Inuence of the fracture geometry on the validity of the LCL for dierent Reynolds numbers

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Lift-induced inertial migration

Both neutrally and non-neutrally buoyant particles can migrate across the ow stream-lines to reach speci c equilibrium positions within the channel (Figure 1.6). In fact, Segr and Silberberg [28] were the rst to witness that particles in a laminar pipe Poiseuille ow 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 e ect. Since then, this phenomenon has been studied extensively using theoretical (Schonberg and Hinch [29], Asmolov [30]), experimental (Karnis et al. [31], Matas et al. [32]) and numerical approaches (Feng et al. [33], Yang et al. [34]). These investigations concluded that inertial migration occurs due to forces that act on particles in inertial ows, known as the inertial lift forces. Recently, lift-induced particle focusing attracted much attention with the development of micro uidics where it has many applications, e.g. in cell separation and isolation in biologi-cal uids (Di Carlo et al. [35], Martel and Toner [36]). The e ect of inertial lift forces must be is explained in detail in the following chapters.

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. [16]). 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 [37]). Zhang et al. [38] 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. [39], Mourzenko et al. [40]). 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 [41]. 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 [42], Belem [43]). 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 [43], Mourzenko et al. [39], Plouraboue [44], Oron & Berkowitz [45], Lefevre [46], Legrain [47]).
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 [48]). 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.

Geometrical description of fractures with corrugated walls

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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 [27], 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)]).

Flow between parallel at walls: the cubic law

In an incompressible laminar ow generated by a pressure gradient along a channel con-sisting of two parallel planes (the parallel-plate model), the inertial terms are identically null and the steady ow can be expressed via equation (2.6). This problem has been thoroughly addressed in the literature. For instance, Zimmerman and Bodvarsson [69] 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: (rP )X = @P = @2VX (2.7).
with rP the applied pressure gradient. If the distance between the two walls is H0, inte-grating equation (2.7) with the no-slip boundary conditions (Vx(0) = Vx(H0) = 0) leads to the following parabolic velocity pro le, similar to the one found in the so called Poiseuille ow: H02 4Z2 Vx = (rP )X (1 ) (2.8).

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 Modeling ow in rough fractures
1.2.2 Inertial eects in fracture ows
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
2.3.2 eect
2.4 Inuence of the fracture geometry on the validity of the LCL for dierent Reynolds numbers
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
2.4.3.b. Inuence of , 0, and x on the relative error between the LCL and NS 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 dierent 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 at walls
3.1.2.d. Channel with sinusoidal walls
3.2 Numerical verication
3.2.1 Simulation procedure
3.2.2 Results
3.2.2.a. Particle focusing
3.2.2.b. Particle trajectories
3.3 Particle transport regime diagrams
3.3.1 Channel with 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
3.3.3 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 ow
4.1.2 Fractures with at 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 at walls
4.3.3.b. Fracture with aat wall and a sinusoidal wall
4.3.3.c. Fracture with two sinusoidal walls
4.3 Conclusion
CONCLUSION AND PERSPECTIVES 

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