Preferential accumulation of particles in periodic channels
Unlike lift-induced particle focusing, accumulation or clustering of particles due to their inertia in liquid ows through channels with corrugated walls, remains a theoretical predic-tion yet to be veri ed experimentally (Nizkaya et al. ). Moreover, particle clustering is only limited to the case of periodic walls corrugations (Figure 1.7) as, in contrast with lift-induced migration, clustering can not occur in channels with at walls. This phenomenon in which particles, with low but nite inertia, are expected to be attracted by a streamline, may occur for speci c channel geometries and ow characteristics.
This phenomenon will be discussed later on with more details. Before that, the ow must be investigated.
Flow in channels with at and corrugated walls
Channels with at 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 ow in fractured geological systems, such as its orientation, its extent, and its interconnection with other fractures (Rasmussen ). Zhang et al.  showed that the hydraulic behavior of a fractured medium is largely in uenced by the characteristic lengths of the single fractures and their number (i.e. fracture density), and fracture orientations. Indeed, the properties of the ow occurring through a network of fractures are strongly controlled by those of the ow 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. , Mourzenko et al. ). However, modeling ow in single fractures remains a key issue that needs to be properly understood before extrapolating to more complex con gurations.
The parameters likely to intervene in the prediction of ow through single fractures have been gathered experimentally by Hakami and Larsson . They noted that, apart from the e ect of the uid properties and pressure conditions on the fracture boundaries, fracture ow depends also on di erent geometrical parameters such as the aperture and spatial cor-relations 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 di erent morphological characteristics such as: amplitude (elevation of points on the surface), angularity (slopes and angles), waviness (periodicity), and curvature (Gentier , Belem ). 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 , Belem , Mourzenko et al. , Plouraboue , Oron & Berkowitz , Lefevre , Legrain ). The roughness in its general sense has a multiplicity of characteristic length scales. How-ever, in the literature, two main scales of roughness are usually highlighted (Figure 1.8). They are characterized and de ned by their e ects on the mechanical and hydraulic behav-ior of a fracture. It is important at this point to distinguish two types of roughness (Louis ). 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 ow 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 ow between two points and the straight distance between these two same points, and, thus, it can has an important e ect the behavior of the ow through rough fractures. The micro-scale roughness is neglected in this thesis and only the e ects of the macro-scale roughness on the behavior of fracture ow are considered.
Modeling ow in rough fractures
To model uid ow in a single fracture, the standard laws of uid mechanics can be used. These are the Navier-Stokes (N S) and continuity equations. Eventually, these equations can be simpli ed according to the channel geometry and/or assumptions regarding the ow, so that analytical solutions can be obtained to determine the pressure and velocity elds.
On the other hand, on a larger scale, the ow can be described by Darcy’s law. Darcy’s law relates the hydraulic gradient to the ow rate using an intrinsic parameter de ned as the permeability. It can be obtained from the N S equation using an upscaling method such as volume averaging an homogenization (Whitaker ).
One of the most important characteristics of a rough fracture is its aperture. The aperture is de ned as the distance between the fracture walls and implicitly depends on the way the roughness is de ned. The aperture can be de ned locally or globally according to geometrical criteria, mechanical criteria, or as a result of hydraulic experiments assuming a law for the ow. Consequently, the geometric mean, the mechanical and the hydraulic apertures can be distinguished (Lin , Davias , Crosnier ).
Early attempts to model ow in single fracture assumed that the ow occurs between two at parallel plates representing the fracture walls (Bear ). This is known as the 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 ow between parallel plates can then be easily obtained. In particular, for a 2D channel with at walls of length L1 and aperture H0, through which a uid of density and dynamic viscosity is owing due to a pressure di erence P = g Z between the inlet and the outlet of the channel, with Z is the hydraulic head and g the gravity acceleration, the volumetric ow rate per unit width is given by: gH03 Z 12 L1 Q = (1.3). The volumetric ow rate per width is Q = V0H0, V0 being the mean velocity of the ow in the channel. In this case, the velocity has a parabolic pro le (Figure 1.9). At the same time, Darcy’s law (Darcy ) gives a linear relation between the volumetric ow rate Qv and the pressure drop P : KA P Qv = (1.4). where K is the hydrodynamic permeability and A is the cross-section on which Qv is com-puted. In order to consider the ow rate per unit width Q, A can be replaced by H0 and equation (1.4) thus becomes: KH0 P Q = (1.5).
Idealized model of fracture geometry
Although real fractures have 3D geometries, 2D geometries can be considered for simpli-cation 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 ow 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 eld in the Y direction and to study the system in the 2D referential (X,Z). In this context, Zimmerman et al.  showed that a 2D model in which the aperture varies only in the ow direction gives qualitatively the same results as a full 3-D model.
In order to study analytically ow and transport in fractures, a geometrical model that best represents the fracture characteristics must be used. Power and Tullis  showed that natural rock fractures, despite having self-a ne 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 con guration in which the pro les 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 e ects of the walls roughness and of the aperture variation on the ow. Actually, Le Borgne et al.  showed that velocity distribution in sinusoidal channels are very similar to that obtained in more complex medium.
The di erence 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 a ect the ow streamlines inside the channel. On the other hand, the sinusoidal model, despite not following exactly the real roughness pro le, can still take into account the aperture variation and the e ect of the channel walls on the ow geometry. These are the reasons why channels with sinusoidal walls have been then widely used to represent rough fractures (Zimmerman et al. , Brown et al. , Zimmerman and Bodvarsson , Waite et al. , Sisavath et al. ,Basha and El Asmar , Yeo and Ge , Nizkaya , Liu and Fan , Renu and Kumar ).
It is convenient also to notice that sinusoidal channels have been used to model homoge-neous porous media. For instance, porous media can be modeled as a set of uniform spherical grains regularly stacked. In this case, a 2D simpli cation of a representative elementary vol-ume can lead to a sinusoidal channel that accounts for the aperture variation inside the medium. This simpli cation has been widely applied to study ow and solute dispersion in porous media (Kitanidis and Dikaar , Edwards et al. , Bolster et al. , Bouquain et al. ).
Throughout the thesis, sinusoidal variation will be considered to represent walls corru-gation. Other than representing fracture roughness, such geometry is convenient for setting up numerical simulations and experimental devices aiming at studying fracture ow.
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 ow direction) and Z to the vertical one. Gravity is taken into account and applies perpendicularly to the main ow direction (along Z).
Following the approach proposed by Nizkaya , we consider a two dimensional fracture having two rough walls with idealized periodic roughness described respectively by the func-tions 1(X) for the lower wall and 2(X) for the upper one (Figure (2.1)). The fracture total length is L1 and is thus de ned by the domain limited by X [0; L1] and Z [ 1(X); 2(X)].
The fracture walls corrugations are smooth so that @ 1;2(X) << 1. The fracture can be @X equally de ned by the local half aperture H(X) = 12 ( 2(X) 1(X)) and the fracture mid-dle line (X) = 12 ( 1(X) + 2(X)). The fracture walls are periodic and have the same corrugation wavelength L0, which is the characteristic length of the ow in the X direction.
The phase shift between the two walls is X and the corrugation amplitude of each wall is: A1;2 = 1 (max[ 1;2(X)] min[ 1;2(X)]). The characteristic length in the Z direction can be de ned as the mean aperture of the fracture given by: 1 ZL1 H0 = L1 0 ( 2(X) 1(X))dX Finally, the fracture aspect-ratio can be de ned as: = H0 L0.
Table of contents :
1 STATE OF THE ART
1.1 Particle-ladenows: Basic concepts
1.1.1 Denition of particle inertia
1.1.2 Particle transport in closed channel ows
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 at and corrugated walls
1.2.1 Modelingow in rough fractures
1.2.2 Inertial eects in fractureows
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 at 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 Inuence of the fracture geometry on its hydraulic aperture
2.3.1 x eect
2.4 Inuence of the fracture geometry on the validity of the LCL for dierent
2.4.1 Numerical Method
2.4.2 Low Re (< 1)
2.4.2.a. Relative error between the LCL and NS solutions for three reference geometries
2.4.2.b. Inuence of , 0, and x on the relative error between the LCL and NS solutions
2.4.3 High Re (> 1)
2.4.3.a. Relative error between the LCL and NS solutions for the reference geometries
uence of , 0,
and x on the relative error between the
LCL and NS solutions
2.5.1 Relation between the hydraulic and the mean apertures
2.5.2 Validity of the local cubic law for dierent Reynolds numbers
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 at walls
3.1.2.d. Channel with sinusoidal walls
3.2 Numerical verication
3.2.1 Simulation procedure
3.2.2.a. Particle focusing
3.2.2.b. Particle trajectories
3.3 Particle transport regime diagrams
3.3.1 Channel with at 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
4 EXPERIMENTAL INVESTIGATION OF PARTICLE TRANSPORT IN FRACTURE FLOWS
4.1 Experimental setup and procedure
4.1.1 Open channel with closed circuitow
4.1.2 Fractures with
at and sinusoidal walls
4.1.3 Liquid properties
4.1.4 Visualization and image treatment
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 at walls
4.3.3.b. Fracture with a at wall and a sinusoidal wall
4.3.3.c. Fracture with two sinusoidal walls
CONCLUSION AND PERSPECTIVES