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How disorder can influence dynamics

How disorder can influence dynamics. Example Diffusion Edwards Wilkinson in a Brownian noise Edwards Wilkinson in a Quenched Noise. TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A A A. How disorder can influence dynamics. Example Diffusion

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How disorder can influence dynamics

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  1. How disorder can influence dynamics • Example • Diffusion • Edwards Wilkinson in a Brownian noise • Edwards Wilkinson in a Quenched Noise TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAA

  2. How disorder can influence dynamics • Example • Diffusion Flat front • Edwards Wilkinson in a Brownian noise Self affine, H=0.5 • Edwards Wilkinson in a Quenched Noise Self affine, H=1.1 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAA

  3. How disorder can influence dynamics • Edwards Wilkinson in a Brownian noise Anomalous subdiffusive scaling ( ) • Edwards Wilkinson in a Quenched Noise Anomalous scaling, different exponent TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAA

  4. Influence of disorder on the flow during fingering in a porous Hele Shaw cell Renaud Toussaint, Gerhard Schaefer, E.A. Fiorentino, J Schmittbuhl, Univ. of Strasbourg Knut Jørgen Måløy, G Løvoll Univ. of Oslo Yves Meheust Geosciences Rennes

  5. Viscous Fingering instability: lowly viscous fluid displaces another one. Paradigm: Hele Shaw cell Experimental setup by Saffman and Taylor. Proc. Roy. Soc. London (1983).

  6. Viscous Fingering in Hele Shaw cell • Movement in layers • Smooth finger • No disorder • Capillary pressure set by interface curvature From Saffman and Taylor Proc. R. Soc. London Ser. A 245, 312, (1958)

  7. Diffusion Limited Aggregation DLA Witten and Sander, PRL 47, 1400, (1981) Particles are released one by one from far field. They move randomly They attach to the central cluster (starts with a seed)

  8. Diffusion Limited Aggregation DLA Witten and Sander, PRL 47, 1400, (1981) Anology between DLA and Viscous Fingering in Porous media Paterson, PRL 52, 1621, (1984) V refers here to a probability of growth, or an average over a number of steps. The BC (Boundary condition) corresponds to P=Cte

  9. Fingering in (random) porous media: Capillary fingering: low Ca (slow drainage) Lenormand and Zarconne, Phys. Rev. Lett 54, 2226, (1985). • Structure controlled by capillary threshold fluctuations • Structured well described by Invasion Percolation model. Wilkinson and Willemsen J. Phys. A 16, 3365, (1983), Viscous fingering: high Ca (fast drainage) • Structure controlled by viscous pressure • Field.

  10. Drainage Defender Invader a Flow rate: R Capillary pressure

  11. Necessary condition for invasion: Defender is capillary threshold pressure Capillary pressure threshold gives cutoff of invasion probability at a single pore. At low Ca: this rules the process:

  12. Invasion percolation simulations • Capillary thresholds distributed at random • Lowest threshold along the front is invaded at each step. • Incompressibility condition and trapping: the mobile front has to be connected by fluid to the outlet. • Compressible case: • film flow, for example

  13. Invasion percolation simulations • Red curve: upwards curvature.

  14. Hele Shaw experiment Experimental setup by Saffman and Taylor. Proc. Roy. Soc. London (1983). Analog model to viscous fingering in porous media ?? Randomness is important for porous media!!

  15. Viscous Fingering in Porous media • Movement in ”a few” single pores • Branched finger structures • Quenched disorder • Capillary threshold set by pore scale. Viscous Fingering in Hele Shaw cell • Movement in layers • Smooth finger • No disorder • Capillary pressure set by interface curvature From Saffman and Taylor Proc. R. Soc. London Ser. A 245, 312, (1958)

  16. Diffusion Limited Aggregation DLA Witten and Sander, PRL 47, 1400, (1981) Anology between DLA and Viscous Fingering in Porous media Paterson, PRL 52, 1621, (1984) • In DLA the growth occurs with one particle at a time. • In Viscous fingering several pores may grow at the same time !!!

  17. Viscous fingering DLA r Måløy,Feder and Jøssang PRL (1985) Henrichsen, Måløy, Feder, Jøsang J. Phys. A 22, L271, (1989) P. Meakin, Phys. Rev. A. 27, 1495 (1983)

  18. Experimental setup Contact paper inlet Mylar film L Fluid: 90% by weight Glycerol - water solution Viscosity: 0.165 Pa s Interface tension: 0.064 N/m Viscosity ratio M=0.0001 W outlet

  19. Capillary number Ca=0.027, Width W=430mm, Total time 89 min.

  20. Capillary number Ca=0.054, Width W=215mm, Total time 20min.

  21. Capillary number Ca=0.120, Width W=215mm, Total time 8 min.

  22. Position of tip Inlet Q

  23. Invasion growth probability density is average number of new filled pores within between each image. and within a delay time Is a normalization constant. Where Gives probability for growth within

  24. MeasuredInvasion probability density Wide model W=430mm Narrow model W=215mm Løvoll, Meheust, Toussaint, Schmittbuhl and Måløy, Phys. Rev. E 70, 026301, (2004) is independent of Invasion probability density Capillary number.

  25. Relation between the mass of the frozen structure and the growth probability density is the average number of filled pores within is the number of invaded pores per time unit Total number of invaded pores in an analysis strip of width at a distance z from the finger tip:

  26. We assume a constant speed of the tip (good approximation): Which gives: • The mass density is proportional to the cumulative growth probability, and the ratio of the injection rate and the tip velocity

  27. Measured mass density and cumulative growth probability

  28. Width of Saffman Taylor finger Saffman and Taylor. Proc. Roy. Soc. London (1958). Ca Saffman and Taylor further calculated the profile of the Finger where the finger width is a free parameter Finger width w/2 is now understood. The selection is due to surface tension:

  29. For DLA it is found that the computed ensemble averaged DLA structures has a smooth profile which coincide with the profile of the Saffman Taylor solution with Arneodo, Couder, Grasseau, Hakim and Rabaud, PRL 63, 984, (1989) Arneodo, Elezgaray, Tabard and Tallet, Phys. Rev. E. 53, 6200, (1996)

  30. Will viscous fingering in 2D models give an average • width consistent with a Saffman Taylor solution? • Is the value of consistent with what is observed • for DLA? We use the same procedure as used by Arneodo et al for DLA Define an average occupancy map: For each time (image) Assign a value 1 to the coordinate (x,y) if air is present and 0 otherwise. Is obtained as the time average of this occupancy function.

  31. Ca=0.22, W/a=110 Ca=0.22, W/a=210 Ca=0.06, W/a=210 Superimposed gray map shows occupancy probability of the invader.

  32. Ocupation density n(z): is normalized version of is a function of z/W and independent of Ca We have used same methods to find the width as used by Arneodo et al: where

  33. Comparison with Saffman Taylor solution: x Clipped for Compared with S. T. solution for . and y

  34. Box counting and Fractal dimension

  35. Dependence of crossover length scale on capillary number Crossover scale is width of capillary threshold distribution. Wt characteristic pressure gradient in growth zone ( Imposed pressure gradient ) (in our case ) Above this scale: differences in pressure value above threshold fluctuations mostly due to viscous pressure drop Below it: Mostly due to fluctuations

  36. Pressure measurements The dependence of the pressure on z/W. Same decay of the pressure difference indicates that the details of the ”fingers” don’t have a strong influence on the pressure field on large scales.

  37. a Capillary and viscous pressure drop between two pores. Requirement for invasion of new pore Where is the capillary pressure threshold Characteristic interface velocity: Step function

  38. Assume a flat Capillary pressure threshold distribution Step function 1 0

  39. Assume that only the average growth rate controls the growth process 1) Moderate flow rates: 2) High flow rates

  40. Our experiments correspond to situation 1) : For DLA : Our situation is closer to a Dielectric Breakdown Model with . .For DBM DBM model Niemeier, Pietronero and Wiesmann, PRL 52, 1033, (1984) Probability for growth of interface site:

  41. Can use the equations 1) 2) 3) and the measured growth probabilityto find a pressure boundary boundary condition that take into acount the change in the capillary pressure.

  42. Conclusion : • The growth probability is a function of z/W and is independent of the capillary number. • Mass of the frozen structure is given by the growth probability : • We compared the width of the ”envelope” structure with DLA and the Saffman Taylor problem: • The cross-over length scale between the viscous and capillary finger patterns inside the ”envelope” structure scales as • Theoretical arguments for the pressure gradient at the cluster surface predicts that the DBM with may better describe the viscous fingering process than the DLA model. • We introduce a new boundary condition for the pressure at the interface of the cluster that uses the measured growth probability which takes into acount the capillary threshold fluctuations. Toussaint, Løvoll, Meheust, Måløy and Schmittbuhl, Europhysics letter 2005. Løvoll, Meheust, Toussaint, Schmittbuhl and Måløy, Phys. Rev. E. 70,026301,2004 Work suported by NFR and CNRS trough a PICS grant , and NFR trough a Petromax and a SUP grant.

  43. Consequences for Hydrology – Flow in soils and porous media • Darcy equation: monophasic flow • Generalized Darcy equation, or Richards equation: diphasic flow

  44. Saturation-pressure models used: • Water retention tests – to determine capillary pressure – saturation relations • Most used models to describe this: Brooks-Corey and Van Genuchten model Tested soil: imposed water pressure, drainage Semi permeable BC

  45. Saturation-pressure models used: • Water retention tests – Brooks-Corey and Van Genuchten model

  46. Pressure fluctuations at low injection rates: Here v is the fluid volume in a typical pore troat, nf the number of interface Throats. We will assume K=constant as zero order approximation Måløy, Furuberg and Feder, PRL 68, 2161 (1992)

  47. Experimental setup: drainage experiments, controlled speed. Contact paper inlet Mylar film L Invading fluid: air. Defending Fluid: Wetting 90% by weight Glycerol - water solution Viscosity: 0.165 Pa s Interface tension: 0.064 N/m Viscosity ratio M=0.0001 W outlet

  48. Slow experiments – drainage – imbibition cycle Pressure imposed at the outlet

  49. Corresponding retention curve. Fit of the drainage part using a Van Genuchten and a Brooks Corey model: Similar behavior to classical 3D hydrological tests

  50. Early stages of the invasion depend on detailed shape of an empty buffer between two plates, after what a step wise invasion process similar to a 3D experiment happens

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