Soutenance de thèse

1. Laboratoire Environnement, Géomécanique & Ouvrages Soutenance de thèse  Transport, dépôt et relargage de particules inertielles dans une fracture à rugosité périodique T. Nizkaya Directeur de thèse: M. Buès Co-directeur de thèse: J.-R. Angilella, LAEGO, Université de Lorraine Ecole doctorale RP2E 1er Octobre 2012 Nancy, Lorraine

2. Particle-laden flows Photo: NASA's Goddard Space Flight Center Particles: air and water pollutants, dust, sprays and aerosols, etc…

3. Particle-laden flows through fractures Hydrogeology: Flows through fractures often carry particles (sediments, organic debris etc.). How to model particle-laden flows?

4. Two models of particles Tracer particles: point particles advected by the fluid (+ brownian motion) Inertial particles: finite size, density different from fluid. Example: sand in the air Example: dye in water

5. Twomodels of particles Tracer particles: point particles advected by the fluid (+ brownian motion) Inertialparticles: finite size, density differentfromfluid. Advection-diffusion equations for particle concentration. Example: sand in the air

6. Twomodels of particles Tracer particles: point particles advected by the fluid (+ brownian motion) Inertialparticles: finite size, density differentfromfluid. Particleinertiais important. Evenweakly-inertialparticles are Advection-diffusion equations for particle concentration. verydifferentfromtracers!

7. Clustering of inertialparticles Inertialparticles tend to cluster in certain zones of the flow. rain initiation Wilkinson & Mehlig (2006) planet formation Barge & Sommeria (1995) aerosol engineering Fernandez de la Mora (1996) Particles in fractures: clusteringcan lead to redistribution of particlesacross the fracture?

8. Clustering of inertialparticles Inertialparticles tend to cluster in certain zones of the flow. rain initiation Wilkinson & Mehlig (2006) planet formation Barge & Sommeria (1995) aerosol engineering Fernandez de la Mora (1996) Inperiodicflowsparticle focus to a single trajectory: Robinson (1955), Maxey&Corrsin (1986), etc.

9. Goal of the thesis Theoreticalstudy of focusingeffect on particle transport in a fracture withperiodic corrugations. Water + particles 0 homogeneos distribution «focusing»

10. Outline of the talk Single-phase flow in a model fracture Focusing of inertial particles in the fracture Influence of lift force on particle focusing Conclusion and perspectives

11. I. Single-phase flow in a thin fracture.

12. I. Single-phase flow in a thin fracture. Goal: Obtain an explicit fluid velocity field for arbitrary fracture shapes Method: Asymptotic expansions

13. Simplified model of a fracture Z X Model fracture: a thin 2D channelwith «slow» corrugation. Typical corrugation lengthL0 >> typical aperture H0. Small parameter:

14. Single-phase flow in fracture Single-phase flow in fracture: 2D, incompressible, stationary Navier-Stokes equations: Streamfunction: Non-dimensional variables: Reynolds number:

15. Equations of inertial lubrication theory Navier-Stokes equations in non-dimensional variables: Boundary conditions: No slip at the walls Hasegawa and Izuchi (1983) Borisov (1982), etc.

16. Equations of inertiallubricationtheory Navier-Stokes equations in non-dimensional variables Boundary conditions: No slipat the walls Small parameterεperturbativemethod

17. Generalization of previousworks } Crosnier (2002) Hasegawa and Izuchi (1983) Presentthesis: full parametrization of the fracture geometry. Borisov (1982)

18. The cross-channel variable: Cross-channel variable : half-aperture of the channel middle-line profile h(x) h(x)

19. Asymptotic solution of 2ndorder 0th : 1st: 2nd:

20. Asymptotic solution of 2ndorder 0th : 1st: 2nd: 3rd… etc. «local cubiclaw» inertial corrections viscous correction

21. Numericalverification: mirror-symmetric --- LCL flow, 2ndorderasymptotics, numerical simulation

22. Numericalverification: flat top wall --- LCL flow, 2ndorderasymptotics, numerical simulation

23. Application: corrections to Darcy’slaw Flow rate depends on pressure drop: Q - curve Darcy’slaw Inertial corrections: analytical expression? Larger flow rates Small flow rates

24. Corrections to Darcy’slaw Pressure drop (from 2ndorderasymptotic solution): No quadraticterm! In accordance withLo Jaconoet al. (2005) and manyothers.

25. Corrections to Darcy’slaw Pressure drop (from 2ndorderasymptotic solution): Geometricalfactors:

26. Corrections to Darcy’slaw Pressure drop (from 2ndorderasymptotic solution): Geometricalfactors: Slope of the linearlawdepends on both aperture and shape of the middle line.

27. Corrections to Darcy’slaw Pressure drop (from 2ndorderasymptotic solution): Geometricalfactors: Cubic correction onlydepends on aperture variation.

28. Numericalverification Pressure drop vs Reynolds number Darcy’slaw numerics (mirror-symmetricchannel) ourasymptotic solution numerics (channelwith flat top wall)

29. II. Transport of particles in the periodic fracture

30. Periodicchannel corrugation period «focusing» Particles: small, non-brownian, non-interacting, passive. Flow: asymptotic solution (leadingorder)

31. Particle motion equations Particledynamics: from Stokes equations around the particle Maxey-Riley equations Gatignol (1983) Maxey and Riley (1983)

32. Particle motion equations Maxey-Riley equations: fluid pressure gradient + gravity drag force added mass Basset’s memory term

33. Typical long-time behaviors (numerics -LCL flow, no gravity) Heavy particles Light particles Q Q Heavy particlescan focus to a single trajectory (or not!) depending on channelgeometry. Q

34. Typical long-time behaviors (numerics -LCL flow, withgravity) Light particles Heavy particles Low Q High Q Focusingpersists in presenceof gravity, if the flow rate Qis high enough (permanent suspension)

35. Goal: Find conditions for particlefocusing depending on channelgeometry and flow rate. Method: Poincaré map + asymptotic motion equations for weakly-inertialparticles

36. SimplifiedMaxey-Riley equations Particleresponse time: Densitycontrast:

37. SimplifiedMaxey-Riley equations Particleresponse time: Densitycontrast: For weakly-inertialparticles: Maxey (1987) particleinertia + weight fluidvelocity fromMaxey-Riley equations

38. Poincaré map for weakly-inertialparticles = rescaled cross-channel variable z after k periods fromsimplified Maxey-Riley equations )

39. Poincaré map for weakly-inertialparticles Poincaré map: ) Stable fixed point: Particles converge to the streamline Focusing!

40. Analytical expression for the Poincaré map Poincaré map for the LCL flow: Fluid/particle density ratio Channel geometry Gravitynumber heavier thanfluid lighter thanfluid

41. Analytical expression for the Poincaré map Poincaré map for the LCL flow: Fluid/particle density ratio Channel geometry Gravitynumber heavier thanfluid lighter thanfluid Attractor position

42. Focusing/sedimentationdiagram Rescaledgravity: (analytical expression) Corrugation asymmetry factor:

43. Focusing/sedimentationdiagram Light particles Heavy particles A Case A:

44. Focusing/sedimentationdiagram B Case B:

45. Focusing/sedimentationdiagram C Case C:

46. Focusing/sedimentationdiagram D Case D:

47. Other applications of Poincaré map Using the Poincaré mapwecancalculate: • Percentage of depositedparticles • Maximal depositionlength • Focusing rate VerifiednumericallyOk

48. Influence of channel geometry on transport properties

49. Shape factors of the channel Aperture-weighted norm: Shape factors: «apparent» aperture aperture variation middle line corrugation differencebetween wall corrugations

50. Single phase flow: geometry influence Pressure drop curve: Slope of the linearlaw: Shape factors: Inertial correction: Weakdependence on channelshape!