t = t r . Then, the variance curve reaches a plateau, and we observe a period of constant variance (f). There are some fluctuations around the plateau, because the plumes are moving inside the bulk. Thus, they could randomly merge creating pattern-free areas. Also, new plumes can randomly emerge on these areas. These effects generate fluctuations on the variance curve.

After the plateau, a second exponential decay starts with a characteristic decay time τ 2 . Here again, the decay is noisy because there are still the random merging and emerging of convective patterns as explained in the precedent paragraph. During this period, the patterns lose contrast (g), as long as they vanish diffusively. The convective downward fingers are still there, so that convection is still present, but diffusion starts to become more efficient. We can say this regime is pseudo-convective.

The second characteristic decay time appears to be larger than the first one. To compare them, we use a fit with two exponential decay times. For one experiment, we normalized the variance by the maximum variance and set the initial time at t = t m .

Shut-down of convection

The second decrease of variance stops when it reaches a new plateau at t = t sd . At this moment, the variance goes back to its baseline level. The convection has shut-down completely and we are again in a pure diffusive regime. The shadowgraph image (h) is similar to the one observed in (b).

Spatial analysis

From the shadowgraph images, we can also perform spatial analysis to get quantitative information on the spatial behaviour of the instability. On the shadowgraph images we see that convective columns can be both interconnected or isolated. In figure 4.2, we present a schematic of the convective patterns. Indeed, when the instability sets-in, the interface starts to destabilize and it take both the form of downward plumes and bridges. When two downward plumes are connected with a bridge, the latter "charges" them with CO 2 and the efficiency of the dissolution is larger. In the figure, we highlighted two characteristic lengths that can be extracted: the plume width w and the mean distance between two plumes Λ. In the following sections, we explain how we measured them.

Plume width

In order to get the plume width, we use the ImageJ software. First, we set the scale of the images but measuring the pixel size of a diameter that corresponds to 12.5 mm. Then, we select a line over the pattern diameter, we plot the gray level over this line. The width of the plumes is defined by the distance between the two peaks on the gray level curve, as shown in the figure 4.3. Here, we observe that the patterns do not have all the same size across the image. Thus, we always perform several measurements to obtain, for each shadowgraph image, the width range of the plumes and its mean value. By repeating this operation on several shadowgraph images all over an entire experiment and for multiple solutal Rayleigh number, we obtain the temporal evolution of the width of the plumes and its dependence with the Ra s .

Wavelength and wave number of the plumes

To get the mean distance between two plumes Λ, which can be seen as a kind of wavelength, one could measure, for each plume in the image, the distance between one plume and its closest neighbours. This is an enormous work that would need too much effort. Instead of doing this, we can measure another number that will give a similar information: the spatial density of the plumes. This number is correlated with 1/Λ 2 .

Spatial density of the plumes

As explained above, the patterns have the shape of plumes connected by tubes. As we will see in further details in the following chapter, the plumes are the patterns that bring the CO 2 downward. Thus, it is interesting to count the number of plumes and to see how this quantity evolves during an experiment. The top of the plumes correspond to the nodes interconnected by the tubes and those that are isolated. To evaluate the spatial density of the plumes in one shadowgraph image, we must count them. In figure 4.4, we display one shadowgraph image on which we placed green circles to highlight the localisation of the plume and to ease their counting. Finally, we count the plumes for the shadowgraph images over one entire experiment to get its evolution with time.

Finger speed

From the images obtained in the experiments performed with the set-up in its horizontal orientation, we see the plumes evolving inside the bulk, and can easily measure their speed. In images as those reported in figure 4.5, we observe the fingers moving downward. We measure the distance of the bottom of the fingers to the interface. Then, knowing the time of each images, we are able to provide the evolution of the speed of the plumes. By measuring at which pixel is the bottom of the fingering, we can get its position, and, knowing the time, its speed.

Thermophysical properties of interest

Thermodynamic properties have a strong impacts on the convective dissolution. As it has been presented in chapter 2, the instability is characterised by the solutal Rayleigh number that depends on the Fick diffusion coefficient D, the density difference between brine and CO 2 -saturated brine ∆ρ, and the dynamic viscosity of the saturated brine η. The calculation of these properties is based on the knowledge of the solubilities in form of the molar fraction, x CO 2 , or by its molality, m CO 2 . We need these values at any temperature, pressure, and brine composition taken in our experimental ranges. This can be obtained by using precise vapour-liquid equilibrium (VLE) data.

In the literature, there are many experimental data and some models available for the calculation of these properties. We also use the software PhreeqC (version 3) [START_REF] Parkhurst | Description of input and examples for PhreeqC version 3-a computer program for speciation, batch-reaction, onedimensional transport, and inverse geochemical calculations[END_REF], that directly integrates the equation of state necessary for the calculations of these thermodynamic properties. In this section, we present the models and the data available in the literature. Finally, we extract simplified models or constant values that we will use in the calculations of the solutal Rayleigh numbers.

Solubilities

The solubility of CO 2 in aqueous solutions has been widely investigated both in pure water [START_REF] Chapoy | Measurement and modeling of gas solubility and literature review of the properties for the carbon dioxide-water system[END_REF][START_REF] Chapoy | On the phase behaviour of the (carbon dioxide + water) systems at low temperatures: Experimental and modelling[END_REF][START_REF] Diamond | Solubility of CO 2 in water from -1.5 to 100 • C and from 0.1 to 100 MPa: Evaluation of literature data and thermodynamic modelling[END_REF][START_REF] Duan | An improved model calculating co 2 solubility in pure water and aqueous NaCl solutions from 273 to 533 K and from 0 to 2000 bar[END_REF][START_REF] Ibrahim | Modeling CO 2 -water mixture thermodynamics using various equations of state (EoSs) with emphasis on the potential of the SPUNG EoS[END_REF][START_REF] Ji | Saft1-rpm approximation extended to phase equilibria and densities of CO 2 -H 2 O and CO 2 -H 2 O-NaCl systems[END_REF][START_REF] Yuanhui | Progress in the study on the phase equilibria of the CO 2 -H 2 O and CO 2 -H 2 O-NaCl systems[END_REF][START_REF] King | The mutual solubilities of water with supercritical and liquid carbon dioxides[END_REF][START_REF] Spycher | CO 2 -H 2 O mixtures in the geological sequestration of CO 2 . I. Assessment and calculation of mutual solubilities from 12 to 100 • C and up to 600 bar[END_REF][START_REF] Valtz | Vapour-liquid equilibria in the carbon dioxide-water system, measurement and modelling from 278.2 to 318.2[END_REF][START_REF] Wiebe | The solubility in water of carbon dioxide at 50, 75 and 100 • C, at pressures to 700 atmospheres[END_REF][START_REF] Wiebe | The solubility of carbon dioxide in water at various temperatures from 12 to 40 • C and at pressures up to 500 atmospheres. critical phenomena[END_REF][START_REF] Wiebe | Vapor phase composition of carbon dioxide-water mixtures at various temperatures and at pressures up to 700 atmospheres[END_REF] and in brine [START_REF] Nikolay | Thermodynamic model of aqueous CO 2 -H 2 O-NaCl solutions from -22 to 100 • C and from 0.1 to 100 MPa[END_REF][START_REF] Bando | Solubility of CO 2 in aqueous solutions of NaCl at (30 to 60) • C and (10 to 20) MPa[END_REF][START_REF] Chabab | Thermodynamic study of the CO 2 -H 2 O-NaCl system: Measurements of CO 2 solubility and modeling of phase equilibria using soreide and whitson, electrolyte CPA and SIT models[END_REF][START_REF] Dubacq | An activity model for phase equilibria in the H 2 O-CO 2 -NaCl system[END_REF][START_REF] Dubessy | Modelling of liquid-vapour equilibria in the H 2 O-CO 2 -NaCl and H 2 O-H 2 S-NaCl Systems to 270[END_REF][START_REF] Ji | Saft1-rpm approximation extended to phase equilibria and densities of CO 2 -H 2 O and CO 2 -H 2 O-NaCl systems[END_REF][START_REF] Yuanhui | Progress in the study on the phase equilibria of the CO 2 -H 2 O and CO 2 -H 2 O-NaCl systems[END_REF][START_REF] Kiepe | Experimental determination and prediction of gas solubility data for CO 2 + H 2 O mixtures containing NaCl or KCl at temperatures between 313 and 393 K and pressures up to 10 MPa[END_REF][START_REF] Rumpf | Solubility of carbon dioxide in aqueous solutions of sodium chloride: Experimental results and correlation[END_REF][START_REF] Spycher | A phase-partitioning model for CO 2 -brine mixtures at elevated temperatures and pressures: Application to CO 2enhanced geothermal systems[END_REF][START_REF] Steele-Macinnis | Fluid inclusions in the system H 2 O-NaCl-CO 2 : an algorithm to determine composition, density and isochore[END_REF][START_REF] Zhao | Phase Equilibria in CO 2 -brine system for CO 2 storage[END_REF]. The solubility, m CO 2 , is defined as the amount of CO 2 dissolved at equilibrium in the liquid phase, depending on temperature, pressure, and the concentration of salt in the solution. It can be expressed in molality, m CO 2 in mol • kg -1 , or in mass fraction, x CO 2 . This value will be used later for the viscosity and density models. Thus, it is necessary to get the solubility at the temperature of interest T = 308.15 K, over the pressure and salt concentration ranges used in our experimental campaigns.

Here, we developed simplified equations for the calculations of the solubility of CO 2 in NaCl solutions at T = 308.15 K. To develop these models, we used experimental data from [START_REF] Wang | Determination of CO 2 solubility in water and NaCl solutions under geological sequestration conditions using a fused silica capillary cell with in situ raman spectroscopy[END_REF] [START_REF] Wang | Determination of CO 2 solubility in water and NaCl solutions under geological sequestration conditions using a fused silica capillary cell with in situ raman spectroscopy[END_REF] and data derived from the model given by [START_REF] Duan | An improved model calculating co 2 solubility in pure water and aqueous NaCl solutions from 273 to 533 K and from 0 to 2000 bar[END_REF] [START_REF] Duan | An improved model calculating co 2 solubility in pure water and aqueous NaCl solutions from 273 to 533 K and from 0 to 2000 bar[END_REF]. As the calculation of solubilities with this model is complicated and time consuming, we derived simplified polynomial models, calculating the solubility for each salt concentration in the pressure range of interest at T = 308.15 K. We derived equations of the solubility m CO 2 for molality of NaCl equals to 0, 1, and 3 mol • kg -1 .

i / (mol • kg -1 ) 0 1 3 a 0i / (mol • kg -1 ) 7.26542331 ×10 -3 -1.08154514 ×10 -2 -1.02425235 ×10 -2 a 1i / (mol • kg -1 • MPa -1 ) 2.71064228 ×10 -1 2.66566855 ×10 -1 1.90911914 ×10 -1 a 2i / (mol • kg -1 • MPa -2 ) -2.07343297 ×10 -3 -2.08510820 ×10 -2 -1.55036523 ×10 -2 a 3i / (mol • kg -1 • MPa -3 ) -2.36577951 ×10 -3 5.25850178 ×10 -4 4.15469500 ×10 -4 a 4i / (mol • kg -1 • MPa -4 ) 1.21197987 ×10 -4 0 0
The equations have the following form :

m CO 2 = a 0i + a 1i p + a 2i p 2 + a 3i p 3 + a 4i p 4 (4.3)
where i is in the range 0-3 and corresponds to the salt concentration of the brine in mol • kg -1 , and p the pressure. All the coefficients a ii are resumed in table 4.1. To obtain the molar fraction of CO 2 in the liquid phase x CO 2 , we need to use the following equation

x CO 2 = m CO 2 M b 1 -m CO 2 (M CO 2 + M b ) (4.4)
where M CO 2 is the molar mass of CO 2 and is equal to 44.01 g • mol -1 and M b is the molar mass of brine that depends on the brine molality m b .

In figure 4.6, we present the extracted models (solid line) and compare them with the experimental data (triangle, diamonds, and squares), and the one obtained with PhreeqC. These results are given for three molality in NaCl: 0 mol • kg -1 (blue), 1 mol • kg -1 (orange), and 3 mol • kg -1 (yellow). The model is in very good agreement with the experimental data and the data given by PhreeqC at m CO 2 = 0 mol • kg -1 . For the two other molalities, the results present a little shift with respect to the data given by PhreeqC that will need to be taken into account in the error measurement. The main aspect of this graph is the fact that the solubility of CO 2 in brine increases almost linearly with the pressure up to 4 MPa, after which it saturates to an almost constant value.

Dynamic viscosities

We report few experimental data for the viscosity of the mixture CO 2 -H 2 O-NaCl. According to the data given by Kestin et al. (1981) [START_REF] Joseph Kestin | Tables of the dynamic and kinematic viscosity of aqueous NaCl solutions in the temperature range 20-150 • C and the pressure range 0[END_REF], the viscosity does not change so much with pressure, that is explained by the small quantity of CO 2 that is dissolved into the liquid phase, even at a high-pressure equilibrium. However, there is a strong dependency of the viscosity on the brine concentration, as it goes from 0.7192 mPa • s for m NaCl = 0 mol • kg -1 , to 1.4104 mPa • s for m NaCl = 6 mol • kg -1 , at T = 308.15 K and p = 0.1 MPa.

From these experimental data, we extract three linear models depending only on the pressure for the three brine concentration of interest m NaCl = (0, 1, and 3) mol • kg -1 . In figure 4.7, the viscosity models are plotted with their equation and the corresponding experimental data points from Kestin et al. (1981) [START_REF] Joseph Kestin | Tables of the dynamic and kinematic viscosity of aqueous NaCl solutions in the temperature range 20-150 • C and the pressure range 0[END_REF]. These experimental data are given with an uncertainty of 0.5% .

Densities

For pure water, we use the data from [START_REF] Hebach | Density of water + carbon dioxide at elevated pressures: Measurements and correlation[END_REF] [START_REF] Hebach | Density of water + carbon dioxide at elevated pressures: Measurements and correlation[END_REF] to extract the values of density for multiple pressures at T = 308.15 K. We complete the extracted data In brine, the quantity of CO 2 dissolved at equilibrium for a given pressure is lower than in pure water. At the same time, the brine density increases with increasing NaCl concentration. This induces variations in the density differences between the initial and final equilibrium. Thus, it is necessary to develop further models for the density of the mixture CO 2 -H 2 O-NaCl, as we did for the solubility. In figure 4.9, we plotted the density of brine, computed with PhreeqC at T = 308.15 K in the pressure range 0-18 MPa for three salt concentrations. From density data we computed with PhreeqC, we derive simplified density models for each salted concentration in order to easily integrate them in the systematic computation of the Rayleigh number. They are given by Figure 4.9: Density of CO 2 -saturated brine at T = 308.15 K in terms of pressure for three molalities of salt: 0 mol • kg -1 (yellow circles), 1 mol • kg -1 (blue triangles), and 3 mol • kg -1 (orange diamonds). These data are all computed with PhreeqC.

ρ(p) = b 0i + b 1i p + b 2i p 2 .
(4.5)

The coefficients b ji are given in table 4.2 Table 4.2: Coefficients for the solubility model described in equation (4.5).

i / (mol.kg -1 ) 0 1 3 b 0i / (mol.kg -1 ) 9.94303592×10 2 1.03271026

×10 3 1.1010904 ×10 3 b 1i / (mol.kg -1 • MPa -1 ) 2.41112224 1.77544333 1.00284022 b 2i / (mol.kg -1 • MPa -2 ) -7.95348998 ×10 -2 -5.34698386 ×10 -2 -2.41366707 ×10 -2

Fick diffusion coefficients

Several studies available in the literature provide measurements of the diffusion coefficient of CO 2 in both pure water and brine. Many of them are based on the pressure decay method [START_REF] Azin | Measurement and modeling of CO 2 diffusion coefficient in saline aquifer at reservoir conditions[END_REF][START_REF] Farajzadeh | Enhanced mass transfer of CO 2 into water: Experiment and modeling[END_REF][START_REF] Mostafa | Measurement of CO 2 diffusivity in synthetic and saline aquifer solutions at reservoir conditions: the role of ion interactions[END_REF][START_REF] Li | Experimental determination of CO 2 diffusion coefficient in a brine-saturated core simulating reservoir condition[END_REF][START_REF] Renner | Measurement and correlation of diffusion coefficients for CO 2 and rich-gas applications[END_REF][START_REF] Shi | Measurement and modeling of CO 2 mass transfer in brine at reservoir conditions[END_REF][START_REF] Wang | The pressure-decay method for nature convection accelerated diffusion of CO 2 in oil and water under elevated pressures[END_REF][START_REF] Zarghami | Diffusion of carbon dioxide in formation water as a result of CO 2 enhanced oil recovery and CO 2 sequestration[END_REF][START_REF] Zhang | The modeling and experimental studies on the diffusion coefficient of CO 2 in saline water[END_REF] that has been introduced in section 2.3.1. It relies on using the diffusive equation both for the convection and the diffusive periods. Thus, this method provides two diffusion coefficients: one effective diffusion coefficient, and a classic molecular diffusion coefficient. Usually, there are one to two orders of magnitude between these two coefficients. However, the data obtained with this method show a lot of dispersion. Other methods are used such as Raman spectroscopy [START_REF] Belgodere | Experimental determination of CO 2 diffusion coefficient in aqueous solutions under pressure at room temperature via raman spectroscopy: Impact of salinity (NaCl)[END_REF][START_REF] Lu | Determination of diffusion coefficients of carbon dioxide in water between 268 and 473 K in a high-pressure capillary optical cell with in situ raman spectroscopic measurements[END_REF], Taylor dispersion methods [START_REF] Cadogan | Diffusion coefficients of CO 2 and N 2 in water at temperatures between 298[END_REF][START_REF] Frank | Diffusion coefficients and viscosities of CO 2 + h 2 O, CO 2 + CH 3 OH, NH 3 + H 2 O, and NH 3 + CH 3 OH liquid mixtures[END_REF], laser induced fluorescence [START_REF] Hirai | Measurement of CO 2 diffusion coefficient and application of LIF in pressurized water[END_REF], microfluidic method [START_REF] Sell | Measurement of CO 2 diffusicity for carbon sequestration: A microfluidic approach for reservoirspecific analysis[END_REF], and liquid-jet method [START_REF] Tamimi | Diffusion coefficients for hydrogen sulfide, carbon dioxide, and nitrous oxide in water over the temperature range 293-368 K[END_REF]. These latter methods provide more precise experimental data points and allow studying the dependence of the diffusion coefficient with several thermodynamic parameters.

While the diffusivity of a binary liquid mixture strongly depends on temperature, see figure 4.10, and on composition [START_REF] Cadogan | Diffusion coefficients of carbon dioxide in brines measured using 13C pulsed-field gradient nuclear magnetic resonance[END_REF], it varies slowly with pressure; see figure 4.11. Thus, we would need to determine the diffusion coefficient at temperature T = 308.15 K, over the experimental pressure range, and for each brine composition we use in this study: pure water and H 2 O-NaCl brine at molality m NaCl = 1 and 3 mol • kg -1 . However, the value of the diffusivity is often given at atmospheric pressure. At VLE, along the bubble line, it has been shown for various gases dissolved in liquids that D can be assumed constant over a pressure range associated with the infinite dilution regime [START_REF] Giraudet | Thermal, mutual, and self-diffusivities of binary liquid mixtures consisting of gases dissolved in n -alkanes at infinite dilution[END_REF][START_REF] Klein | Diffusivities in binary mixtures of [amim][ntf2] ionic liquids with the dissolved gases H 2 , He, N 2 , CO, CO 2 , or Kr close to infinite dilution[END_REF][START_REF] Klein | Influence of liquid structure on fickian diffusion in binary mixtures of n-hexane and carbon dioxide probed by dynamic light scattering, raman spectroscopy, and molecular dynamics simulations[END_REF][START_REF] Wu | Mutual and thermal diffusivities as well as fluid-phase equilibria of mixtures of 1-hexanol and carbon dioxide[END_REF][START_REF] Wu | Diffusivities in 1-alcohols containing dissolved H 2 , He, N 2 , CO, or CO 2 close to infinite dilution[END_REF]. In our pressure range 0.1-10 MPa, the previous works do not show significant dependence of the diffusion coefficient with pressure for the system [START_REF] Cadogan | Diffusion coefficients of CO 2 and N 2 in water at temperatures between 298[END_REF][START_REF] Hirai | Measurement of CO 2 diffusion coefficient and application of LIF in pressurized water[END_REF][START_REF] Lu | Determination of diffusion coefficients of carbon dioxide in water between 268 and 473 K in a high-pressure capillary optical cell with in situ raman spectroscopic measurements[END_REF][START_REF] Sell | Measurement of CO 2 diffusicity for carbon sequestration: A microfluidic approach for reservoirspecific analysis[END_REF], as we can see in figure 4.11. This results are in good agreement with the Wilke-Chang [START_REF] Wilke | Correlation of diffusion coefficients in dilute solutions[END_REF] and Mutoru [START_REF] Mutoru | Modeling infinite dilution and fickian diffusion coefficients of carbon dioxide in water[END_REF] models that give a constant value of the diffusion coefficient of CO 2 in the water-rich liquid phase over the range of CO 2 mass fraction and pressure.

As the quantities of CO 2 dissolved in the brine-rich liquid phase are very low (always less than 5% all over the pressure range used in our experiments), we will not consider the influence of the dissolved CO 2 on the diffusion coefficients. Finally, in this study, we will define a constant value of D for each brine composition in stake, at T = 308.15 K. The experimental data available in the literature for the diffusion coefficient at T = 308.15 K are comprised in the range of 2.18-2.49 × 10 -9 m 2 • s -1 , while the value given by the Arrhenius law at this temperature is D = 2.39 × 10 -9 m 2 • s -1 . For the Arrhenius-like fit, the data points at T smaller than 290 K were not considered. This is because D deviates from an Arrhenius-like behaviour at low temperatures. Since there was not enough data point at T smaller than 290 K, we could not perform a proper fit to the data using Vogel model [START_REF] Thomas | Liquid viscosity and surface tension of n-dodecane, n-octacosane, their mixtures, and a wax between 323 and 573 K by surface light scattering[END_REF]. This is further justified by the large increase in the experimental uncertainties on D from Lu et al. [START_REF] Lu | Determination of diffusion coefficients of carbon dioxide in water between 268 and 473 K in a high-pressure capillary optical cell with in situ raman spectroscopic measurements[END_REF] at temperatures smaller than 295 K.

For pure water, we use the mean value of the experimental data from the literature that is 2.26 × 10 -9 m 2 • s -1 . For the diffusion coefficient of CO 2 in brine, we are using

Computation of the Rayleigh number

The models above can be used to systematically compute the solutal Rayleigh number, Ra s = ∆ρgh 3 /ηD, for any experimental condition. Here, we present experiments at subcritical conditions, i.e. p < 7 MPa, and a constant temperature T = 308.15 K. The thermophysical properties, such as the viscosity and the diffusion coefficient, are taken at the final equilibrium. The density difference ∆ρ is defined as the difference between the density of the initial and final states. Thus, the Rayleigh number range we investigated is comprised between 3.88 × 10 4 and 5.57 × 10 5 for experiments with pure water and brine. Each calculation of the Rayleigh number is resumed in figures A.1, A.2, and A.3 of annex A.

According to the models and commentaries made on the calculation of the thermodynamic properties, the key parameters that strongly influence the value of the Rayleigh number are the density difference between the initial and final state ∆ρ and the brine height level h. In the experiments we performed, we worked at a constant h and changed only the pressure differences imposed to the liquids layer, and, thus, the density differences. Nonetheless, swelling effects were considered in the value of h at different states.

Estimation of the experimental uncertainties

The uncertainties are twofold. First, they come from the computation of the solutal Rayleigh number, induced by the error of the models we are using; second, they come from the uncertainties given by the experimental measurements. In this section we describe the measured uncertainties.

Experimental uncertainties

For every quantity extracted from the shadowgraph images, there are specific measurement uncertainties depending on the experimental conditions.

Uncertainties on the measurement of t c

All the uncertainties are resumed in figures B.1-B.3 in the annex B and are described here. They come both from the experimental protocol, and from the direct measurements of t c on the variance curve.

1) In the shadowgraph images, we cannot see the entire cell. Indeed, the optical clear aperture is equal to 12.5 mm, while the sample has an inner diameter of 20 mm. Thus, in the images, we cannot see the patterns that form in the blind region. In figure 4.12, we display a shadowgraph image showing the two areas of the optical pathway and the cell. To evaluate if it could induce an error on the onset time of convection, we must check where the first convective pattern appears. If the first pattern appears on the optical pathway, there is no error induced by this measurement. Its physical measurement expanded uncertainty corresponds to twice the sample time of the camera, i.e. ∆t c = 100 ms. Owing that the difference in intensity between non convective and convective areas can be small, ∆t c was expanded to 0.5 s. If the first pattern appears outside the optical pathway, there will be an offset in the measured t c value. Since the delay time between the apparition of two patterns in our images is smaller than the overestimated uncertainty, it barely influences t c . Furthermore, most of the first patterns appeared in the centre of our images.

2) The injection of CO 2 is not identical for every pressure difference. To illustrate this, we plotted, in figure 4.13, the pressure, normalized by the maximum pressure, as a function of time. The normalized pressure reaches its final value after a longer time for smaller pressure difference. Indeed, looking at the intersection between the dark dotted line, that is the 95% line, and the normalized pressure curves, we are able to see the time, t p=95%pmax , at which the pressure has reached 95% of its final value. The more the pressure difference, the faster the pressure reached 95% of its initial value. For pressure differences between 1 and 6 MPa, t p=95%pmax is almost constant equal to 5 seconds. When the pressure difference is shorter, t p=95%pmax is larger and reaches 12 seconds for a pressure difference of 0.25 MPa.

This challenge induces an offset on the measurement of t c . Indeed, between t = 0 s and t = t p=95%pmax , the hypothesis that the concentration at the interface is equal to Figure 4.12: Shadowgraph image on which we highlight the size of the optical pathway (red) and the size of the cell (green). The ratio between the two areas depends on the composition of the liquid phase, and the pressure of CO 2 . Figure 4.13: Normalized pressure evolution at early times for multiple pressure differences. For pressure differences ∆p = 6 and 1 MPa, the pressure has risen 95% of its final value after around 5 seconds. For shorter pressure differences, the rise in p is longer, around 7.5 seconds for ∆p = 0.5 MPa, and 11.5 seconds for ∆p = 0.35 MPa. the equilibrium concentration is not valid. The consequence is that t c,measured is smaller than t c,real by few seconds. We measured that during the pressure jump, the mean value of the pressure is equal to 60% of its final value. Thus, during the calculation of of t c , we add a positive error of 0.4 t p=95%pmax .

3) The injection of CO 2 inside the cell is turbulent and induces advective motion in the gaseous phase. This can last for up to 200 seconds, especially at high pressure (larger than 4 MPa). It can cause advective motion in the liquid phase and increase the dissolution rate. Looking at the shadowgraph images for a pressure difference of 6 MPa, in which this effect is the most important, we remark that the convective patterns in the liquid phase do not follow the same motion than the pattern in the gaseous phase. This effect can be neglected in our study. However, for ∆ρ larger than 6 MPa, we could not determine t c .

4) The advective motions due to the injection raise another challenge. As we said just above, there are remaining patterns in the gaseous phase due to the fast pressure increase inside the cell. These patterns make it impossible to use the variance curve to extract the onset time. Thus, for the largest pressure differences, ∆p = 4 and 6 MPa, the onset time of convection is extracted directly from the shadowgraph images. This induces a double uncertainty: one for the measurement by hand that could be less accurate than the measurement with the variance curve; and other for the presence of the patterns in the gaseous phase that makes this measurement difficult.

Uncertainties on the measurement of t m

The exact time at which the contrast is maximum can be easily extracted from the variance curve. Notably, some effects are disturbing the measurement of t m causing experimental uncertainties. In section 4.1.2, we explain that the maximum contrast is close to the moment when the plumes touch the bottom of the cell and, thus, join together. They actually form Voronoï-like structures that increases the contrast. Nevertheless, the patterns do not appear all at the same moment. Indeed, in some experiments, the first plume reaches the bottom of the cell before all the other patterns appeared. This can result in two distinct maxima on the variance curve, as shown in figure 4.14, which makes the measurement of t m undefined. This situation was especially encountered for some experiments performed in pure water and ∆p = (1 and 2) MPa.

In the experimental uncertainties evaluation of t m , we also need to take into account the noise created by the movements of the free surface inside the cell. Furthermore, the advective patterns in the gaseous phase can strongly perturb the measurement of t m , especially for large pressure differences of 4 and 6 MPa. The experimental uncertainties made on the measurement of t m are given in figures B.1-B.3.

Uncertainties on the measurement of the spatial density of plumes

The shape of the convective patterns makes it difficult to spot all the plumes. Although the isolated plumes are easy to recognize, it is much more difficult to distinguish the plumes from the tubes interconnecting them. Even though the width of the plumes is larger than that of the tubes, the difference remains very small. For instance, in figure 4.15, it is difficult to say how many plumes are present on the tube of the image (b). Furthermore, the contrast of the patterns can be very low, especially for small pressure differences. Thus, sometimes we have to guess where are the plumes which induced large uncertainties.

In some cases double plumes appear, as shown in figure 4. 15 (d). This can be caused by the merging of two patterns, or by the presence of a dust particle. In this case, shall we count the double plume of image (d) as two plumes or only one? Here, we still consider two distinctive patterns when the distance between their center is still larger than their radius.

Considering all these effects, we estimated the error made by defining minimum and maximum ranges of the number pattern over one entire series. It appears that the error made in the measurement of the spatial density of plumes is 10% at early times, and 20% after the constant-variance period ended.

Uncertainties on the measurement of the width of the patterns

For highly contrasted patterns, the measurement of the plume width is easy as the maxima in the gray level curve are very well defined; see figure 4.3. For the less contrasted plumes, the definition of the maxima is much less clear which induces huge uncertainties, as it is shown in figure 4.16. This difficulty is encountered especially at low pressure or during the very end of an experiment when the patterns are vanishing and lose contrast. The other big challenge with this measurement is that the patterns appear, merge, and vanish all the experiment long. Thus, in the shadowgraph images, the range of the width of the patterns is large. We try to give it as much precise as possible. Anyway, when presenting the results on the width of the patterns, we discussed only about the mean value of the patterns thus reducing the uncertainties.

Uncertainties in the computation of the Rayleigh number

The calculation of the Rayleigh number is capital for our discussions on the convective dissolution. Thus, it is necessary to know the accuracy of the computation of the Rayleigh number. The computational errors stem from the measurements of the thermodynamic values. In section 4.2.5, we defined the models used for the calculation of the Rayleigh number. These are always given with an uncertainty. They are all given in figures B.1-B.3.

Shape of the interface

From the experimental point of view, there are several sources of uncertainty. First, the height of the liquid layer in the cell is not known precisely. Indeed, from one experiment to another, the shape of the images is not always the same. Sometimes the optical pathway appears shorter on the shadowgraph images, as shown in figure 4.17. This is explained by the shape of the interface. The gas-liquid-solid surface tension creates a meniscus that can form a convex or concave interface. According to [START_REF] Li | Determination of swelling effect in CO 2 -brine systems using microfocus X-ray CT[END_REF] [START_REF] Li | Determination of swelling effect in CO 2 -brine systems using microfocus X-ray CT[END_REF], the contact angle depends on the pressure inside the cell, and increases with an increasing pressure. In figure 4.17, we display four images of the contact angle of CO 2 and brine in their experiment. Thus, the interface plays the role of a concave lens at low pressure, and a convex one at pressure higher than 6 MPa; see figure 4.17. Because of this effect, we consider that the initial liquid layer height, h, is comprised between 1.5 and 2.5 mm, which gives an expanded uncertainty of 25% .

Swelling effect

Another effect induced by the dissolution of CO 2 in water is the swelling effect induced by the adsorption of CO 2 in the liquid phase. The swelling induces an increase in the liquid layer thickness h when the pressure of CO 2 increases. [START_REF] Li | Determination of swelling effect in CO 2 -brine systems using microfocus X-ray CT[END_REF] [START_REF] Li | Determination of swelling effect in CO 2 -brine systems using microfocus X-ray CT[END_REF] performed experiments on this change of the water layer due to the swelling. From this study, we developed a model for a correction factor to be directly included in the computation of the solutal Rayleigh number. This factor, f , is given by the following equation, depending on the pressure of the final equilibrium of an experiment f = 4.7113 × 10 -4 × p eq + 1, (4.6) where p eq the pressure of the experiment at equilibrium. Thus, in the computation of the solutal Rayleigh number, we will add this factor f , and finally Ra s = f • ∆ρgh 3 /ηD.

Convective dissolution of CO 2 in pure water

Our experimental campaign started with a series of convective dissolution experiments of CO 2 in pure water for multiple pressures of injection. Starting always from an equilibrium at 0.1 MPa between a CO 2 -rich gaseous phase and a water-rich liquid phase, we injected CO 2 at 0.35, 0.6, 1.1, 2.1, 3.1, 4.1, or 6.1 MPa. We also performed experiments by injecting 3.6 MPa in a 3.1 MPa equilibrium. In this way, different pressure differences are imposed at the beginning of the experiment, fixing the density difference, and, thus, the solutal Rayleigh number. In this section, we present and discuss the results, to finally describe the entire process of convective dissolution of CO 2 in pure water.

Other experiments have been performed injecting CO 2 at 8, 10, and 12 MPa, i.e. at supercritical state. However, these cannot be used for this investigation. Indeed, after the injection, the CO 2 passes from a gaseous state to a supercritical state because the strength of the turbulent convection hinders our observations so much that these data cannot be used in our study. However, looking at the quantity of CO 2 dissolved at equilibrium for pressure higher than 6 MPa, we notice that the density difference does not vary a lot. From an initial state at 0.1 MPa, ∆ρ = 12.57 kg.m -3 at a thermodynamic equilibrium of 6 MPa, while ∆ρ = 17.07 kg.m -3 for a thermodynamic equilibrium of 12 MPa. Thus, the solutal Rayleigh number increases only by a factor 1.3, which is

Onset time of convection

As explained earlier in this manuscript, the onset time of convection is a crucial parameter defining the efficiency of the convective dissolution. Indeed, from this time on, the dissolution flux becomes much stronger than the pure diffusive one and the dissolution rate increases when t c decreases. Understanding which parameters influence the onset time of convection allows to choose proper injection sites and optimize the storage procedure.

From the performed experiments, we extracted the onset time of convection as explained in section 4.1.2. In figure 4.18, we plotted, in log-log, the obtained onset time t c in terms of the solutal Rayleigh number. From the experimental data points (the yellow squares), we notice a clear tendency between the onset time of convection and the solutal Rayleigh number. To extract a scaling law, we used Matlab and the application "cftool", which provides a fit. With a simple power law, we found the following relation:

t c = (1.06 ± 0.22) • 10 5 × Ra -(0.72±0.02) s .
(4.7)

The method used for the fit is the NonlinearLeastSquares and the parameters of the fit are written in table 4.4. Indeed, considering the experimental uncertainties, the power law matches very well the experimental data, and is kept in the 95% confidence area.

Onset time for various liquid layer thickness

We performed several experiments of convective dissolution, at constant pressure difference, varying the height of the liquid layer, even if we could not control it precisely.

We measured the onset time of convection for all the experimental conditions. Considering the experimental errors, the onset time of convection seems to be constant for all the values of the water layer, as can be seen in figure 4.19. Thus, the onset time of convection seems to be independent of the liquid layer thickness.

t m : maximum variance

A precise knowledge about the evolution of the time at which variance is maximum is somehow more important than the onset time of convection. Indeed, the selection of the best injection site depends on the dissolution rate, which is maximum at t m .

From the experiments, we derived the time of maximum of variance, as explained in section 4.1.2. In figure 4.20, we plot in log-log t m as a function of the solutal Rayleigh number and observe a tendency, from the experimental data points (yellow squares), similar to the one we found for t c . Using the same fitting model as for the onset time of convection, with the same parameters given in table 4.4, we found that t m = (7.57 ± 0.17) • 10 4 × Ra -(0.67±0.02) s .

(4.8)

Therefore, the time of maximum of contrast follows the same trend than the onset time of convection t c .

Characteristic decay times of convection

For all the experiments performed in pure water, we fit the normalized variance curves, using t m as the starting point. Thus, we obtained the two characteristic decay times for each experiment. We computed the mean value of τ 1 and τ 2 of the experiments having the same pressure difference. Then, we plotted those values in terms of the solutal Rayleigh number in figure 4.21.

We remarked that the first characteristic decay time τ 1 is smaller than the second one τ 2 . There is about one order of magnitude difference between the two. Furthermore, we do not detect any dependence of the characteristic decay times with the solutal Rayleigh number outside combined uncertainties.

Shutdown time of convection

We could not measure t sd for the experiments with a pressure difference lower than 1 MPa. However, we still have the trend of the shutdown time in the range of Ra s 100,000 -560,000. In figure 4.22, we see how the shutdown time depends on the solutal Rayleigh number. There are very large uncertainties on the shutdown time as it is difficult to define it precisely from the variance curve. However, we do not see a strong influence of the solutal Rayleigh number on the shutdown time of convection.

The shutdown time of convection is thus independent of the solutal Rayleigh number and the pressure difference. However, this number is strongly dependent on the height h of the water layer. Indeed, the larger the height of the liquid layer, the longer time the plumes touch the bottom of the cell.

Width of the plumes

Figure 4.23 depicts the temporal evolution of the width of the plumes for one experiment and an injection from 0.1 to 6.1 MPa. The plume width has been measured as explained in section 4.1.3. After the convection started and the apparition of the first pattern, there is an increase of the width until t m , when convection has developed over the entire bulk. After that time, the width of the pattern decreases until t r , and reaches a constant value. Then, it decreases again from the end of the plateau until t sd .

In figure 4.24, we display the mean value of the plumes width for several experiments. We measured it for two experiments of multiple pressure differences: ∆p = 1 MPa (circles), ∆p = 2 MPa (diamonds), ∆p = 4 MPa (triangles), and ∆p = 6 MPa (squares). The first thing we notice from this graph is that the plumes are larger for a higher pressure difference and, thus, a larger Ra s .

The behaviour described in the paragraph above for an injection from 0.1 to 6.1 MPa is not the same for all the pressure differences. For ∆p = 4 and 6 MPa, we have similar behaviour with a constant pattern width after t r , followed by a decrease until t sd . For the pressure differences of 1 and 2 MPa, the behaviour is different. The width of the patterns is more or less constant during all the experiment. However, it is difficult to describe a clear behaviour because the data points are very disperse and the repeatability is not good. As for the distance between plumes, their size is rather random. This can be explained by the continuous random emergence and growth of plumes. .25 shows the spatial density of the plumes for an experimental run with an injection from 0.1 to 6.1 MPa. From this graph, we can make several remarks. After t c , the first convective pattern appears with the first plumes. The plume density increases monotonically up to t m , and reaches a first local maximum. After a little drop, it increases again to reach a maximum around t r . Then, the spatial density of plumes decreases quasi exponentially again and reaches a plateau. Finally, it decreases and goes back to zero several seconds after t sd . To see the influence of the pressure difference imposed at the beginning of the experiment, we plot in figure 4.26 four temporal evolutions of the spatial density of plumes for four different pressure differences: 1 MPa (yellow circles), 2MPa (blue diamonds), 4 MPa (orange triangles), and 6 MPa (green squares). First, we notice that the evolution of the spatial density of plumes has the same trend for the four experiments presented here. Thus, the remarks made for the injection from 0.1 to 6.1 MPa are still valid for smaller ∆p and, thus, smaller Ra s . Furthermore, the second remark is that the spatial density of plumes is larger for larger solutal Rayleigh number.

Spatial density of the plumes

This reflects the better efficiency of the convective dissolution process for large Ra s and, thus, ∆ρ. This is in line with the importance of the use of s-CO 2 instead of gaseous CO 2 in CCS.

Convective dissolution of CO 2 in brine

We performed similar experiments and analysis of convective dissolution of CO 2 in brine. Here, the protocol has been slightly modified. We started always from an equilibrium between the CO 2 -rich gaseous phase and the brine-rich liquid phase at 0.1 MPa. Then, we performed successive injections of CO 2 with the same pressure difference. We repeated this protocol for multiple pressure differences from 1 MPa to 6 MPa. The brine used here is a mixture of NaCl and water at two molar concentrations: 1 mol • kg -1 and 3 mol • kg -1 . The experiments presented here were performed by myself and Aziza Nauruzbaeva during her master thesis under my supervision.

In this section, we present the quantities we extracted from the analysis, as we did in the section 4.4, for each brine composition separately. In figure 4.27, we display three variance curves for the three molalities studied and a similar pressure jump ∆p = 2 MPa. We observe that variance differs and, consequently, the extracted quantities differ as well. Thereafter, we will use dimensionless values to compare the results between all the experiments. Thus, we will be in the position of discussing on the influence of the salt, i.e., its type and composition, on the convective dissolution induced by the injection of CO 2 in brine.

Temporal analysis

Onset time of convection

As we did in section 4.4, we extracted the onset time of convection for all the experiments performed with brine. We fitted the experimental data with the same fit used for pure water. The onset time of convection is plotted in figures 4.28 and 4.29 for the two different concentrations. The scaling laws obtained from the fitting are Here, we remark that the uncertainties are much larger than the one obtained with pure water. This is mainly explained by the repeatability of the experiments that is of less good quality. For pure water we repeated 5 times each experiments while we could only do it three time each for the brine. Also, we notice that the experimental data are much more dispersed with brine than with pure water.

t c,1mol•kg -1 = (6.5 ± 12) × 10 4 • Ra -(0.65±0.26) s , ( 4

Maximum variance

Again, from the variance curves of the shadowgraph images, we extracted the times of maximum variance for both concentrations of the brine. The fit provides

t m,1mol•kg -1 = (9.0 ± 20) × 10 4 • Ra -(0.66±0.2) s (4.11) and t m,3mol•kg -1 = (2.9 ± 72) × 10 4 • Ra -(0.56±0.25) s (4.12)
Again, the uncertainties are much larger than the one obtained with pure water and we can have the same remarks than in the previous paragraph.

Shutdown time of convection

As we did with pure water, we measured the shutdown time of variance for the experiments performed with brine. It gives similar results than those obtained with water. Indeed, in figure 4.32, the shutdown times of convection is displayed as a function of the solutal Rayleigh number. We still do not observe a clear dependence of t sd with the solutal Rayleigh number. This confirms what has been said in the previous section for the shutdown time of convection in pure water, i.e. it does not depend on the pressure difference but is should strongly depend on the height of the liquid layer.

Spatial analysis

Width of the plumes

We measured the width of the plumes for several experiments performed in H 2 O-NaCl brine at two concentrations of 1 mol • kg -1 , and 3 mol • kg -1 . For a given concentration, we observe the same trend than for the experiments in pure water, i.e. for a larger pressure difference imposed at the beginning of the experiment, we obtain larger plumes. This is shown in figure 4. [START_REF] De Paoli | Rayleigh-Taylor convective dissolution in confined porous media[END_REF], where the width of the plumes correspond to the experiments performed at 3 mol • kg -1 is plotted.

An interesting result arises from the comparison between the width of the plumes for the three brine concentrations (0, 1, and 3 mol • kg -1 ) at a constant pressure difference. In figure 4.34, we show the temporal evolution of the plumes for the three brine concentrations and a pressure difference of 2 MPa. Although the solutal Rayleigh number decreases when the brine concentration increases, the width of the plumes increases with the brine concentration. This results is counter-intuitive because one can imagine that the width of the patterns depends on the solutal Rayleigh number as t c and t m . Indeed, the solutal Rayleigh number depends strongly on the density difference between the CO 2 high-concentrated plumes and the bulk. Thus, looking only at the experiments performed in pure water, we could draw the conclusion that a larger density difference induces wider plumes. Here, this is not the case anymore as we get a smaller density difference for a brine more concentrated in NaCl and a similar pressure difference imposed at the beginning of the experiment. Other thermophysico-chemical parameters must play a central role on the width of the plumes The parameters that change with the brine concentration are the density difference ∆ρ, the diffusion coefficient D, and the brine viscosity η. We see in the previous section that for a constant brine concentration, the density difference has an influence on the pattern. However, this influence is less strong than the influence of the diffusion coefficient and/or the brine viscosity. There is no reason to think that with a smaller diffusion coefficient we get larger convective patterns. Indeed, a smaller diffusion coefficient induces a slower mass transfer, what does not justify larger plumes. Thus, the parameter that could influence the width of the pattern is the brine viscosity. With a higher viscosity, the plumes will move downward more slowly. We see this because the formation of the Voronoï-like structure in brine takes much more time than for pure water. This also means that the plumes touch the bottom of the cell later. As the brine layer thickness is kept constant, we are able to say that the velocity of the downward plumes is smaller. Furthermore, they need more strength, or mass, to counterbalance the viscosity effects. Indeed, on the images, we see some patterns that try to form, implying that the solutal Rayleigh number exceeds its critical value. The boundary layer is destabilising, but the viscosity effects act against this destabilisation and, finally, there is no formation of new plumes. To form a plume, the destabilisation of the BL needs to be larger to overcome the viscosity effect and, thus, it forms larger patterns and larger plumes.

Spatial density of the plumes

In brine, the spatial density of the patterns follows a similar trend than the one described for the experiments performed in pure water. This is shown in figures 4.35 and 4.36. Furthermore, we can compare the evolution of the spatial density for the same pressure difference ∆p and the three brine concentrations. In figure 4.37, we see the evolu-tion of the spatial density for a pressure difference of 4 MPa and brine concentrations of 0, 1, and 3 mol • kg -1 . This comparison highlights that a higher brine concentration decreases the density difference between the CO 2 high-concentrated plumes and the bulk. The spatial density of the patterns follows the same trend in pure water and in brine, and depends only on Ra s . In the experiments we presented in this dissertation, the water layer was kept constant as much as possible with our experimental set-up. However, preliminary investigations have shown that h has no influence on the onset time of convection. This is in good agreement with the theoretical approach based on the diffusive growth of the boundary layer.

In our experiments, h is relatively small. It induces confinement effects and temporally limits the convective regime. Indeed, as soon as the plumes reach the bottom of the cell, CO 2 starts mixing within the entire bulk. Thus, the density difference between the CO 2 high-concentrated plumes and the CO 2 low-concentrated bulk starts to decrease. When it is smaller than the critical value, the boundary layer cannot destabilise and prevents from the establishment of convection.

For the application at the size of an aquifer, the height of the aquifer defines the volume of brine confined in the reservoir. Thus, it is directly linked with the quantity of CO 2 that can be dissolved in the aquifer.

Conclusion on the influence of the thermodynamic parameters

As explained above, three parameters influence the early-time behaviour of the instability: ∆ρ, D, and η. In CCS projects, these parameters would be determined by the thermodynamic conditions of the aquifer, i.e. its temperature and pressure, and the chemical composition of the brine. From this, we can determine the onset time of convection and the rise of the dissolution rate when convection has set-in. This provides the efficiency of the storage.

At longer times, the behaviour of the instability is determined by the same parameters and the thickness of the aquifer that we consider. These determine the long-term efficiency of the storage. Thus, from the knowledge of the thermophysical, chemical, and geophysical parameters, we are able to select the best site to inject CO 2 .

Modeling of relevant parameters

Here, we will use dimensionless quantities to provide universal scaling laws and easily upscale them to reservoir size. Besides the solutal Rayleigh number, that characterizes the instability, we will use other dimensionless quantities: the solutal Nusselt number N u s , the dimensionless time τ = t/t D , and the dimensionless thickness of the boundary layer δ = d/h.

Boundary layer

The stability of the fluid against convection is determined by the balance between the driving force of buoyancy and the dissipating effects of viscosity and mass diffusion, as expressed by the dimensionless global solutal Rayleigh number

Ra s = g∆ρh 3 ηD . (4.13)
For the present systems, the occurrence of convection is insensitive to the pressure dependence of D and η [START_REF] Cadogan | Diffusion coefficients of CO 2 and N 2 in water at temperatures between 298[END_REF]. This is because the concentration of CO 2 dissolved in brine or pure water remains small enough over the complete pressure range [START_REF] Spycher | CO 2 -H 2 O mixtures in the geological sequestration of CO 2 . I. Assessment and calculation of mutual solubilities from 12 to 100 • C and up to 600 bar[END_REF] such that D and η can be assumed constant [START_REF] Wu | Mutual and thermal diffusivities as well as fluid-phase equilibria of mixtures of 1-hexanol and carbon dioxide[END_REF][START_REF] Wu | Diffusivities in 1-alcohols containing dissolved H 2 , He, N 2 , CO, or CO 2 close to infinite dilution[END_REF]. Above a threshold critical value Ra * s , the system undergoes a symmetry-breaking mechanism leading to the appearance of a convective instability characterised by the formation of density patterns; see figure 4.40.

A quantitative understanding of the scaling relation for the onset time of convection, and in turn of the physical mechanism behind the instability, can be achieved by considering the seminal model developed by Howard [59]. This describes the stability of a horizontal layer of fluid of infinite aspect ratio subjected to a temperature difference. Howard investigated the onset of turbulence in a horizontal layer of fluid heated from below well above its critical Rayleigh number. Under these conditions, heat conduction determines the rapid growth of a temperature boundary layer which becomes unstable when it reaches a critical thickness. The instability determines the emission of plumes, which deplete the boundary layers, and the process iterates until the system reaches a stationary state where the boundary layer thickness is pinned at its critical value. Arguments very similar to those proposed by Howard have been successfully used to understand the onset of solutal boundary layer convection in aqueous suspensions of silica [START_REF] Cerbino | Scaling behavior for the onset of convection in a colloidal suspension[END_REF][START_REF] Giavazzi | Scaling of the spatial power spectrum of excitations at the onset of solutal convection in a nanofluid far from equilibrium[END_REF][START_REF] Mazzoni | Transient oscillations in Soret-driven convection in a colloidal suspension[END_REF] and thermosensitive particles [START_REF] Messlinger | Transient diffusive boundary layers at high Rayleigh numbers in simple and double diffusive fluids: Latency time scaling for the convection onset[END_REF][START_REF] Winkel | Thermal convection in a thermosensitive colloidal suspension[END_REF] triggered by the Soret effect, as well as during the dissolution of a solid into a liquid phase [START_REF] Philippi | Solutal convection induced by dissolution[END_REF].

Here, the density of mixtures of CO 2 and brine is larger than that of brine itself. Thus, the boundary layer is intrinsically unstable against gravity [START_REF] Lindeberg | Vertical convection in an aquifer column under a gas cap of CO 2[END_REF]. Before convection sets-in, the time evolution of the diffusive growth of the boundary layer can be described analytically by solving the diffusion equation ∂c/∂t = D(∂c/∂z) 2 . The CO 2 phase overlying the layer of water can be modelled by adopting a boundary condition of constant concentration of CO 2 dissolved into water at the interface, while for the bottom of the water layer one can assume an impermeable boundary. For times much shorter than the diffusive time t D , or for large sample thickness h, the concentration profile can be approximated by the one obtained in a semi-infinite system, with the top boundary at concentration c eq and the bulk of the fluid at c 0 , yielding c(z

) = c 0 +(c eq -c 0 )erfc[(h- z)/( √ 4Dt)] [23].
Here, z is the vertical coordinate, z = 0 represents the bottom of the sample and the interface is located at z = h.

Dimensionless critical time

The systematic investigation of the injection of CO 2 at different pressure steps shows that the dimensionless time τ c = t c /t D , needed for the onset of convection exhibits a power law behaviour as a function of the Ra s , with exponent -(0.66 ± 0.05); see figure 4.38. We recall that t D = h 2 /πD is the diffusive time; i.e. the time needed to homogeneise the complete solution by means of Fick diffusion. Interestingly, in our lab studies, the time needed for the onset of convection is about two orders of magnitude smaller than t D .

Previous experimental and theoretical works came to the conclusion that the convective dissolution of CO 2 is slowed down or accelerated by the presence of salt when the process occurs into brine, with respect to the times needed in pure water [START_REF] Khosrokhavar | Visualization and investigation of natural convection flow of CO 2 in aqueous and oleic systems[END_REF][START_REF] Loodts | Control of convective dissolution by chemical reactions: General classification and application to CO 2 dissolution in reactive aqueous solutions[END_REF][START_REF] Loodts | Dissolutiondriven convection in reactive systems[END_REF][START_REF] Tang | Experimental study on the density-driven carbon dioxide convective diffusion in formation water at reservoir conditions[END_REF][START_REF] Thomas | Convective dissolution of CO 2 in water and salt solutions[END_REF]. To elucidate this point, we compute τ c for each experiments in pure water and in brine obtained by adding 1 or 3 mol/kg of NaCl to water. As shown in figure 4.38, even though for the same pressure jump the time evolution of the contrast of shadowgraph image differences exhibits a delayed onset of convection for the H 2 O-NaCl brine, once the dimensionless parametrisation is adopted, all data points for pure water and brine collapse on the same master curve expressed by equation (4.16). Therefore, the slowing-down of the onset of convection in brine is only associated with the difference in Ra s associated with the change in the thermophysical properties of the system. This is valid for both, the Rayleigh-Taylor and Rayleigh-Darcy regimes, i.e. in free or in porous media. To model the onset of convection, we assume that the concentration profile in the liquid layer can be divided into two regions, as it was done by Howard [START_REF] Howard | Convection athigh rayleigh number[END_REF]: a boundary layer region at the top, where the concentration varies linearly, and a bulk region at the bottom, where the concentration is essentially uniform. The thickness d of the boundary layer, estimated as the vertical extension of the linear region, changes in time diffusively as d(t) = √ πDt, and in its dimensionless form

δ(τ ) = √ τ . (4.14)
During the initial diffusive growth, the boundary layer is horizontally uniform (except for tiny non-equilibrium concentration fluctuations [START_REF] Croccolo | Non-local fluctuation phenomena in liquids[END_REF]) and the contrast of shadowgraph images does not change significantly. When the boundary layer reaches a critical thickness δ * , it becomes unstable and convection sets-in at time τ * = τ c . Thereafter, plumes start to evolve vertically inside the bulk of the cell. Following Howard [START_REF] Görtler | Applied Mechanics : Proceedings of the Eleventh International Congress of Applied Mechanics Munich[END_REF], the stability of the boundary layer can be characterised by introducing a local solutal Rayleigh number, Ra s,l , defined across the dimensionless thickness δ of the boundary layer. Introducing the dimensionless critical boundary layer thickness δ * = d * /h, when Ra s,l exceeds the critical value Ra * s,l = Ra s δ * 3 . (

convection sets-in in the form of down-welling plumes.

For a biphasic system consisting of a gas on top of a liquid, the bulk geometry of the boundary layer consists of free isomassic boundary conditions. Under these conditions, Ra * s,l = 27π 4 /4 ≈ 657.5 [START_REF] Messlinger | Transient diffusive boundary layers at high Rayleigh numbers in simple and double diffusive fluids: Latency time scaling for the convection onset[END_REF]. Combining equations (4.14) and (4.15), we obtain the scaling relation for the dimensionless critical time needed for the onset of the instability:

τ * = Ra s Ra * s,l -2 3 , (4.16)
which is fully compatible with the experimental results for the onset time shown in figure 4.38.

Investigations on the dissolution rate and convective flux: the

Nusselt number

The common dimensionless number that characterizes the heat transfer during heat convection experiments is the Nusselt number N u. It compares the heat transfer done by convection to the one done by conduction. In solutal convvection, we use similar consideration to define the solutal Nusselt number N u s , also called Sherwood number Sh, that provides the ratio of mass transfer done by convection to the one done by pure diffusion.

Howard mentioned the Malkus' 1954 theory [START_REF] V R Malkus | The heat transport and spectrum of thermal turbulence[END_REF] that is based on equations given by the Boussinesq approximation and on a mathematical development of the convective heat transport as a finite Fourier series of cos(2πnz/d) for n ≤ n 0 . From that, they obtained an equation of the Nusselt number as a function of the Rayleigh number as N u = kRa 1/3 . Using further considerations on the boundary layer development, he provided an equation of the Nusselt number giving N u = (Ra/Ra * ) 1/3 . In our case, to evaluate N u s , we investigate the time occurring from the injection of CO 2 until the formation of the convective patterns. We recall that a CO 2 -rich boundary layer grows diffusively in time until it reaches a critical thickness, leading to the start of convection. We consider that after the onset of the instability, in the presence of a stable driving concentration difference ∆c = c eq -c 0 , the system reaches a stationary convective state where the transfer of mass occurs diffusively in the boundary layer at the critical thickness d * , while it occurs convectively in the bulk of the sample.

The mass conservation through the "interface" between the boundary layer and the bulk imposes that the diffusive flux inside the boundary layer j d = -ρD∆c/d c is equal to the convective flux into the liquid bulk, j c . Thus, by defining a Nusselt number as the ratio between the convective flux, j c , and the diffusive mass flux across the entire thickness, j h = -ρD∆c/h, we can write that

N u s = j c j h = j d j h = h d * . (4.17)
This argumentation is not completely rigorous as it borough considerations of steady state processes and that our is transient. However, this provides a first estimation of the solutal Nusselt number that has been used and has already demonstrated its value, notably in heat convection studies. A more classic and rigorous computation of the Nusselt number consists in measuring the convective flux and divide it by the diffusive one. Although it would bring probably a result slightly different, its order of magnitude would be similar to the one we obtained with the method of Howard.

Thus, the solutal Nusselt number is directly related to the critical thickness of the boundary layer d * that can be estimated from the expression of t c . Combining equations (4.15) and (4.17) leads to the scaling relation

N u s = Ra s Ra * s,l 1 3 , (4.18)
which is fully equivalent to the one proposed by Howard [START_REF] Howard | Convection athigh rayleigh number[END_REF] in the case of heat convection experiments. The analogy with heat convection at large Ra T has been recently exploited to perform fully 3-dimensional simulations of Rayleigh-Darcy convection in a porous medium [START_REF] Pirozzoli | Towards the ultimate regime in Rayleigh-Darcy convection[END_REF]. These simulations showed that the confinement determined by the porous medium strongly affects the scaling relation for the thermal Nusselt number N u T in the large Ra T regime, which becomes a proportionality relation N u T ∝ Ra T , in agreement with theoretical predictions [START_REF] Huppert | The fluid mechanics of carbon dioxide sequestration[END_REF]. An important consequence of equation (4.18) is that the convective mass flux at steady state can be inferred from the onset time τ c needed for the onset of convection. In fact, the onset time τ c allows to determine the critical thickness δ * = δ c of the boundary layer (see equation (4.14)) and, in turn, the solutal Nusselt number. Under the range of the solutal Rayleigh numbers explored in this work, the power law exponent is (0.37 ± 0.02) (figure 4.39), which is compatible with the exponent 1/3 of equation (4.18) and with the experimental exponent of 0.31 reported for turbulent heat convection over a wide range of Rayleigh number [START_REF] Niemela | Turbulent convection at very high Rayleigh numbers[END_REF].

These experimental results have a first interpretation about the efficiency of the process. Indeed, we see that with a larger Rayleigh number, the Nusselt number would be larger. This law is valid in a free media but we can derive a similar theoretical scaling law for porous media giving N u ∝ Ra, and then this remark and the following are valid for both porous and free medium. At a similar pressure, temperature and composition of the brine, the Rayleigh number will only vary with the height of the aquifer. Thus, the convective dissolution will be more efficient in a thicker aquifer, because the part of the mixing made by the convective regime is larger.

Discussion on up-scaling to relevant CCS conditions

At first, it is important to highlight once more that in CCS, CO 2 is injected in supercritical state at pressures around 10 MPa. However, the variation of Ra s for p inj between 6 and 12 MPa is of about 1.35 times larger only. Thus, our results are fully transferable to process relevant conditions.

The availability of a model of the convective dissolution process by means of dimensionless equations allows to project this result at the geological length scales of interest for the convective dissolution of CO 2 into deep saline aquifers. For example, by assuming a thickness h = 100 m for the aquifer, equation (4.18) lets us estimate the solutal Nusselt number of the order of 10 4 .

Going back to dimensioned variables, from equations (4.13) and (4.16) we obtain the remarkable result that the dimensioned onset time t c needed for the onset of convection and the convective flux j c at steady state do not depend on the size h of the system, and represent scale-free features for the convective dissolution of CO 2

t c = 1 πD ρνD g∆ρ Ra * s,l 2 3 , (4.19)
j c = D β g ρνD 1 Ra * s,l 1 3 ∆ρ 4 3 , (4.20)
where β = (1/ρ)(∂ρ/∂c) is the solutal expansion coefficient.

The physical reason behind the absence of a dependence of t c and j d from the typical length scale h of the system is that the boundary layer instability occurs very rapidly, when the boundary layer thickness is still much smaller than that of the laboratory liquid sample or of the aquifer. For this reason, the presence of an impermeable boundary at a depth h below the CO 2 -water interface is still not felt when the boundary layer becomes unstable, and the only relevant length-scale is the critical boundary layer thickness. This feature is akin to thermal turbulence in a layer of fluid, where it is well known that the heat transferred for a given temperature difference is independent of the spacing between the thermalizing plates [START_REF] Elder | The unstable thermal interface[END_REF].

The scaling laws presented in this work are valid only in free media. As stated above, in the presence of a porous medium, or in Hele-Shaw configurations, the Nusselt number becomes proportional to a Rayleigh-Darcy number and, thus, it increases much more rapidly. Thus, the convective dissolution of CO 2 in the brine should be much more efficient than diffusion only.

This approach has been used to perform fully 3-dimensional simulations of heat convection at large Rayleigh-Darcy numbers and infinite Prandtl number [START_REF] Pirozzoli | Towards the ultimate regime in Rayleigh-Darcy convection[END_REF]. It shows that the approach proposed by Howard can be adopted successfully also in the presence of a porous medium, leading to scaling relations similar to equations (4.15)-(4.18), but with different power law exponents [START_REF] Pirozzoli | Towards the ultimate regime in Rayleigh-Darcy convection[END_REF].

The complete phenomenology of the convective dissolution

Starting from an equilibrium between a CO 2 -rich gaseous phase and a brine-rich liquid phase, we inject CO 2 increasing the pressure inside the gaseous phase. The final pressure determines the final vapour-liquid equilibrium and, thus, the final quantity of CO 2 that can be dissolved inside the liquid phase. At the interface, the gaseous CO 2 becomes aqueous and starts to mix with brine. This mixing is, first, purely diffusive and, then, a boundary layer grows, as illustrated in the scheme (i) of figure 4.40. Recalling the definition of the local solutal Rayleigh number, within the boundary layer of thickness d,

Ra s,l = ∆ρgd 3 ηD , (4.21)
the diffusive growth of the boundary layer induces an increase in Ra s,l . If the variation of density ∆ρ is sufficiently large, the boundary layer will destabilise at the onset time of convection when the boundary layer reaches its critical thickness d c = d * . The higher the pressure difference, the higher the density difference and, thus, the lower the critical thickness of the boundary layer and the lower the critical time. After a theoretical analysis, we derive a scaling law giving t c ∝ Ra -2/3 . We performed experiments on convective dissolution over a large range of solutal Rayleigh number and found a similar scaling law with an exponent -0.66 ± 0.05. This value is in excellent agreement with the theoretical one. To the best of our knowledge, this is the first time that this scaling law is experimentally verified in a three-dimensional cell with the system CO 2 -H 2 O-NaCl.

The convective patterns appear at first with a tubular shape (d) along which some downward plumes are emerging (ii). The density, width, and contrast of the plumes depend directly on the pressure difference imposed at the beginning of the experiment and, thus, on Ra s . Indeed, the other key parameter h is kept constant in all the experiments. The variance reaches a maximum, at t m , a few seconds after the onset time of convection, when the convective scheme is complete all over the cell (e). At this moment, we observe the formation of Voronoï-like structures around the patterns, showing that some plumes reach the bottom of the cell and spread around the center of the downward columns. Therefore, the dissolved CO 2 starts to mix in the entire bulk of the cell, the concentration difference between the boundary layer and the bulk decreases, as well as the density difference and, thus, the solutal Rayleigh number. At this moment, the spatial plume density reaches a first local maximum around t m . Most of them are interconnected by tubular patterns that charge them with CO 2 . Also their width is the largest at this moment. The dissolution rate is now much larger than the diffusive one by a factor within the range 5 to 15, as witnessed by investigation on the Nusselt number; see section 4.7.3.

After t m , the convective patterns become less interconnected by the tubular flows and are more and more isolated. This induces a fast decrease in the variance curve with a characteristic decay time τ 1 . Although the variance decreases, some isolated patterns keep emerging from the pattern-free areas due to the vanishing of the tubular patterns. Thus, the spatial density of patterns rises again and reaches a global maximum around t r . Then, it decreases and becomes constant during the constant flux regime. Quickly, the isolated patterns bring the CO 2 downward with a constant flux in what we can call a columnar flow (iii). Here, the patterns are still moving due to the shape of the interface given by the meniscus. Thus, the patterns merge and some new patterns appear in the pattern free areas. This explains the variations of variance around the plateau (f).

After the plateau, the patterns start to be less and less contrasted, fewer and fewer, and smaller and smaller. Here, the solutal Rayleigh number has decreased below its critical value and the plumes vanish diffusively (iv). Indeed, the CO 2 mixed in the entire bulk and the density difference is no longer large enough to create new fingers. However, due to the non flat shape of the interface, the patterns move and can still merge. This creates again pattern-free areas where, locally, the solutal Rayleigh number exceeds anew its critical value, generating new plumes. In spite of this, the spatial density of plumes continues to decrease until it reaches zero some time after t sd . The variance has decreased to its baseline value with a characteristic decay time τ 2 which is about one order of magnitude larger than τ 1 .

The shut-down time of convection t sd is constant in all our experiments, considering experimental uncertainties. Thus, neither the pressure difference nor the solutal Rayleigh number influences t sd . Indeed, with a higher pressure difference, there is a larger amount of CO 2 that need to be dissolved. This is determined by the vapourliquid equilibrium. Thus, we could think that the convective regime will last longer. However, the plumes are more concentrated in CO 2 , and the mixing is stronger, i.e., the dissolution rate is larger. Finally, from these remarks, we can conclude that the only parameter that defines the shut-down time of convection is the height of the liquid layer. Indeed, it is consistent with the following remark. At constant density difference given by the injection of CO 2 , the thicker the aquifer, the higher the quantity of CO 2 to be dissolved, and, thus, the longer the convective dissolution.

Chapter 5

Preliminary investigations on the influence of the Soret effect on carbon storage -Study of the model system 1-hexanol-CO 2 under terrestrial and reduced gravity conditions Saline aquifers are geological formations subjected to the geothermal gradient and extending over several hundred meters. The bottom of the aquifer being warmer than the top, compositional gradients may be influenced by the Soret effect, also called thermodiffusion or thermophoresis. Thanks to the large timescales involved in CO 2 storage, the Soret effect can induce a drift of CO 2 molecules depending on the storage conditions. For instance, molecules of CO 2 could be directed towards the top of the aquifer and, then, causing possible dramatic leakages. It is thus interesting to study the Soret effect but it is not trivial since it can induce further convective motion of the fluid under certain conditions, disrupting or enhancing the dissolution rate of CO 2 . The studies in this chapter have been performed in collaboration with Emma Lisoir within her Bachelor thesis under my supervision.

There exist several experimental and numerical techniques to study the Soret effect [START_REF] Giraudet | Contributions à l'Etude de la Thermodiffusion de Mélanges Binaires en Conditions de Réservoirs[END_REF]. This includes the study of the spatio-temporal behavior of density nonequilibrium fluctuations (NEFs) which can serve as a probing tool for the simultaneous determination of mutliple transport properties, including Fick diffusion and Soret coeffi-cients at the steady state of a thermodiffusion experiment [START_REF] Fabrizio Croccolo | Propagating modes in a binary liquid mixture under thermal stress[END_REF][START_REF] Wu | Simultaneous determination of multiple transport properties over a wide range of temperatures and pressures from the analysis of non-equilibrium fluctuations by the shadowgraph method[END_REF]. NEFs are nowadays routinely analyzed by the dynamic shadowgraph method [START_REF] Croccolo | A light scattering study of non equilibrium fluctuations in liquid mixtures to measure the Soret and mass diffusion coefficient[END_REF][START_REF] Giraudet | High-pressure mass transport properties measured by dynamic near-field scattering of non-equilibrium fluctuations[END_REF][START_REF] Vailati | Giant fluctuations in a free diffusion process[END_REF] combined with the differential dynamic algorithm (DDA) [START_REF] Croccolo | A light scattering study of non equilibrium fluctuations in liquid mixtures to measure the Soret and mass diffusion coefficient[END_REF] for the analysis of images. The latter technique together with the motivation on the choice of a system different than brine and CO 2 are briefly summarized in the following.

During my doctoral studies, I had the big opportunity to take part in a parabolic flight campaign organized by the CNES (Centre National d'Etude Spatial) with the partnership of the company Novespace. Those studies under simulated zero gravity conditions aimed at studying density NEFs when the Soret effect induces convection. The latter situation is typically observable when the temperature or concentration gradients are sufficiently large and unstable in the gravity field. This is what happens for fluid mixtures consisting of CO 2 dissolved in water or 1-hexanol when the fluid mixture is heated from above. Convection masks the signature from NEFs and, thus, hinders a proper study of the influence of the Soret effect on CO 2 storage. After presenting the theory of the shadowgraph method and how it allows to study the NEFs, we will present and discuss the results we obtained on the model system on Earth and under micro-gravity conditions.

Further motivations

Link with the convective dissolution

A proper understanding of the convective dissolution process would require to know and model the mesoscopic force driving the establishment of convection. In other words, we would like to know what happens inside the boundary layer that destabilizes it with or without order at large spatial scales. There are three studies in the literature [START_REF] Oh | Dynamics of fluctuations in a fluid below the onset of rayleigh-bénard convection[END_REF][START_REF] José | Fluctuations in fluids in thermal nonequilibrium states below the convective rayleigh-bénard instability[END_REF][START_REF] Wu | Fluctuations in fluids in thermal nonequilibrium states below the convective rayleigh-bénard instability[END_REF] postulating that convection originates from a rearrangement of NEFs at wavelengths associated with the bifurcation. They are restricted to Rayleigh-Bénard convection. If this result is, as it should be, transferable to solutal convection, studying deeply the NEFs would provide a better understanding of the origin of convective dissolution. Preliminary experiments performed with the very same systems and set-up of the previous chapters, did not work. We faced two challenges: the amount of CO 2 dissolved in the liquid phase is small, leading to a weak optical contrast factor; and, the presence of a gas-liquid interface complicates the analysis of density NEFs from the bulk. To overcome those challenges, we decided to use a mixture consisting of CO 2 dissolved in 1-hexanol and to study NEFs in pre-satured CO 2 solutions at the steady state of a Soret experiment. As it will be shown later, as for water-CO 2 mixtures, 1hexanol-CO 2 mixtures have a negative Soret coefficient and are unstable in the gravity field both by heating them from above and from below. Since Soret-induced convection might give rise to a further challenge, experiments were also performed under reduced gravity conditions. When g → 0, Ra s < Ra s,c and there is no longer convection.

Fluid-phase equilibria of the 1-hexanol-CO 2 system

The 1-hexanol Hexanol is a chemical species of the family of the alcohol and has the formula C 6 H 14 O. In our case, we use the isomer 1-hexanol (or Hexan-1-ol) that is widely used in the pharmaceutical industry. Thus, the properties of this species are very well known, in particular, the properties of the mixture of 1-hexanol and CO 2 .

1-Hexanol-CO 2

According to the classification of van Konynenburg & Scott [START_REF] Van Konynenburg | Critical lines and phase equilibria in binary van der Waals mixtures[END_REF], the mixture 1hexanol-CO 2 is a type III [START_REF] Secuianu | High-pressure phase equilibria in the (carbon dioxide + 1-hexanol) system[END_REF]. This type of mixture has a similar fluid-phase behavior as brine-CO 2 mixtures which are of type V. They present two critical loci between the plait critical points of the pure components. As shown in figure 5.1, critical loci are not linked by a continuous line. The loci of critical points around the lighter component cross a tri-phase vapour-liquid-liquid equilibrium (VLLE) line at the upper critical end point (UCEP). The second critical line is diverging when temperature decreases.

In figure 5.2, we show the p-x CO 2 phase diagram of the system 1-hexanol-CO 2 . The experimental data points are taken from previous studies and for three isotherms at T = 303.15 (orange), 313.15 (green) and 323.15 K (blue). We notice that we can reach mole fractions much higher than for the system CO 2 -H 2 O. For instance, at 6 MPa, we can reach around 40% of CO 2 in the liquid phase at T = 308.15 K, while, at the same pressure and temperature, we reach less than 2% of CO 2 in the liquid phase. The quantity of CO 2 dissolved and, thus, the optical contrast factor associated with the concentration (∂n/∂c) in our experiments, will then be much higher.

Sample preparation and experimental set-ups

Thermodiffusion experiments have been performed both on Earth and under reducedgravity conditions. In the thermodiffusion experiments, the saturated multicomponent mixture, here a mixture of 1-hexanol and CO 2 at a given mole fraction in CO 2 , is injected in the liquid state inside the thermodiffusion cell, which is the very same cell we used in our studies on the convective dissolution of CO 2 in brine.

Vapour-liquid equilibrium (VLE) apparatus

A schematic of the designed VLE apparatus is shown in figure 5.3. It consists in an ISCO pump containing CO 2 connected to the top of a 150 mL vessel, filled beforehand with about 80 mL of 1-hexanol, itself connected to a spindle press. The bottom of the thermodiffusion cell, when positioned vertically, is connected to the spindle press, and [START_REF] Beier | High-pressure phase equilibria of carbon dioxide + 1-hexanol at 303[END_REF][START_REF] Elizalde-Solis | Vapor-liquid equilibria and critical points of the CO 2 + 1-hexanol and CO 2 + 1heptanol systems[END_REF][START_REF] Pereira | High pressure phase equilibrium measurements for binary systems CO 2 + 1-pentanol and CO 2 + 1-hexanol[END_REF][START_REF] Secuianu | High-pressure phase equilibria in the (carbon dioxide + 1-hexanol) system[END_REF][START_REF] Wu | Mutual and thermal diffusivities as well as fluid-phase equilibria of mixtures of 1-hexanol and carbon dioxide[END_REF]. the top to the upper part of the vessel. The connections between the diverse components consist of high-pressure capillary and valves made of stainless-steel. The medium flow of the valves in contact with liquid is free of lubricant for purity reasons. The sample preparation consists in an isobaric transfer of the saturated liquid phase from the vessel to the spindle press and, then, from the spindle press to the sample cell. The different steps used in our protocol are described in the following.

First, 1-hexanol is filtered twice with the help of two stacked PTFE filters with pore sizes of 450 and 200 nm to minimize the presence of particles inside the cell. Indeed, as shown in figure 5.4, a lot of particulate contamination scatters light in the original sample. On the contrary, after filtration, no residual particles are observable. If the filtration is not applied, the scattered light intensity arising from those particles would overwhelm that from density NEFs.

When the VLE vessel is filled with 1-hexanol, we make vacuum in the entire hydraulic circuit down to the vapor pressure of pure 1-hexanol; see figure 5.3. Then, we inject CO 2 at the desired pressure on top of the liquid as well as inside the cell and the spindle press. At this time, CO 2 starts to dissolve in the 1-hexanol liquid phase. To reach the equilibrium determined by the pressure and temperature faster, we man- The second step consists in filling the spindle press with the saturated liquid phase by increasing the volume. The transfer is kept isobaric by keeping the pressure at the corresponding bubble point with the help of the ISCO pump. Then, we pressurize the spindle press and fill the cell, whose pressure has been increased to a pressure slightly smaller than that of the spindle press. During this operation, we must avoid the presence of any bubble inside the cell. The pressure inside the cell is always kept above the bubble pressure of the system.

Description of the shadowgraph set-up

Two shadowgraph apparatuses were used here. Ground-based experiments have been conducted on the very same set-up as the one used for the characterization of the convective dissolution of CO 2 . For the experiments under reduced gravity, we developed a new apparatus that can be fixed inside the A310-zeroG plane. It consists in two experimental racks. The first one contains the experimental set-up; see figure 5.5. The second rack contains the monitoring devices. Both racks have been constructed following the specific requirements given by Novespace which is the company in charge of the parabolic flights.

In the first rack, there are two main parts. One is closed and sealed because it contains the shadowgraph, including the cell that holds the liquid mixture. The complete system is placed inside a waterproof chamber. To keep the gravity center of the apparatus as much low as possible, the shadowgraph set-up is mounted on two optical breadboards assembled in T-shape that are clamped to the top and bottom of the rack through dampers. It includes a superluminescent diode operated at (675 ± 13) nm. The beam is collimated using an achromatic doublet with a focal length of 200 mm and, then, directed through the thermodiffusion cell and to a fast camera ORCA-Flash4.0 v3 positioned in the near field. The cell is inserted in a holding block that maintains it both horizontally and vertically. We also fixed an inclinometer to the breadboard. The design of the sample cell is very similar to the one used in the dissolution experiments. The temperature control is also performed with the help of two Peltier elements placed on the top and bottom of the upper and lower sapphire windows, respectively. However, it was not possible to use a thermostatic bath. The latter was replaced by three fans placed inside the rack. The second (right) part of rack 1 contains the Peltier and inclinometer controllers.

Parabolic flight campaign at Novespace

In the 50s, the parabolic manoeuvre was developed with fighter aircrafts to obtain reduced gravity conditions. From the 60s, larger planes were used to host scientific equipments. In Europe, the CNES (Centre national d'études spatiales) organized, through its Novespace branch (Bordeaux-Mérignac), the first campaigns of parabolic flights to make Europeans autonomous. First, the flights were done by a Caravelle VI-R, then, in 1995, an Airbus A300 Zero G was used. The current plane in use is the Airbus A310 Zero G that made its first flight in April 2015. Today, it is the largest plane in the world making parabolic flights. Even though the scientific research in micro gravity is the main use of the A310 Zero G, some flights are commercialized for private societies and individuals. In 2017, Novespace have cumulated more than 100 hours of weightlessness over more than 17,000 parabola and around 200 parabolic flight campaigns.

A typical campaign lasts two weeks. The first one is dedicated to perform all the necessary checks of the experimental set-ups. The latter and all the experimental protocols constitute a huge upstream work. However, there are still some last-minute preparations to be done on ground, and, finally, the mounting of the experimental racks on the dedicated plane. Figure 5.6: Scheme of one parabola. First, the plane is in stationary flight and starts to gain altitude which induces a hyper-gravity stage that lasts 20 seconds. There, the gravity is 1.8 g. Then, the plane is in a free falling phase where it has a parabolic trajectory and the gravity is close to zero. Finally, another hypergravity phase is necessary to return to the stationary flight.

During the second week, three days are dedicated to the parabolic flights, plus one back-up flight day if the weather is not compatible with flight on one day. Each flight lasts approximatively three hours including the take-off, several series of parabola, and the landing.

Description of the parabola

During a flight, the Airbus A310 Zero G performs 6 series of 5 parabola. The first series has an extra parabola "0" to prepare the new flyers to the micro-gravity conditions. In a parabola, the plane starts to gain altitude starting from a stationary flight condition thus inducing a hyper-gravity phase. The gravity is 1.8 g during this phase that lasts 20 seconds. Then, the pilots cut the gas and the plane enters a free falling phase during which it has a parabolic trajectory. There are reduced-gravity conditions during this phase for around 22 seconds. Finally, the pilots turn the engine on again to go back to the stationary state. In between, a second hyper-gravity phase, similar to the first one, is necessary to reach back the stationary flight at the same altitude than before the parabola. A scheme of one parabola is presented in figure 5.6.

The plane is flown by three pilots, two piloting the plane like in a commercial flight, Figure 5.7: Evolution of the acceleration along the z-axis over the entire flight (a), one parabola (b), and during the micro-gravity period (c). We observe that the microgravity is not perfect and negative accelerations are present. and one controlling the acceleration during the micro-gravity conditions. The microgravity conditions provided by a parabola is not perfectly stable. Indeed, the third pilot needs to adjust the acceleration parallel to the gravity field, and maintain it as close as possible to zero. Despite this, the plane is subject to the external weather conditions. The latter induce some perturbations on the acceleration of the z-axis, and, sometimes, some negative gravity acceleration, as shown in figure 5.7. The resulting acceleration would be probably much better if it was an automatic control process, but, for insurance and security reasons, this is not allowed. However, this parabolic flight still provides reasonably good acceleration conditions similar to those obtained in the ISS (International Space Station).

Fundamentals for the determination of transport prop-

erties from the analysis of NEFs within a thermodiffusion experiment using the shadowgraph method

The Soret effect

Although the Soret effect is widely spread in numerous fields, it is a phenomenon relatively often unknown. It has various names as thermophoresis, thermodiffusion, or the Ludwig-Soret effect, and applications in the fields of aerosols, repartitions of the molecules in reservoirs, or preparations of multicomponent liquids.

The Soret effect consists in the apparition of a concentration gradient induced by the imposition of a thermal gradient. For a binary mixture, a transient state occurs after applying the thermal gradient until the steady-state, and is resumed by an overall mass flux, -→ j 1 of the denser component denominated "1":

- → j 1 = -D --→ ∇c 1 -D T c 1 (1 -c 1 ) -→ ∇T (5.1)
where D is the Fickian diffusion coefficient, D T the thermodiffusion coefficient, and c 1 the average molar fraction of the denser component of the mixture. Here, two processes are in competition: the mutual mass diffusion and the thermodiffusion. Diffusion tends to mix and homogenize the components while thermodiffusion tends to separate them. At steady-state, the flux -→ j 1 is equal to 0, which is translated by:

D T D = - ∆c 1 c 1 (1 -c 1 )∆T = S T (5.2)
and constitutes a definition of the Soret coefficient, S T . The Soret coefficient can be negative or positive. For instance, in a classical experiment of thermodiffusion, in which we heat the sample from the top, a positive Soret coefficient constitutes a stable case, where the thermal and concentration gradients are antiparallel. In the case of a negative Soret coefficient, the denser molecules move towards the hotter regions. This can lead to an unstable case and to the formation of advective motion with mass transfer no longer purely conductive. Although solutal convection induced by the Soret effect could be, in principle, annihilated by inverting the temperature gradient, such procedure could lead to thermal convection. The presence of thermal and concentration gradients induce also the presence of non equilibrium fluctuations (NEFs) inside the bulk.

Density non-equilibrium fluctuations (NEFs)

The topic of NEFs is relatively new. Although the importance of fluctuations has been formulated at the beginning of the 20th by Onsager & Einstein [START_REF] Einstein | Über die von der molekularkinetischen theorie der wärme geforderte bewegung von in ruhenden flüssigkeiten suspendierten teilchen[END_REF][START_REF] Onsager | Reciprocal relations in irreversible processes I[END_REF][START_REF] Onsager | Reciprocal relations in irreversible processes II[END_REF], the NEFs started to be correctly predicted for the first time in 1982 [START_REF] Kirkpatrick | Light scattering by a fluid in a nonequilibrium steady state. II. Large gradients[END_REF][START_REF] Ronis | Nonlinear resonant coupling between shear and heat fluctuations in fluids far from equilibrium[END_REF]. The theory relies on mode coupling theory and fluctuating hydrodynamics [START_REF] Landau | Fluid Mechanics[END_REF]. The analysis of the NEFs is based on the study of the autocorrelation function of the density fluctuations at wave number q and frequency ω, commonly called the dynamic structure factor, S(q, ω). A complete theory was developed by Ortiz de Zárate & Sengers [START_REF] Ortiz De Zarate | Hydrodynamic Fluctuations in Fluids and Fluid Mixtures[END_REF].

Thanks to the local equilibrium hypothesis, the local entropy depends on the same parameters than the global entropy. Then, all the microscopic states of the fluid are at equilibrium and are exchanging information. Therefore, we can consider the thermodynamic properties of the system as spatiotemporal fields. From it, we can define any local variable X as

X( - → r , t) =< X( - → r , t) > +δX( - → r , t) (5.3)
that is now a stochastic value. < X( -→ r , t) > is the mean value and δX is the small difference (fluctuations) from the mean value. Then, using the Boussinesq approximation, the local density does not vary with the pressure but only with thermal and concentration variations. Thus, we write the equation of state

ρ( - → r , t) = ρ(1 + β s δc( - → r , t) -β T δT ( - → r , t)) (5.4)
where β s and β T are the solutal and thermal expansion coefficients, respectively.

As we mentioned in chapter 3, several propagating modes play a role on the evolution of the fluctuations. All the thermodynamic properties influence the variations of the refractive index. Thus, to study the properties of interest, we focus on the light scattering intensity in the Fourier space I(q, t), that is proportional to the power spectrum (the Fourier transform of the correlation function) of the refractive index [START_REF] Segrè | Nonequilibrium fluctuations in liquid mixtures under the influence of gravity[END_REF] I(q, t) ∝< δn q,t , δn * q,t > (5.5)

From the equation of state 5.4, we are able to derive

I(q, t) = A 0 ∂n ∂T 2 p,c
< δT q,t , δT * q,t > + ∂n ∂c 2 p,T < δc q,t , δc * q,t > .

(5.6)

Here, we do not show the cross-correlation term. Indeed, we will make the approximation of large Lewis number, that is valid in the studies made in this thesis. The Lewis number, Le = κ/D compares the thermal diffusivity, κ, with the mass diffusion coefficient. The time autocorrelation functions mentioned above are, at equilibrium [START_REF] Croccolo | Non-local fluctuation phenomena in liquids[END_REF]:

< δT * (q , t), δT (q, 0) >= k B T 2 0 ρc p exp(-D T q 2 t)(2π) 3 δ(q -q), (5.7)
< δc * (q , t), δc(q, 0)

>= k B T 0 ρ exp(-Dq 2 t)(2π) 3 δ(q -q), (5.8)
k B is the Boltzmann's constant and c p the massic heat capacity. At non-equilibrium:

< δT * (q , t), δT (q, 0) >= k B T 2 0 ρc p [(1 + A T )exp(-D T q 2 t) -A ν exp(-νq 2 t)](2π) 3 δ(q -q), (5.9) with A T = c p T0 (ν 2 -D 2 T ) ν D T (∇T 0 ) 2 q 4 , (5.10)
A ν = c p T0 (ν 2 -D 2 T ) (∇T 0 ) 2 q 4 , (5.11)
and < δc * (q , t), δc(q, 0) >= k B T 0 ρ χ p,T (1 + A c )exp(-Dq 2 t)(2π) 3 δ(q -q), (5.12)

with A c = 1 νD χ -1 p,T (∇c 0 ) 2 q 4 . (5.13)
χ p,T = (∂c/∂µ) p,T can be identified with the osmotic compressibility. The establishment of the temperature and concentration gradients does not have the same kinetics. Indeed, in most cases, the thermal diffusivity is much larger than the mass diffusion coefficient. This is exactly stated by the large Lewis number Le 1 approximation.

Analysis of shadowgraph images

The shadowgraph method has been used to observe the convective patterns during the convective dissolution experiments, that we presented in chapter 4. However, this technique is also of major interest to study the NEFs. It is based on heterodyne light scattering in the near field, that relies on the elastic diffusion of radiation. When a parallel monochromatic beam passes through a dense and transparent medium, the major part of the beam is directly transmitted, while a small part is scattered by the density heterogeneities. The techniques of light scattering have several applications at multiple length scales depending on the wavelength of radiation. Here, we study the c-NEFs that happen at the macroscopic scale. Thus, we can use the diffusion of visible light to study them.

In the near field, on the plane of the camera, the total intensity measured by the sensor is the sum of the intensity of the beam and the less intense small modulations or fluctuations of intensity that result from the interferences between the transmitted beam and the beams scattered by the local heterogeneities of refractive index inside the bulk. Thus, the fluctuations of the refractive index δn( -→ r , t) induce fluctuations of intensity δI( -→ r , t) that are observable and can be separated in the Fourier space, δI( -→ q , t). By taking a long series of images at a constant time delay, we use a statistical approach to study the c-NEFs. This provides, only with one experimental acquisition, huge statistics for large ranges of wave vectors and time delays. The range of wave vector available directly depends on the characteristics of the CCD camera. The maximum wave vector is given by a sequence of two pixels q max = 2π/2l pix , while the minimum wave vector is given by the size of the sensor, q min = 2π/l sensor , with l pix the length of one pixel and l sensor the lenght of the sensor, if no magnification is used in the optical set-up.

The differential dynamic algorithm

An efficient method to study the NEFs is to record a series of images with a shadowgraph at constant time delay between two successive images. Then, using a differential dynamic algorithm (DDA), see figure 5.8, we study the time autocorrelation function of intensity differences in the Fourier space. First, we normalize each shadowgraph image by its average intensity. From the normalized shadowgraph images, i( -→ r , t), see figure 5.8 (a), that contains the information on both the transmitted beam and the heterodyne signal of the image, we calculate the image differences, δi( -→ r , t, τ ) of all the combinations of two images separated by a time delay dt ((b) and (c))

δi( -→ r , t, τ ) = i( -→ r , t + τ ) -i( -→ r , t).

(5.14)

The transmitted beam, and the static optical noise are thus subtracted. By this procedure, we isolate the heterodyne part of the signal that contains the information on the c-NEFs. Then, we compute the average Fourier transform, δi( -→ q , t, τ ), of each image difference with a time delay τ (d). We notice that the signal is isotropic. Indeed, the Fourier transform is identical in all -→ q with the same modulus. Thus, we perform an azimuthal average and keep only the amplitude of the wave vector. Finally, we study the power spectra, < |∆I m (q, τ )| 2 >, of these Fourier transforms. According to Croccolo et al. [START_REF] Croccolo | A light scattering study of non equilibrium fluctuations in liquid mixtures to measure the Soret and mass diffusion coefficient[END_REF], we get:

< |∆I m (q, t)| 2 >= 2a{T (q)I s (q)[1 -f (q, τ )] + B(q)}, (5.15)
where I s (q) is the ensemble averaged scattered intensity, f (q, τ ) is the normalized intermediate scattering function (ISF) that gives the temporal evolution of the power spectrum at a given q, a is a renormalization constant, T (q) is the transfer function of the shadowgraph system, and B(q) is a noise term.

f (q, t) = A(q)exp - |τ | τ c (q) + (1 -A(q))exp - |τ | τ T (q) , (5.16)
where A(q) is the overall amplitude of the intensity of concentration and temperature fluctuations, and τ c and τ T are the characteristic times of evolution of a concentration and temperature fluctuation, respectively.

Structure function

I s (q) and f (q, τ ) contain both information on transport properties and other thermophysical properties of the system. For instance, it has been shown [START_REF] Croccolo | A light scattering study of non equilibrium fluctuations in liquid mixtures to measure the Soret and mass diffusion coefficient[END_REF] that at the steady state of a Soret-driven separation process under the influence of gravity, the intensity of light scattered by concentration NEFs is 4 (5.17)

I s (q) = I 0 1 + (q/q c )
where

I 0 = [k B T /(16π 4 ρ)] × [∇c/(βg) -(∂c/∂µ) p,T
] is the value of the plateau due to the effect of gravity for q q c , and q c is the gravitational critical wave vector defined by:

q c = β - → g • - → ∇c νD 1/4
(5.18) where β = 1/ρ(∂ρ/∂c) is the solutal expansion coefficient, k B is the Boltzmann constant, T is the average temperature of the mixture, ρ is the density, µ is the chemical potential, Figure 5.8: Explanation on the DDA method, using an experiment performed by Croccolo et al. [START_REF] Croccolo | A light scattering study of non equilibrium fluctuations in liquid mixtures to measure the Soret and mass diffusion coefficient[END_REF] on a binary liquid mixture (1,2,3,4-tetrahydronaphtalene and isobuthylbenzene) stressed by a thermal gradient (∆T = 16 K). (a) near field image of the sample. (b) and (c) image differences with a time delay of 0.33 s and 33 s. (d) Fourier transform of (c) that appears to be isotropic. (e) Power spectra as a function of wave vector q for τ = 0.33 s (dashed line) and 33 s (continuous line). (f) Power spectra as a function of the delay time t for q = 110 cm -1 (squares) and 192 cm -1 (circles). The scattered intensity I s (q) and its fit to I 0 /[1 + (q/q c ) 4 ] are inserted in (e). and ν is the kinematic viscosity.

From an experimental point of view, the structure factor is measurable by light scattering techniques and the NEFs has been widely investigated to study transport phenomena such as diffusion [12, 13, 27-29, 116, 132, 158, 159], thermodiffusion [START_REF] Croccolo | A light scattering study of non equilibrium fluctuations in liquid mixtures to measure the Soret and mass diffusion coefficient[END_REF], and convection. Indeed, the structure factor is directly related with transport properties such as the diffusion coefficient D, the thermal diffusivity D T , and, thus, the Soret coefficient S T .

Ground-based results

In this section, we present the results we obtained on the system 1-hexanol-CO 2 . We first performed preliminary tests on Earth to select appropriate mixtures and the relevant thermodynamic conditions for the experiments to be conducted later under reduced gravity.

Mixture at x CO 2 ≈ 10%

To study the binary mixture 1-hexanol-CO 2 , we first worked with a molar fraction around 10%. This one was obtained working at a mean temperature of 313.15 K, corresponding to a vapour pressure of 1.073 MPa. To ensure that the system is maintained in its liquid phase, the experiments were conducted at 2.7 and 12 MPa. We applied a mean temperature of 308 K and a temperature gradient of 20 K. For the recording of the images, we worked at three different sample times : 10, 100, and 300 ms. This means that we repeat each experiment three times with different delay times, allowing to study multiple dynamics that does not occur at the same time scale. 10,000 images were recorded for each measurement.

In figure 5.9, we present the results obtained for an experiment performed at 12 MPa with a minimum delay time of 10 ms. A raw image is shown on (a). (b) displays an image difference, taken for a time delay of τ = 100 ms. Image (c) shows the Fourier transform of the image presented in (b). We also show the structure functions, plotted over a large range of wave vector for the three time delays (d). The temporal autocorrelation function is plotted for three different wave vectors in (e).

On the raw image (a), we do not see any signal nor convective pattern. On the image difference (b), we see some NEFs but they are hardly visible. This is more obvious on the Fourier transform (c), where we observe an annulus, characteristic of the NEFs, but

Mixture at x CO 2 ≈ 40%

As we did for the first mixture, we obtained a second solution for a molar fraction of around 40 % , for a vapour pressure of 6 MPa and a mean temperature of 313 K. This time, the results are more clear, and we performed experiment for several pressures: 24, 21, 20.5, 19, 18, 17, 13, 8, 7, and 6 MPa. We worked both at 313 and 303 K for thermal gradients of 18 and 15 K, respectively. We collected the images with a delay time of 10, 150, 400, and 1000 ms. Here, for pressures higher than 8 MPa, we observe convective patterns. For the higher pressures, the convection hinders the analysis of non-equilibrium fluctuations in concentration. By decreasing the pressure, the convection fades away to let space to the fluctuations. The latter are establishing at all the length scales of the system.

Here, we present wisely chosen pressures that illustrate well these remarks. In figure 5.10, for each pressure, we show the raw images (a), two image differences computed for two different time delays τ = 100 and 500 ms (b), as well as the 2D-FFT associated with τ = 100 ms (c).

On the raw images (a), we clearly see the convective patterns, especially for the two first experiments. The intensity and the contrast of these convective rolls are larger when the pressure is increasing, i.e. when we move away from the bubble point. At p = 24 MPa, the high intensity of light masks the non-equlibrium fluctuations that are almost invisible on the two corresponding image differences. They have a preferential orientation as shown in the Fourier transform (c), where we observe a line instead of concentric annulus characteristic of the NEFs. At p = 13 MPa, the convective patterns are less intense, and they appear to be less numerous. Indeed, on the raw image (a) and the image difference at τ = 500 ms (b), the convective roll is almost invisible. At the same time, the intensity of NEFs has slightly increased. We still have a preferential orientation of the convective roll, even if it is less noticeable. The non-equilibrium fluctuations are no longer negligible when the convection tends to vanish.

At the smallest pressure, p = 7 MPa, close to the boiling point, we do not observe any convection, neither on the raw image (a), nor on the image differences (b). Here, there are NEFs that establish at all scales, in an isotropic manner. It is illustrated by the Fourier transform (c), that displays an annulus in all directions, even if they are hardly distinguishable.

To complete the results obtained with this molar fraction, and the conclusion made in figure 5.10, we study the temporal behaviour of the mixture. We keep the same pressures than those presented on figure 5.10, and study now the structure functions. On figure 5.11, we plotted the temporal structure functions in terms of the time difference τ , for q/q min = 15 (on the left), and q/q min = 50 (on the right).

The first graph on the left illustrates that the intensity of the temporal structure function is increasing with pressure. The amplitude of the structure function for the larger pressure, p = 24 MPa (blue), is almost 1000 times larger than the one for the smaller pressure, p = 7 MPa (yellow). This shows the impact of convection to the system. At short times, we observe an inflection point corresponding to the exponential of the theory. On the orange and yellow curves, we observe a second inflection point, corresponding to a second exponential, and characterising the presence of thermal and concentration fluctuations. The wave vectors are q/q min = 15 on the left, and q/q min = 50 on the right On the graph on the left, the dynamics are slightly different, probably due to a change in the hydrodynamic regime. The intensity of the structure functions and the variations of amplitude from an experiment to another are smaller than those at smaller wave vectors. However, the amplitude is still rising with the pressure.

Conclusion on preliminary experiment on Earth

The experiments presented here show interesting results on the system 1-hexanol-CO 2 . Looking at the results obtained with the first mixture, at a molar fraction of 10 %, we draw the conclusion that the NEFs are not observable. The concentration is too small, as the variations of the refractive index. This is in line with what we observed with water. Indeed, we could not see the fluctuations with the systems CO 2 -H 2 O and CO 2 -brine. With these systems, the concentration of CO 2 reached in the liquid phase are always smaller than 10 % in molar fraction. Thus, it is impossible to observe the fluctuations with these systems. This justifies our choice of changing the system of interest. However, we still need to increase the molar fraction of CO 2 inside the liquid phase to study the NEFs and observe convection.

With the second mixture, at a molar fraction of CO 2 of 40 %, we could observe both the convection and the NEFs under certain conditions. As we explained above, the apparition of convection hinders the study of NEFs. Therefore, we want to repeat similar experiments during a parabolic flight campaign. Under micro-gravity conditions, we expect the convection to vanish, letting only the NEFs occurring inside the bulk. Furthermore, on Earth, the gravity affects both the static and the dynamic of the NEFs. Removing gravity, would provide a larger ranger of wave number to study the diffusive behaviour of the NEFs.

Results obtained under reduced-gravity conditions 5.5.1 Experimental protocol

The parabolic flights provide successively an acceleration, according to the z-axis, in the range 0g and 2g. Thus, we observe several transition phases: terrestrial gravity to hyper-gravity, hyper-gravity to reduced-gravity, reduced-gravity to hyper-gravity, and hyper-gravity to terrestrial gravity. During a series of 5 parabola, we set and keep the same thermal gradient. This way, we could test 6 thermal gradients from ∆T = 6 K to ∆T = -6 K in each flight, and have repeatability on the experiments. The mixtures consisted of 26 and 56 mol. % of CO 2 .

Preliminary observations

Looking at the raw shadowgraph images, we observe transitions between the different phases of the parabola. When convection occurs in the 1g phase, it is enhanced when it passes to the hyper-gravity phase, i.e. we observe more convective rolls, that are more contrasted; see figure 5.12. The convection may be turbulent under particular conditions at the 1g phase or after the transition to hyper-gravity. When no convection occurs, no pattern can be observed, but we see some particles inside the bulk sliding/oscillating on each other. Thus, we are able to determine the transition even if it is more difficult than in the case with convection.

When the plane passes to reduced-gravity conditions, we observe two different phenomena depending on the thermodynamic conditions. The convection could either persist or dissipate; see figure 5.13. For the mixture with x CO 2 = 56 mol. %, after the transition to reduced-gravity conditions, the convective patterns are still present in the bulk and we still observe advective motions of these. Thus, we cannot observe and study the fluctuations under those conditions. This is most probably because the time needed for the patterns to vanish by means of diffusion is much larger than the time slot under 0g. In fact, D decreases with increasing x CO 2 . For the mixture concentrated at x CO 2 = 26 mol. % , the convection is vanishing. Thus, the fluctuations become visible and we could study them applying the classical analysis.

Variance analysis

A practical way to observe the transitions between the phases is to compute the temporal evolution of the variance of the images throughout the entire parabola. First, we normalize the images, then we compute the variance of difference of each image with the previous one.

< δi 2 >= var I n (x, y) Īn - I n-1 (x, y) Ī n-1 (5.19)
I n (x, y) is the intensity of the image n, Īn is the mean intensity of the image n. We obtain the curve displayed in the figure 5.14. Here, we observe strong transitions between the different phases of flight. In the experiment presented here, the convection occurs in the 1g phase. Thus, the transitions are easily observable on the variance curve. When we pass to the 2g phase, there is a rise in the variance. Indeed, the patterns are more contrasted and are moving faster in the bulk. The 0g phase presents a lower variance as the convective patterns have vanished. Sometimes, they do not vanish entirely. However, they are less contrasted and they stop moving or are moving more slowly.

In this case, it still induces a lower level of variance during the 0g phase. Finally, the variance quickly rises again when the convective patterns appear anew. x CO 2 = 56 %, ∆T = -2 K.

The analysis of the variance also illustrates other results. First, as shown in figure 5.15, we have a good repeatability. Therein, we displayed five variance curves for the images recorded during the five parabola of one series, performed in the mixture x CO 2 = 56 % and a thermal gradient ∆T = -2 K. Thus, all the experiments where performed under the same conditions.

In figure 5.16, we displayed four variance curves at similar thermodynamic conditions but for four different thermal gradients ∆T = -1 (yellow), -2 (blue), -4 (grey), and -6 K (orange). We can make two remarks from this analysis. First, the transition between the 2g and 0g phases is smoother for a higher negative thermal gradient (∆T = -4 and -6 K). This comes from the fact that the convective patterns vanish more slowly and that they are still moving during the micro-gravity phase when the thermal gradient is larger.

On the contrary, for lower thermal gradients (∆T = -1 and -2 K), we do not observe advective motion anymore during the 0g phase. Furthermore, the convective patterns have almost vanished after the transition. Thus, their contrasts are smaller than those observed with larger thermal gradients.

Results

Several parameters play a role both on the convective patterns and the non-equilibrium fluctuations induced by the Soret effect. Especially, on this section we explain the influence of the thermal gradient and the mixture composition on the three phases of gravity.

x CO 2 = 56 % In each experiment performed with a molar fraction x CO 2 = 56 mol. %, we observed convection both with a positive and a negative thermal gradient. For the positive gradients (∆T = 6 and 2 K), during the 1g phase, convection is observed. The transition to hyper-gravity induces advective oscillations on the bulk and we observe the convective pattern moving. It also enhances the convection as the contrast and the number of patterns is increased. Then, during the reduced-gravity, the convective patterns are less contrasted and their motion due to the oscillation after the transition does not persist as much as in the 2g phase. These effects are stronger when the thermal gradient is larger.

For the negative thermal gradients (∆T = -1, -2, -4 and -6 K), during the 1g phase, we observe a turbulent convection (∆T = -4 and -6 K) or a classic convection (∆T = 1 and -2 K). We call turbulent convection the convection where we observe the convective patterns moving rapidly, i.e. appearing, vanishing, and merging. Their motion is even faster for a larger thermal gradient. This remark is still valid for classic convection. The convection is again enhanced during the 2g phase: the motion of the patterns increases such as their contrast and their number. Then, in reduced-gravity, the patterns are vanishing, i.e. their contrast drops. Furthermore, the motion of the patterns is slowed down for ∆T = -4 and -6 K, or completely stopped for ∆T = 1 and -2 K. Indeed, for larger thermal gradient, the advective motion persists even if the patterns have lost their contrast.

x CO 2 = 26 % There, convection is not observable on all the experiments. For positive thermal gradients, we observe stable images all the parabola long, i.e. without any convection. Looking closer to the images, we observe, at the transition phases, similar oscillations inside the bulk even if there is no convective patterns. We can make similar remarks for ∆T = -1 K.

For negative thermal gradients (∆T = -2, -4, and -6 K), we observe convection during the 1g phase. The convection is much more stable than with the molar fraction Figure 5.17: Image differences computed for images taken during the reduced-gravity phase and delay time τ = 50 ms to 1.6 s. ∆T = -4 K, x CO 2 = 56 mol. % . at x CO 2 = 56 mol. %. This is enhanced when we switch to hyper-gravity. The number of patterns increases and they get more contrastedk. These effects are increasing with increasing thermal gradient, but they are less pronounced than with the previous molar fraction. Then, the convective patterns are vanishing during the hyper-gravity phase before we go back to a convective regime at the 2g phase.

Study of the NEFs

We targeted experiments at which we are able to avoid convection during the microgravity period. Then, we want to observe the NEFs to see if we can study them and if interesting information can be extracted to get from it. Difference of raw images are displayed in figure 5.17 to detect the presence of NEFs. Here, this first rapid analysis does not provide appropriate results because we cannot observe the fluctuations, while it was possible to do with the experiments performed on Earth; see figures 5.9 and 5.10. Indeed, on the image differences, aside from the first one at τ = 50 ms, we observe disturbances that are characteristic of vibrations in the bulk.

Despite this, we tried to study the NEFs with the classical differential dynamic algorithm. From it, we obtain the structure functions that can be plotted both in term of the wave vector q and the time delay τ . In figure 5.18, we observe the structure functions corresponding to the image differences of figure 5.17. We notice that they do not present similar shapes than those we obtained for the NEFs on Earth. We do not observe neither the characteristic oscillations, see figure 5.8 (e), of the transfer function of the shadowgraph on (a), nor any temporal evolution that could present an exponential behaviour characteristic of the thermal and concentrations NEFs (b). Furthermore, on the Fourier tranform (c), we do not observe the characteristic rings that illustrate the transfer function of the shadowgraph in an isotropic manner.

During the reduced-gravity period, we observe sometimes the fluctuations when we compute an image difference. However, this is not stable during all the reduced-gravity period long. We suspect that the vibrations of the plane hinder the analysis of the NEFs. The dampers could not absorb all vibrations. Indeed, in the lab, we use optical tables with a vibration isolation system, to avoid any vibrations due to the activities on the laboratory or the building. Furthermore, during the 2g phases, the plane becomes inclined. It causes advective motion/oscillation just after the transition to the 0g phase.

These two phenomena strongly disturb the analysis of the NEFs. We see on the images some stratified layers that are oscillating/sliding on each other. This causes a loss of the autocorrelation plane in the analysis of the images. Then, it is impossible to observe the NEFs with this set-up during a parabolic flight.

Discussion

The study of the system 1-hexanol-CO 2 is interesting as it has been studied many times in the literature [START_REF] Beier | High-pressure phase equilibria of carbon dioxide + 1-hexanol at 303[END_REF][START_REF] Elizalde-Solis | Vapor-liquid equilibria and critical points of the CO 2 + 1-hexanol and CO 2 + 1heptanol systems[END_REF][START_REF] Pereira | High pressure phase equilibrium measurements for binary systems CO 2 + 1-pentanol and CO 2 + 1-hexanol[END_REF][START_REF] Secuianu | High-pressure phase equilibria in the (carbon dioxide + 1-hexanol) system[END_REF]. Thus, a solid basis is available to compare to our results. On Earth, the preliminary experiments we performed in our laboratory allowed us to determine the favourable experimental conditions to obtain convection. The objective was then to study the NEFs during the reduced-gravity phase while the convective patterns present in the 1g and 2g phases have vanished.

When the plane is in free fall to mimic the micro-gravity conditions, the convection vanishes when it is not to strong during hyper and normal gravity phases. This was the case for all the experiments with the molar fraction x CO 2 = 26% . For the larger molar fraction, x CO 2 = 56%, the convection completely vanishes for thermal gradients ∆T = -2 and -1 K. We were optimistic about the possibility to study the NEFs during the reduced-gravity phase in these cases. However, after looking at the image differences, Figure 5.18: Structure functions during the micro-gravity phase, in terms of q for τ = 50 ms to 1.6 s (a), and in terms of τ for q/q min = 10, 25, and 50 (b). The graphs correspond to the image differences presented in figure 5.17. In (c) we displayed a typical Fourier transform of an image difference.

we remark that the NEFs are not visible. This is confirmed after a deeper analysis through the differential dynamic algorithm. The vibrations of the plane combined with its trajectory to achieve the parabola prevent us from studying the non-equilibrium fluctuations in such conditions, due to the loss of the plane of autocorrelation.

Other solutions exist to mimic the microgravity conditions. For instance, our research group has already performed experiments on-board the International Space Station [START_REF] Mialdun | Preliminary analysis of Diffusion Coefficient Measurements in ternary mIXtures 4 (DCMIX4) experiment on board the international space station[END_REF][START_REF] Triller | Thermodiffusion in ternary mixtures of water/ethanol/triethylene glycol: First report on the DCMIX3-experiments performed on the international space station[END_REF]. While it solves the most of the vibration issues, some are persisting due to the activity of astronauts, g-jitters, and the movements of the ISS. Micro-gravity drop towers reduce these vibrations but the available time slot under micro-gravity conditions is too small to study the NEFs. Another solution is the use of micro-gravity rockets. However, the best solution to reproduce this experiment successfully can be the use of a satellite dedicated to micro-gravity [START_REF] Vailati | Fractal fronts of diffusion in microgravity[END_REF]. In that case, the vibrations would be better controlled than in any other solution. It also has the benefits to offer larger micro-gravity periods with an excellent stability of the acceleration along the z-axis.

Chapter 6 Conclusion and perspectives

The recent growing concerns about climate change pushed the scientists to develop mitigation solutions. The storage of CO 2 in deep saline aquifers is a promising solution to reduce CO 2 emissions. It involves convective dissolution that largely enhances the dissolution rate of CO 2 in brine. Indeed, after the injection of CO 2 , there is a stratification of a CO 2 -rich gaseous layer above a brine-rich liquid layer in which CO 2 starts dissolving. An intermediate CO 2 -enriched layer of brine, denser than brine, is formed at the gas-liquid interface and could destabilise in the shape of fingerings. These latter bring huge quantity of CO 2 downward. In the literature, we noticed a lack of experimental studies conducted in a three-dimensional cell, and under thermodynamic conditions relevant to the saline aquifers. Therefore, this thesis aimed at developing a reliable set-up to investigate the behaviour of the convective dissolution of CO 2 in brine in three dimensions to understand the influence of several physico-chemical and thermodynamic parameters on this phenomenon.

We have developed an experimental set-up consisting in a three-dimensional cylindrical cell embedded in a shadowgraph system, to study the convective dissolution. Inside it, CO 2 dissolves into the brine and forms downward plumes that locally vary the refractive index. This results in darker areas on the shadowgraph images. From them, we are able to study the convective dissolution both spatially and temporally, and the influence of chemical and thermophysical parameters, such as the salt concentration and the pressure difference imposed at the start of the experiment.

From the analysis of the variance of shadowgraph image differences as a function of time, we have shown that we are able to extract numerous information: the onset time of convection, t c , the time at which the dissolution rate is maximum t m , a latency time after which the variance reaches a plateau t r , and t sd , the shut-down time of convection. Despite a turbulent injection of CO 2 , it dissolves inside the liquid phase between t = 0 and t c . Here, the regime is purely diffusive. An intermediate mixing liquid layer composed of CO 2 and brine grows diffusively between the gas-liquid interface and the bulk. After t c , the latter starts to destabilise, leading to convective patterns in the form of tubes interconnected by downward plumes. The variance increases as the patterns appear all over the entire bulk and gain contrast, to finally reach a maximum of variance at t m . At this moment, the dissolution rate is maximum. Thereafter, two exponential relaxations are observable in the variance curve. The first one is explained by the decrease of the density difference between the highly concentrated intermediate layer and the bulk, as the mixing has already started. Here, the convective patterns lose contrast, but their spatial density remains constant. The second relaxation is caused by the diffusive vanishing of the convective patterns until the complete shut-down of convection. During this period, the contrast and the spatial density of the patterns decrease down to its initial value and zero, respectively.

We also developed an analysis method to characterise the spatial behaviour of the convective dissolution. Since we could not perform a systematic analysis that provides us with quantitative values directly, like it is case for the variance in the temporal analysis, we determined the width and the spatial density of plumes for relevant experiments at different times. This was done for carefully selected experiments. The spatial analysis completes the temporal one allowing us to build the complete history of the convective dissolution of CO 2 in brine.

We find remarkable results that improve our general understanding on the process in three-dimension. First, we demonstrated both theoretically and experimentally that the onset time of convection is defined by an universal scaling law, which is independent of the type of brine, brine concentration, and system geometry. The dimensionless convective time τ c = t c /t D allows us to transfer our findings to any type of system in free medium. Furthermore, our conclusion provides a different view with respect to the one reported in the literature. Indeed, it is commonly accepted that the presence of a salt in the brine hinders the development of the convective instability. This results is true in some cases, but there is one study that prove the opposite [START_REF] Loodts | Dissolutiondriven convection in reactive systems[END_REF]. In our view, the presence of a salt is changing the thermodynamic properties of the system, explaining the shift towards the longer or shorter time for the onset time of convection. This conclusion includes now every type of brine and every system geometry.

From a deeper analysis of the boundary layer and the diffusive and convective fluxes, we brought further information on the solutal Nusselt number. The latter characterises the efficiency of the convective flux with respect to the diffusive one in dissolving the CO 2 into the brine. It gives direct information on the efficiency of the storage process. Again, we provided a scaling law only depending on the solutal Rayleigh number, showing that, the larger the height of the aquifer, the more efficient the convective dissolution process.

During this Ph.D., we also studied the 1-hexanol-CO 2 system. The latter mimics the CO 2 -water system with the advantage of having access to a broader ranger of dissolved CO 2 , providing more optically contrasted results. This is an important feature to analyse the spatio-temporal behaviour of density non-equilibrium fluctuations (NEFs) occurring naturally in presence of gradients. We know that the NEFs, combined with the gravity field, are at the origin of the convection, but the NEFs were not visible with the system CO 2 -brine due to their weak optical contrast and the fast appearance of convection hindering their visualisation. Then, the idea was to study the system 1-hexanol-CO 2 under terrestrial and reduced gravity conditions to check the interplay between NEFs and Soret-induced convection. All measurements were performed in thermodiffusion experiments. The experiments cited here consist in a first step in the investigation on this system, before performing convective dissolution. However, we had mitigated results under reduced-gravity. Indeed, the airplane produces a lot of vibrations that make us lose the autocorrelation plane in the shadowgraph images. Other solutions such as sending rockets or satellites are envisaged. Despite this, we can realise similar experiments in a satellite that reproduces reduced gravity conditions with much less vibrations than on the airplane, and for a longer period. It would be quite interesting to perform convective dissolution with the 1-hexanol-CO 2 system. With the latter, we dissolve more CO 2 in the liquid phase and, thus, we could reach larger solutal Rayleigh numbers without injecting CO 2 at supercritical state, thus avoiding the turbulent flow induced by the phase change that hinders the study of convective dissolution. This can be interesting to check weither or not there is a break of the scaling law at very large Rayleigh numbers. This doctoral work represents a solid basis for the analysis of the convective dissolution of CO 2 in brine in a three-dimensional system. However, there are questions that remain open and some improvements can be made. The first optimisation that must be done concerns the sample cell. In order to vary the fluid layer thickness and facilitate the insertion of a porous medium, a thicker cell is needed. This would also allow to study the double convective dissolution process in a combined free and porous geometry. A second point with respect to the design of a new sample cell is to increase by up to five times the inner diameter to prevent from meniscus effects with the benefit of increasing the optical path area which will, in turn, increase of statistics and decrease uncertainties. For large optical accesses, intra-or extra-focal shadowgraph method is needed.

If another cell is developed, we can implement also a multi-shadowgraph system. Currently, our cell provides an optical access in one direction. We can flip the set-up to observe the instability in another direction, but the geometry would change and the measurements are not simultaneous. An idea would be to develop an hexagonal cell, with three optical access, providing a direct three-dimensional vision of the instability. Thus, we could realise three dimensional reconstruction and obtain the flow of convective patterns in the entire bulk. It would also have the benefit to observe directly the position of the gas-liquid interface and, thus, measure the liquid layer thickness, that was not possible in the vertical configuration of our current set-up.

The experiments reported here have been performed in free-medium while the saline aquifers are made of porous media. That was a necessary step to precisely understand the instability behaviour in a three-dimension geometry. Obviously, the next step involves 3D porous media. To do so, we can model a transparent porous medium, e.g., glass beads, and realise refractive index matching between the transparent beads and the liquid phase. Here, by varying the size of glass spheres and their size dispersion, the influence of pore size and pore size distribution will be studied. Real porous media, in form of silica monoliths, show a double porosity centred at about 10 nm and 10 µm and would, for sure, provide a breakthrough in our studies

In our study, we used pure water and brine composed of NaCl at multiple composition. However, the brine in the saline aquifers generally contains several salts. The multicomponent brine could form a reactive system with the dissolved CO 2 . This could have the effect to hinder or enhance the convective dissolution and the dissolution rate. Thus, it is important to investigate how multicomponent brine mixture and reactive systems influence the scaling law derived in this thesis.

All these perspectives would enhance our understanding of the convective dissolution process. Many parameters would be tested to observe their influence at the laboratory scale. Then, all the results would be implemented in models at the reservoir scale. That will help to better evaluate the influence of convective dissolution on carbon storage projects, from the choice of the saline aquifer to the optimisation of the injection of CO 2 during the storage phase. Finally, our research results will contribute in the reduction of our emissions of CO 2 and mitigate the global warming.

1. 1

 1 Evolution of the global surface temperature relative to 1850-1900 . . . 1.2 Evolution of the globally averaged sea level change . . . . . . . . . . . 1.3 Atmospheric greenhouse gas concentrations for multiple greenhouse gases 1.4 Future annual GHGs emissions and their contribution to global warming 1.5 Global CO 2 emissions reductions by energy sector . . . . . . . . . . . . 1.6 Time scale of the trapping mechanisms . . . . . . . . . . . . . . . . . . 2.1 p-T phase diagram of CO 2 . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 p-T phase diagram of the mixture CO 2 -H 2 O . . . . . . . . . . . . . . . 2.3 p-x projection of phase diagram of the mixture CO 2 -H 2 O . . . . . . . 2.4 Schematic cross section of the sample . . . . . . . . . . . . . . . . . . . 2.5 Evolution of p, ln(p i -p eq ), and the pseudo-diffusive time . . . . . . . . 3.1 Schematics of the experimental set-up . . . . . . . . . . . . . . . . . . . 3.2 Explicative schematics of the effects of a concave or convex gas-liquid interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Schematic of the main hydraulic circuit of the experimental set-up . . . 3.4 Pressure evolution with time . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Pictures showing the three insulation layers . . . . . . . . . . . . . . . 3.6 Pressure decay for an injection from 0.1 to 2.0 MPa . . . . . . . . . . . 3.7 Pictures of the experimental set-up . . . . . . . . . . . . . . . . . . . . 3.8 Shadowgraph images in two configurations . . . . . . . . . . . . . . . . 4.1 Temporal evolution of the variance . . . . . . . . . . . . . . . . . . . . 4.2 Schematic of the plumes . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Width of the pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Number of nodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Fingering speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Solubility of CO 2 as a function of its pressure . . . . . . . . . . . . . . 4.7 Viscosity of CO 2 -brine mixtures as a function of pressure . . . . . . . . 4.8 Density of CO 2 -saturated pure water as a function of pressure . . . . . 4.9 Density of CO 2 -saturated brine in terms of pressure . . . . . . . . . . . 4.10 Diffusion coefficient of CO 2 in pure water vs. temperature . . . . . . . 4.11 Diffusion coefficient of CO 2 in pure water vs. pressure . . . . . . . . . . v 4.12 Error due to the size of the optical pathway . . . . . . . . . . . . . . . 4.13 Normalized pressure evolution at early times . . . . . . . . . . . . . . . 4.14 Variance curve showing two maxima . . . . . . . . . . . . . . . . . . . 4.15 Visual explanation of the error made on the spatial density of plume . . 4.16 Explanation of the error made on the measurement of the width of the plumes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.17 Variations of the contact angle CO 2 -brine due to variation of pressure . 4.18 Onset time vs. Ra s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.19 Onset time of convection vs. h . . . . . . . . . . . . . . . . . . . . . . . 4.20 Maximum variance time as a function of Ra s . . . . . . . . . . . . . . . 4.21 Characteristic decay times . . . . . . . . . . . . . . . . . . . . . . . . . 4.22 Shut down time vs. Ra s . . . . . . . . . . . . . . . . . . . . . . . . . . 4.23 Plumes width vs. time (∆p = 6 MPa) . . . . . . . . . . . . . . . . . . . 4.24 Plumes width vs. time (several ∆p) . . . . . . . . . . . . . . . . . . . . 4.25 Plume density vs. time (∆p = 6 MPa) . . . . . . . . . . . . . . . . . . 4.26 Plume density vs. time (several ∆p) . . . . . . . . . . . . . . . . . . . . 4.27 Variance curves for three molalities and a similar pressure jump . . . . 4.28 Onset time of convection vs. Ra s , (H 2 O + NaCl) brine at 1 mol • kg -1 . 4.29 Onset time of convection vs. Ra s , (H 2 O + NaCl) brine at 3 mol • kg -1 . 4.30 Time of the maximum variance vs. Ra s , (H 2 O + NaCl) brine at 1 mol • kg -1 4.31 Time of the maximum of variance vs. Ra s , (H 2 O + NaCl) brine at 3 mol • kg -1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.32 Shutdown time of convection vs. Ra s , (H 2 O + NaCl) brine at 1 and 3 mol • kg -1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.33 Width of the plumes in a 3 mol • kg -1 H 2 O-NaCl brine . . . . . . . . . 4.34 Width of the plumes for three brine concentrations at ∆p = 2 MPa . . 4.35 Evolution of the spatial density of plumes for brine concentrated at 1 mol • kg -1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.36 Evolution of the spatial density of plumes for brine concentrated at 3 mol • kg -1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.37 Evolution of the spatial density of plumes for brine concentrations of 0, 1, and 3 mol • kg -1 and ∆p= 4 MPa . . . . . . . . . . . . . . . . . . . . 4.38 Dimensionless onset time of convection for experiments performed both in pure water and in brine . . . . . . . . . . . . . . . . . . . . . . . . . 4.39 N u s vs. Ra s for all brine concentrations . . . . . . . . . . . . . . . . . . 4.40 Temporal evolution of the variance with precision about te regimes. . .

. 4 . 4 4 . 4 . 6

 44446 Convective dissolution of CO 2 in pure water . . . . . . . . . . . . . . . 4.4.1 Onset time of convection . . . . . . . . . . . . . . . . . . . . . . 4.4.2 t m : maximum variance . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Characteristic decay times of convection . . . . . . . . . . . . . 4.4.4 Shutdown time of convection . . . . . . . . . . . . . . . . . . . . 4.4.5 Width of the plumes . . . . . . . . . . . . . . . . . . . . . . . . x Spatial density of the plumes . . . . . . . . . . . . . . . . . . . 4.5 Convective dissolution of CO2 in brine . . . . . . . . . . . . . . . . . . 4.5.1 Temporal analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Spatial analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Influence of the thermodynamic and physico-chemical parameters . . . 4.6.1 Density difference . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.2 Diffusion coefficient . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.3 Viscosity of the brine-rich liquid phase . . . . . . . . . . . . . . 4.6.4 Brine thickness . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.5 Conclusion on the influence of the thermodynamic parameters . 4.7 Modeling of relevant parameters . . . . . . . . . . . . . . . . . . . . . . 4.7.1 Boundary layer . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.2 Dimensionless critical time . . . . . . . . . . . . . . . . . . . . . 4.7.3 Investigations on the dissolution rate and convective flux: the Nusselt number . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.4 Discussion on up-scaling to relevant CCS conditions . . . . . . . 4.8 The complete phenomenology of the convective dissolution . . . . . . . 5 Preliminary investigations on the influence of the Soret effect on carbon storage -Study of the model system 1-hexanol-CO 2 under terrestrial and reduced gravity conditions 5.1 Further motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Link with the convective dissolution . . . . . . . . . . . . . . . . 5.1.2 Fluid-phase equilibria of the 1-hexanol-CO 2 system . . . . . . . 5.2 Sample preparation and experimental set-ups . . . . . . . . . . . . . . 5.2.1 Vapour-liquid equilibrium (VLE) apparatus . . . . . . . . . . . 5.2.2 Description of the shadowgraph set-up . . . . . . . . . . . . . . 5.2.3 Parabolic flight campaign at Novespace . . . . . . . . . . . . . . 5.3 Fundamentals for the determination of transport properties from the analysis of NEFs within a thermodiffusion experiment using the shadowgraph method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 The Soret effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Density non-equilibrium fluctuations (NEFs) . . . . . . . . . . . 5.3.3 Analysis of shadowgraph images . . . . . . . . . . . . . . . . . . 5.4 Ground-based results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Mixture at x CO 2 ≈ 10% . . . . . . . . . . . . . . . . . . . . . . 5.4.2 Mixture at x CO 2 ≈ 40% . . . . . . . . . . . . . . . . . . . . . . 5.4.3 Conclusion on preliminary experiment on Earth . . . . . . . . . 5.5 Results obtained under reduced-gravity conditions . . . . . . . . . . . . 5.5.1 Experimental protocol . . . . . . . . . . . . . . . . . . . . . . . 5.5.2 Preliminary observations . . . . . . . . . . . . . . . . . . . . . . 5.5.3 Variance analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi Chapter 1

Figure 1 . 1 :Figure 1 . 2 :

 1112 Figure 1.1: Evolution of the global surface temperature relative to the period 1850-1900. Panel (a): Decadally averaged global surface temperatures relative to 1850-1900 obtained from paleoclimate archives and from direct observation. The bar on the left shows the estimated temperature during the warmest period in the last 100,000 years. The very likely ranges of the temperature reconstruction is highlighted here with the grey shading with white lines. Panel (b): Annually averaged global surface temperature relative to 1850-1900 (black line). There are two simulations : one taking into account the human and natural factors (brown line), and the other the natural factor only (blue-green line).The confidence level of the simulations are given with the coloured shades.[START_REF] Masson-Delmotte | Summary for policymakers[END_REF]

Figure 1 . 4 :

 14 Figure 1.4: Future annual GHGs emissions and their contribution to global warming across the five scenarios: SSP1-1.9, SSP1-2.6, SSP2-4.5, SSP3-7.0, and SSP5-8.5 [101].

Figure 1 . 6 :

 16 Figure 1.6: Graphical illustration of the time scale of the trapping mechanisms and their contribution to the storage [104].

Figure 2 . 1 :

 21 Figure 2.1: p-T phase diagram of CO 2 using data from Span and Wagner (1996) [141].In yellow, we highlighted the critical temperature and pressure.

Figure 2 . 2 :

 22 Figure 2.2: p-T phase diagram of the mixture CO 2 -H 2 O in the temperature range 273-313 K, using data from Spycher et al. (2002) [143].

Figure 2 . 3 :

 23 Figure 2.3: p-x projection of phase diagram of the mixture CO 2 -H 2 O in the pressure range 0-10 MPa. The solubility for isotherms are given for temperature between 293.15 and 323.15 K (dashed lines). The data come from Diamond & Akinfiev (2003) [35].

Figure 2 . 4 :

 24 Figure 2.4: Schematic cross section of the sample showing the simplified geometry of our problem at different times, where two layers of gaseous CO 2 and liquid brine are superimposed. (a) and (c) are at equilibrium for pressures p 0 and p eq , respectively. (b)is a transient step where the pressure of the gaseous phase has been risen and CO 2 starts to dissolve in the liquid phase forming a boundary layer. c 0 and c eq are the equilibrium concentrations at p 0 and p eq , respectively. Here, the concentration in the boundary layer is given as a constant equal to the concentration at the gas-liquid interface, which is true in most of the boundary layer. g is the acceleration of gravity, h the height of the liquid layer, z the axis parallel to the gravity acceleration pointing downward, and d(t) the height of the boundary layer growing with time.

Figure 2 . 5 :

 25 Figure 2.5: Evolution of the pressure (in black), the non dimensionless pressure ln(p i -p eq ) (in red), and the extracted pseudo diffusive coefficient (in blue) (figure from Zhao et al. (2017) [183]).

  [START_REF] Zhang | Experimental and numerical investigation on interfacial mass transfer mechanism for rayleigh convection in Hele-Shaw cell[END_REF] [START_REF] Zhang | Experimental and numerical investigation on interfacial mass transfer mechanism for rayleigh convection in Hele-Shaw cell[END_REF] developed a cell that has a Hele-Shaw-like geometry. However, the gap between the two plates (e = 3 mm) is too large to consider this cell as a real Hele-Shaw cell. Khosrokavar et al. (2014)[START_REF] Khosrokhavar | Visualization and investigation of natural convection flow of CO 2 in aqueous and oleic systems[END_REF] and[START_REF] Mahmoodpour | Convective dissolution of carbon dioxide in deep saline aquifers: Insights from engineering a high-pressure porous Hele-Shaw cell[END_REF] [START_REF] Mahmoodpour | Convective dissolution of carbon dioxide in deep saline aquifers: Insights from engineering a high-pressure porous Hele-Shaw cell[END_REF] also performed experiments in a three-dimensional bulk.[START_REF] Nadal | Stationary plume induced by carbon dioxide dissolution[END_REF] [START_REF] Nadal | Stationary plume induced by carbon dioxide dissolution[END_REF] studied the dynamics of a single plume in a cylindrical cell. Scaling laws are given for the width and velocity of the plume and for the Nusselt number in terms of Rayleigh number.

Figure 3 . 1 :

 31 Figure 3.1: Schematics of the optical set-up, showing the part dedicated to the shadowgraph (left), and the high-pressure cell (right). The white transparent rectangle with the dashed perimeter corresponds to the optical pathway.

Figure 3 . 2 :

 32 Figure 3.2: Explicative schematics of the effects of a concave or convex gas-liquid interface (right) with the corresponding shadowgraph images (left).

Figure 3 . 3 :

 33 Figure 3.3: Schematic of the hydraulic circuit. It shows the two systems to inject CO 2 and brine with the help of a combination of two Teledyne ISCO pumps and a syringe pump, respectively.

Figure 3 . 4 :

 34 Figure 3.4: Pressure evolution with time for three injection of CO 2 around 4.0 MPa.

Figure 3 . 5 :

 35 Figure 3.5: Pictures showing the three insulation layers of the containing cell. The first layer is in contact with the Peltier elements and the aluminium plates, the second layer is placed on the first one and within the stainless steel blocks, and the third layer is placed around the entire block.

Figure 3 . 6 :

 36 Figure 3.6: Pressure decay of two experiments for an injection from 0.1 to 2.0 MPa after we enhanced the set-up with insulation layers.

Figure 3 . 7 :

 37 Figure 3.7: Pictures of the experimental set-up in both horizontal and vertical configurations.

Figure 3 . 8 :

 38 Figure 3.8: Shadowgraph images obtained in the horizontal configuration (a) and vertical configuration (b) after the injection of CO 2 in the sample cell at 4.1 MPa. In (a), plumes are superimposed in the x-direction since their mean diameter is much smaller than the thickness of the cell.

Figure 4 . 2 :

 42 Figure 4.2: Schematic of the plumes both represented when they are connected by tubes or isolated. On the left part of the scheme, we highlighted two characteristic numbers: w, the width of one plume, and Λ, the mean distance between two plumes.

Figure 4 . 3 :

 43 Figure 4.3: Width of the convective pattern. On (a) we displayed an entire shadowgraph image. On (b) we keep only one pattern given by the green square on (a). We select a line over the diameter of the pattern in (b) with ImageJ. Then, we plot the gray level over this line (c), and measure the width of the pattern, w, as the distance between the two gray level maxima. Pure water, ∆p = 2 MPa, t = 42.5 s.

Figure 4 . 4 :

 44 Figure 4.4: Number of nodes in one shadowgraph image. Pure water, ∆p = 6 MPa, t = 17.5 s.

Figure 4 . 5 :

 45 Figure 4.5: Shadowgraph images showing how we measure the fingering speed with the set-up in its horizontal configuration. The position is taken at the bottom of the plume and is highlighted by the black line on the images.

Figure 4 . 6 :

 46 Figure 4.6: Solubility of CO 2 as a function of the pressure, at T = 308.15 K, for three molalities of salt: 0 mol • kg -1 (blue triangles), 1 mol • kg -1 (orange diamonds), and 3 mol • kg -1 (yellow squares). The experimental data are taken from Wang et al. (2019)[START_REF] Wang | Determination of CO 2 solubility in water and NaCl solutions under geological sequestration conditions using a fused silica capillary cell with in situ raman spectroscopy[END_REF] and derived from the model of[START_REF] Duan | An improved model calculating co 2 solubility in pure water and aqueous NaCl solutions from 273 to 533 K and from 0 to 2000 bar[END_REF] [START_REF] Duan | An improved model calculating co 2 solubility in pure water and aqueous NaCl solutions from 273 to 533 K and from 0 to 2000 bar[END_REF]. The solid lines show the models extracted for our study. The crosses are data computed with PhreeqC.

Figure 4 . 7 :

 47 Figure 4.7: Viscosity of CO 2 -brine mixtures as a function of pressure at T = 308.15 K, for three molalities of salt: 0 mol • kg -1 (yellow squares), 1 mol • kg -1 (blue triangles), and 3 mol • kg -1 (orange diamonds). The experimental data are taken from Kestin et al. (1981) [77]. The solid lines correspond to the linear fit experimental data.

Figure 4 . 8 :

 48 Figure 4.8: Density of CO 2 -saturated pure water as a function of pressure. The blue squares are derived from experimental data points from Hebach et al. (2004) [62] at T = 308.15 K. The orange line is the model derived from the blue squares. The gray triangles are data computed with PhreeqC, at T = 308.15 K.

Figure 4 . 10 :

 410 Figure 4.10: Literature data of the diffusion coefficients of CO 2 in pure water as a function of the temperature. The blue solid line represents the Arrhenius-like fit to the data.

Figure 4 . 11 :

 411 Figure 4.11: Experimental data of the diffusion coefficients of CO 2 in pure water as a function of the pressure. These data do not shows any clear dependence of the diffusion coefficient with the pressure.

Figure 4 . 14 :

 414 Figure 4.14: Variance curve showing two maxima delayed by 6 s.

Figure 4 . 15 :

 415 Figure 4.15: Visual explanation of the error made on the spatial density of plume. (b)-(e) show zoomed areas where we can see patterns along a tube (b), distinct patterns (c), or merging patterns that are counted as two distinct patterns (d) and as one pattern (e).

Figure 4 . 16 :

 416 Figure 4.16: Visual explanation of the error made on the measurement of the plume width. On the left, an entire shadowgraph image of an experiments made in pure water for ∆p = 1 MPa is displayed. A zoom on one pattern (up right) with a green line on it and the related the gray level. It shows the difficulty in defining two maxima due to the blurred pattern.

Figure 4 . 17 :

 417 Figure 4.17: Images giving the contact angle of the CO 2 -rich gaseous phase and the brine-rich liquid phase for multiple pressure of CO 2 : 0.1 MPa (a), 2 MPa (b), 5 MPa (c), and 8 MPa (d). The brine used here is deionized water and 7 wt % of potassium iodide (KI). Images from Li et al. (2017) [90].

Figure 4 . 18 :

 418 Figure 4.18: Log-log plot of the onset time of convection vs. the solutal Rayleigh number. The experimental data points are given in yellow squares with their measured uncertainties. A fit obtained with Matlab is displayed (black line) with a 95% confidence area given by the blue area. The fit is given by equation (4.7).

Figure 4 . 19 :

 419 Figure 4.19: Onset time of convection for multiple water layer thickness. The yellow squares are the experimental data. The black zone on the left is the zone where no convection can appear because h < δ c . The pressure goes from 0.1 to 6.1 MPa.

Figure 4 . 20 :

 420 Figure 4.20: Maximum variance time as a function of Ra s . It follows a similar trend than t c . A power law behaviour is visible also in this case, as given by equation (4.8).

Figure 4 .

 4 Figure 4.21: Semi-log plot of the characteristic decay times of the variance, τ 1 and τ 2 , in terms of Ra s . The values displayed here are the mean value of the experiments having the same pressure difference, for ∆p varying between 1 to 6 MPa.

Figure 4 .

 4 Figure 4.22: Semi-log plot of the shutdown time of convection vs. the solutal Rayleigh number. The experimental data points are given in yellow squares with their measured uncertainties.

Figure 4 . 23 :

 423 Figure 4.23: Temporal evolution of the width of the plumes for an injection from 0.1 to 6.1 MPa. The transparent zones shows the width range of the plumes, and the squares the width mean values. The timings given in figure 4.1 are indicated in the abscissa.

Figure 4 . 24 :

 424 Figure 4.24: Temporal evolution of the width of the plumes for several ∆p: 1 MPa (circles), 2 MPa (diamonds), 4 MPa (triangles), and 6 MPa (squares). For each pressure difference, we measured the width of the pattern for 2 different experiments and show their average.

Figure 4

 4 Figure 4.[START_REF] Croccolo | Non-local fluctuation phenomena in liquids[END_REF] shows the spatial density of the plumes for an experimental run with an injection from 0.1 to 6.1 MPa. From this graph, we can make several remarks. After t c , the first convective pattern appears with the first plumes. The plume density increases monotonically up to t m , and reaches a first local maximum. After a little drop, it increases again to reach a maximum around t r . Then, the spatial density of plumes decreases quasi exponentially again and reaches a plateau. Finally, it decreases and goes back to zero several seconds after t sd .

Figure 4 . 25 :

 425 Figure 4.25: Temporal evolution of the spatial density of the plumes, for an injection from 0.1 to 6.1 MPa. The timings given in figure 4.1 are indicated in the abscissa.

Figure 4 . 26 :

 426 Figure 4.26: Temporal evolution of the spatial density of the plumes for four pressure differences: 1 MPa (yellow circles), 2 MPa (blue diamonds), 4 MPa (orange triangles), and 6 MPa (green squares). The timings given in figure 4.1 are indicated in the abscissa.

  .9) and t c,3mol•kg -1 = (9.2 ± 32) × 10 4 • Ra -(0.68±0.23)

Figure 4 . 27 :

 427 Figure 4.27: Variance curves for three molalities (pure water in blue, NaCl brine at 1 mol • kg -1 in orange and NaCl brine at 3 mol • kg -1 in gray) and a similar pressure jump ∆p = 2 MPa.

Figure 4 . 28 :

 428 Figure 4.28: Onset time of convection vs. Ra s . The experiments have been performed in a (H 2 O + NaCl) brine at 1 mol • kg -1 . The fit is given in equation (4.9).

Figure 4 . 29 :

 429 Figure 4.29: Onset time of convection vs. Ra s . The experiments have been performed in a (H 2 O + NaCl) brine at 3 mol • kg -1 . The fit is given in equation (4.10).

Figure 4 . 30 :

 430 Figure 4.30: Time of the maximum variance vs. Ra s . The experiments have been performed in a (H 2 O + NaCl) brine at 1 mol • kg -1 . The fit is given in equation (4.11).

Figure 4 . 31 :

 431 Figure 4.31: Time of the maximum of variance vs. Ra s . The experiments have been performed in a (H 2 O + NaCl) brine at 3 mol • kg -1 . The fit is given in equation (4.12).

Figure 4 . 32 :

 432 Figure 4.32: Shutdown time of convection vs. the solutal Rayleigh number. The experiments have been performed in a (H 2 O + NaCl) brine at 1 mol • kg -1 (blue triangles), and 3 mol • kg -1 (orange diamonds).

Figure 4 . 33 :

 433 Figure 4.33: Width of the plumes in a 3 mol • kg -1 H 2 O-NaCl brine. Multiple pressure differences, imposed at the beginning of the experiment, are presented here: 4 MPa (yellow squares), 2 MPa (blue triangles), and 1 MPa (orange diamonds).

Figure 4 . 34 :

 434 Figure 4.34: Width of the plumes for an injection from 0.1 MPa to 2.1 MPa in NaCl brine concentrated at 0 mol • kg -1 (yellow squares), 1 mol • kg -1 (blue triangles), and 3 mol • kg -1 (orange diamonds).

Figure 4 . 35 :

 435 Figure 4.35: Evolution of the spatial density of plumes for H 2 O-NaCl brine concentrated at 1 mol • kg -1 at multiple pressure difference: 4 MPa (orange triangles), 2MPa (blue diamonds), and 1 MPa (yellow squares).

Figure 4 . 36 :

 436 Figure 4.36: Evolution of the spatial density of plumes for H 2 O-NaCl brine concentrated at 3 mol • kg -1 at multiple pressure difference: 4 MPa (orange triangles), 2MPa (blue diamonds), and 1 MPa (yellow squares).

Figure 4 . 37 :

 437 Figure 4.37: Evolution of the spatial density of plumes for H 2 O-NaCl brine concentrated at 0 (yellow squares), 1 (blue triangles), and 3 mol • kg -1 (orange diamonds), at ∆p = 4 MPa.

Figure 4 . 38 :

 438 Figure 4.38: Dimensionless onset time of convection for experiments performed both in pure water and in H 2 O-NaCl brine at concentrations of 1 mol • kg -1 and 3 mol • kg -1 .

Figure 4 . 39 :

 439 Figure 4.39: Evolution of the solutal Nusselt number as a function of the solutal Rayleigh number for the brine concentrated at 0 (yellow squares), 1 (blue triangles), and 3 mol • kg -1 (orange diamonds).

Figure 4 . 40 :

 440 Figure 4.40: Temporal evolution of the variance of the shadowgraph images. Multiple regimes have been highlighted on this figure. First, the dissolved CO 2 spreads inside the liquid layer by pure Fickian's diffusion. Then, convection starts with the shape of fingerings first, and, next, columnar flow at constant flux, as shown on the images obtained with the set-up in its horizontal configuration. Then, the convective patterns vanish diffusively. Finally, we go back to a pure diffusive regime once the patterns have vanished entirely to reach the final equilibrium. To illustrate the different regimes, the same images than in figure 4.1 have been displayed, such as schemes of two-dimensional transversal sections explaining what is happening in all the regimes.

Figure 5 . 1 :

 51 Figure 5.1: p -T phase diagram of the system 1-hexanol-CO 2 . It shows the two critical lines (---) starting from the critical points of the pure components (CP1 and CP2).The UCEP is crossing the critical line linked to the CP of CO 2 and the tri-phase line (-• -). The solid lines are the curve of pressure of the pure components. Data taken from Beier et al.[START_REF] Beier | High-pressure phase equilibria of carbon dioxide + 1-hexanol at 303[END_REF]

Figure 5 . 2 :

 52 Figure 5.2: p-x CO 2 phase diagram of the system 1-hexanol-CO 2 . It shows experimental data for the vapor-liquid equilibrium and the vapor-liquid-liquid equilibrium at three isotherms: T = 303.15 K (orange), T = 313.15 K (green), and T = 323.15 K. Data taken from [9, 42, 123, 131, 174].

Figure 5 . 3 :

 53 Figure 5.3: Scheme of the hydraulic circuit made to prepare the mixtures of 1-hexanol-CO 2 and their injection inside the cell.

Figure 5 . 4 :

 54 Figure 5.4: Top view of two cuvettes containing non-filtered and filtered 1-hexanol. A laser beam positioned normal to the windows passes through the bulk.

Figure 5 . 5 :

 55 Figure 5.5: Picture of the set-up "rack 1" used under reduced gravity.

Figure 5 . 9 :

 59 Figure 5.9: Raw image (a), image difference for τ = 100 ms (b), Fourier transform associated to the previous image (c), normalized structure functions plotted in terms of q for various τ (d), and in terms of τ for various q (e), at p = 15 MPa and ∆T = 20 K.

Figure 5 . 10 :

 510 Figure 5.10: Raw image (a), normalized image difference for τ = 100 and 500 ms (b), Fourier transform associated to the previous image for τ = 100 ms (c). Three thermodynamic conditions are shown here, from top to bottom: p = 24 MPa, ∆T = 18 K, and T = 313 K; p = 13 MPa, ∆T = 15 K, and T = 303 K; p = 7 MPa, ∆T = 15 K, and T = 303 K.

Figure 5 . 11 :

 511 Figure 5.11: Temporal structure function vs. the time difference τ , for 3 pressures p = 24 (blue), 13 (orange), and 7(yellow) MPa.The wave vectors are q/q min = 15 on the left, and q/q min = 50 on the right

Figure 5 . 12 :

 512 Figure 5.12: Comparison between images in 2g and 0g. x CO 2 = 56 mol. % , ∆T = 6 K, T = 298 K, p = 7.69 MPa.

Figure 5 . 13 :

 513 Figure 5.13: Evolution of the convective patterns during transition to 0g phase. Top row, the higher concentration of CO 2 in the mixture induces more contrasted convective patterns that persist during the micro-gravity phase. Bottom row, the concentration in CO 2 is lower and the patterns have vanished after the transition to reduced-gravity.

Figure 5 . 14 :

 514 Figure 5.14: Evolution of the image variance during a parabola. On the left, we display the images denoted by the letters (a-i) on the variance curve on the right.

Figure 5 . 15 :

 515 Figure 5.15: Multiple variance curves computed for experiments performed under similar thermodynamic conditions. In blue is displayed the micro-gravity period. x CO 2 = 56 %, ∆T = -2 K.

Figure 5 . 16 :

 516 Figure 5.16: Multiple variance curves computed for multiple thermal gradients, ∆T = -1 (yellow), -2 (blue), -4 (grey), and -6 K (orange). In blue is displayed the micro-gravity period.

Figure B. 1 :

 1 Figure B.1: Experimental Ra s , t c , t m , and N u s and their measurement errors for experiments conducted in pure water.

Figure B. 2 :

 2 Figure B.2: Experimental Ra s , t c , t m , and N u s and their measurement errors for experiments conducted in brine concentrated at 1 mol • kg -1 .

Figure B. 3 :

 3 Figure B.3: Experimental Ra s , t c , t m , and N u s and their measurement errors for experiments conducted in brine concentrated at 3 mol • kg -1 .

  Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.1 Ecological and socio-economical context . . . . . . . . . . . . . 2 1.1.2 Mitigation solutions . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2 Carbon Capture and Storage (CCS) . . . . . . . . . . . . . . . . . . . . 8 1.2.1 General concept . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2.2 Capture techniques . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2.3 Transportation . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.4 Injection sites . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.5 Trapping processes . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.6 Risks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 CO2ES Chair . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 This thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Fundamentals and state-of-the-art on convective dissolution 2.1 Fluid-phase equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Thermodynamic fundamentals . . . . . . . . . . . . . . . . . . . 2.1.2 Fugacities in liquid mixtures . . . . . . . . . . . . . . . . . . . . Convective dissolution . . . . . . . . . . . . . . . . . . . . . . . . . . . ix 2.2.1 Fickian's diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Description of the convective dissolution process . . . . . . . . . 2.3 State-of-the-art on convective dissolution of CO 2 in water and salted solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Experimental studies . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Position of this thesis in the literature . . . . . . . . . . . . . . Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 The experimental cell . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Hydraulic circuit and pressure control . . . . . . . . . . . . . . . 3.1.3 Temperature control . . . . . . . . . . . . . . . . . . . . . . . . 3.1.4 Optical table . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.5 Optical arrangement . . . . . . . . . . . . . . . . . . . . . . . . 3.1.6 Horizontal and vertical configurations . . . . . . . . . . . . . . . 3.1.7 Acquisition of housekeeping data and images . . . . . . . . . . . 3.2 The shadowgraph method . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Experimental protocol . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Densities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4 Fick diffusion coefficients . . . . . . . . . . . . . . . . . . . . . . 4.2.5 Computation of the Rayleigh number . . . . . . . . . . . . . . . 4.3 Estimation of the experimental uncertainties . . . . . . . . . . . . . . . 4.3.1 Experimental uncertainties . . . . . . . . . . . . . . . . . . . . . 4.3.2 Uncertainties in the computation of the Rayleigh number . . .

	Table of contents
	Abstract Résumé List of figures List of tables Table of contents 1 Introduction 1.1 3 Experimental methods 3.1	i ii iv vii ix 1

2.1.3 The solubility of gases in liquids . . . . . . . . . . . . . . . . . . 2.1.4 The CO 2 -H 2 O system . . . . . . . . . . . . . . . . . . . . . . . . 2.2 3.3.1 Cleaning protocol . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Assembling protocol . . . . . . . . . . . . . . . . . . . . . . . . 3.3.3 Injection of fluids . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 4 Convective dissolution of CO 2 in brine 4.1 Analysis methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Primary qualitative remarks . . . . . . . . . . . . . . . . . . . . 4.1.2 Temporal analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3 Spatial analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Thermophysical properties of interest . . . . . . . . . . . . . . . . . . . 4.2.1 Solubilities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Dynamic viscosities . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3

Table 4 . 1 :

 41 Coefficients for the solubility model described in equation (4.3).

Table 4 .

 4 3: Diffusion coefficients for multiple brine composition and concentration. type of the brine Salt concentration / (mol • kg -1 ) D / (10 -9 × m 2 • s -1 )

	Salt none	0	2.26 ± 0.2
	NaCl	1 3	2.06 ± 0.2 1.7 ± 0.2
	data from [14]. All the values of the diffusion coefficient are summarized in table 4.3.

Table 4 . 4 :

 44 Parameters of the fit used to determine the relation between t c and Ra s , given by equation (4.7). very small compared to the factor of around 22 between the experiments performed at 0.35 MPa and 6 MPa. The results presented here can, therefore, directly be extrapolated to real injection pressures where CO 2 is in supercritical state.

	Method	NonlinearLeastSquares
	Robust	Off
	Algorithm	Trust-Region
	DiffMinChange	1.0 ×10 -8
	DiffMaxChange	0.1
	MaxFunEvals	600
	MaxIter	400
	TolFun	1.0 ×10 -6
	TolX	1.0 ×10 -6

Acknowledgements

iii

4.6 Influence of the thermodynamic and physico-chemical parameters 4. [START_REF] Backhaus | Convective instability and mass transport of diffusion layers in a Hele-Shaw geometry[END_REF]

.1 Density difference

The density difference is undoubtedly the key parameter driving the instability. Indeed, in the gravity field, it is the motor of convection inducing the destabilisation of the boundary layer, that leads to the formation of downward convective patterns. This quantity is directly computed from the quantity of CO 2 dissolved in the brine-rich liquid phase at both the initial and final equilibrium. Thus, it depends directly on the pressure difference imposed in the gaseous phase at the beginning of the experiment and the composition of the brine.

Diffusion coefficient

The Fick diffusion coefficient is another key parameter, that influences the behaviour of the instability at early times. Indeed, a higher diffusion coefficient induces a faster dissolution rate during the pure diffusive regime. The larger the Fick diffusion coefficient is, the faster the boundary layer grows and gets destabilised. Although D is associated with the tracer diffusivity of CO 2 and, thus, independent on the amount of CO 2 dissolved in brine, it varies with brine composition.

Viscosity of the brine-rich liquid phase

The dynamic viscosity of the brine-rich liquid phase has a strong influence on the speed of the downward plumes. An increase in the viscosity decreases strongly the speed of the patterns. It has a second consequence. Indeed, the viscosity effects are stronger when the dynamic viscosity increases, hindering the formation of downward plumes at the liquid-liquid interface. Thus, to form the downward plumes, the patterns need to be larger.

Brine thickness

The brine layer thickness h is another key parameter of the instability since it appears to the power three in the definition of the solutal Rayleigh number. Thus, a change

Appendix B

Experimental data and measurement errors