Keywords: Particle-laden flow, Particle trajectory, Corrugated walls, Rough Fracture, Local Cubic Law

Analytical, numerical, and experimental investigations of particle transport in fractures with flat and corrugated walls The aim of the present thesis is to study the transport and deposition of small solid particles in fracture flows. First, single-phase fracture flow is investigated in order to assess the validity of the local cubic law for modeling flow in corrugated fractures. Channels with sinusoidal walls having different geometrical properties are considered to represent different fracture geometries. It is analytically shown that the hydraulic aperture of the fracture clearly deviates from its mean aperture when the walls roughness is relatively high. The finite element method is then used to solve the continuity and the Navier-Stokes equations and to simulate fracture flow in order to compare with the theoretical predictions of the local cubic law for Reynolds numbers Re in the range 6.7 × 10 -2 -6.7 × 10 1 . The results show that for low Re, typically less than 15, the local cubic law can properly describe the fracture flow, especially when the fracture walls have small corrugation amplitudes. For Re higher than 15, the local cubic law can still be valid under the conditions that the fracture presents a low aspect ratio, small corrugation amplitude, and moderate phase lag between its walls. Second, particle-laden flows are studied. An analytical approach has been developed to show how particles sparsely distributed in steady and laminar fracture flows can be transported for long distances or conversely deposited inside the channel. More precisely, a rather simple particle trajectory equation is established. Based on this equation, it is demonstrated that when particles' inertia is negligible, their behavior is characterized by the fracture geometry and by a dimensionless number W that relates the ratio of the particles sedimentation terminal velocity to the flow mean velocity. The proposed particle trajectory equation is verified by comparing its predictions to particle tracking numerical simulations taking into account particle inertia and resolving the full Navier-Stokes equations. The equation is shown to be valid under the conditions that flow inertial effects are limited. Based on this trajectory equation, regime diagrams that can predict the behavior of particles entering closed channel flows are built. These diagrams enable to forecast if the particles entering the channel will be either deposited or transported till the channel outlet. Finally, an experimental apparatus that was designed to have a practical assessment of the analytical model is presented. Preliminary experimental results tend to verify the analytical model. Overall, the work presented in this thesis give new insights on the behavior of small particles in fracture flows, which may improve our prediction and control of underground contamination, and may have applications in the development of new water filtration and mineral separation techniques.

I would like to thank my thesis supervisor, Prof. Michel Buès, for the confidence and the freedom he granted me to prepare this thesis, for his support, and for giving me the opportunity to express my taste for teaching.

I am grateful to my thesis co-supervisor, Dr. Luc Scholtès, for his guidance, his availability, and his constructive and thoughtful suggestions throughout my time as a PhD student. I am also very thankful to him for his careful editing and proofreading work, which contributed enormously to the productivity of this thesis.

I would like to thank Dr. Constantin Oltéan for the constructive discussions we had together and for providing me continuously with new ideas to improve my work. I thank him also for accepting to participate in the jury.

I would like to express my gratitude to Jean-Rgis Angilella, Professor at Université de Caen, and to Valeri Mourzenko, Director of Research at Institut Pprime, for the honor they gave me by agreeing to review my thesis dissertation, as well as for their interesting comments and constructive criticism. I am equally grateful to Prof. Anne Tanire, to Dr. Armelle Jarno-Druaux, and to Prof. Mohammed Souhar for agreeing to participate in the jury and for making my defense an enjoyable moment.

I would also like to thank Mr. Éric Lefèvre for his technical assistance and his good humor, and for all the GeoRessources employees for their kindness and for creating a nice working environment.

In my thanks, I can not forget my kind colleagues, PhD students and Postdocs, with whom I shared the most pleasant moments.

I do not forget to thank all my former teachers at Lycée Pascal, the school in which I spent fifteen years of my life. I thank them for molding me into the person I am today.

My deep gratitude is due to my brothers Mohammad and Mahmoud, to my parents-inlaw, and to my brothers-in-law for their support despite the distance.

Last but not least, I address my special and affectionate thanks and my deep indebtedness to my wife Aseel, for her unconditional support and patience. I could not have completed this work without her. I would like also to express my love to my precious daughter, Asmahan, whose presence in my life made the hardship of this task bearable.

Résumé

Études analytique, numérique, et expérimentale du transport de particules dans des fractures parois plates et ondulées Le but de cette thèse est d'étudier le transport et le dépôt de particules solides dans les écoulements à travers les fractures. Dans un premier temps, l'écoulement monophasique à travers les fractures est étudié afin d'évaluer la validité de la loi cubique locale comme mod le de l'écoulement. Des canaux à parois sinsoïdales à géométrie variable sont utilisés pour représenter différents types de fractures. Un premier développement analytique montre que l'ouverture hydraulique de la fracture diffère de son ouverture moyenne lorsque la rugosité des parois est élevée. La méthode des éléments finis est ensuite utilisée pour résoudre les équations de continuité et de Navier-Stokes et comparer les solutions numériques aux prédictions théoriques de la loi cubique locale sur une gamme relativement étendue de nombres de Reynolds Re. Pour de faibles Re, typiquement inférieurs à 15, la loi cubique locale décrit raisonnablement l'écoulement, surtout lorsque la rugosité et le déphasage entre les parois sont relativement faibles. Dans un deuxième temps, les écoulements chargés de particules sont étudiés. Une approche analytique est d'abord développée pour montrer comment des particules distribuées dans un écoulement stationnaire et laminaire à travers une fracture peuvent être transportées sur de longues distances ou au contraire se déposer à l'intérieur. Plus précisément, une équation simple décrivant la trajectoire d'une particule est établie. Sur la base de cette équation, il est démontré que, quand l'inertie des particules est négligeable, leur comportement dépend directement de la géométrie de la fracture et d'un nombre adimensionnel W qui relie la vitesse de sédimentation des particules à la vitesse moyenne de l'écoulement. L'équation proposée est vérifiée en comparant ses prédictions à des simulations numériques de suivi de particules prenant en compte l'inertie des particules et résolvent complètement les équations de Navier-Stokes. Il est montré que l'équation est valide lorsque l'inertie du fluide est faible. Des diagrammes de régimes, permettant de prévoir le comportement des particules à travers la fracture sont proposés. Enfin, un appareil expérimental conçu dans le but d'effectuer une évaluation pratique du modèle analytique est présenté et les résultats préliminaires sont discutés. Les rèsultats expérimentaux préliminaires tendent valider le modèle analytique. De façon plus générale, les résultats obtenus à travers ce travail de thèse font progresser nos connaissances du comportement des petites particules transportées dans les écoulements de fractures. Potentiellement, ce travail devrait permettre d'améliorer notre prévision de la pollution souterraine, et peut avoir des applications dans le développement de nouvelles techniques de filtration de l'eau et de séparation des minéraux.

Mots-clefs: Transport de particules, Fracture rugueuse, Parois ondulées, Loi cubique locale 9

Résumé étendu

La modélisation des fluides chargés en particules à travers des canaux d'écoulement interne est fondamentale afin de mieux appréhender divers applications environnementales, telles que le transport de sédiments et la pollution souterraine ou industrielles, comme la filtration de l'eau et la séparation des minéraux, ou encore, l'exploitation des ressources pétrolières. Le transport de contaminants à travers les fractures rugueuses est aussi un sujet de recherche important de par sa relation directe avec la contamination des formations aquifères.

Dans ce contexte, la présente thèse est consacrée à l'étude du transport et de la déposition de particules solides dans des écoulements à travers des canaux fermés, avec une application aux fractures rugueuses. En particulier, on considère des fractures à parois planes et ondulées (Figure 1). L'objectif principal de ce travail est de déterminer les conditions pour lesquelles les particules se déposeront à l'intérieur de la fracture ou, au contraire, seront transportées sur de grandes distances. Plusieurs paramètres doivent être pris en compte pour étudier le comportement des particules immergées dans un fluide en mouvement. Tout d'abord, les propriétés des particules, telles que leur taille et leur densité, doivent être connues pour déterminer les forces pouvant agir sur leur déplacement. Par exemple, lorsque la taille des particules est inférieure au micron, leur comportement est dominé par la diffusion brownienne. En revanche, dans le cas de particules plus grosses, leur mouvement est insensible à la diffusion brownienne et leur transport dépend uniquement des forces macroscopiques extérieures, comme les forces gravitationnelles et hydrodynamiques. Normalement, l'augmentation de la taille et/ou de la densité des particules tend à favoriser leur déposition en raison de la prédominance des effets gravitationnels sur leur comportement. Deuxièmement, les caractéristiques de l'écoulement, telles que la vitesse, la viscosité et la Dans tout écoulement chargé de particules, il est important d'avoir une description précise de l'écoulement du fluide avant de modéliser le transport de particules. Dans le cas des fractures rugueuses, un modèle bien connu et souvent utilisé pour décrire l'écoulement est la loi cubique locale (LCL), qui est une solution analytique approximative des équations de Navier-Stokes (N S) pour les écoulements laminaires visqueux à travers les fractures.

Cependant, l'applicabilité de la LCL reste discutable. En fait, un certain désaccord est évoqué dans les critères proposés par différents auteurs pour valider cette loi. Ceci est dû est lié aux études précédentes réalisées avec des fractures ayant des géométries spécifiques. Afin de pallier à ce problème, une étude numérique approfondie a étv menée. Ainsi, la première partie de notre travail est dédiée à l'effet induit par la géométrie de la fracture sur la validité de la LCL, sous différentes conditions liées à la géométrie et à l'écoulement. Cette étude est plus que nécessaire puisque la LCL constitue la base du modèle de transport de particules dans les fractures.

Pour l'étude du transport des particules, trois approches ont été adoptées:

• Approche analytique : En supposant que l'inertie des particules soit négligeable, une forme simplifiée de l'équation du mouvement des particules est couplée à la LCL, et, par conséquent, une équation décrivant les trajectoires des particules est développée.

Les particules peuvent alors être suivies analytiquement et la distance à laquelle elles se déposeront peut être calculée. Cette équation relie un nombre sans dimension W à la géométrie de la fracture. W dépend des propriétés des particules et des caractéristiques de l'écoulement. Basé sur W et sur les propriétés géométriques de la fracture, des régimes de transport et de sédimentation sont définis, et des diagrammes de régime sont établis.

• Approche numérique : En prenant en compte l'inertie des particules et en résolvant les équations complètes de N S, des simulations numériques sont menées pour confirmer la capacité du modèle analytique à prédire le comportement des particules dans les fractures. Les distances auxquelles les particules sédimentent à l'intérieur de la fracture sont calculées numériquement et comparées aux solutions de l'équation des trajectoires déterminée analytiquement. Des expériences numériques sont ensuite menées afin d'évaluer la pertinence des diagrammes de régime.

• Approche expérimentale : Un dispositif expérimental a été conçu et construit dans le but principal de vérifier le modèle analytique. Des tests préliminaires utilisant des graines de pavot comme particules ont été conduits, et les résultats expérimentaux ont été comparés aux prédictions du modèle analytique.

Cette thèse est divisée en quatre chapitres:

Dans le chapitre 1, on présente les concepts de base des écoulements chargés en particules sont présentés, ainsi qu'un état de l'art sur l'écoulement et le transport des particules dans des fractures.

Le chapitre 2 est consacré à l'étude des écoulements monophasiques dans des fractures à parois sinusoïdales. Les simulations numériques visant à évaluer la validité de la LCL sont présentées et les résultats sont discutés et comparés aux travaux précédents.

Dans le chapitre 3, l'intérê est proté sur le modèle analytique décrivant le transport des particules faiblement inertielles dans les canaux fermés. Les expériences numériques visant à vérifier le modèle analytique sont également présentées et discutées.

Dans le chapitre 4, le dispositif expérimental conçu pour une évaluation pratique du modèle analytique est décrit. Des résultats expérimentaux préliminaires utilisant des graines de pavot sont présentés.

Enfin, les principaux résultats obtenus tout au long de la thèse sont résumés et les perspectives du travail sont discutées.

Chapitre 2

Différents modèles ont été utilisés en hydrogéologie pour étudier l'écoulement à travers des fractures à parois rugueuses. L'idéalisation de la fracture en tant que canal à deux parois plates simplifie grandement le problème et permet de trouver une solution analytique pour le champ des vitesses, appelée la loi cubique (CL). En tenant compte de la rugosité des parois et en considérant une faible variation de l'ouverture dans la direction de l'écoulement, on peut utiliser l'équation de Reynolds qui conduit à la loi cubique locale (LCL), où les composantes du vecteur vitesse sont exprimées en fonction de la géométrie de la fracture.

Cependant, la validité de la CL et de la LCL reste discutable. En effet, il existe des critères, strictement liés à la géométrie de la fracture, permettant l'applicabilité de ces deux lois. Dans ce chapitre, une étude numérique visant à évaluer la validité de la CL et de la LCL, en considérant des fractures avec différentes géométries, est menée. Les fractures sont représentées par des canaux à parois sinusoïdales ayant des propriétés géométriques différentes définissant l'ouverture du canal, l'amplitude et la longueur d'onde des ondulations des parois, l'asymétrie entre les ondulations des deux parois, et le déphasage entre les deux parois. La validité de la LCL est évaluée pour des nombres de Reynolds dans la gamme [6.7 × 10 -2 , 6.7 × 10 1 ], en comparant ses prédictions à la solution numérique des équations de Navier-Stokes (N S). Cette dernière est obtenue en utilisant la méthode des éléments finis, implémentée dans le logiciel COMSOL Multiphysics.

Les résultats obtenus confirment que la CL, basée sur l'ouverture moyenne de la fracture, peut remplacer la LCL, basée sur l'ouverture hydraulique, tant que les ondulations des parois sont relativement petites ou lorsque les parois sont identiques et parallèles. Par contre, elle ules sont non-browniennes, passives et de dimensions largement plus faibles que l'ouverture de la fracture. Comme au chapitre 2, des canaux à parois planes et sinusoïdales sont considérés.

L'inertie des particules est considérée comme faible de telle sorte qu'elle peut être négligée dans l'équation du mouvement. On montre que, sous cette condition, le comportement des particules peut être caractérisé par la géométrie du canal et par un nombre sans dimension W qui représente le rapport entre la vitesse de sédimentation des particules et la vitesse moyenne de l'écoulement. Une équation différentielle définissant la trajectoire des particules dans les canaux à parois ondulées et une équation exacte de cette trajectoire dans les canaux à parois planes ont été dérivées sous l'hypothèse que la vitesse de l'écoulement peut être explicitement paroi inférieure tend à diminuer la zone de transport et à augmenter la zone de sédimentation.

W cr1 /h * W cr2 /h * W h *
Le diagramme de régimes et les effets des paramètres géométriques sur la variation de ses zones sont vérifiés par des expériences numériques menées en injectant 100 particules dans le canal et en calculant les pourcentages de particules qui se déposent à l'intérieur du canal.

Les principaux résultats de ce chapitre ont été publiés dans le "European Journal of Mechanics B/Fluids" (Hajjar et al. [START_REF] Hajjar | Transport and deposition of weakly inertial particles in closed channel flows at low reynolds number[END_REF]).

Chapitre 4

Le modèle analytique proposé au chapitre 3, sous l'hypothèse que l'inertie des particules est négligée et que l'écoulement suit la loi cubique locale (LCL), a été vérifié numériquement via la résolution numérique des équations de N S et en prenant en compte l'inertie des particules. Pour aller plus loin, une validation expérimentale est nécessaire, afin de considérer des situations réelles et d'évaluer la validité du modèle analytique sur une base pratique. Par conséquent, ce chapitre est dédié au modèle expérimental. La première partie du chapitre est consacrée à la présentation de la conception et à la mise en place du modèle physique.

Ensuite, la procédure expérimentale et la méthodologie utilisée pour traiter les données expérimentales sont décrites. Enfin, plusieurs résultats préliminaires sont présentés et discutés vis-à-vis des objectifs initiaux de l'étude. Le dispositif est utilisé pour effectuer une étude préliminaire du transport des particules dans les fractures avec des parois planes et/ou sinusoïdales, ayant des dimensions conformes aux hypothèses théoriques formulées dans les chapitres précédents. En conclusion, les résultats expérimentaux préliminaires confirment le modèle analytique développé dans cette thèse. En plus, ces résultats démontrent la capacité du dispositif expérimental à étudier le transport des particules dans les écoulements en canaux fermés.

Il peut être judicieusement utilisé pour de futures études expérimentales sur le transport de particules, ce qui peut améliorer notre compréhension du comportement des particules et valider les modèles déjà développés.

Conclusion

Dans l'ensemble, les résultats obtenus dans cette thèse améliorent notre compréhension du comportement de petites particules immergées dans les écoulements à travers des canaux fermés, avec une application directe au transport des contaminants dans les fractures. Par exemple, on peut identifier, en fonction de leur taille et de leur densité, les contaminants susceptibles de se déposer à l'intérieur de la fracture ou être en suspension et transportés sur de longues distances. Ces résultats ont d'autres applications dans la filtration de l'eau et dans la séparation des minéraux. En effet, sur la base de nos diagrammes de régime, un système de séparation basé sur la sédimentation de particules dans des canaux à parois sinusoïdales pourrait être envisagé. Cela permettrait de séparer les particules en fonction de leur taille et/ou de leur densité en fonction de la distance à laquelle elles se déposent dans le canal.

Comme l'écoulement dans le canal peut simplement être créé par une différence de charge hydraulique, l'avantage d'un tel système par rapport aux techniques de séparation actuelles est qu'il est passif et ne nécessite pas une importante alimentation en énergie. Une autre application concerne la focalisation inertielle qui peut trouver des échos en microfluidique.

Tout d'abord, les résultats obtenus peuvent conduire à une quantification des conditions (Re et taille des particules) dans lesquelles la focalisation devient efficace. Deuxièmement, on a pu observer la focalisation inertielle dans les canaux à parois sinusoïdales. Des analyses supplémentaires pourraient révéler de nouvelles caractéristiques à l'origine du phénomène de focalisation, comme par exemple l'effet de la courbure des parois des canaux sur les forces In such a context, the present thesis is devoted to the investigation of the transport and deposition of small solid particles in closed channel flows, with application to fracture flows. In particular, fractures with flat and corrugated periodic walls are considered. The main objective is to determine the conditions under which the particles will settle inside the fracture or, on the contrary, be transported over long distances. Several parameters must be considered in order to study the behavior of particles immersed in a moving fluid. First, the physical properties of the particles such as their size and density must be known to determine the forces acting on them. For instance, for sub-micron particles, Brownian diffusion dominates particle behavior and can be considered for studying particle transport. On the other hand, the transport of larger particles, which are insensitive to Brownian diffusion, depends directly on the forces acting on the particles due to gravitational and hydrodynamical effects.

H 0 fracture aperture L ∞ fracture length L 0 wavelength of corrugations A 1,
Normally, increasing the particle size and/or density tends to favor particle deposition due to the predominance of gravitational effects on their behavior. Second, the characteristics of the flow such as its velocity and the fluid viscosity and density are also important factors that must be taken into account for modeling correctly particle behavior. Increasing the fluid viscosity, for example, tends to enhance the transport of particles for longer distances due to greater friction forces between the fluid and the particles. Finally, the effects of the channel geometrical properties related to its aperture and to the wall corrugations on particle behavior must be comprehended.

Before addressing particle-laden flows, it is important to have a precise description of the fluid flow itself. In the case of rough fractures, a well-known model commonly used to describe the flow is the local cubic law (LCL), which is an approximate analytical solution of the Navier-Stokes (N S) equations for viscous laminar flows through thin channels. However, the LCL applicability remains arguable. In fact, a certain discrepancy emerged in the criteria proposed by different authors for its validity. This is due to the fact that the previous studies have been performed with specific fracture geometries. This discrepancy motivated the first part of our work. In particular, a thorough numerical study is conducted in order to to investigate the effect of the fracture geometry on the validity of the LCL under different geometrical and kinematic conditions. This investigation is necessary since the LCL constitutes the basis of our study aiming to model particle transport and depositions in fracture flows.

To sum up, this thesis is an attempt to answer the following questions:

• Is the LCL a suitable model of fracture flow and what are the effects of the fracture geometry on its validity?

• What are the effects of the particle properties and of the flow characteristics on the behavior of particles when immersed in fracture flows?

• How do the the geometrical properties of the fracture affect the particle behavior?

For the study of particle transport and deposition, we adopt three approaches:

• Analytical approach: Assuming that particle inertia is negligible, a simplified form of the particle motion equation is coupled to the LCL and an equation describing particle trajectories is developed. Particles can be tracked analytically and the distance at which the particle may deposit can be calculated. This equation relates a dimensionless number W to the fracture geometry. W depends on the particle properties and on the flow characteristics. Based on W and on the geometrical properties of the fracture, arbitrary regimes of transport and deposition are defined, and regime diagrams are established.

• Numerical approach: Taking into account particle inertia, and solving the full N S equations, numerical simulations are conducted to confirm the ability of the analytical model to predict the behavior of particles immersed in fracture flows. The distances at which particles deposit inside the fracture are computed numerically and compared to the trajectory equation determined analytically. Numerical experiments are then conducted to assess the relevance of the regime diagrams.

• Experimental approach: An experimental apparatus has been designed and constructed, with the main aim of verifying the analytical model. Preliminary tests using poppy seeds as moving particles are conducted and experimental results are compared to the analytical predictions.

Outline of the thesis

This thesis consists of four chapters:

In chapter 1, we present the basic concepts of particle-laden flows and a bibliographic review is made regarding flow and particle transport in fractures.

Chapter 2 is devoted to the study of single-phase flows in fractures with sinusoidal walls.

The numerical simulations aiming to assess the validity of the local cubic law are presented, and the results are discussed and compared to previous works.

In chapter 3, we introduce the analytical model describing the transport of weakly-inertial particles in closed channel flows. Numerical experiments aiming to verify the analytical model are also presented and discussed.

In chapter 4, the experimental apparatus that was designed for further validation and practical assessment of the analytical model is described. Preliminary experimental results using poppy seeds are presented.

Finally, the main results obtained throughout the thesis are summarized and the perspectives of the work are discussed.

29

Chapter 1 Examples of particle-laden flows include solid particles in a liquid or a gas, gas bubbles in a liquid, or liquid particles in a gas. They are ubiquitous in many situations, whether at the natural and environmental level like rain droplets in clouds, transport of sediments, and dust inhalation, or at the industrial level like solid-particle separation, bubble column reactors, and sprays.

STATE OF THE ART

The complete set of equations describing particle-laden flows are in general very complicated to be solved analytically or require high computation costs to be solved numerically.

Therefore, some approximations are usually made to simplify the problem such as point-force particles or mixed multiphase flow. It is also typical to assume that the dispersed particles size is very small compared to the flow domain so that the suspension is diluted. This assumption can greatly simplify the problem by neglecting particle-particle interactions (like collisions) with respect to particle-fluid interactions.

The concentration of particles certainly affects the particle dynamics in the flow. Nonethe- In the latter case, as the concentration of the particles is high, particle motion is dominated by particle-particle interactions. On the other hand, in a dispersed system, particle dynamics is governed by the fluid hydrodynamical forces that prevail over particle-particle interactions.

In this case, the concentration also defines the level of coupling between the dispersed and continuous phases. Generally, one-way coupling is considered when the particle concentration is very low so that their effect on the carrier fluid is neglected. The flow equations can then be solved independently from particle motion equations, which can be then solved using the corresponding flow fields at the particle position. For higher particle concentrations, particles can affect the flow by changing the local density and/or viscosity or the velocity field. In this case, the flow and particle motion equations are mutually coupled. In any case, to correctly predict the behavior of particle-laden flows, it is important to have an accurate description of the particle properties and flow characteristics. In addition, depending on the particle size, a distinction has to be made between colloidal and non-colloidal particles though some studies suggest extending their size up to 10µm (Khilar and Fogler [START_REF] Khilar | Migrations of fines in porous media[END_REF], Sen and Khilar [START_REF] Sen | Review on subsurface colloids and colloid-associated contaminant transport in saturated porous media[END_REF]). Colloids are sensible to Brownian diffusion, i.e. the mechanism by which particles move (diffuse) from zones of higher concentration to zones of lower concentration (Jones [START_REF] Jones | Soft condensed matter[END_REF]). Because Brownian diffusion dominates the other mechanisms in colloidal transport, colloids may be suspended in the fluid for long time periods. On the other hand, when the particle size increases, typically above 1µm, Brownian effects become negligible and particle behavior is driven by external forces due to the interaction with the fluid and to potential external fields such as, for instance, gravity in the case of dense particles and electricity in the case of charged particles. An example of the distinction between particles based on their size can be found in the processes of water filtration. In figure 1.3, the different types of water contaminants which can be encountered in water filtration are illustrated, highlighting the difference between colloidal and non-colloidal particles.

Throughout this thesis, Brownian effects are neglected so that only non-colloidal particles are considered.

Definition of particle inertia

To understand particle inertia, one can consider a case in which inertia is negligible. A good example for the latter situation is flow tracers, which are commonly utilized to measure flow velocity when combined with imaging techniques such as PIV and PTV. In fact, a tracer particle follows exactly the flow streamlines due to its small size and to having a density matching that of the carrier fluid.

The motion of a non-colloidal tracer can be simply described by:

v p = v f (1.1)
where v p and v f are respectively the particle and fluid dimensionless velocities rescaled based on the typical velocity scale of the flow.

When their size and/or density increase, particles have a proper dynamic and their trajectories deviate from the flow streamlines. These particles are called inertial particles. The motion of inertial particles can be very complex even when particles are passive and have non Brownian dynamics (Babiano et al. [START_REF] Babiano | Dynamics of a small neutrally buoyant sphere in a fluid and targeting in hamiltonian systems[END_REF], Haller and Sapsis [START_REF] Haller | Where do inertial particles go in fluid flows?[END_REF], Cartwright et al. [START_REF] Cartwright | Dynamics of finite-size particles in chaotic fluid flows[END_REF],

Balkovsky et al. [START_REF] Balkovsky | Intermittent distribution of inertial particles in turbulent flows[END_REF]). This characteristic behavior of inertial particles has a great importance in many practical situations in earth sciences like oceanology (Lunau et al. [START_REF] Lunau | Physical and biogeochemical controls of microaggregate dynamics in a tidally affected coastal ecosystem[END_REF]) and atmospheric sciences (Shaw [12]).

Another interesting feature of inertial particles is that, under the effect of their inertia, they tend to cluster or accumulate in well defined regions of the flow. This phenomena has been widely studied for different types of fluid flows (e.g. Eaton and Fessler [START_REF] Eaton | Preferential concentration of particles by turbulence[END_REF] and Squires and Eaton [START_REF] Squires | Preferential concentration of particles by turbulence[END_REF] in turbulent flows, Bec [START_REF] Bec | Fractal clustering of inertial particles in random flows[END_REF] in random flows, Nizkaya et al. [START_REF] Nizkaya | Inertial focusing of small particles in wavy channels: Asymptotic analysis at weak particle inertia[END_REF] in laminar spatially periodic flows, Angilella [START_REF] Angilella | Dust trapping in vortex pairs[END_REF] and Angilella et al. [START_REF] Angilella | Inertial particle trapping in an open vortical flow[END_REF] in vortex flows). This clustering ability of inertial particles can explain, for example, how rain in turbulent clouds can be enhanced by the accumulation of water droplets (e.g. Falkovich et al. [START_REF] Falkovich | Accleration of rain initiation by cloud turbulence[END_REF]).

Moreover, particle inertia has a great effect on particle trajectories in fluid flows. For instance, Stommel [START_REF] Stommel | Trajectories of small bodies sinking slowly through convection cells[END_REF] studied particle motion in cellular flow fields without taking into account particle inertia. He found that the particles' trajectories can be calculated according to the ratio between the particles settling velocity and the flows vortex velocity. Maxey [START_REF] Maxey | The motion of small spherical particles in a cellular flow field[END_REF] extended this concept by including the effect of particle inertia and found that two dimensionless numbers must be considered: Stokes number St, which characterizes particle inertia, and the particle to fluid density ratio.

By definition, St is the ratio between particle relaxation time t p and the flow characteristic time t 0 . t p can be seen as the characteristic time of particle reaction to a change in the fluid velocity. If St << 1, then the particle will quickly adapt itself to the fluid velocity and acts as a tracer. On the other hand, for higher St, the particle will take longer time to respond to any change in the fluid velocity and will continue along its initial trajectory.

Unlike tracers, inertial particle motion is governed by a second order equation that can be written in the following general form:

d v p dt = - 1 St ( v p -v f ) + f (1.2)
the term -1 St ( v pv f ) being related to the drag force while f includes forces resulting from externally applied fields, such as gravity or other forces (electrical, magnetic...). If St → 0 and if f is negligible (e.g. a neutrally buoyant particle in a gravity field), then equation (1.2) reduces to equation (1.1) and the particle is simply a tracer. When St increases and/or when f is not negligible, deviation from equation 1.1 appears and solving the particle motion equation is more complex. However, for small inertia, equation 1.2 can be solved based on an asymptotic expansion, St being the perturbation parameter.

Particle transport in closed channel flows

Closed channel flows define configurations where a fluid is moving inside closed conduits such as tubes, pipes or confined channels like fractures (Figure 1.4).

In this thesis, We consider 2D flows occurring in closed channels with flat and corrugated walls (Figure 1.5). In addition, we consider that the fluid fills the channel cross-section and there is no free surface of the fluid. In a hydrogeological context, such channels are commonly used to model rough fractures for example. Flows through these fractures can carry tiny particles such as rock sediments or organic debris. Depending on the particle physical properties, on the flow characteristics, and on the fracture geometry, these particles can have different behaviors. Most of previous works emphasizing on fracture flows focused either on solute transport (Therrien and Sudicky [START_REF] Therrien | Three-dimensional analysis of variably-saturated flow and solute transport in discretely-fractured porous media[END_REF], Bouqain et al. [START_REF] Bouquain | The impact of inertial effects on solute dispersion in a channel with periodically varying aperture[END_REF], Oltéan et al. [START_REF] Oltéan | Numerical and experimental investigation of buoyancy-driven dissolution in vertical fracture[END_REF]) or on the transport of colloidal particles (Boutt et al. [START_REF] Boutt | Trapping zones: The effect of fracture roughness on the directional anisotropy of fluid flow and colloid transport in a single fracture[END_REF]).

Generally, for particle-laden flows through fractures, the macroscopic behavior of the particulate phase can be described by breakthrough curves, i.e. the plot of the variation of particle relative concentration as a function of time, where the relative concentration is defined as the ratio of the actual concentration to the one at the source. Breakthrough curves in previous experimental works have shown that particles are not simply advected like tracers and that particles velocity deviates from that of the fluid (Novawski et al. [START_REF] Novakowski | The analysis of tracer experiments conducted in divergent radial flow fields[END_REF]). This phenomenon was shown to be due to the effect of Brownian diffusion which leads to the redistribution of particles along the channel cross-section. However, this is not valid for larger particles that are not affected by diffusion. For example, the transport of such particles has been investigated by Nizkaya [START_REF] Nizkaya | Transport and deposition of inertial particles in a fracture with periodic corrugation[END_REF] and Nizkaya et al. [START_REF] Nizkaya | Inertial focusing of small particles in wavy channels: Asymptotic analysis at weak particle inertia[END_REF] who showed how they can be accumulated in preferential regions and focus towards specific streamlines inside the flow. Another focusing phenomenon that is expected to occur in closed channels is particle focusing due to the lift forces that emerge when flow inertial effects are important. These two distinct phenomena of particle focusing are briefly described in the following section. and Silberberg [START_REF] Segré | Behaviour of macroscopic rigid spheres in poiseuille flow part 2. experimental results and interpretation[END_REF] were the first to witness that particles in a laminar pipe Poiseuille flow congregate on an annulus located at a certain distance from the pipe centerline equal to 0.6 times the pipe radius. This phenomenon is known as the tubular pinch effect. Since then, this phenomenon has been studied extensively using theoretical (Schonberg and Hinch [START_REF] Schonberg | Inertial migration of a sphere in poiseuille flow[END_REF],

Asmolov [START_REF] Asmolov | The inertial lift on a spherical particle in a plane poiseuille flow at large channel reynolds number[END_REF]), experimental (Karnis et al. [START_REF] Karnis | The flow of suspensions through tubes: V. inertial effects[END_REF], Matas et al. [START_REF] Matas | Inertial migration of rigid spherical particles in poiseuille flow[END_REF]) and numerical approaches (Feng et al. [33], Yang et al. [START_REF] Yang | Migration of a sphere in tube flow[END_REF]). These investigations concluded that inertial migration occurs due to forces that act on particles in inertial flows, known as the inertial lift forces.

Recently, lift-induced particle focusing attracted much attention with the development of microfluidics where it has many applications, e.g. in cell separation and isolation in biological fluids (Di Carlo et al. [START_REF] Di Carlo | Continuous inertial focusing, ordering, and separation of particles in microchannels[END_REF], Martel and Toner [START_REF] Martel | Inertial focusing in microfluidics[END_REF]). The effect of inertial lift forces must be is explained in detail in the following chapters.

1.1.3.b. Preferential accumulation of particles in periodic channels

Unlike lift-induced particle focusing, accumulation or clustering of particles due to their inertia in liquid flows through channels with corrugated walls, remains a theoretical prediction yet to be verified experimentally (Nizkaya et al. [START_REF] Nizkaya | Inertial focusing of small particles in wavy channels: Asymptotic analysis at weak particle inertia[END_REF]). Moreover, particle clustering is only limited to the case of periodic walls corrugations (Figure 1.7) as, in contrast with liftinduced migration, clustering can not occur in channels with flat walls. This phenomenon in which particles, with low but finite inertia, are expected to be attracted by a streamline, may occur for specific channel geometries and flow characteristics. This phenomenon will be discussed later on with more details. Before that, the flow must be investigated.

Flow in channels with flat and corrugated walls

Channels with flat and corrugated walls have been studied extensively in earth sciences as a model of single rough fractures.

In fact, many studies have shown the importance of the fracture characteristics when considering flow in fractured geological systems, such as its orientation, its extent, and its interconnection with other fractures (Rasmussen [START_REF] Rasmussen | Computer simulation model of steady fluid flow and solute transport through three-dimensional networks of variably saturated, discrete fractures[END_REF]). Zhang et al. [START_REF] Zhang | Evaluation of the 2-d permeability tensor for fractured rock masses[END_REF] showed that the hydraulic behavior of a fractured medium is largely influenced by the characteristic lengths of the single fractures and their number (i.e. fracture density), and fracture orientations. Indeed, the properties of the flow occurring through a network of fractures are strongly controlled by those of the flow occurring through single or discrete fractures. For a network of fractures, the percolation theory is an appropriate technique for solving the permeability problems (Mourzenko et al. [START_REF] Mourzenko | Percolation and conductivity of self-affine fractures[END_REF], Mourzenko et al. [START_REF] Mourzenko | Permeability of self-affine fractures[END_REF]). However, modeling flow in single fractures remains a key issue that needs to be properly understood before extrapolating to more complex configurations.

The parameters likely to intervene in the prediction of flow through single fractures have been gathered experimentally by Hakami and Larsson [START_REF] Hakami | Aperture measurements and flow experiments on a single natural fracture[END_REF]. They noted that, apart from the effect of the fluid properties and pressure conditions on the fracture boundaries, fracture flow depends also on different geometrical parameters such as the aperture and spatial correlations related to the walls roughness. The roughness characterizes the morphology of the fracture walls, their general shape and their surface state. The term roughness encompasses very different morphological characteristics such as: amplitude (elevation of points on the surface), angularity (slopes and angles), waviness (periodicity), and curvature (Gentier [START_REF] Gentier | Morphologie et comportement hydromécanique d'une fracture naturelle dans le granite sous conrainte normale[END_REF],

Belem [START_REF] Belem | Morphologie et comportement mécanique des discontinuités rocheuses[END_REF]). The geometric description of the roughness and morphological characteristics of fractures is based on empirical, geometrical and statistical analyzes, which can be either geostatistical or fractal (Gentier [42], Belem [START_REF] Belem | Morphologie et comportement mécanique des discontinuités rocheuses[END_REF], Mourzenko et al. [START_REF] Mourzenko | Percolation and conductivity of self-affine fractures[END_REF], Plouraboué [START_REF] Plouraboue | Propriétés géométriques et propriétés de transport des fractures à parois rugueuses[END_REF],

Oron & Berkowitz [START_REF] Oron | Flow in rock fractures: The local cubic law assumption reexamined[END_REF], Lefèvre [START_REF] Lefêvre | Comportement en cisaillement et évolution de la morphologie des discontinuités rocheuses[END_REF], Legrain [START_REF] Legrain | Etude de l'influence de la rugosité sur l'écoulement de fluide dans les fissures rocheuses[END_REF]).

The roughness in its general sense has a multiplicity of characteristic length scales. However, in the literature, two main scales of roughness are usually highlighted (Figure 1.8).

They are characterized and defined by their effects on the mechanical and hydraulic behavior of a fracture. It is important at this point to distinguish two types of roughness (Louis [START_REF] Louis | Etude des écoulements d'eau dans les roches fissurées et de leurs influences sur la stabilité des massifs rocheux[END_REF]). At the micro-scale level, roughness is related to irregularities in the surface of the walls.

It may slightly increase the linear head loss inside the fracture. The macro-scale roughness characterizes the overall shape of the walls. It causes changes in flow direction and the shape of the streamlines. The concept of tortuosity is then often used in the literature. Tortuosity represents the ratio of the length of the trajectory of the flow between two points and the straight distance between these two same points, and, thus, it can has an important effect the behavior of the flow through rough fractures. The micro-scale roughness is neglected in Figure 1.8: Different scales of roughness according to Dippenaar and Van Rooy [START_REF] Dippenaar | On the cubic law and variably saturated flow through discrete open rough-walled discontinuities[END_REF]. i 1 represents first order waviness corresponding to the macro-roughness. i 2 represents second order asperities corresponding to the micro-roughness. this thesis and only the effects of the macro-scale roughness on the behavior of fracture flow are considered.

Modeling flow in rough fractures

To model fluid flow in a single fracture, the standard laws of fluid mechanics can be used.

These are the Navier-Stokes (N S) and continuity equations. Eventually, these equations can be simplified according to the channel geometry and/or assumptions regarding the flow, so that analytical solutions can be obtained to determine the pressure and velocity fields.

On the other hand, on a larger scale, the flow can be described by Darcy's law. Darcy's law relates the hydraulic gradient to the flow rate using an intrinsic parameter defined as the permeability. It can be obtained from the N S equation using an upscaling method such as volume averaging an homogenization (Whitaker [50]).

One of the most important characteristics of a rough fracture is its aperture. The aperture is defined as the distance between the fracture walls and implicitly depends on the way the roughness is defined. The aperture can be defined locally or globally according to geometrical criteria, mechanical criteria, or as a result of hydraulic experiments assuming a law for the flow. Consequently, the geometric mean, the mechanical and the hydraulic apertures can be distinguished (Lin [START_REF] Lin | Étude du comportement hydromécanique d'une fracture rocheuse sous contrainte normale: développement d'un modèle numérique[END_REF], Davias [START_REF] Davias | Modélisation numérique d'écoulements en massif rocheux fracturé: contribution à la modélisation du comportement hydromécanique des milieux fracturés[END_REF], Crosnier [START_REF] Crosnier | Écoulement dans une fracture de géométrie radiale: influence d'une rugosité périodique[END_REF]).

Early attempts to model flow in single fracture assumed that the flow occurs between two flat parallel plates representing the fracture walls (Bear [54]). This is known as the Figure 1.9: The parallel-plate model parallel-plate model (Figure 1.9). This model is based on the observations stating that most natural fractures are approximately plane at the fracture length scale. Analytical solutions to the problem of laminar flow between parallel plates can then be easily obtained. In particular, for a 2D channel with flat walls of length L ∞ and aperture H 0 , through which a fluid of density ρ and dynamic viscosity µ is flowing due to a pressure difference ∆P = ρg∆Z between the inlet and the outlet of the channel, with ∆Z is the hydraulic head and g the gravity acceleration, the volumetric flow rate per unit width is given by:

Q = - ρgH 0 3 12µ ∆Z L ∞ (1.
3)

The volumetric flow rate per width is Q = V 0 H 0 , V 0 being the mean velocity of the flow in the channel. In this case, the velocity has a parabolic profile (Figure 1.9)

At the same time, Darcy's law (Darcy [START_REF] Darcy | Les fontaines publiques de la ville de Dijon: exposition et application[END_REF]) gives a linear relation between the volumetric flow rate Q v and the pressure drop ∆P :

Q v = - KA µ ∆P L (1.4)
where K is the hydrodynamic permeability and A is the cross-section on which Q v is computed. In order to consider the flow rate per unit width Q, A can be replaced by H 0 and equation (1.4) thus becomes:

Q = - KH 0 µ ∆P L (1.5)
Replacing equation (1.3) in equation (1.5), the permeability of the channel is equal to

K = H 2 0 12
, and, therefore, the channel transmissivity can be computed as T = KH 0 = H 3 0 12 . Since T is proportional to the cube of the aperture, equation (1.3) is known as the cubic law (CL) (D.M. Brown [START_REF] Brown | Stochastic analysis of flow and solute transport in a variable-aperture rock fracture[END_REF], S.R. Brown [START_REF] Brown | Fluid flow through rock joints: the effect of surface roughness[END_REF], Silliman [START_REF] Silliman | An interpretation of the difference between aperture estimates derived from hydraulic and tracer tests in a single fracture[END_REF], Chen et al. [START_REF] Chen | An experimental investigation of hydraulic behaviour of fractures and joints in granitic rock[END_REF], Konzuk and Kueper [START_REF] Konzuk | Evaluation of cubic law based models describing single-phase flow through a rough-walled fracture[END_REF], Dippenaar and Van Rooy [START_REF] Dippenaar | On the cubic law and variably saturated flow through discrete open rough-walled discontinuities[END_REF]). If the CL is valid for large fracture apertures, i.e. when walls macro-roughness can be neglected, it tends to overestimate the flow rate if the roughness and the aperture have the same order of magnitude. In this case, the two walls can not be modeled as flat plates and the local aperture varies along the flow direction, an effect that is not taken into account in equation (1.3). However, the CL can accurately predict the flow rate in a rough fracture when the apparent aperture H 0 is replaced by a fitting parameter calculated from experimental data and obeying equation (1.3) (Whitherspoon et al. [START_REF] Witherspoon | Validity of cubic law for fluid flow in a deformable rock fracture[END_REF]). This parameter is called the hydraulic aperture H h . Attempts to match the measured flow rates to the hydraulic head as predicted by the CL have been made using different aperture definitions such as the arithmetic mean (Brown [START_REF] Brown | Fluid flow through rock joints: the effect of surface roughness[END_REF]), the geometric mean (Tsang and Tsang [START_REF] Tsang | Hydrological characterization of variable-aperture fractures[END_REF], Renshaw [START_REF] Renshaw | On the relationship between mechanical and hydraulic apertures in rough-walled fractures[END_REF]), the harmonic mean (Unger and Mase [START_REF] Unger | Numerical study of the hydromechanical behavior of two rough fracture surfaces in contact[END_REF]) and the volume-averaged mean (Hakami and Barton [65]) of local apertures or by applying correction factors to include other information about the fracture (Whitherspoon et al. [START_REF] Witherspoon | Validity of cubic law for fluid flow in a deformable rock fracture[END_REF], Gutfraind and Hansen [START_REF] Gutfraind | Study of fracture permeability using lattice gas automata[END_REF], Neuman [START_REF] Neuman | Trends, prospects and challenges in quantifying flow and transport through fractured rocks[END_REF], De Vallejo and Ferrer [START_REF] Vallejo | Geological engineering[END_REF]). A comparative evaluation of the different definitions of the aperture can be found in the work presented by Konzuk and Kueper [START_REF] Konzuk | Evaluation of cubic law based models describing single-phase flow through a rough-walled fracture[END_REF].

Instead of matching the measured flow rates to the predictions of the CL, a different approach consists in considering the spatial variation of the aperture along the flow direction, implying that the CL is valid locally along the fracture length and leading to the known local cubic law LCL. In fact, fluid flow through rough fractures can be fully described by the N S equations (Zimmerman and Bodvarsson [START_REF] Zimmerman | Hydraulic conductivity of rock fractures[END_REF]). However, the non-linearity of the inertial term in these equations makes them difficult to be solved analytically without the use of perturbation expansions (Hasegama and Izuchi [START_REF] Hasegawa | On steady flow through a channel consisting of an uneven wall and a plane wall: Part 1. case of no relative motion in two walls[END_REF], Basha and El Asmar [START_REF] Basha | The fracture flow equation and its perturbation solution[END_REF]). When the flow inertial effects are negligible, the N S equations reduce to the Stokes equation. Furthermore, when the channel aperture varies slowly along the flow direction, Stokes equation can be further reduced to Reynolds equation also known as the LCL (Zimmerman et al. [START_REF] Zimmerman | Fluid flow in rock fractures: From the navier-stokes equations to the cubic law[END_REF]).

The development of the LCL based on the N S equation is detailed and discussed in the next chapter.

Inertial effects in fracture flows

At the macroscopic level, fracture flow is characterized by the relation between the volumetric flow rate Q v and the pressure drop ∆P applied between the fracture inlet and outlet. At low flow rates, this relation is linear and is described by Darcy's law (equation (1.4)).

For low Q v , the permeability K depends only on the fracture geometry. When Q v increases, experimental investigations showed that the variation of ∆h as a function of Q v deviates from the linear defined in Darcy's law (e.g. Firdaouss et al. [START_REF] Firdaouss | Nonlinear corrections to darcy's law at low reynolds numbers[END_REF]). This deviation is due to the inertial effects developing in the flow and to the walls corrugation (Bear [54], Dybbs and Edwards [START_REF] Dybbs | A new look at porous media fluid mechanicsdarcy to turbulent[END_REF]). The intensity of flow inertial effects allows the separation between Darcian and non-Darcian regimes and is characterized by the Reynolds number Re = V 0 H 0 ν , V 0 = Q A being the flow mean velocity, H 0 is the characteristic length, and ν is the fluid kinematic viscosity.

For low and moderate Re, when the inertial and viscous forces in the liquid are of the same order of magnitude, the deviation of Darcy's law is cubic in Q v (proportional to Q 3 v ). This cubic deviation was later confirmed by many theoretical investigations (Firdaouss et al. [START_REF] Firdaouss | Nonlinear corrections to darcy's law at low reynolds numbers[END_REF], Skjetne and Auriault [START_REF] Skjetne | New insights on steady, non-linear flow in porous media[END_REF], Jacono et al. [START_REF] Jacono | Weak-inertial flow between two rough surfaces[END_REF]). For higher Re, inertial effects can lead to the appearance of recirculation zones in the flow (Figure 1.10). They are due to the viscous shear stresses resulting from the fluid momentum mismatch in the channel center and near-wall regions. The central fluid moves faster than the quasi-stagnant fluid close to the channel walls, causing thus fluid recirculations within the channel furrows. Such zones can theoretically appear even in Stokes flows, but only when the aperture variations are important (Kitanidis and Dikaar [START_REF] Kitanidis | Stokes flow in a slowly varying two-dimensional periodic pore[END_REF], Malevich et al. [START_REF] Malevich | Stokes flow through a channel with wavy walls[END_REF]). Due to the presence of recirculation zones at high Re, the deviation from Darcy's law is quadratic in Q (Forchheimer regime [START_REF] Forchheimer | Wasserbewegung durch boden[END_REF]). In this case, the deviation is due to energy dissipation that may be explained by the loss of kinetic energy occurring when single jets tend to penetrate into the separating line between the main flow and the recirculation zone (Lucas et al. [START_REF] Lucas | High velocity flow through fractured and porous media: the role of flow non-periodicity[END_REF]). Therefore, it is convenient to define the flow regimes as follows:

• Viscous regime: inertial forces are negligible with respect to viscous forces. This is a Figure 1.10: Recirculation zones appearing in a fracture with rough walls as identified by Boutt el al. [START_REF] Boutt | Trapping zones: The effect of fracture roughness on the directional anisotropy of fluid flow and colloid transport in a single fracture[END_REF].

Darcian regime in which the variation of ∆P as a function of Q v is linear.

• Transition regime: viscous and inertial forces are of the same order of magnitude. The deviation from Darcy law is cubic in Q v . This regime is also known as weakly inertial regime.

• Inertial regime: viscous forces are negligible with respect to inertial forces. The deviation from Darcy law is quadratic in Q v .

In this thesis, fracture flow is investigated for low and moderate Re. Investigating the onset of creation of recirculation zones as well as the effect of the channel geometry on their appearance in the inertial regime is not part of the study and will thus not be addressed.

The CL and LCL provide simple relationships relating the hydraulic aperture of a fracture to its permeability, given that the flow is Darcian. For high Re, when the flow becomes non-Darcian, the validity of the CL and of the LCL becomes questionable. The applicability of these models must then be assessed before considering them to model fracture flow.

Idealized model of fracture geometry

Although real fractures have 3D geometries, 2D geometries can be considered for simplification sake. Indeed, the fracture aperture is practically many orders of magnitude smaller than its width. If the fracture is presented in a reference frame (X, Y, Z), where X is in the main flow direction (along the length), Y is in the orthogonal direction (along the width) and Z is in the vertical direction (along the aperture), it is reasonable to assume an invariance of the velocity field in the Y direction and to study the system in the 2D referential (X,Z).

In this context, Zimmerman et al. [START_REF] Zimmerman | Lubrication theory analysis of the permeability of rough-walled fractures[END_REF] showed that a 2D model in which the aperture varies only in the flow direction gives qualitatively the same results as a full 3-D model.

In order to study analytically flow and transport in fractures, a geometrical model that best represents the fracture characteristics must be used. Power and Tullis [START_REF] Power | Euclidean and fractal models for the description of rock surface roughness[END_REF] showed that natural rock fractures, despite having self-affine properties, can be described by the sum of multiple sine waves with equal amplitude to wavelength ratios. This means that the walls roughness, even if irregular, has an oscillatory nature and can be represented by a regular corrugation, which leads to the sinusoidal fracture model.

The configuration in which the profiles of the walls vary sinusoidally along the length can be used as a model of real fractures because, it captures their oscillatory nature, as well as the effects of the walls roughness and of the aperture variation on the flow. Actually, Le Borgne et al. [START_REF] Le Borgne | Effective porescale dispersion upscaling with a correlated continuous time random walk approach[END_REF] showed that velocity distribution in sinusoidal channels are very similar to that obtained in more complex medium.

The difference between using the parallel plate model and the sinusoidal model is shown in Figure 1.11. On the one hand, it is obvious that the parallel plate model does not take into account surface roughness. It presents a constant aperture and a geometry that would not affect the flow streamlines inside the channel. On the other hand, the sinusoidal model, despite not following exactly the real roughness profile, can still take into account the aperture variation and the effect of the channel walls on the flow geometry. These are the reasons why channels with sinusoidal walls have been then widely used to represent rough fractures (Zimmerman et al. [START_REF] Zimmerman | Lubrication theory analysis of the permeability of rough-walled fractures[END_REF], Brown et al. [START_REF] Brown | Applicability of the reynolds equation for modeling fluid flow between rough surfaces[END_REF], Zimmerman and Bodvarsson [START_REF] Zimmerman | Hydraulic conductivity of rock fractures[END_REF], Waite et al. [START_REF] Waite | The effect of surface geometry on fracture permeability: A case study using a sinusoidal fracture[END_REF], Sisavath et al. [START_REF] Sisavath | A simple model for deviations from the cubic law for a fracture undergoing dilation or closure[END_REF],Basha and El Asmar [START_REF] Basha | The fracture flow equation and its perturbation solution[END_REF], Yeo and Ge [START_REF] Yeo | Applicable range of the reynolds equation for fluid flow in a rock fracture[END_REF], Nizkaya [START_REF] Nizkaya | Transport and deposition of inertial particles in a fracture with periodic corrugation[END_REF], Liu and Fan [START_REF] Liu | The characteristics and estimation of flow through a single rough-walled fracture[END_REF], Renu and Kumar [START_REF] Renu | Benzene dissolution and transport in a saturated sinusoidal fracture with non-uniform flow: Numerical investigation and sensitivity analysis[END_REF]) It is convenient also to notice that sinusoidal channels have been used to model homogeneous porous media. For instance, porous media can be modeled as a set of uniform spherical grains regularly stacked. In this case, a 2D simplification of a representative elementary volume can lead to a sinusoidal channel that accounts for the aperture variation inside the medium. This simplification has been widely applied to study flow and solute dispersion in porous media (Kitanidis and Dikaar [START_REF] Kitanidis | Stokes flow in a slowly varying two-dimensional periodic pore[END_REF], Edwards et al. [START_REF] Edwards | Dispersion of inert solutes in spatially periodic, two-dimensional model porous media[END_REF], Bolster et al. [START_REF] Bolster | Solute dispersion in channels with periodically varying apertures[END_REF], Bouquain et al. [START_REF] Bouquain | The impact of inertial effects on solute dispersion in a channel with periodically varying aperture[END_REF]).

Throughout the thesis, sinusoidal variation will be considered to represent walls corrugation. Other than representing fracture roughness, such geometry is convenient for setting up numerical simulations and experimental devices aiming at studying fracture flow.

Chapter 2 SINGLE PHASE FLOW THROUGH FRACTURES

In order to study particle transport, it is crucial to consider a model that can describe accurately the fluid flow. Different models were used in hydrogeology for investigating flow through single rough-walled fractures. Idealizing the fracture as a channel with two flat walls simplifies greatly the problem and enables to find an analytical solution for the velocity field, known as the cubic law (CL).

Taking into account walls corrugation and considering slowly varying apertures, Reynolds equation can be used and leads to the local cubic law (LCL), which gives expressions of the velocity components depending the fracture geometry.

However, the validity of the CL and the LCL remains questionable. Indeed, there exist some criteria, strictly related to the fracture geometry, for the applicability of these two laws. In this chapter, we propose a numerical study aiming to assess the validity of the CL and of the LCL by considering different fracture geometries. The fractures are represented by channels with sinusoidal walls having different geometrical properties defined the channel aperture, the amplitude and the wavelength of the walls corrugation, the corrugations asymmetry and the phase shift between the two walls. The validity of the CL and of the LCL is evaluated for Reynolds number in the range [6.7 × 10 -2 , 6.7 × 10 1 ].

Introduction

Flow in rough-walled fractures is governed by the Navier-Stokes (N S) equations (Zimmerman and Bodvarsson [START_REF] Zimmerman | Hydraulic conductivity of rock fractures[END_REF]. However, the non-linearity of the inertial term in these equations makes their solution very difficult to be obtained without the use of perturbation expansions that may give approximate solutions (Hasegaw and Izuchi [START_REF] Hasegawa | On steady flow through a channel consisting of an uneven wall and a plane wall: Part 1. case of no relative motion in two walls[END_REF], Basha and El Asmar [START_REF] Basha | The fracture flow equation and its perturbation solution[END_REF]).

As pointed out in chapter 1, a fracture can be approximated as a channel with two flat parallel walls. This approximation simplifies the problem and leads to an analytical solution of the N S equations known as the CL. Another approximation consists in considering that the fracture aperture varies slowly and the Reynolds equation, i.e. the LCL, can then be used.

However, several authors have shown that Reynolds equation tends overestimate the flow rate in rough-walled and corrugated fractures depending on their geometry and the flow characteristics (Mourzenko et al. [START_REF] Mourzenko | Permeability of a single fracture; validity of the reynolds equation[END_REF], Brown et al. [START_REF] Brown | Applicability of the reynolds equation for modeling fluid flow between rough surfaces[END_REF], Nicholl et al. [START_REF] Nicholl | Saturated flow in a single fracture: Evaluation of the reynolds equation in measured aperture fields[END_REF], Lee et al. [START_REF] Lee | Assessment of the validity of stokes and reynolds equations for fluid flow through a rough-walled fracture with flow imaging[END_REF]). Indeed, there are some restrictions related to the flow and to the fracture geometry for the Reynolds equation, and thus for the LCL, to be valid. First, different geometrical criteria have been proposed for the validity of the LCL. For instance, Zimmerman et al. [START_REF] Zimmerman | Lubrication theory analysis of the permeability of rough-walled fractures[END_REF], by applying the perturbation approach of Hasegawa and Izuchi [START_REF] Hasegawa | On steady flow through a channel consisting of an uneven wall and a plane wall: Part 1. case of no relative motion in two walls[END_REF] to fractures with mirror-symmetric sinusoidal walls, found that Stokes equation can be replaced by Reynolds equation only if the ratio between the wavelength of the wall and the standard deviation of the local apertures is higher than five. Using a similar approach with a fracture consisting of a flat wall and a sinusoidal wall, Zimmerman et al. [START_REF] Zimmerman | Fluid flow in rock fractures: From the navier-stokes equations to the cubic law[END_REF] suggested that the wavelength of the walls must be higher than three times the arithmetic mean of local apertures for Reynolds equation to be valid. Later on, Yeo and Ge [START_REF] Yeo | Applicable range of the reynolds equation for fluid flow in a rock fracture[END_REF] performed numerical simulations of flow in fractures with sinusoidal parallel walls with different corrugation amplitudes, and identified a criterion relating flow tortuosity and walls roughness for Reynolds equation to be applicable. On the other hand, the influence of the flow characteristics on the validity of the LCL has been investigated. The LCL is generally considered valid for viscous flows when Re < 1 ( [START_REF] Renshaw | Measuring fracture apertures: A comparison of methods[END_REF], [START_REF] Brush | Fluid flow in synthetic rough-walled fractures: Navier-stokes, stokes, and local cubic law simulations[END_REF], [START_REF] Konzuk | Evaluation of cubic law based models describing single-phase flow through a rough-walled fracture[END_REF]). When Re increases, flow inertial effects become significant and thus the deviation of the LCL from the N S equations increases. The first inertial corrections to the solution given by the lubrication theory were found by [START_REF] Van Dyke | Slow variations in continuum mechanics[END_REF]. Second order corrections were then calculated for specific geometries, like for instance, the case of a fracture having a flat wall and a corrugated wall ( [START_REF] Hasegawa | On steady flow through a channel consisting of an uneven wall and a plane wall: Part 1. case of no relative motion in two walls[END_REF]), the case of a fracture with parallel walls ( [START_REF] Borisov | Viscous liquid flow in a channel with sinusoidal walls[END_REF]) or for mirror-symmetric fractures ( [START_REF] Crosnier | Écoulement dans une fracture de géométrie radiale: influence d'une rugosité périodique[END_REF], [START_REF] Buès | Macroscale model and viscousinertia effects for navier-stokes flow in a radial fracture with corrugated walls[END_REF] in the case of a radial fracture). [START_REF] Basha | The fracture flow equation and its perturbation solution[END_REF] and [START_REF] Nizkaya | Transport and deposition of inertial particles in a fracture with periodic corrugation[END_REF] generalized these results by calculating the inertial corrections for corrugated fractures with arbitrary walls.

Besides, numerous studies have focused on the determination of the hydraulic aperture, i.e. the aperture of a fracture with flat walls that would have the same permeability as the actual rough fracture. For example, in order to relate precisely the hydraulic aperture to the walls roughness and to the fracture mean aperture (known also as the mechanical aperture), Sisavath et al. [START_REF] Sisavath | A simple model for deviations from the cubic law for a fracture undergoing dilation or closure[END_REF], following the method of Van Dyke [START_REF] Van Dyke | Slow variations in continuum mechanics[END_REF], used a perturbation approach to study fluid flow in a sinusoidal fracture undergoing dilatation or closure, and developed an expression for the hydraulic aperture taking into account a resistance term that can not be simply deduced from Reynolds equation. This result is valid for fractures with mirror-symmetric walls and is similar to the one obtained by Hasegawa and Izuchi [START_REF] Hasegawa | On steady flow through a channel consisting of an uneven wall and a plane wall: Part 1. case of no relative motion in two walls[END_REF] for flow through fractures consisting of a corrugated wall and a flat wall. Later on, Liu et al. [START_REF] Liu | The characteristics and estimation of flow through a single rough-walled fracture[END_REF] improved this expression of the hydraulic aperture by taking into account the roughness, the mean aperture, and the phase shift between the two walls. All these results clearly demonstrated that the hydraulic aperture deviates from the mean aperture in rough fractures.

Several attempts were made to modify the LCL in order to take into account the surface roughness, the tortuosity, the friction loss and the flow inertial effects, with different definitions of the mean aperture, like the geometric mean, the arithmetic mean and the true aperture (Ge [START_REF] Ge | A governing equation for fluid flow in rough fractures[END_REF], Konzuk and Kueper [START_REF] Konzuk | Evaluation of cubic law based models describing single-phase flow through a rough-walled fracture[END_REF], Mallikamas and Rajaram [START_REF] Mallikamas | An improved two-dimensional depth-integrated flow equation for rough-walled fractures[END_REF], Qian et al.

[102], Wang et al. [START_REF] Wang | Modification of the local cubic law of fracture flow for weak inertia, tortuosity, and roughness[END_REF], Chen et al. [START_REF] Chen | Effect of roughness on water flow through a synthetic single rough fracture[END_REF]). These modifications were able to reduce the over-prediction of the flow rate by the LCL.

Nevertheless, it would be convenient, when studying fracture flow in rough fracture with applications ranging from particle and solute transport to heat and mass transfer, to use the LCL, or the CL. Indeed, they give a direct and relatively simple relationship between the flow rate and the pressure difference and lead to an explicit expression of the velocity components inside the fracture. However, many issues must be resolved before using the LCL for modeling the fracture flow. First, the criteria proposed in the literature regarding the applicability of the LCL present certain discrepancy among authors. Most likely, the discrepancy arise from the specific fracture geometry considered by each one of these authors.

In addition, to our knowledge, none of the available studies provides a quantification of the relative error between the LCL and the N S equations for arbitrary geometries, i.e. when the two walls are shifted or when they present different corrugation amplitudes. Such assessment is nonetheless crucial before considering further complexity in the modeling of fracture flow processes (particle transport, heat and mass transfer, etc...). Finally, a fine analysis of the mechanisms involved at the local scale is required to better understand the influence of the fracture geometry on the flow field and how inertial effects may affect the validity of the LCL.

To estimate the validity of the LCL and the CL for modeling flow in rough fractures, we present here a thorough numerical investigation aiming at characterizing flow in rough fractures by studying the effect of their geometry on their hydraulic aperture. For convenience, fractures are modeled as channels with sinusoidal walls.

In this chapter, we do not study a specific fracture geometry. Instead, we consider all the possible configurations involving sinusoidal walls to investigate the influence of the fracture mean aperture, walls corrugation wavelength and amplitude, asymmetry between walls corrugations, and horizontal phase shift between the walls on the flow characteristics.

Instead of developing analytical solutions for the flow, we verify the validity of the LCL to model flow by comparing its predictions to numerical solutions of the N S equations for different values of Re.

Chapter outline:

In section 1, we present the fracture configuration as well as the different parameters that define its geometry. In section 2, the equations governing the flow in corrugated fractures are recalled and the LCL model is developed in the dimensionless form. In section 3, these equations are expoited to evaluate the effect of the fracture geometry on the difference between the hydraulic aperture predicted by the LCL and the mean or mechanical aperture used in the CL . In section 4, the numerical method used to solve the N S equations is described. The validity of the LCL is then assessed for Re ranging from 6.7×10 -2 to 6.7×10 1 . In particular, these predictions given by the LCL are compared to the numerical results, and the relative error between the two solutions systematically quantified for different geometrical configurations. In section 5, a discussion is provided based on the results of the study.

Geometrical description of fractures with corrugated walls

The domain is represented in a reference frame (X, Z) where X corresponds to the horizontal direction (the main flow direction) and Z to the vertical one. Gravity is taken into account and applies perpendicularly to the main flow direction (along Z).

Following the approach proposed by Nizkaya [START_REF] Nizkaya | Transport and deposition of inertial particles in a fracture with periodic corrugation[END_REF], we consider a two dimensional fracture having two rough walls with idealized periodic roughness described respectively by the functions Φ 1 (X) for the lower wall and Φ 2 (X) for the upper one (Figure (2.1)). The fracture total length is L ∞ and is thus defined by the domain limited by

X [0, L ∞ ] and Z [Φ 1 (X), Φ 2 (X)].
The fracture walls corrugations are smooth so that ∂Φ 1,2 (X) ∂X << 1. The fracture can be equally defined by the local half aperture H(X) = 1 2 (Φ 2 (X) -Φ 1 (X)) and the fracture middle line Φ(X) = 1 2 (Φ 1 (X) + Φ 2 (X)). The fracture walls are periodic and have the same corrugation wavelength L 0 , which is the characteristic length of the flow in the X direction.

The phase shift between the two walls is ∆X and the corrugation amplitude of each wall is:

A 1,2 = 1 2 (max[Φ 1,2 (X)] -min[Φ 1,2 (X)])
The characteristic length in the Z direction can be defined as the mean aperture of the fracture given by:

H 0 = 1 L ∞ L∞ 0 (Φ 2 (X) -Φ 1 (X))dX
Finally, the fracture aspect-ratio can be defined as:

= H 0 L 0
At this point, it is important to note that there are two types of channels that can be found in literature depending on the order of magnitude of . Channels with ∼ 1 are commonly used to model flow and heat transfer in heat exchangers tubes (e.g. Yin et al.

[105], Mohammed et al. [START_REF] Mohammed | The effects of geometrical parameters of a corrugated channel with in out-of-phase arrangement[END_REF]) and channels with << 1 are commonly considered to represent rough fractures.

It is more convenient to study the fracture flow using dimensionless parameters. The length scales L 0 and H 0 are thus used to define the dimensionless variables:

z = Z H 0 , φ 1,2 (x) = Φ 1,2 (X) H 0 , φ(x) = Φ(X) H 0 , v x = V X V 0 , v z = 1 V Z V 0 , ψ = Ψ Q
In addition, the following dimensionless parameters are introduced:

• δ 0 = A 1 + A 2 2H 0
is the dimensionless average corrugation defined by the ratio of the mean corrugation amplitude of the walls to the fracture mean aperture. This parameter is an indicator of the roughness of the walls.

• γ = A 2 -A 1 A 2 + A 1
represents the asymmetry between the walls corrugations defining the difference of corrugation level between the two walls. γ > 0 means that the upper wall is more corrugated than the lower one and vice versa. γ = 0 corresponds to upper and bottom walls with the same corrugation amplitude and γ = 1 to a fracture with flat bottom wall.

• ∆x = ∆X L 0 is the dimensionless phase shift
The fracture geometry is thus defined by four parameters , δ 0 , γ and ∆x.

In this thesis, the periodic corrugations of the walls are represented by a sinusoidal variation (cf. chapter 1). For a fracture with sinusoidal walls, the walls can be defined by the following equations:

φ 1 (x) = - 1 2 + δ 0 (1 -γ)sin[2π(x - ∆x 2 )] and φ 2 (x) = 1 2 + δ 0 (1 + γ)sin[2π(x + ∆x 2 )] (2.1)
The equation of the dimensionless local half-aperture of the fracture is given by:

h(x) = 1 2 + δ 0 (cos(2πx)sin(π∆x) + γ(sin(2πx)cos(π∆x))) (2.2)
and the equation of the fracture middle-line is:

φ(x) = δ 0 (sin(2πx)cos(π∆x) + γ(cos(2πx)sin(π∆x))) (2.3)
In such a case, the effect of δ 0 , γ and ∆x on the fracture geometry is illustrated in Table 2.1.

Governing equations

Steady flows in closed channels are governed by the N S equations:

ρ(V.∇)V = -∇P + ρg + µ∇ 2 V (2.4)
where V is the velocity of the fluid, ρ is the fluid density, µ is the fluid dynamic viscosity, and P is the hydrodynamic pressure. This equation is an application of the fundamental principle of dynamics to a fluid particle. The rate change of the linear momentum of the fluid particle (left-hand side of equation (2.4)) is equal to the sum of the external forces:

• ρ(V.∇)V : describes the inertial acceleration. As steady flow is independent of time, acceleration is only due to spatial variations of the velocity field by convection. This term is a source of non-linearity in the flow as it presents the square of the velocity vector.

• -∇P : represents the pressure gradient.

• ρg: represents the external body forces. In our case, these forces are solely due to gravity.

• µ∇ 2 V : accounts for the energy dissipation due to viscous friction occurring inside the fluid causing momentum diffusion. Its contribution is more important in zones of high shear, near walls for example.

Equation (2.4), when written in the scalar form, consists in a system of three equations with four variables, the three velocity vector components and the pressure. The system of equations is closed using the continuity equation (mass conservation) which, in the case of an incompressible fluid flow, is written as:

∇.V = 0 (2.5)
The relative importance between viscous and inertial forces is the ratio between the orders of magnitude of the inertial acceleration (ρ(V.∇)V ) and of the viscous friction (µ∇ 2 V ), which is the Reynolds number Re = ρV 0 H 0 µ . When Re >> 1, viscous friction becomes negligible (perfect fluid case). On the other hand, when Re << 1, viscosity dominates the flow, and equation (2.4) can be simplified into the Stokes equation:

µ∇ 2 V -∇P + ρg = 0 (2.6)

Flow between parallel flat walls: the cubic law

In an incompressible laminar flow generated by a pressure gradient along a channel consisting of two parallel planes (the parallel-plate model), the inertial terms are identically null and the steady flow can be expressed via equation (2.6). This problem has been thoroughly addressed in the literature. For instance, Zimmerman and Bodvarsson [START_REF] Zimmerman | Hydraulic conductivity of rock fractures[END_REF] recalled in detail all the equations and conditions Of the problem. We therefore recall here only the major conclusions.

If the wall length is much greater than the distance separating them, one can assume that only the X-component of the velocity vector is non-zero. In such case, the X-component of equation (2.6) is written as:

(∇P ) X = ∂P ∂X = µ ∂ 2 V X ∂X 2 (2.7)
with ∇P the applied pressure gradient. If the distance between the two walls is H 0 , integrating equation (2.7) with the no-slip boundary conditions (V x (0) = V x (H 0 ) = 0) leads to the following parabolic velocity profile, similar to the one found in the so called Poiseuille flow:

V x = - H 2 0 8µ (∇P ) X (1 - 4Z 2 H 2 0 ) (2.8)
For a difference ∆h in the hydraulic head h between the ends of the two plates, (∇P ) X = ρg∆h L 0 . The volumetric flow rate per unit width can thus be expressed as ( [START_REF] Bear | Dynamics of fluids in porous media[END_REF]):

Q = H0/2 -H 0 /2 V X dZ = - ρgH 0 3 12µ ∆h/L 0 (2.9)

Flow between corrugated walls: the local cubic law

We consider here the fractures defined in section 1. The viscous flow in the fracture is calculated under the assumption that the simplification of equation (2.4) into equation (2.6) is valid. The no-slip boundary conditions on the fracture walls yield:

V (X, Φ 1 (X)) = 0 V (X, Φ 2 (X)) = 0 (2.10)
and the volumetric flow rate per unit width is given by:

Q = Φ 2 (X) Φ 1 (X) V X (X, Z)dZ (2.11)
Writing equation (2.6) in scalar form and estimating

∂ 2 V X ∂X 2 proportional to V 0 L 2 0 and ∂ 2 V X ∂Z 2 proportional to V 0 H 2 0
, one can see that when << 1, the derivative of the velocity with respect to X can be neglected compared to that with respect to Z. Thus, the projection of equation (2.6) on X can be reduced to:

µ ∂ 2 V X ∂Z 2 = ∂P ∂X (2.12)
Integrating equation (2.12) with respect to Z, and taking into account the boundary conditions (equation (2.10)), leads to:

V X (X, Z) = 1 2µ ∂P ∂X (Z -Φ 1 (X))(Z -Φ 2 (X)) (2.13) Defining H(X) = 1 2 (Φ 2 (X) -Φ 1 (X))
as the local half aperture, the average velocity can then be calculated as:

V X = 1 2H(X) Φ 2 (X) Φ 1 (X) V X (X, Z)dZ = -H 2 (X) 3µ ∂P ∂X (2.14)
Recalling the no-slip boundary conditions

V X (X, Φ 1 (X)) = V X (X, Φ 2 (X)) = 0, equation (2.5)
, which is applied to local velocities V X , can be also applied to the flux 2H(X)V X (Zimmerman and Bodvarsson [START_REF] Zimmerman | Hydraulic conductivity of rock fractures[END_REF]), thus leading to:

∇.[H 3 (X)∇P ] = d dX [H 3 (X) dP dX ] = 0 (2.15)
which corresponds to the Reynolds equation for a steady incompressible flow ( [START_REF] Brown | Fluid flow through rock joints: the effect of surface roughness[END_REF]). Reynolds equation gives an approximate description of the flow between two slightly non parallel and non-planar surfaces. Equation (2.15) implies that the CL, which is valid for flows occurring between two parallel planar walls, can be applied locally (at a point of coordinate X) to a channel with non parallel walls, leading to the LCL model. To proceed from the local level to the fracture level, equation (2.15) can be integrated between two points X 1 and X 2 along the channel ( [START_REF] Silliman | An interpretation of the difference between aperture estimates derived from hydraulic and tracer tests in a single fracture[END_REF]):

∆P = -12µQ X 2 X 1 1 H 3 (X) dX (2.16)
with ∆P the pressure drop between these points. When the walls of the fracture are periodic with a wavelength L 0 , equation (2.16) gives ( [START_REF] Brown | Fluid flow through rock joints: the effect of surface roughness[END_REF], [START_REF] Zimmerman | Lubrication theory analysis of the permeability of rough-walled fractures[END_REF]):

∆P L 0 = - 12µQ H h 3 (2.17)
where H h represents the hydraulic aperture of the fracture.

To write the LCL in dimensionless form, x = X L 0 and h(x) = H(X) H 0 are defined. The hydraulic aperture of the fracture is thus:

H h = 2H 0 < h -3 > -1 3 (2.18)
with:

< h -3 > = 1 0 1 h 3 (x) dx (2.19)
the dimensionless hydraulic aperture.

Taking the viscous lubrication pressure P * = µV 0 H 0 as a pressure scale, equation (2.17) leads to:

p = - 3 2 < h -3 > (2.20)
with p = ∆P/P * the dimensionless pressure drop.

Flow velocity components in corrugated channels

To calculate the flow velocity components, equation (2.6) is rewritten as a function of the stream function Ψ(X, Z) ([70], [START_REF] Basha | The fracture flow equation and its perturbation solution[END_REF], [START_REF] Nizkaya | Transport and deposition of inertial particles in a fracture with periodic corrugation[END_REF]), which satisfies

V X = ∂Ψ ∂Z and V Z = - ∂Ψ ∂X : ∇ 4 Ψ = ∂ 4 Ψ ∂X 4 + 2 ∂ 4 Ψ ∂X 2 ∂Z 2 + ∂ 4 Ψ ∂Z 4 = 0 (2.21)
Choosing the walls corrugation wavelength L 0 and the channel mean aperture H 0 as length scales and the mean flow velocity V 0 as a velocity scale in the X direction, the following dimensionless parameters can be defined:

z = Z H 0 , φ 1,2 (x) = Φ 1,2 (X) H 0 , φ(x) = Φ(X) H 0 , v x = V X V 0 , v z = 1 V Z V 0 , ψ = Ψ Q (2.22)
with v x = ∂ψ ∂z and v z = -∂ψ ∂x the velocity components in the dimensionless form. Equation

(2.21) can then be rewritten as:

4 ∂ 4 ψ ∂x 4 + 2 2 ∂ 4 ψ ∂x 2 ∂z 2 + ∂ 4 ψ ∂z 4 = 0 (2.23)
When << 1, the terms proportional to 2 and 4 can be neglected (zeroth-order when the stream function is developed in the form of an asymptotic expansion), and equation (2.23) reduces to:

∂ 4 ψ ∂z 4 = 0 (2.24)
In order to simplify the problem, a cross-channel coordinate is introduced as ([69], [START_REF] Nizkaya | Inertial focusing of small particles in wavy channels: Asymptotic analysis at weak particle inertia[END_REF]):

η = z -φ(x) h(x) η enables to map the channel interior into a rectangle (z [φ 1 (x), φ 2 (x)] → η [-1, 1]).
Equation (2.24) becomes:

1 h 4 (x) ∂ 4 ψ ∂z 4 = 0 (2.25)
with the boundary conditions (equation 2.10) expressed as:

ψ(x, -1) = - 1 2 , ψ(x, 1) = 1 2 , ∂ψ ∂η (x, -1) = ∂ψ ∂η (x, 1) = 0 (2.26)
The solution of equation (2.25) is thus:

ψ(x, η) = 1 4 η(3 -η 2 ) (2.27)
and the flow velocity components are respectively equal to ( [START_REF] Zimmerman | Hydraulic conductivity of rock fractures[END_REF], [START_REF] Nizkaya | Inertial focusing of small particles in wavy channels: Asymptotic analysis at weak particle inertia[END_REF]):

v x = 3 4h(x) (1 -η 2 ) and v z = 3(φ (x) + ηh (x)) 4h(x) (1 -η 2 ) (2.28)
The validity of these expressions to describe the flow velocity components in corrugated channels will be assessed later on. Note that replacing the dimensional form of equation (2.28) in Equation 2.12 also leads to the LCL in the form given by equation (2.16).

Influence of the fracture geometry on its hydraulic aperture

The CL, established with respect to the mean aperture, tends to overestimate the flow rate in channels with corrugated walls ( [START_REF] Konzuk | Evaluation of cubic law based models describing single-phase flow through a rough-walled fracture[END_REF], [START_REF] Dippenaar | On the cubic law and variably saturated flow through discrete open rough-walled discontinuities[END_REF]) and thus needs to be replaced by the LCL for a better description of the flow. To evaluate this overestimation, we examine the effect of the fracture geometry on the difference between the hydraulic aperture H h (on which the LCL is based) and the mean aperture H 0 (on which the CL is based).

In a channel with corrugated walls, the ratio H h H 0 calculated from the dimensionless hydraulic aperture < h

∆x effect

The variation of

H h H 0 as a function of ∆x is evaluated for ∆x ∈ [0, 1] (Figure (2.2)
). This range covers all the possible phase shifts between the top and bottom walls. ∆x = 0 and ∆x = 1 correspond to in phase walls whereas ∆x = 0.5 corresponds to the case where the phase shift is maximum between the two walls. Different values of δ 0 (0.1, 0.2 and 0.4) and γ (0, 0.5 and 1) are considered to investigate multiple geometrical configurations. γ = 0 corresponds to a channel having two walls with the same corrugation amplitude and γ = 1

to a channel with one flat wall.

H h H 0 is very close to 1 for δ 0 = 0.1 in all the considered configurations. This means that H h does not deviate strongly from H 0 in the case of low corrugation amplitudes. On the other hand, for δ 0 = 0.2 and δ 0 = 0.4, H h H 0 decreases with the increase of the phase shift between the two walls and reaches its minimum when this shift is maximum (∆x = 0.5).

Note that when γ = 0 and ∆x = 0, corresponding to parallel identical walls, H h is equal to H 0 for all the values of δ 0 . Therefore, the deviation of H h from H 0 increases for higher wall corrugations amplitudes and when the walls are shifted. For γ = 1, H h H 0 does not vary as a function of ∆x as there is, by definition, no phase shift for this specific configuration.

However, when γ = 1, H h H 0 decreases significantly when δ 0 increases. between its walls has the same hydraulic aperture as a channel having a flat wall and a corrugated wall.

H h H 0 γ = 1 δ 0 = 0.1 δ 0 = 0.2 δ 0 = 0.4

Influence of the fracture geometry on the validity of the LCL for different Reynolds numbers

In this section, numerical simulations using the finite element method (F EM ) are conducted to study the effect of different geometrical properties of fractures with sinusoidal walls on the validity of the LCL, for Re values ranging from 6.7 × 10 -2 to 6.7 × 10 1 . To do so, we systemically compare the theoretical predictions of the LCL to the numerical solution obtained by solving the full N S equations.

Numerical Method

The F EM is used to compute numerical solutions of the N S equations. The numerical solutions are obtained using COMSOL Multiphysics software. Like most of the numerical methods, the F EM is based on the discretization of the geometrical domain into small and simple cells such as triangles or quadrilaterals in 2D. Each one of these cells is called an element. Each element contains nodes, which are points in the domain on which approximated solutions are computed. These approximated solutions are represented by basis functions that are usually selected as polynomials of order 1 to 3. For each element, the solver searches values at the nodes and the polynomials constants that can match the differential equation in the most accurate way.

Numerical simulation procedure

The aim of the simulations is to verify the validity of the LCL represented in the dimensionless form by equation (2.20). To do so, the pressure difference between two sections perpendicular to the main flow and distant of a wavelength, computed numerically via the resolution of the N S equation, is compared to the pressure difference estimated by the LCL.

The procedure of the numerical simulations is as follows:

• The geometry of the fracture is first defined. The fracture is built by setting the amplitude and the wavelength of its walls, and the horizontal phase shift between them according to the expressions of Φ 1 (X) and Φ 2 (X). A flat part is constructed upstream to ensure a parabolic velocity profile at the fracture inlet.

• The computational domain is discretized using a mesh consisting of triangular elements.

The mesh is refined in the near wall regions and gradually coarsened when moving towards the fracture center (Figure (2.4)). A mesh objectivity test was performed and the spatial discretization considered sufficiently fine when the same problem, treated with a finer mesh (1.5 times more nodes) led to the same solution for each considered field (velocity and pressure).

• A parabolic velocity profile with a mean flow velocity V 0 is defined at the inlet. V 0 is defined according to the value of the corresponding Re. A zero pressure is imposed at the outlet P outlet = 0, and the no-slip boundary conditions are imposed on the fracture walls ( V f = 0).

• The continuity and N S equations governing the steady incompressible flow in the fracture are solved, and the pressure and velocity fields are computed.

Low Re (< 1)

The LCL is generally considered valid for Re < 1. In this section we set Re = 0.1 for all the simulations. One can notice that the relative error increases with the increase of for all the geometrical configurations. This error is less than 2 % for < 0.1 in all the geometrical configurations.

When increases, the error is maximum in the channel with parallel walls. Moreover, this error and consequently the deviation of the LCL from N S are maximal in a channel with parallel walls and minimal in a channel with mirror-symmetrical configuration.

To compare the two solutions on a local level, the dimensionless velocity profiles are plotted on two cut lines inside the channel located at x = 1.25 and x = 1.5 (Figure 2.4) for = 0.1 and = 0.4, in the cases of channels with parallel and mirror symmetric walls (Figures (2.6) and (2.7)). For = 0.1, the velocity profiles seem to be identical in the case of parallel The relative error between the LCL and N S solutions does not increase significantly in the mirror symmetrical configuration for all possible values of δ 0 and remains around 1 %.

We conclude that the variation of δ 0 has a small effect on the deviation of the LCL in this configuration. Nonetheless, the relative error increases with the increase of δ 0 in the channel with arbitrary walls, and in a greater extent in the channel with parallel walls. The deviation of the LCL with the respect to the N S solutions due to the increase of δ 0 is then maximum in a channel with parallel walls and minimum in a channel with mirror symmetrical walls.

The variation of the relative error between the LCL and N S solutions as a function of γ is plotted in Figure (2.9). γ was varied from 0 (a channel having two identical walls) to 1 (a channel having a flat wall, representing the maximal possible asymmetry between the two walls). For the three channels, the relative error between the two solutions remains between 1.5 % and 2.5 % when γ increases. Thus, increasing γ has a negligible effect on the deviation of 70

-~-•-•-•- ------. ~ . . . . ~ . . . . ..... ----------- -~ .-.. -
the LCL solution with respect to that of N S equations.

The dimensionless phase shift ∆x was varied between 0 and 1 to take into account all the possible phase shifts between the two walls. The variation of the relative error between The variation of the pressure as a function of ∆x presents a symmetry around ∆x = 0.5. This is due, as shown in section 3, to the fact that the hydraulic aperture is independent of the direction of the phase shift between the two walls. The flow in a channel where the bottom wall is offset from the top wall is thus identical to that in a channel where the top wall is offset from the bottom one, if the offset is equal. One can notice that the relative error is minimum for ∆x = 0.5 and maximum for ∆x = 0.

High Re (> 1)

When Re increases, the relation between the pressure gradient along the fracture and the flow rate does not remain linear (as in equation (2.16)) due to the effect of fluid inertia that becomes more important. For the same applied pressure gradient, the flow rate decreases, or vice versa, maintaining a given flow rate through the fracture requires increasing the applied pressure gradient.

On the other hand, we recall that the LCL was established under the assumption of a Stokes flow inside the fracture and therefore for Re << 1. In its dimensional form (equa-tion(2.20)), the pressure is independent of Re although it is expected to increase as Re increases. The validity of the LCL for high Re must then be assessed. Moreover, the velocity profiles are no more ideally parabolic., and depending on the channel geometry, recirculation zones may appear inside the fracture. They are due to the viscous shear stresses resulting from the fluid momentum mismatch in the fracture center and the near-wall regions. The fluid particles located in the central part of the fracture move faster than the quasi-stagnant ones located in the channel hollows. This results in the formation of fluid recirculations near the rough walls. These recirculation zones can also appear under Stokes flow conditions (Kitanidis and Dykaar [START_REF] Kitanidis | Stokes flow in a slowly varying two-dimensional periodic pore[END_REF], Malevich et al. [START_REF] Malevich | Stokes flow through a channel with wavy walls[END_REF]) but under stronger geometrical constraints.

Such recirculation zones greatly affect the validity of the LCL. Investigating the onset of creation of these zones as well as the effect of the fracture geometry on their appearance are out of the scope of this study. Instead, we focus on studying the range of Re to which the LCL can still be used to model flow in fractures, and the effect of the fracture geometry on this validity.

To do so, we use the reference fractures defined in the previous section (Table 2.2), and calculate the dimensionless pressure gradient p to compare the LCL and N S solutions for Re varying in the range [6.7 × 10 -2 , 6.7 × 10 1 ]. We then compare the relative error between the two solutions for several geometrical configurations corresponding to different values of , δ 0 , γ and ∆X.

2.4.3.a. Relative error between the LCL and N S solutions for the reference geometries

First, we study the variation of the dimensionless pressure p as a function of Re according to the LCL and to N S solutions for each reference geometry (Table 2.2). The variation of the relative error between the two solutions is plotted in Figure (2.11). For high Re (i.e.

Re > 40), the deviation between the LCL and N S solutions is maximal in the mirrorsymmetrical configuration and minimal in the channel with parallel walls. This result is different than the one found for low Re (section 4.2.1). It can also be noted that the relative error remains constant for Re < 15 in all the geometrical configurations. It then increases progressively, although remaining relatively low, i.e. below 2 %, for Re values less than 30. 2.2).

To further investigate this effect, the dimensionless velocity profiles are compared for Re = 6.6 × 10 -2 and Re = 6.6 × 10 1 , in the cases of fractures with parallel and mirror symmetric walls (Figures (2.12) and (2.13)). One can see that the velocity profile calculated by the LCL coincides exactly with the profile predicted by N S solution when Re = 6.6×10 -2 , for the two cases. For Re = 6.6 × 10 1 , the two profiles also coincide for the fracture with parallel walls but a deviation appears for the fracture with mirror symmetric walls, which confirms that the error is more important in the latter case (Figure 2.11).

In order to study the effect of the fracture geometry on the validity of the LCL for higher values of Re, we computed the variation of the relative error between the two solutions as a function of Re for each geometry, changing a parameter each time.

2.4.3.b. Influence of , δ 0 , γ and ∆x on the relative error between the LCL and

N S solutions

In the reference geometries (Table 2.2), was fixed equal to 0.1. Here, the LCL and N S solutions are computed for the same geometries but with = 0.2 to study the effect of the aspect ratio on the relative error. The variation of the relative error as a function of Re for each value of and for each geometry is plotted in Figure (2.14).

In all fractures, the relative error between the LCL and N S solutions is higher for = 0. 4.2.1). However, at high Re, the deviation becomes more important in the case of a channel with mirror symmetrical walls. In the case of a channel with arbitrary walls, the relative error tends to increase with the increase of and Re but remains nonetheless inferior to the error obtained for mirror symmetrical walls.

To study the effect of δ 0 , different values are tested and compared to the reference geometries where δ 0 was fixed equal to 0.2. To investigate the effect of the walls corrugation amplitude, and therefore the walls roughness, on the deviation between the LCL and N S solutions, the two solutions are now computed for the same geometries but with δ 0 = 0.4. Re = 66 For all the geometries, the relative error is higher for δ 0 = 0.4, whatever the value of Re. However, for small Re, increasing δ 0 does not affect the relative error, while the error is 75

Re= 0.066

(a)
Re = 66 higher for δ 0 = 0.4 when Re is higher than 15.

In the reference geometries (Table 2.2), when the channels have parallel and mirror symmetric walls, γ = 0 because the walls have the same corrugation amplitude. We investigate the influence of the corrugation asymmetry by setting γ = 1 3 . In addition, we compare these two cases to the case of a fracture having a flat wall (γ = 1) to evaluate the effect of the asymmetry between the two walls on the validity of the LCL for high Re. The variation of the relative error as a function of Re for each geometry and for each reference geometry is plotted in Figure (2.16). When the phase shift between the two walls is maximum (∆x = 0.5), the variation of γ has no effect on the relative error whatever the value of Re. When the two walls are in phase (∆x = 0), the relative error increases slightly when Re increased. This increase is more important in the case of a channel having a flat wall.

Once again, it is important to note that in all the cases studied in this section, the relative error remains relatively constant for Re < 15, meaning that the results obtained in section 4.2., on the effects of the fracture geometry on the validity of the LCL at low Re, remain valid for Re up to 15.

Discussions

Relation between the hydraulic and the mean apertures

The effect of the geometrical properties of a fracture on the discrepancy between its hydraulic aperture H h and its mean aperture H 0 has been investigated. H h and H 0 were compared to estimate the conditions under which the LCL, developed with respect to the hydraulic aperture H h , can be replaced by the CL, developed with respect to the mean aperture H 0 .

Analytical expressions of H h based on Reynolds equation or the zeroth order solution of N S equations exist for specific geometries, like fractures having mirror-symmetric walls (Zimmerman et al. [START_REF] Zimmerman | Lubrication theory analysis of the permeability of rough-walled fractures[END_REF]), parallel and shifted walls (Basha and El Asmar [START_REF] Basha | The fracture flow equation and its perturbation solution[END_REF]), and nonidentical walls (Nizkaya [START_REF] Nizkaya | Transport and deposition of inertial particles in a fracture with periodic corrugation[END_REF]). These expressions show that H h deviates from H 0 and depends clearly on the fracture geometry. Instead of considering specific geometries, we directly calculated H h using equations (2.18) and (2. [START_REF] Falkovich | Accleration of rain initiation by cloud turbulence[END_REF], and plotted the variation of H h H 0 as a function of the average corrugation amplitude δ 0 , the corrugations asymmetry γ and the horizontal phase shift ∆x between the bottom and top walls (Figures (2. The results show clearly that, in most cases, H h is smaller than H 0 . This leads to a systematic overestimation of the flow rate by the CL compared to the LCL. However, these two apertures are equal in fractures with identical parallel walls (γ = 0 and ∆x = 0) independently of the corrugation amplitude. This is consistent with the result of Basha and El Asmar [START_REF] Basha | The fracture flow equation and its perturbation solution[END_REF] who found that the zeroth order solution of N S equations for such fractures does not depend on any geometrical parameter. In fact, in this particular case, even if δ 0 increases, the local aperture remains constant along the fracture length and is always equal to the mean aperture (Table 2.1(a)).

The deviation of H h from H 0 increases when the walls are shifted and is maximum when the phase lag between the two walls is maximum (∆x = 0.5). This can be explained by the narrowing that appears inside the fracture with the increase of the phase shift (Table 2.1(b)). This narrowing leads to a decrease in the local aperture and therefore to a higher deviation between H h and H 0 . The same narrowing effect explains the diminution of H h with the increase of the walls roughness and with the increase of the asymmetry between the two walls corrugations (Table 2.1(c)).

Therefore, the deviation of H h from H 0 is not only due to the surface roughness, as these two apertures are equal in fractures with rough parallel walls. The deviation appears when the two walls are not in phase or when the two walls do not have the same corrugation amplitude, as a result of the variation in the local aperture.

It is important to note that H h is calculated here using Reynolds equation (or the zeroth order solution of N S equations). Considering higher order solutions leads to additional terms that can not be considered using Reynolds equation alone (Sisavath et al. [START_REF] Sisavath | A simple model for deviations from the cubic law for a fracture undergoing dilation or closure[END_REF], Liu et al. [START_REF] Liu | The characteristics and estimation of flow through a single rough-walled fracture[END_REF]). These terms are not taken into account in the formulation of H h in the current study.

Validity of the local cubic law for different Reynolds numbers

The validity of the LCL for modeling flow in rough fractures has been assessed for Re ranging from 6.7 × 10 -2 to 6.7 × 10 1 by comparing its solution to the numerical solution of N S equations and considering different geometries.

• Low Reynolds number: Although Reynolds equation is generally considered valid for Re < 1, the numerical results obtained cosidering Re = 0.1 show that the LCL solution can deviate significantly from the N S solution depending on the fracture geometry. This is consistent with the experimental results of Nicholl [START_REF] Nicholl | Saturated flow in a single fracture: Evaluation of the reynolds equation in measured aperture fields[END_REF] who showed that Reynolds equation can overestimate the flow rate for Re as low as 0.06, and Lee et al. [START_REF] Lee | Assessment of the validity of stokes and reynolds equations for fluid flow through a rough-walled fracture with flow imaging[END_REF] who found that Reynolds equation is not always valid even for Re between 0.014 and 0.86.

The numerical results show that for Re = 0.1, the relative error between the LCL and the N S solutions increases for higher and δ 0 . However, this error is minimal when the walls are totally shifted and maximal when the walls are in phase. The asymmetry between the two walls γ has little effect on the deviation between the two solutions. This confirms that the validity of the LCL is highly dependent on the fracture geometry. This may also explain the discrepancy in the previous geometrical criterion proposed for the applicability of Reynolds equation ( δ 0 < 0.14 suggested by Zimmerman et al. [START_REF] Zimmerman | Lubrication theory analysis of the permeability of rough-walled fractures[END_REF] for a mirror-symmetric fracture, 2 δ 0 < 0.01 by Yeo and Ge [START_REF] Yeo | Applicable range of the reynolds equation for fluid flow in a rock fracture[END_REF] for a fracture with parallel walls and < 0.33 by Zimmerman et al. [START_REF] Zimmerman | Fluid flow in rock fractures: From the navier-stokes equations to the cubic law[END_REF] for a fracture with one flat wall and one sinusoidal one).

To understand the origin of this discrepancy, Lee et al. [START_REF] Lee | Assessment of the validity of stokes and reynolds equations for fluid flow through a rough-walled fracture with flow imaging[END_REF] performed direct visualizations of flow velocity through rough-walled fractures for Re ranging between 0.014 and 0.86, and found that inertial forces are negligible with respect to viscous forces. This means that the fracture geometrical properties are solely responsible for the deviation between the LCL and N S solutions for low Re. In fact, walls roughness causes curvature in the flow streamlines that induces flow tortuosity and energy dissipation. In order to analyze the impact of the fracture geometry on the flow streamlines, the flow streamlines obtained numerically for Re = 0.1 considering four different geometries are plotted in Figure 2.17. For all the cases, was fixed equal to 0.1, so that he assumption << 1 is verified. Case (d) corresponds to a fracture having a bottom flat wall and a top corrugated wall (γ = 1), with δ 0 = 0.25. Near the bottom wall, the streamlines are straight and their curvature increases when they are closer to the top wall. This is similar to case (a) where ∆x = 0.5 but different than (b) where ∆x = 0. This explains why increasing γ has a stronger effect on the deviation of the LCL solution for lower ∆x than for higher ∆x (Figure (2.9)).

These results suggest that the deviation between the LCL and N S solutions increases when the overall curvature of the streamlines increases. This is due to several effects. First, the curvature of the streamlines leads to a tortuous flow. Tortuosity represents the ratio of the length of the trajectory of the flow between two points and the distance between these two same points. In fact, equation (2.17) is established assuming that a fluid particle travels a distance equal to L 0 , which would be valid in a channel with flat walls. However, in rough channels, because of the curvature of the streamlines (especially near the walls), fluid particles travel longer distances. Second, the viscous dissipation is more significant when the streamlines are curved due to their disruption especially near the channel throats (Basha and El Asmar [START_REF] Basha | The fracture flow equation and its perturbation solution[END_REF]). In fact, as the viscous dissipation is proportional to the velocity gradient, it increases when the streamlines are narrowed. All these effects point to the fact that the curvature of the streamlines is the source of deviation between the LCL and N S solutions at low Re, as confirmed by the numerical results.

• High Reynolds number: If Re increases, flow inertial effects become significant and the fracture flow does not exhibit a Darcian behavior ( [START_REF] Qian | Experimental study of the effect of roughness and reynolds number on fluid flow in rough-walled single fractures: a check of local cubic law[END_REF], [START_REF] Zimmerman | Non-linear regimes of fluid flow in rock fractures[END_REF]). This leads to a higher deviation of the LCL with respect to the N S solution.

From our study, it appears that, for every case studied, the relative error between the two solutions remained quasi-constant for Re < 15, implying that the conclusions drawn for low Re can be valid for Re values up to 15. It confirms the analytical solution obtained by Nizkaya [START_REF] Nizkaya | Transport and deposition of inertial particles in a fracture with periodic corrugation[END_REF] for mirror-symmetric channels. It is somewhat consistent with the result obtained by Zimmerman et al. [START_REF] Zimmerman | Fluid flow in rock fractures: From the navier-stokes equations to the cubic law[END_REF] who, using an order of magnitude analysis of N S equations for a channel having one flat wall and one sinusoidal wall, argued that inertial effects become significant for Re > 10. It is also in agreement with the results suggested by Zimmerman et al. [START_REF] Zimmerman | Non-linear regimes of fluid flow in rock fractures[END_REF] who conducted experimental and numerical simulations on flow in a natural fracture and showed that inertial effects become important for Re starting between 10 and 20. As the condition Re > 15 was found in all the geometries that we considered (Figures 2.14, 2.15 and 2.16), one can generalize the result for any fracture geometry.

Our numerical results show that the relative error does not change as a function of Re in the case of parallel walls. This is in agreement with the analytical results of Basha and El Asmar [START_REF] Basha | The fracture flow equation and its perturbation solution[END_REF] and Nizkaya [START_REF] Nizkaya | Transport and deposition of inertial particles in a fracture with periodic corrugation[END_REF] who found that inertial terms are null in the first and second order terms of the N S expansion solution. On the other hand, this error is maximum when the phase shift between the two walls ∆x is maximum.

In all cases, the relative error between the LCL and N S solutions for higher Re increases with increasing . This error is higher when the walls are shifted than when they are in phase.

A similar tendency appears when δ 0 and/or γ increase. The deviation is more important when the asymmetry between the walls increases, and reaches its maximum in channels having one flat wall.

These results prove that for high Re, the fracture geometrical properties should be taken into account to describe the deviation between LCL and N S . This confirms the result obtained by Qian et al. [START_REF] Qian | Experimental study of the effect of roughness and reynolds number on fluid flow in rough-walled single fractures: a check of local cubic law[END_REF] who showed that fracture flow depends both on Re and on the walls roughness.

In order to study the impact of the fracture geometry on the flow inertial effects, the flow streamlines obtained numerically for Re = 26.66 considering four different geometries are plotted in Figure 2.18. Comparing cases (a) and (b) that have the same δ 0 and γ, shows that when the horizontal phase shift ∆x = 0, the velocity remains constant on the same streamline so that the velocity variation is only in the vertical direction, like in the case of flat walls. In case (a) where ∆x = 0.5, one can see that the flow is slower in the channel when Comparing cases (a) and (c) shows that for the same , γ and ∆x, the alternation in the fluid movement is more pronounced when the dimensionless amplitude δ 0 is higher. This is due to the fact that the flow becomes faster as the channels aperture is narrowing. This explains why the deviation of the LCL from N S solution is greater for higher δ 0 (Figure (cl

(d )
high Re in rough-walled fractures is due to the increase of inertial effects that result directly from the fracture geometry.

To sum up, the fracture geometrical properties affect greatly the flow. At low Re, the curvature of the streamlines induced by the fracture geometry increases tortuosity and energy dissipation, leading to a deviation between the LCL and N S solutions. At high Re, typically higher than 15, this deviation is due to inertial effects that are more or less pronounced depending on the fracture geometry.

Conclusion

In this chapter, the effects of the geometry of rough fractures on the applicability of the CL and the LCL have been investigated, considering fractures with sinusoidal walls.

Our results confirm that care must be taken when using the CL to model flow in roughwalled fractures. More specifically, if the CL can replace the LCL as long as the walls corrugations are relatively small or when the walls are identical and parallel, it clearly overestimates the flow once the corrugations amplitude become significant in fractures having walls that are shifted and/or with different corrugation amplitudes.

The LCL is valid to model fluid flow in rough-walled fractures at low Re, especially in fractures with small aspect ratios and low corrugation amplitudes. However, the deviation between the LCL and N S solutions increases even at low Re when the walls are in phase and when both walls present high corrugation amplitude. This deviation is due to the curvature of the streamlines which increases the tortuosity and energy dissipation inside the fracture, depending on the geometrical properties of the fracture.

When Re increases, inertial effects become significant for Re > 15. This means that the conclusions drawn for low Re can be valid for Re up to 15. Above this limit, the LCL can still be valid to model the flow provided that the fracture presents small aspect ratio, corrugation amplitudes, and small variations in the local aperture along the flow direction.

When these conditions are not respected, the repeated acceleration and deceleration of the flow due to the variation of the local aperture tends to promote the inertial effects and therefore enhances the deviation between the LCL and N S solutions.

In conclusion, a quantitative estimation of the error made when using the LCL, and even the CL, to model fluid flow in rough fractures was given. It is shown that, because fracture flows are strongly dependent on the fracture geometry, the criteria proposed in the literature for applying the cubic law and the local cubic law are not sufficient to be generalized for random fracture geometries. However, the LCL is valid to model the flow for low Re in channels with low aspect ratio .

Chapter 3 TRANSPORT AND DEPOSITION OF WEAKLY-INERTIAL PARTICLES IN FRACTURE FLOWS

This chapter is devoted to the study of particle transport in fractures, assuming that the flow can be described by the LCL as discussed in chapter 2. The particles are non-Brownian, passive, and much smaller than the fracture aperture. Their inertia is considered very small so that it can be neglected in the equation of motion. The equation of motion is coupled to the flow velocity field as predicted by the LCL to provide a model that can predict the particles' behavior inside the fracture (transport and/or deposition) as well as their trajectories.

In order to validate this simplified model, its predictions are systemically verified against numerical simulations where the full N S equations are solved, taking into account particle inertia and flow inertial effects. Based on the trajectory equation, regime diagrams that can predict the behavior of particles entering closed fracture flows are built. These diagrams enable to forecast if the particles entering the flow will be either deposited or transported along the fracture. The influence of the fracture geometry on the particle behavior is then investigated by considering channels with flat and sinusoidal walls. In particular, the effect of the corrugation amplitude, of the asymmetry and of the phase lag between the walls on the extent of the transport and deposition regimes is evaluated.

Here again, the results are systemically verified against numerical experiments taking into account particle inertia and fully resolving the N S equations. The main results of this chapter have been published in the European Journal of Mechanics B/Fluids (Hajjar et al. [START_REF] Hajjar | Transport and deposition of weakly inertial particles in closed channel flows at low reynolds number[END_REF]).

Introduction

As shown in chapter 1, the motion of particles immersed in a fluid flow is very difficult to predict due to different effects related to the particle properties and flow characteristics.

Many previous works on particle motion in closed channel flows focused on the phenomena of lift-induced particle migration across the streamlines, principally because of its crucial implication in microfluidics (Di Carlo et al. [START_REF] Di Carlo | Continuous inertial focusing, ordering, and separation of particles in microchannels[END_REF], Martel and Toner [START_REF] Martel | Inertial focusing in microfluidics[END_REF]). These works targeted the effect of the particle size and flow characteristics on the order of magnitude of the different inertial lift forces. Because lift-induced particle migration occurs when particle motion is dominated by the inertial lift forces, particle inertia has not been generally considered.

Taking into account particle inertia, Jebakumar et al. [START_REF] Jebakumar | Lattice boltzmann method simulations of stokes number effects on particle trajectories in a wall-bounded flow[END_REF] investigated the effects of Stokes number on particle trajectories in wall-bounded vertical channel flow and found that for small Stokes number, particles behave similarly to neutrally buoyant particles. However, few works have investigated the effect of gravity on the behavior of non neutrally buoyant particles in horizontal channel flows. For example, Chen et al. [START_REF] Chen | Deposition of charged particles in a channel[END_REF], [START_REF] Chen | Deposition of particles in a convergent channel[END_REF] studied the deposition of charged particles in straight and convergent channels, investigating the contribution of gravity and image forces in the particle deposition process. Even less works considered particle transport and deposition in fractures and in closed channels with corrugated walls.

For instance, based on migration conditions formulated initially by Sapsis and Haller [START_REF] Sapsis | Clustering criterion for inertial particles in two-dimensional time-periodic and three-dimensional steady flows[END_REF],

Nizkaya et al. [START_REF] Nizkaya | Inertial focusing of small particles in wavy channels: Asymptotic analysis at weak particle inertia[END_REF] neglected fluid inertia and studied particle focusing in channels with periodic corrugated walls. In particular, they showed that, even if the fluid inertial effects are neglected, focusing can occur due to particle inertia and to the waviness of the streamlines imposed by the walls corrugation. These authors also defined a trapping diagram that predicts the presence of inertial focusing as a function of the channel geometry and of the flow Froude number.

In the present work, particle inertia, which is defined by Stokes number and the particle response time, is so small that the focusing phenomenon predicted by Nizkaya et al. [START_REF] Nizkaya | Inertial focusing of small particles in wavy channels: Asymptotic analysis at weak particle inertia[END_REF] is not expected to occur. The flow is unidirectional, laminar and follows the LCL. In addition, the particles are assumed to be sparsely distributed inside the fluid so that the flow affects the motion of the particles, but not vice versa. In other words, only one way coupling between the solid and fluid phases is considered.

Chapter outline: In section 1, the trajectory equations of particles immersed in fluid flows occurring in channels with both flat and corrugated walls are developed. In section 2, these equations are verified through numerical simulations using a hybrid technique combining a finite element method and a Lagrangian particle tracking method. In section 3, the diagram defining different particle transport regimes is established, and its validity is demonstrated by comparing their predictions against numerical experiments. In section 4, the main results obtained in this chapter are summarized and discussed.

Governing equations

We consider 2D flows. The domain is represented in a reference frame (X, Z) where X corresponds to the horizontal direction (the main flow direction) and Z to the vertical one.

Gravity is taken into account and applies perpendicularly to the main flow direction (along Z). To simplify the problem, We consider a solid spherical particle of radius a and density ρ p moving at a velocity V p in a fluid of density ρ f and dynamic viscosity µ flowing at a velocity V f .

Forces acting on each particle

The forces involved when a particle is being driven by a closed channel flow are:

• Gravity and buoyancy forces F g : they are significant when there is a noticeable dif-ference between the particle and the fluid respective densities. The fluid applies an upward thrust upon the particle which is equal to the weight of an equal volume of the liquid displaced. Combining the weight and thrust forces, the resulting force is equal to:

F g = 4 3 πa 3 (ρ p -ρ f )g (3.1)
g being the gravity acceleration. This force is equal to zero for non-buoyant particles.

• Drag force F d : this force results from the pressure and the viscous frictional forces exerted by the fluid on the surface of the particle, in the case of uniform continuous flow. The mathematical expression of the drag force depends on the flow regime around the particle. It is characterized by the particle Reynolds number Re p = ρ f (Vp-V f )2a µ . In the Stokes or viscous regime, when Re p << 1, the resolution of Stokes equations leads to the following mathematical expression of the drag force:

F d = -6πµa(V p -V f ) (3.2)
• Added mass force F am (Auton et al. [START_REF] Auton | The force exerted on a body in inviscid unsteady non-uniform rotational flow[END_REF]): during its motion, the particle displaces a volume of the surrounding fluid. An additional inertia therefore arises from the acceleration of the displaced fluid mass. This phenomenon produces the added mass force of which the expression, in the case of a spherical particle, is given by: 3 being the mass of the fluid displaced by the sphere and

F am = m f 2 ( DV f Dt - dV p dt ) (3.3) m f = 4 3 ρ f πa
DV f
Dt the total derivative with respect to time.

• Tchen Force F T chen ( [START_REF] Tchen | Mean Value and Correlation Problems Connected with the Motion of Small Particles Suspended in a Turbulent Fluid[END_REF]): it represents the action exerted by the undisturbed flow on the volume of fluid occupied by the particle. It is calculated by integrating the pressure gradient exerted by the undisturbed fluid on the particle volume and by subtracting the hydrostatic contribution (which corresponds to the buoyancy thrust). This force involves only the characteristics of the fluid and is expressed as follows:

F T chen = m f DV f Dt (3.4)
• History force F h : known also as the Basset-Boussinesq force ( [START_REF] Candelier | On the effect of the boussinesq-basset force on the radial migration of a stokes particle in a vortex[END_REF]), it is due to the non-stationarity of the disturbance flow around the particle and reflects the effect of the delay caused by the viscous diffusion of the momentum. Indeed, the particle displaces a quantity of fluid, which disturbs the established flow and a time is therefore necessary for the stationary regime to be recovered. This force is expressed in an integral form retaining the whole history of the particle acceleration, as follows:

F h = -6πa 2 √ µρ f t -∞ 1 π(t -t ) d dt [ V p -V f ]dt (3.5)
In this thesis, we consider that the disturbance flow around the particle is steady so that the history force is neglected.

• Lift forces F l : these inertial forces occur when there is a pressure gradient in the direction orthogonal to that of the particle motion. This pressure gradient results from a rotational movement or the shear in the flow of the carrier fluid. Lift forces are nil in the case of Stokes regime (Re p = 0). The lift forces acting on a particle in the case of a channel flow are detailed below.

Four types of lift forces develop in channel flows: rotation-induced (Magnus), shear-slip (Saffman), shear gradient, and wall-induced lift forces (Figure 3.1). The reviews of Zhang et al. [START_REF] Zhang | Fundamentals and applications of inertial microfluidics: a review[END_REF] and Martel and Toner [START_REF] Martel | Inertial focusing in microfluidics[END_REF] explain in detail the effect of each one of these forces and how they are scaled to Re and to the ratio of the particle diameter to the channel height.

We recall here a brief summary of these forces and their effects.

• Magnus rotation-induced lift force: this force appears when a particle is rotating in a uniform flow, so it is not limited to bounded or channel flows. Suppose that the rotational velocity ω of a spherical particle is in the direction of the flow, the particle perturbs the surrounding velocity field asymmetrically and drags the fluid faster around pressure decreases. Therefore, the pressure difference which appears between the lower and upper sides of the spherical particle creates a force in the direction of the pressure gradient. This force, perpendicular to the flow direction, is called the Magnus force ( [START_REF] Magnus | On the deviation of projectiles, and: on a sinking phenomenon among rotating bodies[END_REF]).

• Saffman shear-slip lift force: this force acts on particles that lead or lag the fluid in shear flows. The difference between the particle and the fluid velocities generates a force directed towards the high velocity region. The Saffman force is significant when external forces (gravitational, electric, magnetic) affect the particle relative velocity with respect to the fluid. Examples include the case of non-neutrally buoyant particles transported in a vertical flow, where the gravity can accelerate or decelerate the particle according to the relative difference between the particle and the fluid respective densities.

• Shear-gradient lift force: This force acts on particles immersed in Poiseuille flows and is induced by the shear gradient resulting from the curvature of the velocity profile in the channel. In fact, due to the presence of the walls, the velocity is zero at the channel walls (no-slip) and increases gradually to reach its maximum at the channel center, leading to a parabolic velocity profile. In the presence of a particle, the relative velocity of the fluid to the particle is lower in the channel center region than the near-wall region, and thus leading to a force directed toward the channel walls.

• Wall-induced lift force: This force acts on the particles in the near-wall region. excess pressure is created in the constriction between the particle and the wall, causing a force directed away from the wall. In fact, the flow streamlines are more curved on the side of the particle directed toward the channel center than in the constriction between the particle and the wall leading to higher velocity in the curved region and therefore to a higher pressure in the constriction. The resulting lift force is directed toward the channel center.

Net inertial lift force: the effect of inertial lift forces can be understood by their implication in the appearance of inertial focusing behavior. Ho and Leal [START_REF] Ho | Inertial migration of rigid spheres in two-dimensional unidirectional flows[END_REF] and Vasseur and Cox [START_REF] Vasseur | The lateral migration of a spherical particle in two-dimensional shear flows[END_REF] found analytically that the balance between the shear-gradient lift force and the wallinduced lift force explains the particle migration toward equilibrium positions inside the channel. Their results further showed that the Saffman shear-slip and Magnus rotationinduced lift forces are respectively one order and three orders of magnitude smaller than the shear-gradient lift force. Saffman and Magnus forces are therefore generally neglected when studying of inertial focusing.

The main effect of inertial lift forces on particle transport in closed channel flows is to produce a lateral migration of particles with respect to the flow direction, especially for non-buoyant particles. Bretherton [START_REF] Bretherton | The motion of rigid particles in a shear flow at low reynolds number[END_REF] demonstrated theoretically that lateral migration can not occur under Stokes flow conditions. Lift forces can thus be neglected when no inertial migration is expected to occur. Di carlo et al. [START_REF] Di Carlo | Continuous inertial focusing, ordering, and separation of particles in microchannels[END_REF] have shown through experimental observations that for very low particle Reynolds number (defined as R p U 0 a 2 νH 0 in this case), non-buoyant particles do not migrate across streamlines and therefore inertial lift forces are negligible with respect to other forces. They also showed that in serpentine channels, even for higher R p , lift forces remain negligible for particle radius to channel width ratios a/H 0 < 0.035. This result was later verified by Zhang et al. [START_REF] Zhang | Particle inertial focusing and its mechanism in a serpentine microchannel[END_REF] who performed numerical simulations without considering the inertial lift forces and found a very good agreement with their experimental observations. These results confirm that for very low Rp and small a H 0 , it is reasonable to neglect inertial lift forces. Therefore, in the developments presented throughout the following sections, we neglect the inertial lift forces by making the assumption that flow inertial effects are negligible (LCL model) and that the particles are much smaller than the channel aperture.

Particle motion equation and particle trajectory equation

The particle motion equation is obtained by applying Newton's second law of motion to a single particle, stating that the sum of forces acting on a particle is equal to the product of its mass by its acceleration. When the particle diameter is small and when Re as well as the velocity gradient around the particle are small, the equation of motion of a solid spherical particle of radius a and density ρ p moving in a fluid of density ρ f and dynamic viscosity µ, as derived by Maxey and Riley [START_REF] Maxey | Equation of motion for a small rigid sphere in a nonuniform flow[END_REF] and Gatignol [START_REF] Gatignol | The faxén formulas for a rigid particle in an unsteady non-uniform stokes-flow[END_REF], with corrected added mass term (Auton et al. [START_REF] Auton | The force exerted on a body in inviscid unsteady non-uniform rotational flow[END_REF]) can be written, in the dimensionless form, as:

d v p dt = (1 - 3R 2 ) G (3.6) - 1 τ ( v p -v f - a 2 6L 0 ∇ 2 v f ) - 3R 2 D Dt [( v f + a 2 20L 0 ∇ 2 v f )] 3 √ R √ 2τ t -∞ 1 π(t -t ) d dt [ v p -v f - a 2 6L 0 ∇ 2 V f ]dt
where the total length of the fracture L ∞ is selected as a horizontal length scale, and V 0 as a velocity scale, so that the following dimensionless parameters were defined:

h * = H 0 L ∞ , x = X L ∞ , z = Z H 0 , v x f = V x f V 0 and v z f = 1 h * V z f V 0
. d dt denotes the derivative with respect to time taken along the particle trajectory, and G = e z /F r, F r = V 2 0 L 0 g being Froude's number. The forces taken into account in this equation are (according to their order of appearance): gravity and buoyancy forces, drag force, added mass contribution, force due to the pressure gradient of the unperturbed flow, and Basset history force due to unsteadiness of the disturbance flow around the particle. We assume that this disturbance flow is steady so that Basset force can be neglected. The terms proportional to the Laplacian of the velocity field, known as Faxen corrections ( [START_REF] Faxén | Der widerstand gegen die bewegung einer starren kugel in einer zähen flüssigkeit, die zwischen zwei parallelen ebenen wänden eingeschlossen ist[END_REF]), are due to the non-uniformity of the fluid flow at the particle scale. They are neglected under the assumption that the particle radius is substantially small compared to the characteristic length L 0 of the flow.

Under these conditions, i.e. neglecting Basset force and Faxen corrections terms, and developing the terms of the right-hand side of equation ( 1), we can write:

L 0 V 0 d V p dt = 1 τ ( V f -V p + 2 9 a 2 (1 -k)g ν g |g| ) + L 0 V 0 3R 2 D Dt ( V f ) (3.7) In this case, τ = 2 9R a 2 V 0 νL 0 = St R
V X f = 3V 0 H 0 4H(X) (1 -η 2 ) and V Z f = 3V 0 (φ (X) + ηH (X)) 4H(X) (1 -η 2 ) (3.8)
where H (x) and Φ (x) correspond respectively to the variations of the dimensionless fracture aperture and fracture middle line along the flow direction Particle inertia can be measured by the dimensionless response time τ , defined as the ratio between the particle characteristic time (particle relaxation time) and the flow characteristic time T 0 = L 0 V 0 . In this study, we consider weakly inertial particles i.e. τ << 1. A similar assumption was made by Nizkaya [START_REF] Nizkaya | Transport and deposition of inertial particles in a fracture with periodic corrugation[END_REF] to study particle focusing in channels with periodic walls. Nonetheless, in the current thesis, particle inertia will be neglected (in the analytical developments) so that focusing is not expected to occur. In the next section, we recall briefly the result obtained by Nizkaya [START_REF] Nizkaya | Transport and deposition of inertial particles in a fracture with periodic corrugation[END_REF] to highlight the difference between her approach and the approach adopted in the present thesis. Then, we define, according to the order of magnitude of each term in the particle motion equation, the value of τ under which we assume that particle inertia can be neglected.

3.1.2.a. Focusing of weakly inertial particles in channels with periodic walls L 0 and H 0 are set as length scales, and V 0 as velocity scale. The solution of equation (3.7) is known to converge exponentially to the first order equation ( [START_REF] Maxey | Equation of motion for a small rigid sphere in a nonuniform flow[END_REF], [START_REF] Haller | Where do inertial particles go in fluid flows?[END_REF]):

v p = v f + τ ( 3R 2 -1)( D vf Dt -G) + O(τ 2 ) (3.9)
The components of equation (3.9) are respectively equal to:

v x p = v x f -τ ( 3R 2 -1) D Dt (v x f ) and v z p = v z f -τ ( 3R 2 -1)G -τ ( 3R 2 -1) D Dt (v z f ) (3.10)
In order for particle focusing to occur, the trajectories of moving particles are assumed to coincide with an attracting streamline. A flow streamline is defined by

dz f dx f = v z f v x f
whereas the particle trajectory is defined by

dz p dx p = v z p v x p
. When the particle trajectory and a flow streamline coincide, the particle follows the streamline and its trajectory is defined by the streamline equation, meaning that:

v z p v x p = v z f v x f (3.11)
Replacing the particle velocity components using equation (3.10) leads to:

v z f -τ ( 3R 2 -1)G -τ ( 3R 2 -1) D Dt (v z f ) v x f -τ ( 3R 2 -1) D Dt (v x f ) = v z f v x f (3.12)
which gives:

D Dt (v z f ) -G D Dt (v x f ) = v z f v x f (3.13)
In the case of stationary flow, equation (3.13) can be written as:

v x f d dx (v z f ) + v z f d dz (v z f ) -G v x f d dx (v x f ) + v z f d dz (v x f ) = v z f v x f (3.14) Noting β(x, z) = v z f v x f
, equation 3.14 can be developed, for the case of a steady flow, and leads to:

(v x f ) 2 ( dβ dx + β dβ dz ) -G = 0 (3.15)
Replacing the expressions of v x f , β, and G into equation (3.15) leads to a relation between z and x which defines the preferential trajectory.

For a flow following the LCL, β = (φ (x) + ηh (x)) and equation (3.15) gives:

16h(x) 2 G 9(1 -η 2 ) 2 -[ (φ (x) + ηh (x)) + 2 h (x) h(x) (φ (x) + ηh (x))] = 0 (3.16)
The η coordinate of the attracting streamline is the solution of equation (3.16). In a specific channel, the expressions of h(x) and φ(x) are known and their derivatives can be calculated.

Equation (3.16) can be numerically solved to find the expression of η. To do so, equation (3.16) is integrated along one wavelength (x between 0 and 1), with φ (0) = φ (1) and h (0) = h (1), and gives:

γ 1 z -(1 -η 2 ) 2 (∆ + η) = 0 (3.17) γ 1 z = 16G 9(2 -2 )J h
being the inverse Froude number, and ∆ =

J φh J h , with J h = 1 0 h (x) 2 dx h(x) 3 and J φh = 1 0 h (x)φ (x)dx h(x) 3
two shape factors that characterize the channel geometry.

3.1.2.b. Trajectory equation of inertia-free particles

We consider the case where the particle response time τ is very small so that the term proportional to 1/τ is dominant in equation (3.7) compared to the two other terms, and equation (3.7) can thus be simplified to:

V p = V f + 2 9 a 2 (k -1)g ν g |g| (3.18)
where g |g| is a vertical unit vector pointing in the downward direction. This equation applies to inertia-free sedimenting particles and has been already used in the past (e.g. Stommel [START_REF] Stommel | Trajectories of small bodies sinking slowly through convection cells[END_REF]). Bergougnoux et al. [START_REF] Bergougnoux | The motion of solid spherical particles falling in a cellular flow field at low stokes number[END_REF]) studied experimentally the motion of solid particles in a spherical flow field and found that particle inertia can be safely neglected for τ < 10 -2 .

Similarly, we assume that the model that we will develop for inertia-free particles is valid for τ ≤ 10 -3 . In order to ensure this hypothesis, we consider particles such that τ < 10 -4 throughout the present study.

According to equation (3.18), in 2D, the two components of the particle velocity are thus:

V x p = V x f = dX p dt and V z p = V z f + 2 9 a 2 (1 -k)g ν = dZ p dt (3.19)
where X p and Z p are the particle coordinates. This equation shows that neutrally buoyant inertia-free particles (k = 1 and τ → 0) follow exactly the flow streamlines. For heavy particles (denser than the fluid such as k > 1), we have V z p < V z f , and for particles lighter than the fluid such as k < 1, we have V z p > V z f . For the general case, the trajectory of the particle is then defined by:

dZ p dX p = dZ p dt dt dX p = V z f + 2 9 a 2 (1-k)g ν V x f (3.20)
The total length of the fracture L ∞ is selected as a horizontal length scale. The following dimensionless parameters are introduced:

h * = H 0 L ∞ , x = X L ∞ , z = Z H 0 , v x f = V x f V 0 and v z f = 1 h * V z f V 0 .
Equation (3.20) can then be rewritten in the dimensionless form:

dz p dx p = dz p dt dt dx p = v z f - W h * v x f (3.21) with W = 2 9
a 2 (k -1)g νV 0 a dimensionless number that represents the ratio between the particle sedimentation Stokes velocity and the flow mean velocity V 0 .

Knowing the particle initial position (x 0 p , z 0 p ), and knowing the components of the fluid velocity (v x f and v z f ) at each point (x, z) of the flow domain, the integration of equation (3.21) gives the position of the particle. In the next section, v f will be calculated for a closed channel flow, assuming that the flow follows the LCL.

Using equations (3.21) and (3.8), the trajectory of an inertia-free particle in a closed channel LCL flow can be defined by:

dz p dx p = -4h(x) 3(1 -η 2 ) W h * + (φ (x) + ηh (x)) (3.22)
The trajectory of the particle thus depends on the channel geometrical parameters h(x) and φ(x), and on the ratio W h * . Equation (3.22) can then be exploited to determine particle trajectories in channels with flat and sinusoidal walls.

3.1.2.c. Channel with flat walls

We consider the simple case of a horizontal channel of total length L ∞ made up by two parallel flat walls. The aperture between the walls is H 0 . In this case,

φ 1 (x) = -1/2, φ 2 (x) = 1/2, h(x) = 1/2, φ(x) = 0, h (x) = φ (x) = 0, η = 2z,
dz p dx p = -2 3(1 -4z 2 ) W h * (3.23)
Integrating equation (3.23) between the initial position of the particle (x 0 p , z 0 p ) and its actual position (x p , z p ), the equation of the particle trajectory in a channel with flat walls can be expressed as:

3(z p -z 0 p ) -4(z 3 p -(z 0 p ) 3 ) = 2 W h * (x 0 p -x p ) (3.24)
In the case of a closed channel flow, equation (3.24) enables to predict the final position of a particle injected at a defined initial position inside the channel. More specifically, it allows to know if the particle will settle inside the channel or if it will be transported till the outlet. In fact, by replacing x p by 1 (which corresponds to X p = L ∞ ) and by searching for the corresponding z p , the particle will exit the channel if z p > -1 2 , it will settle inside if z p < -1 2 .

3.1.2.d. Channel with sinusoidal walls

For the case of a channel with sinusoidal walls, in the dimensional form, the walls are defined by:

Φ 1 (X) = H 0 2 + A 1 Sin( 2π L 0 (X - ∆X 2 )) and Φ 2 (X) = H 0 2 + A 2 Sin( 2π L 0 (X + ∆X 2 )) (3.25)
where A 1 is the lower wall amplitude, A 2 is the upper wall amplitude, and ∆X is the horizontal shift between the two walls (Figure 1 in chapter 2). The dimensionless form of Φ 1 (X) and Φ 2 (X) are given in chapter 2 (equation (2.1)). In this case, the expressions of the dimensionless local aperture h(x) and channel middle line φ(x) and their derivatives are more complex so an equation in the form of equation (3.24) is difficult to obtain analytically.

Nonetheless, it is possible to find the position of a particle by numerically integrating equation (3.20) over [x 0 p , 1] considering z(x 0 p ) = z 0 p as the initial condition. For the simple case of channels having two parallel walls, making the substitution z * (x) = z(x)φ(x) leads to an equation relating z * (x) and x similar to equation (3.24). Therefore, the behavior of particles transported in corrugated channels with parallel walls is identical to the one occurring in channels with parallel walls, when studied with the channel middle line as reference.

Numerical verification

In this section, we verify equations (3.22) and (3.24) by performing numerical simulations. These equations were developed under the assumption that particle inertia can be neglected (equation (3.18)) and the flow velocity components were calculated using the LCL which assumes that the flow is dominated by viscous forces. The numerical simulations presented here aim to challenge the validity of these assumptions and to verify equations vector component (2 equations are solved in 2D) using a fourth order Runge-Kutta method with an adaptive time step. At each time step, the forces acting on each particle are obtained from the calculated fields at the current particle position. The positions of the particles are then updated. This process is repeated until the preset simulation time is reached or when the particle comes in contact with a boundary.

The ODE considered in the simulations is equation (3.7). It corresponds to equation (3.7), with Faxen corrections and Basset history force neglected under the assumptions that particle size is small compared to the channel aperture and that the disturbance flow due to the particle is quasi-steady.

Simulation procedure

The geometry of the channel is first defined. In the case of a channel with flat walls, it is represented by a horizontal rectangle defined by the channel aperture as its width and the channel total length as its length. In the case of a channel with sinusoidal walls, the two walls are constructed according to the expressions of Φ 1 (X) and Φ 2 (X) (cf. Chapter 2, equation (2.1)). The channel is then built by defining its mean aperture and the horizontal shift. The channel inlet consists of a square part to ensure a parabolic velocity profile.

The computational domain is discretized with a triangular mesh (Figure (3.3)). The mesh is refined in the near wall regions and gradually coarsened when moving towards the channel center. A preliminary grid independence test has been performed to determine the optimal number of elements with regards to computational cost and numerical precision.

On a total meshed area of 0.0049 m 2 , the average element surface area is 0.0192 mm 2 . The mesh chosen for this case consists of 252364 elements.

A parabolic velocity profile with a mean flow velocity V 0 is defined at the inlet, a zero pressure is imposed at the outlet P outlet = 0 (Figure ( The last step of the simulation consists in calculating the particle trajectories by solving the equation of motion for each particle (equation (3.7)). When a particle comes in contact with a wall, it sticks to it and it is considered as deposited. The calculation of its trajectory is then terminated.

Results

Before validating the analytical model (equations (3.22) and (3.24) for τ < 10 -4 , numerical simulations were performed for τ = 0.01 to verify the presence of inertial focusing as predicted by equation (3.17). The aim of these first simulations is to check the ability of the numerical method that we used to verify analytical models and more particularly to take into account particle inertia. In fact, the focusing in this case is only due to inertia. Therefore, if the numerical simulations can capture the focusing effect then particle inertia is indeed taken into account. It is clear that particles tend to follow the same trajectory after a certain distance and that particle focusing indeed occurred inside the channel. This result confirms that equation (3.17) is valid and that particle inertia is indeed taken into account in its formulation.

3.2.2.b. Particle trajectories

In this section, particle trajectories are calculated such ensuring that τ < 10 -4 .

i. Channel with flat walls

To verify equation (3.24), we computed numerically the trajectories of particles in a channel with flat walls considering the full equation (equation (3.7)). We set up a numerical experiment with a channel of total length L ∞ = 1 m and of aperture H 0 = 5 mm. 5 particles of radius a = 5 µm and density ratio k = 2.5 are injected in the channel at the same initial horizontal position x 0 p but at different initial vertical positions z 0 p . The fluid used is water and the mean flow velocity V 0 fixed equal to 0.01 m/s. In this case W = 0.00817 and τ = 1.66 * 10 -7 . The final distance x f p traveled by each particle until they come in contact with the bottom wall was calculated for each particle initial vertical position z 0 p and comparisons were made with respect to the predictions of equation (3.24). As shown in Figure (3.5), the two approaches are in very good agreement with a maximum relative error between the predicted final particle positions equal to 0.2 %.

ii. Channel with corrugated walls

To verify equation (3.22) for the case of corrugated channels, we considered a channel made up by two sinusoidal walls with H 0 = 5mm, L 0 = 5 cm, L ∞ = 0.98 m, A 1 = 1.5 mm, A 2 = 2 mm, and ∆X = 0.25 L 0 . This configuration corresponds to a general case of a random geometry because the channel walls are neither identical nor parallel. The particle properties and initial positions were chosen equal to the ones used in the previous section (case of channel with flat walls) and the flow mean velocity was also fixed equal to V 0 = 0.01 m/s.

In this case W = 0.00817 and τ = 3.33 * 10 -6 . The distances x f p traveled by the particles for different initial vertical positions z 0 p are plotted in Figure (3.6), considering the solutions predicted by equation (3.22) and by the numerical model. The two approaches are here again in good agreement with a maximum relative error of about 3.3 % between their respective predictions. The origin of this relative error is further investigated in section 3.2.2.d.

To further compare the two approaches, the trajectories predicted by the analytical model or the numerical model for the particle with z 0 p = -0. that the trajectories are quasi-identical and have exactly the same corrugated shape, which further demonstrates the validity of the proposed analytical approach. In the two previous sections, we verified the particle trajectory equations in channels with flat and corrugated walls for specific geometries. However, according to equation (3.22), the trajectory depends on the geometry of the channel (φ (x) and h (x)) and on the ratio W h * . For a specified channel geometry (h * is constant), W is the only dimensionless parameter affecting particle trajectories. More precisely, if, in the same channel, two particles with different properties are injected in flows with different characteristics (velocity, viscosity), the particles should follow the same trajectory if they present the same dimensionless number W .

In order to verify the dependency of particle trajectories on W , we conducted a sensitivity analysis. To do so, we first studied 5 cases (cases 1 to 5 in Table ( To further verify this dependency on W , 5 additional cases (cases 6 to 10 in Table 3.1) were tested for a value of W ≈ 0.0081. The results of this second series of tests are plotted in Figure (3.9). Here again, the dependency of particle trajectory on W in the non inertial regime is verified, validating furthermore its ability for determining the behavior of particles immersed in channel flows.

iv. Relative error between the numerical and analytical solutions

The error between the numerical and the analytical solutions can be due to: (i) errors in the numerical simulations, (ii) the assumptions under which the analytical model was developed such as the use of the LCL as a solution of the N S equations or, (iii) the negligence of particle inertia. To investigate the origin of this error, the numerical solution obtained for case 5, where the maximum relative error is equal to 5.11 %, was recalculated 4 times under different conditions. The outcomes of this series of tests are as follows:

• When the mesh size is reduced by half, the maximum relative error remains equal to 5.11 %.

• When the time step used to update the particle position is reduced by a factor of 2, the maximum relative error remains equal to 5.11 %.

• When the particle inertia is neglected, the maximum relative error decreases slightly to 5 %. Taking into account these results, it is clear that the numerical errors are limited since the relative error between the two solutions remains similar whatever the mesh or time step refinement. On the other hand, neglecting particle and fluid inertia seems to reduce the relative error between the two approaches. We can certainly argue that the error is due to the use of the LCL in equation (3.22) as it cannot ensure a perfect representation of the velocity field. Nonetheless, The error is expected to remain limited if and Re are small (cf.

chapter 2)

Particle transport regime diagrams

As illustrated in the previous section, the trajectories of small particles with very low inertia, in channel flows satisfying the LCL approximation, depend on the channel geometry and on the dimensionless number W . In this section, we try to characterize the different transport regimes of these particles. To do so, we assume a uniform distribution of the particles at the inlet, and we define three arbitrary regimes that can occur based on the amount of particles that settle inside the channel:

• Transport: at least 75 % of the particles are transported in the channel till the outlet.

All the particles having an initial vertical position z 0 p ∈ [-1/4, 1/2[ are transported till the outlet of the channel, and particles with z 0 p ] -1/2, -1/4] settle inside the channel.

• Sedimentation: at least 75 % of the particles settle inside the channel. Particles with

z 0 p ∈ ]1/4, 1/2[
exit the channel, and the ones with z 0 p ∈ ] -1/2, 1/4] settle inside the channel.

• Transition: less than 75 % but more than 25 % of the particles exit the channel. This regime describes the transition between the transport and the sedimentation regimes described previously.

The arbitrary selection of the separating percentages 25 % and 75 % defining the different regimes will be discussed later on. The dependence of these regimes on W is studied first in a channel with flat walls, and then in corrugated channels presenting different geometrical properties (wavelength, amplitude, horizontal shift between the walls).

Channel with flat walls

The channel geometry is defined by the dimensionless parameter h * = H 0 L∞ . Based on equation (3.24), the three arbitrary regimes defined previously correspond respectively to different values of W h * : Consequently, we keep the 2D representation along the chapter as it gives a better graphical representation of the effects of either the channel geometry (h*) or the particle and flow properties (W) on the transport regimes. This will enable us to highlight the effects of the geometrical parameters of the channel on the variation of the different zones in the diagram. Similar diagrams will be plotted subsequently to highlight the effect of the channel geometrical parameters on the transport regimes.

• Transport: W h * < 0.156 = W cr1 h * • Sedimentation: W h * > 0.84 = W cr2 h * • Transition: 0.156 < W h * < 0.
The particle transport regimes were arbitrarily defined so that the diagram depends on the percentages of particles that can be transported or deposited inside the channel. When changing these percentages, the variation of W cr1 and W cr2 as functions of h * remains constant so that the regimes diagram keeps the same form but the transport and sedimentation zones will be wider or narrower. The variation of the ratios W cr 1 /h * and W cr 2 /h * as functions of the percentages of particles transported or sedimented is plotted in Figure (3.11).

W cr1 /h * W cr2 /h * W h *
The critical ratio W cr 2 /h * tends to increase when the arbitrary sedimentation regime corresponds to a higher fraction of transported particles. Inversely, W cr 1 /h * tends to decrease when the fraction of transported particles is bigger.

To verify numerically the regime diagram obtained for channels with flat walls, a numerical experiment was conducted using the particle tracking technique presented in section 3. As illustrated in Table (3.2), the percentages of sedimented particles obtained numerically for each case are in good agreement with the regimes predicted by the regime diagram and thus confirms the validity of the proposed approach.

Corrugated channel with sinusoidal walls

The channel is here defined by the following dimensionless parameters: the average cor-

rugation amplitude δ 0 = A 0 H 0 = A 1 +A 2 2H 0 , the phase shift α = 2π∆X
L 0 , the asymmetry between the wall corrugations γ = A 2 -A 1 A 1 +A 2 and l * = L 0 L∞ . In this case, h * = l * . The study for this type of channel is slightly more complex than for channels with flat walls because three additional geometrical parameters need to be taken into account (δ 0 , α, γ). To do so, we first rewrote equation (3.22) as a function of these parameters. Then, we applied the criteria defining the three transport regimes using this equation. The strategy consists in varying one parameters while keeping the other two constant so that its influence on the regime diagram can be assessed.

The main emerging result is that, whatever the walls amplitudes and horizontal shift are, the variations of W cr1 and W cr2 as functions of h * remain linear. Therefore, even for channels with sinusoidal walls, the diagram is similar to the one obtained for channels with flat walls (Figure (3.10)). The difference lies in the extension or reduction of the transport and sedimentation zones depending on the channel geometry. In order to assess the influence of the channel geometry on the regime diagram, we studied three main configurations:

• channel having in phase walls (α = 0, γ = 0)

• channel having out of phase identical walls (α = 0, γ = 0)

• channel having walls with maximum phase lag (α = π, γ = 0)

3.3.2.a. Channel with in phase walls

In this case, there is no horizontal shift ∆X between the two walls and α = 0. The two walls do not have necessarily the same amplitude (Figure (3.12)). Figure 3.12: Channel with in phase sinusoidal walls. Here, the upper wall is more corrugated than the lower wall meaning that γ = 0.

By varying the amplitudes of the walls A 1 and A 2 , the corrugations wavelength L 0 and the channel aperture H 0 , we find that in all cases, W cr1 = 0.156h * and W cr2 = 0.84h * . These values are identical to the ones obtained in section 3.3.1. Consequently, the diagram obtained for channels with flat walls (Figure (3.10)) can also be used to characterize transport regimes in corrugated channels without horizontal shift between the walls. To verify this hypothesis, the numerical experiments presented in section 3.3.1 were conducted in a channel with sinusoidal in phase walls using the same values of (h * , W ). The percentages of sedimented particles obtained numerically are summarized in Table (3.3). Again, the percentages of sedimented particles agree well with the predictions of the regime diagram, confirming therefore the validity of the regime diagram for corrugated channels with in phase walls. Interestingly, the diagram is identical to the one obtained for channels with flat walls.

% of sedimented particles (h * , W )

3.3.2.b. Channel with out of phase identical walls

In this case, the two walls are identical, i.e. they have the same corrugation amplitude A 1 = A 2 and thus γ = 0. α and δ 0 can vary respectively such as:

• δ 0 will be modified by changing the walls amplitude and/or the channel mean aperture.

• α will be modified by changing the phase lag between the upper wall and the lower wall. α = 0 corresponds to channels with parallel walls and α = π corresponds to channels with mirror-symmetrical walls When δ 0 increases, c 1 increases as well, which means that the transport zone increases.

Conversely, c 2 decreases when δ 0 increases, leading to a decrease of the sedimentation zone.

To further illustrate these variations, we compared two different cases considering δ 0 equal respectively to 0.1 and 0.4 while keeping α = π constant and plotted the associated Effect of increasing the dimensionless corrugation amplitude δ 0 on particle transport regimes for channels having out of phase identical walls. The red solid lines present the limits between the regimes for a channel with δ 0 = 0.1 and the red dotted lines present the limits between the regimes after increasing the walls corrugations (δ 0 = 0.4). The white zones present the variations in the transport and sedimentation zones respectively.

Here again, numerical experiments were conducted to verify the diagram's predictions.

Two configurations corresponding to two pairs of (h * , W ) located respectively in the two variation zones of the diagram were used considering δ 0 = 0.1 and δ 0 = 0.4. The results are summarized in Table (3.4).

For (h * = 0.01, W = 0.0021), when δ 0 = 0.1, the percentage of the deposited particles is % of sedimented particles (h * , W ) equal to 30 % (transition), while this percentage is equal to 10% when δ 0 = 0.4 (transport).

For (h * = 0.01, W = 0.0074), when δ 0 = 0.1, the percentage of the sedimented particles is equal to 70 % (transition), while this percentage is equal to 80% when δ 0 = 0.4 (sedimentation). The numerical experiments confirm the effect of δ 0 on the regime diagram.

ii. Influence of α

To investigate the effect of the phase lag between the walls on particle transport, α was varied from 0 to 2π for two cases: δ 0 = 0.1 and δ 0 = 0. Firstly, one can see that the behavior of the particles does not change when the phase lag between the walls is in the opposite direction (± α). For example, if the lower wall is shifted from the upper wall by ∆X = L 0 /4 (α = π/2), the particles settle at the same distance as if the shift was ∆X = -L 0 /4 (α = 3π/2) (Figure (3.16)).

Secondly, c 1 reaches its maximal value when the phase lag between the walls is maximum (α = π). This means that the transport zone increases with the increase of the phase lag.

Similarly, c 2 is minimum for α = π and therefore the sedimentation zone also increases when the phase lag increases.

To further illustrate these variations, we compared two different cases considering α equal respectively to 0 and π while keeping δ = 0.2 constant (Figure (3.17)). One can see that increasing the phase lag between the walls leads to an increase of both the transport and sedimentation zones in the diagram, their maximum extents being reached when the channel walls are mirror symmetric (α = π).

Here again, numerical experiments were conducted to verify the diagram's predictions.

Two configurations corresponding to two pairs of (h * , W ) located respectively in the two variation zones of the diagram were used considering α = 0 and α = π. The results are summarized in Table (3.5). 3.5: Percentages of sedimented particles in sinusoidal channels with out of phase identical walls defined respectively by α = 0 and α = π, for two configurations corresponding to (h * = 0.009, W = 0.0016) and (h * = 0.0085, W = 0.0065). Comparison between the regime diagram predictions (Figure 3.17) and particle tracking numerical simulations results.

% of sedimented particles (h * , W )

It is shown that for (h * = 0.009, W = 0.0016), when α = 0, the percentage of the .17: Effect of increasing α on particle transport regimes for channels having out of phase identical walls. The red solid lines present the limits between the regimes for a channel with α = 0 and the red dotted lines present the limits between the regimes after increasing the phase lag to α 0 = π. The white zones present the variations in the transport and sedimentation zones respectively. deposited particles is equal to 26 % (transition), while this percentage is equal to 20% when α = π (transport). For (h * = 0.0085, W = 0.0065), when α = 0 the percentage of the sedimented particles is 66 % (transition), while this percentage is 80% when α = π (sedimentation). The effect of α on the regime diagram is again confirmed by the numerical experiment.

3.3.2.c. Channel with maximum phase lag between the walls

In this type of channel, the horizontal shift between the walls is maximum (∆X = L 0 /2 or α = π). When the two walls have the same corrugation amplitude (γ = 0), the channel presents a mirror-symmetrical geometry (Figure (3.18)). When the walls are not identical, γ varies between -1 when the upper wall is flat (A 2 = 0) and 1 when the lower wall is flat (A 1 = 0). γ < 0 means that the lower wall is more corrugated than the upper wall and vice versa.

To assess the influence of the corrugation amplitude asymmetry on particle transport, we studied the variation of c and minimal when the lower wall is flat (γ = 1). This means that the transport zone in the diagram is maximal when the upper wall is flat. The transport zone decreases when increasing the upper wall corrugation or decreasing the lower wall corrugation, to become minimal when the lower wall is flat. However, when γ = 0.5 (the upper wall corrugation is equal to three times the lower wall corrugation), the value of c 1 is the same for the two values of δ 0 . Further calculations showed that for this case (γ = 0.5), c 1 is constant and independent of δ 0 . When γ < 0.5, c 1 increases when δ 0 increases and, as a result, the transport zone increases. γ > 0.5 leads to the opposite result. Here again, we verified the predictions of the regime diagram against the numerical experiment performed using 100 particles. Two configurations corresponding to two pairs of (h * , W ) located respectively in the two variation zones of the diagram were used considering α = 0 and α = π. The results are summarized in Table 3.6. For (h * = 0.01, W = 0.001), when γ = -1, no particles are deposited inside the channel (transport), while 30% of the particles are deposited when γ = 1 (transition). For (h * = 0.01, W = 0.0075), when γ = -1, the percentage of the sedimented particles is equal to 53 % (transition), while this percentage is equal to 80% when γ = 1 (sedimentation). The effect of γ on the regime diagram is once again verified by the numerical experiments.

% of sedimented particles (h * , W )

Summary

The behavior of weakly inertial particles injected in a channel laminar flow depends on the dimensionless number W and on the channel geometry. The separation between the transport, sedimentation and transition regimes is characterized by the linear variation of two critical values of W , W cr1 and W cr2 respectively. These two values are functions of the ratio between the aperture and the total length of the channel, which is characterized by the dimensionless number h * = H 0 L ∞ . W cr1 = c 1 h * defines the limit between the transport and transition zones, while W cr2 = c 2 h * defines the limit between the transport and the sedimentation zones. This leads to the definition of a regime diagram that is valid for arbitrary channel geometries.

It was found that configurations involving a channel with flat walls and a channel with in phase sinusoidal walls (α = 0) share the same values of c 1 and c 2 . When the top and bottom walls are shifted, the direction of this shift does not affect the distance traveled by the particles. However, transport and sedimentation zones are optimum when the phase lag between the two walls is maximum(α = π).

In channels with identical walls (γ = 0), increasing the wall corrugation (δ 0 ) or the horizontal shift (α) tends to enlarge both the transport and the sedimentation zones. In channels having nonidentical walls (γ = 0) and a maximum phase lag (α = π), the increase of the upper wall corrugation compared to the lower one leads to a narrower transport zone and to a larger sedimentation zone. The transport zone increases when the upper wall corrugation decreases, and reaches its maximum size when the upper wall becomes flat (γ = -1). Conversely, increasing the upper wall corrugation promotes sedimentation, and the sedimentation zone reaches its maximum size when the lower wall becomes flat (γ = 1).

Conclusion

This chapter was devoted to the study of the transport regimes of particles with low inertia, in 2D fracture flows. The flow was assumed to be laminar and dominated by viscous forces, characterized by low values of and Re, being the fracture aspect ratio and Re the flow Reynolds number estimated on the fracture mean aperture. Channels with flat and sinusoidal walls were considered to represent different fracture geometries.

We showed that when particle inertia is neglected, the particle behavior can be characterized by the channel geometry and a dimensionless number W which represents the ratio between the particle settling velocity and the flow mean velocity. A differential equation defining the particle trajectory in corrugated channels and an exact equation of this trajectory in channels with flat walls were derived under the assumption that particle inertia can be neglected and that the flow velocity components can be explicitly calculated using the LCL. These equations were verified against numerical experiments based on a particle tracking technique combining the particle motion equations together with the flow field obtained by solving the N S equations. The numerical simulations were performed taking into account both particle and fluid inertias. The numerical results confirmed the assumptions under which the analytical approach was developed. Furthermore, it confirmed that the particle trajectory can be directly predicted according to the value of W and to the channel geometry without the need for further calculations or numerical simulations.

Based on these developments, a regime diagram was established, which predicts the transport or the sedimentation of particles as a function of W and of a geometrical parameter h * , representing the ratio between the channel mean aperture and its total length. For channels with corrugated walls, the regime diagram is similar to the one obtained for channels with flat walls, but the zones of transport and sedimentation tend to increase or decrease depending on the channel geometry, related to the corrugation wavelength and amplitude and to the phase lag between the channel walls. It is shown that for a corrugated channel having two in phase walls, the regime diagram is identical to the one obtained for a channel with flat walls. When the two walls are out of phase, increasing the wall corrugation leads to an increase of the zones of transport and sedimentation in the diagram. Taking into account the asymmetry between the two walls corrugations, increasing the upper wall corrugation relatively to the lower one tends to decrease the transport zone and to increase the sedimentation zone. The regime diagram and the effects of the geometrical parameters on its zones variation were verified by numerical experiments which were conducted by releasing 100 particles in the channel and by computing the percentages of particles that deposited inside the channel.

The results presented in this chapter describe in which way the geometry of a fracture affect the transport and the deposition of particles as well as the distance at which they will deposit inside the fracture.

In the following chapter, the experimental device that was designed to verify the analytical results is presented.

121

Introduction

The experimental apparatus is first described and explanations are given regarding the design of each one of its components. In the past, numerous experiments have been performed to investigate flow and particle transport in open and closed channel flows. Laboratory scale open channels are commonly used to reproduce phenomena occurring in earth-surface liquid flows such as sediment transport in rivers (Best [125],Yilmaz [START_REF] Yilmaz | Experimental study of sediment transport in meandering channels[END_REF], Wang et al. [START_REF] Wang | Effects of bed load movement on mean flow characteristics in mobile gravel beds[END_REF]) and sand transport in near-shore regions (Jarno-Druaux et al. [START_REF] Jarno-Druaux | Dynamical evolution of ripples in a wave channel[END_REF], Chu et al. [START_REF] Chu | Particle trajectories and size sorting above a rippled bed under standing water waves[END_REF]).

Similarly, laboratory scale closed channels have been used to model flows in either natural (e.g. Hakami and Larsson [START_REF] Hakami | Aperture measurements and flow experiments on a single natural fracture[END_REF]) or synthetic (e.g. Qian et al. [START_REF] Qian | Experimental study of the effect of roughness and reynolds number on fluid flow in rough-walled single fractures: a check of local cubic law[END_REF]) fractures. Synthetic fractures can be made either by creating surface roughness replicating that of real fractures (Yeo et al. [START_REF] Yeo | Effect of shear displacement on the aperture and permeability of a rock fracture[END_REF], Ju et al. [START_REF] Ju | An experimental investigation on the mechanism of fluid flow through single rough fracture of rock[END_REF]), or by considering fracture walls with idealized roughness.

For instance, Qian et al. [START_REF] Qian | Experimental study of the effect of roughness and reynolds number on fluid flow in rough-walled single fractures: a check of local cubic law[END_REF] glued small square plates on flat surfaces to generate surface roughness. Crosnier [START_REF] Crosnier | Écoulement dans une fracture de géométrie radiale: influence d'une rugosité périodique[END_REF] used disks with crenelated surface to study flow in radial fractures.

In the present work, we made the choice to model rough fractures as channels with sinusoidal walls (cf. chapters 1 and 2). Our experimental apparatus and its schematic representation Since the aim of this apparatus is to assess the analytical solution developed in chapter 3, the experimental setup was designed in such a manner that it satisfies the theoretical assumptions. In particular, two main constraints were considered to define the dimensions of the fracture walls. First, the flow through the fracture must be unidirectional and must follow the LCL. It is obvious that this latter condition implies a certain constraint on the flow rate through the cross-section of the fracture. Second, the corrugations of the fracture walls must be smooth (cf. chapter 2), and it should be possible to adjust the aperture and the phase shift between the fracture walls so as to investigate different fracture configurations..

A detailed description of the experimental apparatus as well as the choice of the fracture walls dimensions are presented in the next sections. The channel has a total length of 400 cm and a cross section of 16 × 34 cm 2 (width × height). The side walls are made of transparent reinforced glass, to ease the visualization of the flow. All the other components in contact with the liquid are made of stainless steel or glass reinforced plastic to prevent corrosion. The inclination of the channel can be finely adjusted using a screw system and inclinometer that indicates the angle of inclination. In our experiments, the channel has been always placed in a horizontal position. All the equipment is set up on an steel coated frame. The frame is equipped with adjustable rubber feet to avoid vibrations.

Upstream, a polyethylene stilling tank is connected to the centrifugal pumps, with a drain valve in its lower part. A honeycomb (flow straightener) is placed in the tank at the channel inlet to reduce turbulence (Figure 4.3(c)). Downstream, another tank is placed with direct evacuation into the main reservoir. The reservoir is a tank made of fiberglass with a capacity of 500 L and is connected to the liquid pumps. It has a transparent lid and a drain valve and is equipped with a low-level sensor that turns off the pumps as soon as the liquid level is lower than the inlet of the pumps.

Liquid circulation between the reservoir and the channel is ensured using a pumping system. This system consists of three centrifugal pumps, an electromagnetic flow meter, an electrical control box, and different valves ( between the upper and lower walls of the fracture, which corresponds to the aperture, can be adjusted. In the same way, by turning the bolts in the opposite direction, the fracture aperture can be decreased. The mean aperture of the fracture H 0 can be varied between 0 (When the bolts are fully inserted and the two walls are completely in contact one with each other) and 2 cm (when the bolts are pulled out to the maximum). Similarly to the method used to move the slab vertically, two bolts were inserted on the right side of the slab and allowed it to move horizontally. In this way, the phase shift between the upper and lower wall can be modified in order to control the phase lag between them (cf. chapter 2).

4.2.2.a. Fracture geometrical properties

Fractures with flat walls are only defined by their mean aperture H 0 which, in our case, can vary between 0 and 2 cm as mentioned in the previous section. The total length of the walls is equal to L ∞ = 69 cm. For sinusoidal walls, the fracture roughness is defined by the wavelength L 0 and the amplitude A of the wall corrugations (A 0 is the mean corrugation amplitude between the two walls), the horizontal phase shift between the two walls ∆X.

Two identical sinusoidal walls were machined with A = 1.25 mm and L 0 = 2.5 cm (Figure 4.7). These two parameters were selected in order to have smooth wall corrugations, which is a condition for the LCL to be valid and for eliminating the risk of having recirculation zones in the fracture furrows (cf. chpater 2).

By combining the flat and sinusoidal walls, 4 geometrical configurations of the fracture can be studied respectively:

i with two flat walls ii with a lower flat wall and an upper sinusoidal wall iii with a lower sinusoidal wall and an upper flat wall iv with two sinusoidal walls

The geometrical properties of the fracture can be adjusted in the following way:

• = H 0 L 0 can be varied between 0 (when the two walls are in contact) and 0.8 (when H 0 is maximum). Practically, is varied between 0.1 and 0.4 to ensure that ≤ 0.4 (cf.

chapter 2). • δ 0 = A 0 H 0 can be varied between 0.03 when one of the walls is flat and the aperture is maximum, and 0.5 when the two walls are sinusoidal and the aperture is minimum.

• γ = A 2 -A 1 A 2 + A 1
(A 2 corresponds to the upper wall and A 1 to the lower one) can be varied between -1 (in the case (iii)) above, 0 (case (iv)) and 1 (case (ii)).

• ∆x = ∆X L 0 can be varied between 0 when the walls are in phase, and 0.5 when the phase shift is maximum.

The walls inlet follows an exponential profile instead of a rectangular one to ensure a smooth laminar flow and to avoid any stagnant zone that may appear at the lower wall inlet (Figure 4.7). The flow through the fracture is created by a hydraulic head difference ∆h (pressure difference ∆P = ρg∆h) between the inlet and the outlet.

4.2.2.b. Particle injection

In order to verify the theoretical results, the particles must be small, typically less than 1 mm in diameter. In addition, the analytical model was developed in 2D. To be able to reproduce the 2D conditions experimentally, particles must stay away from the side walls of the open channel to avoid any particle-wall interaction that may affect their behavior.

The first technique adopted for injecting the particles in the flow consisted in dropping them off directly in the closed circuit and tracking them as soon as they enter the fracture.

This technique is the less restrictive and is eased by the fact that the centrifugal pumps allow the passage of suspended solids up to 10 mm in diameter. However, many difficulties arose when using this technique. First, the particles are randomly dispersed in the flow, meaning that they can enter the fracture at different positions along the width and at different times.

Particles can then be very close to the open channel side walls which may thus affect their behavior. This technique would be fine for studying particle transport in tubes (e.g. Segré

and Silberberg [START_REF] Segré | Behaviour of macroscopic rigid spheres in poiseuille flow part 2. experimental results and interpretation[END_REF], Matas et al. [START_REF] Matas | Inertial migration of rigid spherical particles in poiseuille flow[END_REF]), but it is not convenient in our case. Second, as will be discussed later on, some particles can be located out of the camera focus. Finally, the moment at which particles enter the fracture can not be precisely estimated and the recording start time is thus unknown.

To overcome these difficulties, another technique was chosen. It consists in injecting the particles manually and directly at the fracture inlet, at the center of the flow path. To make this possible, a rubber hose was fixed on the slab holding the fracture upper wall (cf. section

Liquid properties

At first, we intended to work with water as a liquid. The problem that appeared was that this option could only provide too high Re with regards to the assumptions made concerning the study of particle transport (cf. chapter 3). As an example, to assure Re = 5 in a fracture having two flat walls with H 0 = 5 mm, the hydraulic head must be ∆h = 0.03 mm which is, from an practical point of view, impossible to set up. Therefore, the use of another fluid has been considered to match the purpose of the study. Nonetheless, we performed some preliminary tests with water to verify the capability of our experimental device to produce lift induced particle focusing, as a validation step.

In order to reduce inertial effects and Re, a liquid with higher viscosity was considered.

We chose to work with glycerin which is odorless, colorless and has a viscosity theoretically about 1000 times higher than water. The problem with the glycerin alone is that its very high viscosity risks to reduce significantly the hydraulic characteristics of the centrifugal pumps.

As glycerin is soluble in water, we decided to mix glycerin and water with a volumetric ratio 1L of water/4L of glycerin. This ratio was selected because according to the empirical formula of Cheng [START_REF] Cheng | Formula for the viscosity of a glycerol-water mixture[END_REF], the viscosity of the mixture at 20 Moreover, as the density varies slightly with the temperature, the mixture density ρ f was measured at room temperature (20 • C). This was done by measuring the weight m f of a predefined volume V f . To take into account the measurement errors that may occur, the weights of different volumes were measured (Table 4.1).

Throughout the experimental study, we considered ρ f = 1211kg/m 3 . To improve the tracking of the particles with the camera, two light projectors were placed above the fracture. These projectors were adjusted to shed white light on the fracture, so that particles with different color can be easily detected. To isolate the fracture walls from the rest of the channel, two black cardboards were glued to the side walls above the upper wall and below the lower wall so that only the fracture walls and the gap between them are visible (Figure 4.10).

Experimental procedure and image treatment

The first components to install in the experimental apparatus are the fracture walls.

According to the predefined geometrical properties (flat or sinusoidal walls), two walls among the four available are selected. The lower wall is first fixed. Then, the upper wall is fixed on the holding slab, and the slab is then placed and adjusted according to the desired mean aperture and, when applicable, to the phase shift between the two walls.

The camera is fixed on the belt conveyor. Particles are inserted into the syringe which is then connected to the inlet hose. When using the water-glycerin mixture, the pumps are turned on for at least one hour to ensure that the glycerin is completely dissolved in water and that the mixture is thus homogeneous.

In the experiments where the camera is fixed at a specified point alongside the fracture, recording is started directly once the particles are injected and is turned off when all the moving particles have passed.

When the camera is moving along the fracture, a first test is performed to determine the velocity at which the belt conveyor should be set. Then, the experimental procedure is as follows:

1. The camera is positioned at the fracture inlet.

2. The video recording is turned on and particles are injected immediately after.

3. As soon as the particles get out of the hose end and enter the fracture, the camera is translated at the velocity previously determined.

4. When the camera reaches the fracture outlet, recording is turned off when all the particles have passed.

5. The video is then post-processed by applying an image treatment procedure described hereafter.

The image treatment procedure consists first in transforming the recorded video into a series of images. For that purpose, we used the software MPlayer. The number of images depends on the duration of the video, knowing that the frame rate of the camera is set at 50 frames/second. As an example, a video of 10 s duration will be transformed into 500 images.

After that, the number of pictures is reduced by keeping only those which feature particles. For instance, when the camera starts recording before the particles are injected, the first images without particles are deleted, and the same goes for the last images that were recorded after all the particles have exited the fracture.

The images are then edited using the open-source image editor GIMP. A simple Batch processing plugin, David's Batch Processor, is used to automatically perform operations on all the images because treating each image alone would be time-consuming and practically impossible. The image treatment procedure is as follows:

1. The images are cropped by selecting only the zone between the two walls. A random image is selected before, and cropped in GIMP in order to extract the crop required values (width and height) and apply them to the Batch Processor.

2. The contrast of the images is increased to make the particles more distinguishable.

3. This step is optional because it is possible at point (2) to directly process the images in order to track the particles and to find the distance to deposition, or to study particle trajectories. However, it is possible also to convert the colored images into black and white either directly or by applying a threshold.

in the water-glycerin mixture.

Figure 4.12: Image treatment applied in the case of a black poppy seed immersed in the water-glycerin mixture.

In addition to this procedure, it is possible to draw the trajectories of single particles when the camera is fixed, using a Matlab script, by adding successive photos of the moving particle and combining them into one. This will be shown later on when single particle trajectories are discussed.

Preliminary results with poppy seeds

The ability of the experimental apparatus to investigate particle transport in closed channels must be assessed before going into further investigations related to the analytical model presented in chapter 3. This was done by performing simple experiments using small particles that can be easily found on a daily basis like, for instance, poppy seeds. In fact, poppy seeds are relatively small, have a quasi-spherical shape and a gray to black color, making them convenient for experiments and image treatment. However, poppy seeds are polydisperse. They have slightly different sizes and densities. This tends to slightly hinder the validation of the analytical model but it also provides a better representation of practical situations in which particles are neither perfectly spherical nor monodisperse. The considered measured density of poppy seeds is equal to ρ p = 1070 kg/m 3 . This means that the particles are slightly denser than water and lighter than the water-glycerin mixture (ρ p = 1211 kg/m 3 ). This does not modify the analytical model verification, but instead of considering sedimentation or deposition, floating will be studied. In fact, the particle trajectories depend on the ratio k between the particle and liquid densities. In the case of poppy seeds immersed in water-glycerin mixture, k < 1 and the vertical position of the particles increases along the fracture length until the particles reach the upper wall.

Particle properties

Hence, trajectories are investigated from the point at which particles are injected until they reach the upper wall.

In addition, all the particles do not have the same size. At first, using sieves, particle diameter d p was found to range from 500 µm to 1 mm. In order to find a more specific range of d p , an digital image analysis was performed. A portion of the particles was dispersed on a homogeneous paper and photographed. Using the software ImageJ, particles were converted into circular black dots on a white background and their surfaces were determined (Figure 4.13). d p was then calculated.

Transport with water as the operating liquid

At first, experiments were performed using water as a liquid. In these experiments, the hydraulic head difference between the fracture inlet and outlet was set to ∆h = 9 mm. As mentioned above, in such a configuration, inertial effects dominate due to the low viscosity of the fluid. Therefore, lift forces can not be neglected, meaning that the analytical model presented in chapter 3 can not be assessed. On the other hand, one can expect the occurrence of inertial focusing in such configuration, meaning that particles injected randomly should focus on the same trajectory due to the lift forces acting on them (cf. chapter 3). Even though poppy seeds are slightly denser than water, gravity forces are dominated by lift forces and therefore particles do not deposit on the lower wall. In order to verify the suitability of the experimental setup to deal with particle transport, we performed tests with water so that inertial focusing could be observed. Two experiments are proposed. In the first one, we injected a single particle. In the second one, we injected 2 particles. These particles were injected into a fracture with flat walls.

4.3.2.a. Trajectory of a single particles

First tests were performed by injecting a single particle into the fracture in order to study their trajectories. In such a configuration, particle-particle interactions are avoided and inertial focusing, if it is observed, is solely due to the effect of lift forces.

The camera was fixed right before the fracture outlet, capturing the last 15 cm of the fracture. One particle was injected at the fracture inlet, and it was filmed when it was exiting the fracture. The succession of particle positions after image treatment, allows to plot its trajectory as illustrated in The analysis of this figure shows a slight difference between the two trajectories which, however, remain almost parallel to the fracture wall. This small difference could be assigned to the respective physical properties of the particles that are not identical (they have a different size). Under these conditions, we can consider that the particles converge to an equilibrium position and that inertial focusing occurs. This result gives a first insight about the ability of the experimental apparatus to capture some of the physical mechanisms that take place in closed channel flows. It also confirms that inertial lift forces cannot be neglected in our experiment when water is used and thus that the theoretical assumptions of our analytical model cannot be fulfilled.

Transport with water-glycerin mixture as the operating liquid

To investigate the particle deposition case, we replaced water by a water-glycerin mixture.

This mixture presents a higher dynamic viscosity than that of water permitting to reduce, with a reasonable hydraulic head difference (∆h), the flow inertial effects. Therefore, we expect that the particles would float towards the upper wall. Comparisons can then be made with the predictions of the analytical model presented in chapter 3, in terms of distance traveled. For convenience, the term "deposition" is used to refer to the fact that the particle comes in contact with the upper wall. Deposition is normally used for particles that sediment on the bottom wall, but as the mechanism is the same, it will be used for particles that come in contact with the upper wall.

Another important advantage of choosing the water-glycerin mixture instead of the water alone, is that by increasing ∆h, we can study non-inertial regimes at low Re (for small ∆h), in which the analytical model is valid, as well as inertial regimes at high Re (for high ∆h), in which the inertial effects are significant. This can thus give us insights into the limitations of the analytical model for high Re.

We recall that the analytical model was developed with the aim to predict particle trajectories and positions without knowing the exact flow velocity field. Instead, an approximated velocity is calculated using the LCL (cf. chapter 3). In fact, based on the hydraulic aperture H h (which depends on the fracture geometry) and on the pressure difference ∆P , the flow mean velocity V 0 can be estimated. Then, depending on the particle radius a and on the density ratio k, the dimensionless number W is determined and the particle trajectory equation as well as the distance at which the particle will deposit can be calculated. This can be done under the assumption that the particles have a small inertia (particle response time τ < 10 -3 ) and that Re = V 0 H 0 ν is small.

To sum up, in each case, ∆h is set and Re is calculated. For relatively low Re, W and τ are determined, and the distance to deposition can be calculated for each particle. This distance is compared to that found experimentally. For high Re, the presence of inertial focusing is investigated.

The mixture and particle properties are defined in sections 2.3 and 3.1. The mixture temperature was measured at different positions in the channel and at different times, using a mercury-in-glass thermometer and was found to be equal to 21.5 • C. The kinematic viscosity is approximately equal to ν f = 55 cSt and the density ratio is k = 0.88. As the particle diameter was defined in a given range, the same applies for W and τ . In the experiments, H 0 was varied between 5 mm and 10 mm, and ∆h between 0.8 cm and 15 cm. In the following, we present selected significant results which show specific particle behaviors. According to this figure, the majority of the particles floated and were moving on the upper wall. This means that the inertial lift forces are negligible with respect to gravity (buoyancy) forces. To assess the predictions of the analytical model, a random particle with an initial vertical position z 0 p 0 was selected. For this particle The distance to deposition (in the dimensionless form with L ∞ as a length scale x f = X f L∞ ) was found experimentally (Figure 4.18) to be x f exp = 0.075 (or X f exp = 0.051 m in the dimensional form). Note that in the case of a fracture with a corrugated wall, the distance to deposition can be more easily determined by counting the number of the wavelengths that the particle has traveled before touching the wall. Using the analytical model, the range of the distance to deposition Although this experimental observation can be disputed due to the heterogeneity of the physical properties of the particles (size, density), it is nonetheless in relatively good agreement with the analytical solution, confirming thus that the analytical model gives a good approximation of the distance at which polydisperse particles would deposit in fractures with flat walls.

ii. Inertial regime: Lift-induced particle focusing

The mean aperture was set to H 0 = 9 mm and the hydraulic head difference to ∆h = 5 cm. In this case, V 0 = 0.171 m/s and Re = 27.81. Thus, flow inertial effects should be dominant. Several particles were injected at the fracture inlet and the camera was fixed in the same way as previously (section 3.2.1), i.e. at the fracture outlet. ). This means that the distance to deposition found experimentally belongs to the range predicted by the analytical model. This result proves that even when the fracture has a corrugated wall, the analytical model gives a good approximation of the range of the distance at which polydisperse particles would deposit.

ii. Inertial regime: Lift-induced particle focusing As discussed in chapters 1 and 3, previous investigations on inertial focusing have mainly considered channels with flat walls (e.g. Di Carlo et al. [START_REF] Di Carlo | Continuous inertial focusing, ordering, and separation of particles in microchannels[END_REF]) or other geometries such as serpentine channels (Zhang et al. [START_REF] Zhang | Particle inertial focusing and its mechanism in a serpentine microchannel[END_REF]) and spiral channels (Russom et al. [START_REF] Russom | Differential inertial focusing of particles in curved low-aspect-ratio microchannels[END_REF]). Even though sinusoidal walls have not been considered before, one can expect the same effect to occur due to the equilibrium between the different lift forces acting on the particles. To this end, a test was performed with relatively high ∆h.

We set the mean aperture H 0 = 9 mm (the hydraulic aperture was then calculated H h = 8.82 mm), and ∆h = 6 cm leading to Re = 18. Several particles were injected at the to what was observed with two flat walls, the particles do not float but instead remain at a certain distance from the upper wall. This can also be interpreted as inertial focusing. An approximate line on which particles focused was drawn and it shows a sinusoidal shape. This experiment proves that inertial focusing can occur even when the fracture has a sinusoidal wall.

4.3.3.c. Fracture with two sinusoidal walls

The final tests were performed in the fracture with two sinusoidal walls. The walls were mounted in a mirror-symmetric configuration for which the variation of the aperture is maximum, and consequently the deviation of the LCL from the N S solution is maximum (cf. chapter 2). This configuration is, therefore, the most convenient to validate the analytical model.

i. Non-inertial regime: verification of the analytical model

The mean aperture was set equal to H 0 = 7 mm (H h = 6.12 mm). To minimize inertial ii. Inertial regime: Lift-induced particle focusing This test was performed with relatively high ∆h to investigate the presence of inertial focusing when the two walls of the fracture are sinusoidal. Similarly to the two previous cases, one can expect inertial lift forces to dominate the buoyancy forces due to the increase of the flow inertial effects, so that the analytical model does not remain valid.

We set the mean aperture H 0 = 8 mm (the hydraulic aperture was then calculated Once again, the particles do not float but remain at a certain distance from the upper wall, which can be interpreted as inertial focusing. An approximate line on which particles focused was drawn and it also shows a sinusoidal shape. This experiment confirms that inertial focusing can occur even when the fracture has sinusoidal walls, and that the analytical model is not valid when the flow inertial effects are relatively important.

Conclusion

In this chapter, the experimental apparatus that was designed and built up to study particle transport in closed channels was presented. A detailed description of the experimental setup and procedure was given. The apparatus was used to conduct a preliminary study of particle transport in fractures with flat and/or sinusoidal walls with dimensions in accordance with the theoretical assumptions made in the previous chapters.

The experimental results show that the apparatus is able to capture the different behaviors of particles in fractures, such as transport and deposition as well as inertial focusing.

Many tests were performed using water and a water-glycerin mixture as liquid and poppy seeds as polydisperse particles moving in the liquid. By choosing an optimal difference in piezometric load, flow inertial effects can be reduced and the experimental results are in agreement with the analytical solution. The distance traveled by the particles from the inlet till they deposit fall in the range predicted by the analytical model developed in chapter 3. This suggests that the analytical solution can be used to evaluate deposition distance of polydisperse particles in fractures with flat and/or sinusoidal walls. When the hydraulic head was increased, i.e. for higher Re, it was shown that in fractures with both flat and sinusoidal walls, the particles focused on a single trajectory, verifying thus the presence of inertial focusing and confirming that inertial lift forces govern the particle behavior. This result verifies the assumption that the analytical model is valid only for low Re.

As a conclusion, the preliminary experimental results verify the analytical model developed in this thesis. In addition, these results demonstrate the ability of the experimental device for investigating particle transport in closed channel flows. It can be soundly used for future experimental studies of particle transport, which may improve our understanding of particle behavior and further validate the previously developed models.

CONCLUSION AND PERSPECTIVES

Single-phase flow and particle transport through fractures with flat and corrugated periodic walls have been investigated.

For single-phase flow, a thorough numerical study has been conducted to investigate the effects of the fracture geometry on the validity of the local cubic law (LCL). Numerical solutions of the Navier-Stokes (N S) equations have been compared to the predictions of the LCL. The results suggest that the validity of the LCL depends strongly on the geometrical properties of the fracture, defined by its aperture, the roughness (corrugation amplitude) of its walls, as well as the phase shift between the walls, and the asymmetry of the walls corrugations. The results further suggest that the criteria previously proposed for the applicability of the LCL to fracture flow, developed considering specific fracture geometries, can not be generalized for fractures with arbitrary geometries. In all the cases tested in the present manuscript, the LCL was found to be valid for modeling fracture flow, under the condition that the walls roughness as well as the Reynolds number Re are relatively small. Another numerical result, obtained in all the configurations tested in the simulations, suggests that the flow remains in the non-inertial regime for Re ≤ 15.

To investigate particle transport through fractures, an approach combining analytical, numerical, and experimental developments has been adopted. In the analytical approach, the LCL was used to model the flow and was coupled to a simplified form of the particle motion equation in which particle inertia is neglected. A model describing the trajectories of the particles transported in channels with flat and corrugated walls was developed. It was found that when particle inertia is negligible, the particle behavior depends on a dimen-sionless number W relating the particle properties and the flow characteristics, as well as on the geometry of the fracture. Using the proposed trajectory equation, and considering a uniform distribution of particles at the channel inlet, different transport regimes could be identified: transport, sedimentation and transition. According to the channel geometry, different diagrams that can predict the transport regime of particles entering a channel were then established. Moreover, numerical simulations combining the finite element method (F EM ) to a Lagrangian particle tracking technique were conducted to assess the relevance of the analytical approach. Contrary to the analytical model, these simulations were conducted solving the full N S equations and taking into account particle inertia. Particles were injected at different positions at the inlet of channels with flat and corrugated walls, and the distance to deposition was computed numerically for each particle. This distance was then compared to the distance predicted by the analytical model. First, the dependence of the particle behavior on W and on the channel geometry was verified. Numerical experiments were then performed considering 100 particles. The results confirmed the ability of the regime diagrams to predict the behavior of the moving particles.

Finally, an apparatus was constructed to study particle behavior in fractures with flat and sinusoidal walls. The aim of this apparatus was to evaluate the pertinence of the analytical model when applied to practical configurations. To reduce the flow inertial effects, a waterglycerin mixture was used as the operating liquid. Preliminary tests conducted using poppy seeds seem to confirm the validity of the analytical model. For high pressure difference, the analytical model was no longer valid: The behavior of the particles is dominated by inertial lift forces, leading to inertial focusing in both the flat and sinusoidal walls configurations.

Overall, the results obtained in this thesis improve our understanding of the behavior of small particles immersed in closed channel flows, with a direct application to the transport of contaminants transported through fractures. For instance, we can identify, based on their size and density, the contaminants that are likely to deposit inside the fracture or be suspended and transported for long distances. Our results have other applications in water filtration and in mineral separation. In fact, based on our regime diagrams, a system of separation based on particle deposition in channels with sinusoidal walls could be designed.

It would enable to separate particles based on their size and/or density depending on the distance at which they would deposit in the channel. As the flow in the channel can simply be created by a hydraulic head difference, the advantage of such system with respect to present separation techniques is that it is passive and does not require important energy supply.

Another application concerns inertial focusing, and may find some echoes in microfluidics.

First, the results obtained may lead to a quantification of the conditions (Re and particle size) under which focusing becomes efficient. Second, we were able to observe inertial focusing in channels with sinusoidal walls. Further analyses may reveal new features at the origin of the focusing phenomenon, such as, for instance, the effect of the curvature of the channel walls on the inertial lift forces acting on the particles.

Perspectives

The ability of the experimental apparatus to study flow and particle transport in closed channels has been confirmed. It can now be used for further investigations. First, singlephase flow can be addressed. As the apparatus can be utilized over a wide range of flow rates and pressure differences, the validity of the LCL can be assessed experimentally. The results described in the second chapter of the present document could thus be practically evaluated.

Concerning particle transport, the most important perspective would consist in conducting experiments with spherical particles of different sizes and/or densities, in order to have an exact comparison with the analytical model and more particularly to validate the regime diagrams established in chapter 3. This can be done by injecting particles slightly denser than the liquid at the fracture inlet and by calculating the percentage of particles exiting the fracture. Distances to deposition could also be measured and compared to the analytical predictions, similar to what was done with poppy seeds in this thesis. Moreover, our experimental results suggest that the analytical model is not valid anymore when the flow inertial effects are important. The apparatus could therefore help to quantify the range of validity of the theoretical assumptions, i.e. Re and the ratio of particle size to channel aperture above which inertial lift forces start to be dominant with respect to other forces acting on the particles.

Regarding now the analytical and numerical modeling presented in this manuscript, sev-eral developments could be considered. We neglected inertial effects so that our analytical model is valid only for small Re. For high Re, flow inertial effects may lead to the appearance of recirculation zones, especially when the walls corrugations are not smooth. These recirculation zones can trap the particles under specific conditions. Investigating this trapping mechanism requires an exact description of the fracture flow, and the LCL can no longer be used. A numerical analysis based on the N S equations could be applied.

Another perspective would consist in considering dense particle suspensions. In such configuration, two-way coupling must be taken into account, as well as particle-particle interactions. A numerical model combining the lattice Boltzmann method for describing the fluid flow and a discrete element method for describing particle motion has recently been developed in GeoRessources. This model enables to consider the effect of the particles on the flow and the particle-particle interactions. This couples approach could be used in combination with the experimental investigations to verify the analytical developments in both dilute and dense suspensions. This numerical model could also help to generalize the results concerning particle transport and deposition for the case of non-spherical particles.

In addition, different particle shapes such as oblate and prolate spheroids, or parallelepipeds can be considered and the effect of particle orientation on the particle behavior may be eventually addressed.

Finally, another perspective would consist in assessing the validity of the analytical results obtained in this thesis in the case of fractures with non-periodic and non-smooth rough walls. Indeed, the analytical models proposed in this thesis do not require the fracture walls to be periodic. However, these analytical models incorporate expressions for the variation of the fracture walls, and more precisely, expressions for the variation of the fracture middle line and the fracture half-aperture along the fracture length (H(X) and Φ(X) in chapters 2 and 3). Numerical and experimental verifications using fractures with non-periodic walls could be conducted to evaluate the validity of the analytical predictions, and more particularly the distance at which injected particles would deposit inside the fracture. The only limitation for such generalization would be the singularities that may appear in the wall shapes, which may give rise to singularities in the N S equations. This final point would require additional investigations.

Résumé

Études analytique, numérique, et expérimentale du transport de particules dans des fractures parois plates et ondulées Le but de cette thèse est d'étudier le transport et le dépôt de particules solides dans les écoulements à travers les fractures. Dans un premier temps, l'écoulement monophasique à travers les fractures est étudié afin d'évaluer la validité de la loi cubique locale comme mod le de l'écoulement. Des canaux à parois sinsoïdales à géométrie variable sont utilisés pour représenter différents types de fractures. Un premier développement analytique montre que l'ouverture hydraulique de la fracture diffère de son ouverture moyenne lorsque la rugosité des parois est élevée. La méthode des éléments finis est ensuite utilisée pour résoudre les équations de continuité et de Navier-Stokes et comparer les solutions numériques aux prédictions théoriques de la loi cubique locale sur une gamme relativement étendue de nombres de Reynolds Re. Pour de faibles Re, typiquement inférieurs à 15, la loi cubique locale décrit raisonnablement l'écoulement, surtout lorsque la rugosité et le déphasage entre les parois sont relativement faibles. Dans un deuxième temps, les écoulements chargés de particules sont étudiés. Une approche analytique est d'abord développée pour montrer comment des particules distribuées dans un écoulement stationnaire et laminaire à travers une fracture peuvent être transportées sur de longues distances ou au contraire se déposer à l'intérieur. Plus précisément, une équation simple décrivant la trajectoire d'une particule est établie. Sur la base de cette équation, il est démontré que, quand l'inertie des particules est négligeable, leur comportement dépend directement de la géométrie de la fracture et d'un nombre adimensionnel W qui relie la vitesse de sédimentation des particules à la vitesse moyenne de l'écoulement. L'équation proposée est vérifiée en comparant ses prédictions à des simulations numériques de suivi de particules prenant en compte l'inertie des particules et résolvent complètement les équations de Navier-Stokes. Il est montré que l'équation est valide lorsque l'inertie du fluide est faible. Des diagrammes de régimes, permettant de prévoir le comportement des particules à travers la fracture sont proposés. Enfin, un appareil expérimental conçu dans le but d'effectuer une évaluation pratique du modèle analytique est présenté et les résultats préliminaires sont discutés. Les rèsultats expérimentaux préliminaires tendent valider le modèle analytique. De façon plus générale, les résultats obtenus à travers ce travail de thèse font progresser nos connaissances du comportement des petites particules transportées dans les écoulements de fractures. Potentiellement, ce travail devrait permettre d'améliorer notre prévision de la pollution souterraine, et peut avoir des applications dans le développement de nouvelles techniques de filtration de l'eau et de séparation des minéraux.

Mots-clefs: Transport de particules, Fracture rugueuse, Parois ondulées, Loi cubique locale

Figure 1 :

 1 Figure 1: Fractures à parois planes et ondulées considérées dans cette thèse

Figure 2 :

 2 Figure 2: Diagramme de régimes de transport de particules dans un canal á parois planes. (a) représentation 2D des différentes zones selon la variation de W en fonction de h * (b) représentation 1D de ces zones selon W h * .

Figure 3 :

 3 Figure 3: Modèle d'une fracture à parois sinusoïdales utilisées dans les expériences

1 . 1

 11 Particle-laden flows: Basic concepts . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Definition of particle inertia . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Particle transport in closed channel flows . . . . . . . . . . . . . . . . 1.1.3 Focusing phenomena in closed channels . . . . . . . . . . . . . . . . . 1.1.3.a. Lift-induced inertial migration . . . . . . . . . . . . . . . . . 1.1.3.b. Preferential accumulation of particles in periodic channels . . 1.2 Flow in channels with flat and corrugated walls . . . . . . . . . . . . . . . .

2 Particle and fluid properties a particle radius ρ p particle density ρ f fluid density ν fluid kinematic viscosity µ fluid dynamic viscosity g gravity acceleration Particle and fluid velocities x p particle position v p particle velocity v f fluid velocity field ψ fluid stream function fracture geometry φ 1

 21 Walls corrugation amplitudes A 0 mean corrugation amplitude V 0 flow mean velocity T 0 time scale and deposition of small particles in closed channel flows is of fundamental importance in many environmental issues, such as underground pollution and sediment transport, and in several industrial applications, like water filtration and mineral separation. Other applications concern energy extraction processes like the injection of proppants in petroleum reservoirs, and medical and biological research like the deposition of inhaled particles in human airways and cell sorting and separation in microfluidics. In earth sciences, the transport of contaminants in rough fractures is a crucial research topic due to its tight relation with water contamination in aquifers.

1. 1

 1 Particle-laden flows: Basic concepts The term particle-laden flows refers to two phase flows in which a carrier fluid contains suspensions of particles. Such flows consist of a fluid phase, called continuous phase, and the collection of all the particles in the flow, called dispersed phase. They represent a complex medium where multiple interactions are developed between phases which may have different physical properties.

Figure 1 . 1 :

 11 Figure 1.1: Schematic representation of the difference between dispersed and dense systems according to particle concentration

Figure 1 . 2 :

 12 Figure 1.2: Suspended colloids with random fluctuations (Brownian motion) and non colloidal particles sedimenting due to their density.

Figure 1 . 3 :

 13 Figure 1.3: Different types of water contaminants encountered in water filtration and their sizes, including underground sediments (white rectangles). Adaptation of Water Quality Association source material.

Figure 1 . 4 :

 14 Figure 1.4: Examples of closed and open channels

1. 1 . 3

 13 Focusing phenomena in closed channels 1.1.3.a. Lift-induced inertial migration Both neutrally and non-neutrally buoyant particles can migrate across the flow streamlines to reach specific equilibrium positions within the channel (Figure 1.6). In fact, Segré

Figure 1 . 6 :

 16 Figure 1.6: Lift-induced inertial migration in a serpentine channel (Di Carlo et al. [35]).

Figure 1 . 7 :

 17 Figure 1.7: Accumulation of particles in a channel with corrugated walls as predicted by Nizkaya [27].

Figure 1 . 11 :

 111 Figure 1.11: Fracture with irregular roughness modeled as a channel with parallel flat walls (a) and a channel with sinusoidal walls (b).

Figure 2 . 1 :

 21 Figure 2.1: Schematic diagram of a 2D fracture having sinusoidal walls.

Table 2 . 1 :

 21 Geometry of the fracture walls corresponding to different values of δ 0 (a), ∆x (b) and γ (c). δ 0 = 0.2 , ∆x = 0.5 γ = 0.2 γ = 0.8

Figure 2 . 2 : 63 - 4 Figure 2 . 3 :

 2263423 Figure 2.2: Variation of H h H 0 as a function of the phase shift ∆x for different values of the dimensionless corrugation amplitude δ 0 , and for corrugations asymmetry γ equal to 0, 0.5 and 1 respectively

Figure 2 . 4 :

 24 Figure 2.4: Example of a fracture with sinusoidal walls used in the simulations where the pressure difference P 1 -P 2 is calculated along the third wavelength (left), where the flow is developed and far from the inlet and outlet boundaries. The mesh (adaptive) used for the numerical simulations (right). The two dashed lines represent the two cut lines on which the velocity profile is plotted

2. 4

 4 .2.a. Relative error between the LCL and N S solutions for three reference geometries To study the effect of each one of the geometrical parameters on the validity of the LCL, we consider three reference geometries (Table (2.2)).

Table 2 . 2 :ForFigure 2 . 5 :

 2225 Figure 2.5: Variation of the relative error between the LCL and N S solutions as a function of the aspect ratio for channels with parallel walls, with mirror symmetric walls and with arbitrary walls (Table (2.2)).

Figure 2 . 6 :Figure 2 . 7 :

 2627 Figure 2.6: Dimensionless velocity profiles given by the LCL (dashed lines) and N S (solid lines) solutions for aspect ratios = 0.1 and = 0.4 in the case of a channel with parallel walls. (a) velocity profiles on cut line 1. (b) velocity profiles on cut line 2

Figure 2 . 9 :

 29 Figure 2.9: Variation of the relative error between the LCL and N S solutions as a function of the corrugations asymmetry γ for channels with parallel walls, with mirror symmetric walls and with arbitrary walls (Table (2.2)).

Figure 2 . 10 :

 210 Figure 2.10: Variation of the relative error between the LCL and N S solutions as a function of the phase shift ∆x.

72

 72

Figure 2 . 11 :

 211 Figure 2.11: Variation of the relative error between the LCL and N S solutions as a function of Re for the reference geometries (Table2.2).

73 -Figure 2 . 12 :

 73212 Figure 2.12: Dimensionless velocity profiles given by the LCL and N S solutions for Re = 6.6 × 10 -2 and Re = 6.6 × 10 1 in the case of a fracture with parallel walls. (a) velocity profiles on cut line 1. (b) velocity profiles on cut line 2

74 Re

 74 The variation of the relative error as a function of Re for each value of δ 0 and for each geometry is plotted in Figure(2.15).

Figure 2 . 13 : 2 Figure 2 . 14 :

 2132214 Figure 2.13: Dimensionless velocity profiles given by the LCL and N S solutions for Re = 6.6 × 10 -2 and Re = 6.6 × 10 1 in the case of a fracture with mirror symmetric walls. (a) velocity profiles on cut line 1. (b) velocity profiles on cut line 2

2 Figure 2 . 15 :

 2215 Figure 2.15: Variation of the relative error between the LCL and N S solutions as a function of Re. Comparison between δ 0 = 0.2 and δ 0 = 0.4 for fractures with parallel walls, mirror symmetric walls, and arbitrary walls.

Figure 2 . 16 :

 216 Figure 2.16: Variation of the relative error between the LCL and N S solutions as a function of Re for different values of the corrugations asymmetry γ.

  2) and (2.3)).

Figure 2 . 17 :

 217 Figure 2.17: Flow streamlines plotted for four different fracture geometries: (a) ∆x = 0.5, δ 0 = 0.25 and γ = 0. (b) ∆x = 0, δ 0 = 0.25 and γ = 0. (c) ∆x = 0.5, δ 0 = 0.4 and γ = 0. (d) δ 0 = 0.25 and γ = 1. Re = 0.1 and = 0.1 are fixed in all the cases

Figure 2 . 18 :

 218 Figure 2.18: Flow streamlines in four different fracture geometries for Re = 26.66 and = 0.1 (a) ∆x = 0.5, δ 0 = 0.25 and γ = 0. (b) ∆x = 0, δ 0 = 0.25 and γ = 0. (c) ∆x = 0.5, δ 0 = 0.4 and γ = 0. (d) δ 0 = 0.25 and γ = 1. The streamlines are colored according to the velocity of the fluid flow. The blue color corresponds to low flow velocities and red color corresponds to high velocities

( 2 .

 2 15)).Case (d) corresponds to a fracture having a lower flat wall and a top corrugated wall (γ = 1), with δ 0 = 0.25. The repeated acceleration-deceleration of the flow in this case is similar to case (a) and it is also due to the aperture variation. This explains why for higher Re, a higher value of γ increases the deviation of the LCL from N S solution (Figure(2.16)). These results suggest that for higher Re, inertial effects become significant due to repeated acceleration and deceleration of the fluid inside the fracture. The fracture geometry gives rise to this alternation in the fluid movement by inducing variation in the local aperture along the flow direction. Therefore, the deviation between the LCL and the N S solutions at 82

Figure 3 . 1 :

 31 Figure 3.1: Different types of lift forces: (a) Magnus rotation-induced, (b) Saffman shear-slip, (c) shear-gradient and (d) wall-induced forces

  represents the particle dimensionless response time, St being the Stokes number and R = 2ρ f 2ρp+ρ f = 2 2k+1 a dimensionless number where k = ρp ρ f is the ratio of particle density to fluid density, V 0 is the flow mean velocity and ν is the fluid kinematic viscosity. The fluid flow is governed by the LCL. As shown in chapter 2, the flow velocity components are equal to:

  and equation (3.22) becomes:

Figure 3 . 2 :

 32 Figure 3.2: The algorithm used in the numerical simulations.

Figure 3 . 3 :

 33 Figure 3.3: Boundary conditions applied on a channel with sinusoidal walls (left) and the triangular mesh used in the numerical simulations (right).

3. 2

 2 .2.a. Particle focusing A first numerical simulation was performed under the following conditions: 10 particles are injected at different initial vertical positions in a mirror-symmetrical channel with H 0 = 0.3mm, = 0.1, and δ 0 = 0.25. The particle properties and the flow characteristics are set to ensure that τ = 0.01 and that equation (3.17) is resolved so that an attracting streamline is 101 P vulNt • 0 expected to appear in the flow. The variation of the particle vertical positions with respect to time is plotted in Figure 3.4.

Figure 3 . 4 :

 34 Figure 3.4: Evolution of the vertical position of 10 particles initially located at different heights in a sinusoidal channel for τ = 0.01 as predicted by F EM numerical simulations.

Figure 3 . 5 :

 35 Figure 3.5: Distances x f p traveled by particles with different initial vertical positions z 0 p in a channel with flat walls such as W = 0.00817 and τ = 1.66 * 10 -7 . Comparison between equation (3.24) (solid line) and particle tracking numerical simulations (symbols).

2 Figure 3 . 6 :

 236 Figure 3.6: Distances x f p traveled by particles with different initial vertical position z 0 p in a corrugated channel such as W = 0.00817 and τ = 3.33 * 10 -6 . Comparison between equation (3.22) (solid line) and particle tracking numerical simulations (symbols).

Figure 3 . 7 :

 37 Figure 3.7: Comparison between particle trajectories in the same channel as found in the numerical simulation and as predicted by the analytical model

  3.1)) in which the particle properties and fluid mean velocities were different but W was kept quasi constant (≈ 0.0043). For each case, 8 particles were injected in the channel at different initial vertical positions z 0 p . The distances x f p traveled by the particles are plotted in Figure (3.8) as a function of their initial positions for different initial vertical positions. The maximum relative error between the numerical and the analytical solutions of the distances traveled by the particles is 5.1 %. To sum up, particles injected at the same initial altitude travel approximately the same distance before sedimentation if W

Figure 3 . 8 :

 38 Figure 3.8: Distances x f p traveled by particles with different initial positions z 0 p for different configurations presenting same W 0.004. Comparison between equation (3.22) (solid line) and particle tracking numerical simulations (symbols).

•

  When Stokes equation is solved instead of the N S equations (to neglect fluid inertia), the maximum relative error decreases to 4.08 %.

84 whereW

 84 cr1 and W cr2 are critical values of W characterizing the transition between the respective regimes. W cr1 and W cr2 vary linearly as a function of h * . Diagrams of transport regimes can thus be established for channels with flat walls. In Figure (3.10) (a), a 2D diagram is built where three different regimes corresponding to the transport, transition and sedimentation zones are delimited by the linear variation of W cr1 and W cr2 as functions of h * . In Figure (3.10) (b), a 1D diagram is built where these zones are presented according to the single parameter W h * . Although the 1D representation could be considered more relevant in terms of delimitation of the three zones than the 2D one, it does not reflect clearly the respective contribution of parameters W and h* on the variation of the zones in the diagram.

Figure 3 . 10 :

 310 Figure 3.10: Transport regimes diagram for weakly inertial particles in a closed channel with flat walls. (a) 2D representation of the different zones according to the variation of W as a function of h * . (b) 1D representation of these zones according to W h * .

2 ,Figure 3 . 11 : 2 :

 23112 Figure 3.11: Variation of W cr 1 /h * and W cr 2 /h * as a function of the percentage of particles transported inside the channel.

3 :

 3 Percentages of sedimented particles in a sinusoidal channel with in phase walls for configurations corresponding respectively to the transport regime (h * = 0.008, W = 0.001), the transition regime (h * = 0.008, W = 0.0035), and the sedimentation regime (h * = 0.008, W = 0.0075) presented in Figure(3.10). Comparison between the regime diagram predictions (Figure3.10) and particle tracking numerical simulations results.

i. Influence of δ 0

 0 Let's consider W cr1 = c 1 h * and W cr2 = c 2 h * with c 1 and c 2 two constants describing their linear variation. The variations of c 1 and c 2 as functions of δ 0 are plotted in Figure(3.13).

Figure 3 . 13 :Figure 3 .

 3133 Figure 3.13: Variation of c 1 (left) and c 2 (right) as a function of δ 0 for two cases: α = π/2 and α = π.

4 :

 4 Percentages of sedimented particles in sinusoidal channels with out of phase identical walls defined respectively by δ 0 = 0.1 and δ 0 = 0.4, for two configurations corresponding to (h * = 0.01, W = 0.0021) and (h * = 0.01, W = 0.0074). Comparison between the regime diagram predictions (Figure3.14) and particle tracking numerical simulations results.

2 .

 2 The variation of c 1 and c 2 as a function of α is plotted in Figure (3.15).

2 Figure 3 . 15 :

 2315 Figure 3.15: Variation of c 1 (left) and c 2 (right) as a function of α for two cases: δ = 0.1 and δ = 0.2.

Figure 3 . 16 :

 316 Figure 3.16: Channel with sinusoidal walls with positive horizontal shift α = π/2 (left) and negative horizontal shift α = 3π/2 (right).

Figure 3

 3 Figure3.17: Effect of increasing α on particle transport regimes for channels having out of phase identical walls. The red solid lines present the limits between the regimes for a channel with α = 0 and the red dotted lines present the limits between the regimes after increasing the phase lag to α 0 = π. The white zones present the variations in the transport and sedimentation zones respectively.

  1 and c 2 as a function of γ for two different average corrugation amplitudes δ 0 = 0.1 and δ 0 = 0.2 (Figure (3.19)).

Figure 3 . 18 :

 318 Figure 3.18: Channels with lower flat wall (left), mirror-symmetrical walls (center) and upper flat wall (right).

2 Figure 3 . 19 :

 2319 Figure 3.19: Variation of c 1 (left) and c 2 (right) as a function of γ for two different average corrugation amplitudes δ 0 = 0.1 and δ 0 = 0.2.

c 2 Figure 3 . 20 :

 2320 figure shows that increasing the upper wall corrugation relatively to the lower wall leads to a decrease of the transport zone and to an increase of the sedimentation zone.

Table 3 . 6 :

 36 Percentages of sedimented particles in sinusoidal channels with out of phase identical walls defined respectively by γ = -1 and γ = 1, for two configurations corresponding to (h * = 0.01, W = 0.001) and (h * = 0.01, W = 0.0075). Comparison between the regime diagram predictions (Figure3.20) and particle tracking numerical simulations results.

are shown respectively in Figures 4 .1 and 4 . 2 . 124 Figure 4 . 1 :Figure 4 . 2 :

 4421244142 Figure 4.1: Picture of the experimental apparatus

4. 1

 1 Experimental setup and procedure 4.1.1 Open channel with closed circuit flow The channel is designed and manufactured by DIDATEC Technologies. It was initially dedicated to study open channel systems but we modified it by inserting a fracture model into the flume (Figure 4.3).

Figure 4 . 3 :

 43 Figure 4.3: The open channel used in the experiments. (a) Side view showing the channel walls. (b) Inside view of the channel. (c) The honeycomb (flow straightener) inserted at the channel inlet.

Figure 4 . 4 )

 44 . Each pump has a flow rate that can be adjusted between 50 and 350 L/min (3 and 21 m 3 /h). The speed of the pumps is controlled by a manual set point frequency. A flow meter measures the total flow rate of the

Figure 4 . 4 :

 44 Figure 4.4: Pumping system consisting of centrifugal pumps, a flow meter, and electrical control box.

Figure 4 . 5 :

 45 Figure 4.5: Top and side views of the sinusoidal walls used in the experiments.

Figure 4 . 6 :

 46 Figure 4.6: The hollow slab holding the upper wall and the bolts used to raise and lower the slab.

Figure 4 . 7 :

 47 Figure 4.7: The dimensions of the sinusoidal walls (in mm).

2. 2 )

 2 with one end directly positioned at the fracture inlet(Figure 4.8). The other end is connected to a plastic syringe used to inject the particles. The syringe has a volume of 50 mL and the internal diameter of its tip is 2 mm. To avoid the appearance of air bubbles in the flow, the particles are placed in the syringe without the liquid and the piston is fully pushed in. The syringe is then attached to the hose and the liquid is sucked up directly from the open channel. In this way, the nozzle only contains liquid without any air bubbles. Finally, the syringe is shaken to disperse the particles in the liquid before injecting the suspension into the fracture.

Figure 4 . 8 :

 48 Figure 4.8: The syringe and the rubber hose used to inject particles at the fracture inlet.

Figure 4 . 9 :

 49 Figure 4.9: Variation of the kinematic viscosity of the water-glycerin mixture as a function of the temperature, as measured by the glass capillary viscometer.

4. 2

 2 .4.b. Camera and bench The camera utilized in the experiments (Canon EOS 6D) is a 20.2-Megapixel full-frame CMOS digital single-lens reflex camera. It is equipped with a 24 -105 mm zoom lens with optical image stabilization. Throughout the experiments, the lens was set with a maximum zoom in. Each setting of the camera is controlled by the application EOS Utility which is also used for remote shooting and for download and display. Videos recorded by the camera have a 1920 × 1080 full high definition recording quality and the frame rate was set to 50 f ps.The camera is fixed on a bench equipped with a motor-driven belt conveyor (Figure4.11).The total length of the bench is 1.7 m. The stepping motor allows to define a moving velocity up to 1.5 m/s. It is controlled by the application MINIMOVE which allows to set the position and the displacement velocity. The camera can be then fixed at any position alongside the fracture. It can be also moved along the fracture length to follow the displacement of the

Figure 4 . 10 :

 410 Figure 4.10: The black cardboards placed on the open channel wall to isolate the fracture and to ease visualization. The visible part of the fracture is also shown.

Figure 4 . 11 :

 411 Figure 4.11: The bench and the belt conveyor on which the camera is fixed.

Figure 4 .

 4 Figure 4.13: (a) A sample of poppy seeds dispersed on a homogeneous surface. (b) Particles converted into black circular dots on a white background.

Figure 4 .

 4 Figure 4.14: A histogram of particle size distribution of a representative sample of 30 poppy seeds.

Figure 4 . 15 .

 415 According to this figure, it is clear that the particle trajectory is a straight horizontal line parallel to the fracture wall. By definition, this behavior is characteristic of inertial focusing.

Figure 4 . 15 :

 415 Figure 4.15: Single particle trajectory plotted at the fracture outlet as a succession of particle positions. (a) Positions of the particle at different times of the experiment. (b) Deduced particle trajectory

Figure 4 . 16 :

 416 Figure 4.16: Trajectories of the two particles plotted at the fracture outlet as a succession of particle positions.(a) Positions of the two particles at different times of the experiment. (b) Deduced particle trajectories

4. 3

 3 .3.a. Fracture with two flat walls The configuration with two flat walls was used with different values of H 0 and ∆h. i. Non-inertial regime: verification of the analytical model The mean aperture was set to H 0 = 5 mm and the hydraulic head difference to ∆h = 2.5 cm, leading to V 0 = 0.013 m/s (using the LCL) and Re = 1.184. In this case, the dimensionless numbers are τ ∈ [9.6 × 10 -6 ; 1.7 × 10 -5 ] and W ∈ [-0.058; -0.032]. A picture illustrating the particles right before they exit the fracture is shown in Figure 4.17.

Figure 4 . 17 :

 417 Figure 4.17: Multiple particles filmed moving on the upper wall, ahead of the fracture outlet for ∆h = 0.8cm (Re = 1.184, τ ∈ [9.6 × 10 -6 ; 1.7 × 10 -5 ] and W ∈ [-0.058; -0.032]).

  x f ana is equal to x f ana ∈ [0.043; 0.078] (or X f ana ∈ [0.029 m; 0.053 m] ) considering the range of W mentioned above. This means that the distance to deposition found experimentally falls in the range predicted by the analytical model.

Figure 4 . 18 :

 418 Figure 4.18: The distance X f exp traveled by a particle entering the fracture with two flat walls at Z 0 p

Figure 4 .

 4 [START_REF] Falkovich | Accleration of rain initiation by cloud turbulence[END_REF] shows the

Figure 4 . 20 :

 420 Figure 4.20: Multiple particles floated to the upper sinusoidal wall, ahead the fracture outlet for ∆h = 2.5cm (Re = 1.184, τ ∈ [9.6 × 10 -6 ; 1.7 × 10 -5 ] and W ∈ [-0.058; -0.032]).

Figure 4 .

 4 Figure 4.20 shows the particles right before they exited the fracture. Most of the particles floated to the upper wall due to the density difference, suggesting that the inertial lift forces are overcome by the buoyancy forces so that inertial focusing can not occur. To verify the predictions of the analytical model, a random particle with an initial vertical position z 0 p

Figure 4 . 21 :

 421 Figure 4.21: Particle with Z 0 p 0 traveling through the fracture until it touches the upper wall.

Figure 4 .

 4 [START_REF] Therrien | Three-dimensional analysis of variably-saturated flow and solute transport in discretely-fractured porous media[END_REF] shows the particles right before they exited the fracture. Similarly

Figure 4 .

 4 Figure 4.22: (a) Multiple particles filmed ahead the fracture outlet for ∆h = 6cm (Re = 18). (b) Approximated line on which the particles have focused is plotted in red by connecting the particles. (c) Schematic representation of the fracture and the approximated focusing trajectory.

Figure 4 . 23 :

 423 Figure 4.23: Particle with Z 0 p 0 traveling through the fracture until it touches the upper wall.

H h = 7 .

 7 22 mm), and ∆h = 10 cm leading to Re = 14.5. Several particles were injected at the fracture inlet.

Figure 4 .

 4 [START_REF] Oltéan | Numerical and experimental investigation of buoyancy-driven dissolution in vertical fracture[END_REF] shows the particles right before they exited the fracture.

Figure 4 .

 4 Figure 4.24: (a) Multiple particles filmed ahead the fracture outlet for ∆h = 10cm (Re = 14.5). (b) Approximated line on which the particles have focused.

  Influence of the fracture geometry on the validity of the LCL for different Reynolds numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Numerical Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Low Re (< 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2.a. Relative error between the LCL and N S solutions for three reference geometries . . . . . . . . . . . . . . . . . . . . . . 2.4.2.b. Influence of , δ 0 , γ and ∆x on the relative error between the LCL and N S solutions . . . . . . . . . . . . . . . . . . . . . 2.4.3 High Re (> 1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2.a. Particle focusing . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2.b. Particle trajectories . . . . . . . . . . . . . . . . . . . . . . . 3.3 Particle transport regime diagrams . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Channel with flat walls . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Corrugated channel with sinusoidal walls . . . . . . . . . . . . . . . . 3.3.2.a. Channel with in phase walls . . . . . . . . . . . . . . . . . . 3.3.2.b. Channel with out of phase identical walls . . . . . . . . . . . 3.3.2.c. Channel with maximum phase lag between the walls . . . . . .3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

	CONCLUSION AND PERSPECTIVES
	List of symbols
	Length, velocity, and time

1.2.1 Modeling flow in rough fractures . . . . . . . . . . . . . . . . . . . . . 1.2.2 Inertial effects in fracture flows . . . . . . . . . . . . . . . . . . . . . 1.2.3 Idealized model of fracture geometry . . . . . . . . . . . . . . . . . . 2 SINGLE PHASE FLOW THROUGH FRACTURES 2.1 Geometrical description of fractures with corrugated walls . . . . . . . . . . 2.2 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Flow between parallel flat walls: the cubic law . . . . . . . . . . . . . 2.2.2 Flow between corrugated walls: the local cubic law . . . . . . . . . . 2.2.3 Flow velocity components in corrugated channels . . . . . . . . . . . 2.3 Influence of the fracture geometry on its hydraulic aperture . . . . . . . . . . 2.3.1 ∆x effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 γ effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.4 2.4.3.a. Relative error between the LCL and N S solutions for the reference geometries . . . . . . . . . . . . . . . . . . . . . . 2.4.3.b. Influence of , δ 0 , γ and ∆x on the relative error between the LCL and N S solutions . . . . . . . . . . . . . . . . . . . . . 2.5 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Relation between the hydraulic and the mean apertures . . . . . . . . 2.5.2 Validity of the local cubic law for different Reynolds numbers . . . . 2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 TRANSPORT AND DEPOSITION OF WEAKLY-INERTIAL PARTI-CLES IN FRACTURE FLOWS 3.1 Governing equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Forces acting on each particle . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Particle motion equation and particle trajectory equation . . . . . . . 3.1.2.a. Focusing of weakly inertial particles in channels with periodic walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2.b. Trajectory equation of inertia-free particles . . . . . . . . . . 3.1.2.c. Channel with flat walls . . . . . . . . . . . . . . . . . . . . . 3.1.2.d. Channel with sinusoidal walls . . . . . . . . . . . . . . . . . . 3.2 Numerical verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Simulation procedure . . . . . . . . . . . . . . . . . . . . . . . . . . .

33.3.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 EXPERIMENTAL INVESTIGATION OF PARTICLE TRANSPORT IN FRACTURE FLOWS 4.1 Experimental setup and procedure . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Open channel with closed circuit flow . . . . . . . . . . . . . . . . . . 4.1.2 Fractures with flat and sinusoidal walls . . . . . . . . . . . . . . . . . 4.1.3 Liquid properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.4 Visualization and image treatment . . . . . . . . . . . . . . . . . . . 4.2.4.a. Lighting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4.b. Camera and bench . . . . . . . . . . . . . . . . . . . . . . . . 4.1.5 Experimental procedure and image treatment . . . . . . . . . . . . . 4.2 Preliminary results with poppy seeds . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Particle properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Transport with water as the operating liquid . . . . . . . . . . . . . . 4.3.2.a. Trajectory of a single particles . . . . . . . . . . . . . . . . . 4.3.2.b. Inertial focusing of two particles . . . . . . . . . . . . . . . . 4.2.3 Transport with water-glycerin mixture as the operating liquid . . . . 4.3.3.a. Fracture with two flat walls . . . . . . . . . . . . . . . . . . . 4.3.3.b. Fracture with a flat wall and a sinusoidal wall . . . . . . . . .

4.3.3.c. Fracture with two sinusoidal walls . . . . . . . . . . . . . . . 4

  -3 > (equation(2.18)) depends only on the corrugation amplitude representing walls roughness defined by δ 0 , the corrugations asymmetry defined by γ, and the dimensionless phase shift between the two walls ∆x (equations(2.19) and (2.2)). As H h H 0 does not depend on the channel aspect ratio , if the channel mean aperture H 0 is fixed, then two channels having walls with different wavelengths should have the same hydraulic aperture H h .

  Figure 2.8: Variation of the relative error between the LCL and N S solutions as a function of the dimensionless corrugation amplitude δ 0 for for channels with parallel walls, with mirror symmetric walls and with arbitrary walls (Table (2.2)).

			€ = 0.1 2 8 4 relative			(a)				€ = 0.4
	1.80 1.60 1.40 1.20 1.00 0.80 0.60 0.40			2 4 6 0 relative error %	0			1.80 1.60 1.40 1.20 1.00 0.80 0.60 0.40 0.1			0.2			0.3
	0.20 0.00 •0.20			0	0	0.1	0.2	0.20 0.00 •0.20	0.3	0.4		δ 0
	•0.4		•0.2 --NS --• LO. 0 l parallel 0.2 parallel 0.4 δ 0 mirror-symmetric •0.4	.0.2 --NS --• LO. 0 l arbitrary mirror-symmetric 0.2 0.4	a
	L20 LOO		€ = 0.1			(b}	1.20 1.00			€ = 0.4
	0.80							0.80
	0.60							0.60
	0.40							0.40
	0.20							0.20
	0.00							0.00
	.0.20							•0.20
	.o.6	.0.4	•0.2	0	0.2	0.4	0.6		•0.6	.0.4	.0.2	0	0.2	0.4	0.6
			l --NS --• LO.						l --NS --• LO.
							69

Table 3 .

 3 is kept constant.

		Parameter W	τ	Re H a(µm) k	V 0 (m/s)
		case 1 case 2 case 3 case 4 case 5 case 6 case 7 case 8 case 9 case 10	0.0043 4.26 * 10 -7 2.5 0.00429 1.89 * 10 -6 4 0.00431 1.55 * 10 -7 1 0.00429 4.62 * 10 -6 5 10 0.00432 1.8 * 10 -5 0.00809 5.35 * 10 -7 1.5 0.00809 9.45 * 10 -7 2.5 0.00809 6.97 * 10 -6 3.75 0.00809 7.07 * 10 -6 7.5 0.00809 1.47 * 10 -5 10	2.5 5 3 7.5 10.5 4.5 4 11 8 7	2.58 0.005 1.63 0.008 1.44 0.002 1.35 0.01 1.36 0.02 1.55 0.003 2.16 0.005 1.23 0.0075 1.87 0.015 2.59 0.021
	0.4	z 0 p					case 1 : max. error = 3.77 %
	0.2						case2 : max. error = 4.08 % case 3 : max. error =4.18 %
					x f p		case 4 : max. error = 4.12 % case 5 : max. error = 5.11 %
		0.2	0.4	0.6	0.8	1	Equation 8
	-0.2
	-0.4

1: Flow characteristics and particles properties used in the simulation to ensure W 0.0043 (cases 1 to 5) and W 0.00809 (cases 6 to 10).

  • C is expected to be in the order of 65cSt, which is a moderately high viscosity with respect to water. The liquid reservoir was then filled approximately with 50 L of water and 200 L of glycerin. The selected glycerin has a degree of purity higher than 99.5% and a water content inferior to 0.5%. The kinematic viscosity of the mixture ν f was measured with a glass capillary (U-tube) viscometer, for temperatures ranging from 20 • C to 31 • C. The variation of ν f as a function of the temperature is plotted in Figure4.9.

		60
	f (cSt)	50
	ν	40
		30
		20	20	22	24	26	28	30	32
				Temperature ( • C)

Table 4 . 1 :

 41 Density of the mixture as the ratio between measured weights m f and volumes V f .V f (mL) m f (g) ρ f (g/L or kg/m 3 )

	50	60.33	1206.6
	100	121.02	1210.2
	150	182	1213.3
	200	242	1213.5
	Mean density	1210.9
	4.1.4 Visualization and image treatment
	4.2.4.a. Lighting

Table 4 .

 4 The average density ρ p of poppy seeds was first measured by weighing samples and by soaking them in a water volume V f in order to measure the change in volume ∆V , which is equal to the particles volume V p . To take into account measurement errors, four samples with different weights (m p ) were considered. The results are presented in table 4.2. 2: Geometrical properties of the three fractures used as references in the numerical simulations.

	m p (g) ∆V = V p (mL) ρ p (kg/m 3 )
	5	10	1068
	10	100	1072
	20	150	1069
	30	200	1070
	Average density	1069.75

calculée en utilisant la LCL. Ces équations ont été vérifiées à travers les solutions numériques basées sur une technique de suivi des particules impliquant les équations du mouvement des particules et le champ d'écoulement obtenu par la résolution des équations de N S. Les simulations numériques ont été réalisées en tenant compte à la fois de l'inertie des particules et de celle du fluide. Les résultats numériques ont confirmé les hypothèses sous lesquelles l'approche analytique a été développée. De plus, ils ont confirmé que la trajectoire des particules peut être prédite directement en fonction de la valeur de W et de la géométrie du canal, sans avoir besoin de calculs ou de simulations numériques supplémentaires.En se basant sur ces développements, un diagramme de régime qui prédit le transport ou la sédimentation des particules en fonction de W et d'un paramètre géométrique h * , représentant le rapport entre l'ouverture moyenne du canal et sa longueur totale, a été proposé (Figure2).Pour les canaux à parois ondulées, le diagramme de régimes est similaire à celui obtenu pour les canaux à parois planes, mais les zones de transport et de sédimentation ont tendance à augmenter ou diminuer en fonction de la période et l'amplitude des ondulations, et le déphasage entre les parois. Quand les deux parois sont en phase, le diagramme de régimes est identique à celui obtenu pour un canal à parois planes. Lorsque les deux parois sont décalées, l'augmentation de l'ondulation de la paroi entraîne une augmentation des zones de transport et de sédimentation dans le diagramme. En considérant l'asymétrie entre les ondulations des deux parois, l'augmentation de l'ondulation de la paroi supérieure par rapport à celle de la

Acknowledgments

Chapter 4 EXPERIMENTAL INVESTIGATION OF PARTICLE TRANSPORT IN FRACTURE FLOWS

An analytical model describing the transport of weakly inertial particles in closed channels with flat and corrugated walls has been proposed in chapter 3, under the assumptions that particle inertia is neglected and that the flow follows the local cubic law (LCL). This analytical model has been verified numerically taking into account particle inertia and fully solving the Navier-Stokes (N S) equations. To go further, an experimental validation is required. In order to consider real situations and to assess the validity of the analytical model on a practical basis. In this chapter, the experimental apparatus that was designed and built up to this end is presented. The first part of the chapter is devoted to the presentation of the design and the setup of the physical model. Then, the experimental procedure and the methodology used to process the experimental data are described. Finally, several preliminary results are presented and discussed with respect to the initial objectives of the study. The slight deviation between the particle positions is due to their polydispersity. However, one can conclude that inertial focusing occurs this experiment.

4.3.3.b. Fracture with a flat wall and a sinusoidal wall

In this case, the fracture consists of a bottom flat wall and an upper sinusoidal one.

The tests performed are similar to those done with the flat fracture (section 3.3.1). We first performed experiments with low ∆h to assess the analytical model for this particular configuration. ∆h was then increased to generate high Re in order to examine the effect of the fluid inertia on the particle behavior and the potential presence of inertial focusing.

i. Non-inertial regime: verification of the analytical model

The mean aperture was set equal to H 0 = 5 mm (the hydraulic aperture was then calculated H h = 4.9mm). In order to minimize the inertial effects, the hydraulic head was fixed equal to ∆h = 2.5 cm. Using the LCL, V 0 = 0.013 m/s and Re = 1.184. In this case, the dimensionless numbers are τ ∈ [9.6 × 10 -6 ; 1.7 × 10 -5 ] and W ∈ [-0.058; -0.032].

Abstract

Analytical, numerical, and experimental investigations of particle transport in fractures with flat and corrugated walls The aim of the present thesis is to study the transport and deposition of small solid particles in fracture flows. First, single-phase fracture flow is investigated in order to assess the validity of the local cubic law for modeling flow in corrugated fractures. Channels with sinusoidal walls having different geometrical properties are considered to represent different fracture geometries. It is analytically shown that the hydraulic aperture of the fracture clearly deviates from its mean aperture when the walls roughness is relatively high. The finite element method is then used to solve the continuity and the Navier-Stokes equations and to simulate fracture flow in order to compare with the theoretical predictions of the local cubic law for Reynolds numbers Re in the range 6.7 × 10 -2 -6.7 × 10 1 . The results show that for low Re, typically less than 15, the local cubic law can properly describe the fracture flow, especially when the fracture walls have small corrugation amplitudes. For Re higher than 15, the local cubic law can still be valid under the conditions that the fracture presents a low aspect ratio, small corrugation amplitude, and moderate phase lag between its walls. Second, particle-laden flows are studied. An analytical approach has been developed to show how particles sparsely distributed in steady and laminar fracture flows can be transported for long distances or conversely deposited inside the channel. More precisely, a rather simple particle trajectory equation is established. Based on this equation, it is demonstrated that when particles' inertia is negligible, their behavior is characterized by the fracture geometry and by a dimensionless number W that relates the ratio of the particles sedimentation terminal velocity to the flow mean velocity. The proposed particle trajectory equation is verified by comparing its predictions to particle tracking numerical simulations taking into account particle inertia and resolving the full Navier-Stokes equations. The equation is shown to be valid under the conditions that flow inertial effects are limited. Based on this trajectory equation, regime diagrams that can predict the behavior of particles entering closed channel flows are built. These diagrams enable to forecast if the particles entering the channel will be either deposited or transported till the channel outlet. Finally, an experimental apparatus that was designed to have a practical assessment of the analytical model is presented. Preliminary experimental results tend to verify the analytical model. Overall, the work presented in this thesis give new insights on the behavior of small particles in fracture flows, which may improve our prediction and control of underground contamination, and may have applications in the development of new water filtration and mineral separation techniques.