How disorder can influence dynamics

• Example

• Diffusion

• Edwards Wilkinson in a Brownian noise

• Edwards Wilkinson in a Quenched Noise

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How disorder can influence dynamics

• Example

• DiffusionFlat front• Edwards Wilkinson in a Brownian noiseSelf affine, H=0.5• Edwards Wilkinson in a Quenched NoiseSelf affine, H=1.1

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How disorder can influence dynamics

• Edwards Wilkinson in a Brownian noise

Anomalous subdiffusive scaling ( )

• Edwards Wilkinson in a Quenched Noise

Anomalous scaling, different exponent

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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

Experimental setup by Saffman and Taylor. Proc. Roy. Soc. London (1983).

Viscous Fingering instability: lowly viscous fluid displaces another one.

Paradigm:

Hele Shaw cell

Viscous Fingering in Hele Shaw cell

•Movement in layers

•Smooth finger

•No disorder

•Capillary pressure set by interface curvatureFrom Saffman and Taylor

Proc. R. Soc. LondonSer. A 245, 312, (1958)

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)

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

Capillary fingering: low Ca (slow drainage)

•Structure controlled by capillary threshold fluctuations•Structured well described by Invasion Percolation model. Wilkinson and Willemsen J. Phys. A 16, 3365, (1983),

Lenormand and Zarconne, Phys. Rev. Lett54, 2226, (1985).

Viscous fingering: high Ca(fast drainage)

•Structure controlled by viscous pressure Field.

Fingering in (random) porous media:

a

Defender Drainage

Invader

R

Flow rate:

Capillary pressure

Defender

Necessary condition for invasion:

Capillary pressure threshold gives cutoff of invasion probability at a single pore.

is capillary threshold pressure

At low Ca: this rules the process:

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

Invasion percolation simulations

• Red curve: upwards curvature.

Experimental setup by Saffman and Taylor. Proc. Roy. Soc. London (1983).

Hele Shaw experiment

Analog model to viscous fingering in porous media ??

Randomness is important for porous media!!

•Movement in ”a few” single pores

•Branched finger structures

•Quenched disorder

•Capillary threshold set by pore scale.

Viscous Fingering in Porous media

Viscous Fingering in Hele Shaw cell

•Movement in layers

•Smooth finger

•No disorder

•Capillary pressure set by interface curvatureFrom Saffman and Taylor

Proc. R. Soc. LondonSer. A 245, 312, (1958)

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 !!!

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)

Experimental setup

L

W

Contact paper

Mylar film

Fluid: 90% by weight Glycerol - water solutionViscosity: 0.165 Pa sInterface tension: 0.064 N/mViscosity ratio M=0.0001

inlet

outlet

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

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

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

Position of tip

Q

Inlet

Invasion growth probability density

is average number of new filled pores within

Where Is a normalization constant.

Gives probability for growth within

and within a delay time between each image.

Measured Invasion probability density

Wide model W=430mm Narrow model W=215mm

Løvoll, Meheust, Toussaint, Schmittbuhl and Måløy, Phys. Rev. E 70, 026301, (2004)

Invasion probability density is independent of

Capillary number.

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:

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

Measured mass density and cumulative growth probability

Width of Saffman Taylor finger

Saffman and Taylor. Proc. Roy. Soc. London (1958).

Finger width w/2 is now understood. The selection is due to surface tension:

Saffman and Taylor further calculated the profile of the Finger where the finger width is a free parameter

Ca

For DLA it is found that the computed ensemble averagedDLA 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)

•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.

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.

Ocupation density n(z):

is normalized version of

We have used same methods to find the width as used by Arneodo et al:

where

is a function of z/W and independent of Ca

x

y

Comparison with Saffman Taylor solution:

Clipped for

Compared with S. T. solution for . and

Box counting and Fractal dimension

is width of capillary threshold distribution. characteristic pressure gradient in growth zone

Crossover scale

Dependence of crossover length scale on capillary number

Wt

( 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

Pressure measurements

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.

The dependence of the pressure on z/W.

Where is the capillary pressure threshold

Characteristic interface velocity:

Requirement for invasion of new pore

Capillary and viscous pressure drop between two pores.

a

Step function

Assume a flat Capillary pressure threshold distribution

Step function

1

0

Assume that only the average growth rate controls the growth process

1) Moderate flow rates:

2) High flow rates

Our experiments correspond to situation 1) :

For DLA :

Our situation is closer to a Dielectric Breakdown Model with . .For DBM

Probability for growth of interface site:

DBM model Niemeier, Pietronero and Wiesmann, PRL 52, 1033, (1984)

Can use the equations

and the measured growth probability to find a pressure boundary boundary condition that take into acount the change in the capillary pressure.

1)

2)

3)

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.

Work suported by NFR and CNRS trough a PICS grant , and NFR trough a Petromax and a SUP grant.

Toussaint, Løvoll, Meheust, Måløy and Schmittbuhl, Europhysics letter 2005.Løvoll, Meheust, Toussaint, Schmittbuhl and Måløy, Phys. Rev. E. 70,026301,2004

Consequences for Hydrology – Flow in soils and porous media

• Darcy equation: monophasic flow

• Generalized Darcy equation, or Richards equation: diphasic flow

P1 ¡ P2 = f (S)

µ= f (h)

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

Saturation-pressure models used:

• Water retention tests – Brooks-Corey and Van Genuchten model

Pressure fluctuations at low injection rates:

Måløy, Furuberg and Feder, PRL 68, 2161 (1992)

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

Experimental setup: drainage experiments, controlled speed.

L

W

Contact paper

Mylar film

Invading fluid: air.Defending Fluid: Wetting 90% by weight Glycerol - water solutionViscosity: 0.165 Pa sInterface tension: 0.064 N/mViscosity ratio M=0.0001

inlet

outlet

Slow experiments – drainage – imbibition cycle

Pressure imposed at the outlet

Corresponding retention curve.

Fit of the drainage part using a Van Genuchten and a Brooks Corey model:

Similar behavior to classical 3D hydrological tests

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

Faster drainage experiments at imposed flux:

Retention curves depend on the flux, as well as the residucal saturation: dynamic effect.

Pressure drop across the cell:

• “dynamic capillary” pressure drop across the cell corresponds to capillary pressure, plus viscous pressure drop in the viscous fluid:

Constant capillary pressure:

Mean field viscous pressure:

Check: data collapse:

• Possible to relate and using the geometry of the invader:

• This leads to a general relationship for the dynamic pressure drop as function of the saturation.

• It also produces a collapse for the residual saturation as function of the saturation

• Key: analyzing the geometry of the invading structure: fractal dimensions, cutoff dimensions

•Fingering is observed.•What is the structure most often occupied in the cell? •Is it consistent with a Saffman Taylor solution?

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.

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.

Most occupied region: width

Box counting and Fractal dimension

Consequences of the scaling

• This leads to:

• And a relation between and the saturation:

• So we expect a collapse of the retention curve:

Which also gives a collapse for the residual saturation at breakthrough, with above

1-

Final collapse of the dynamic pressure measured as function of the wetting saturation:

• collapse of the retention curve:

Which also gives a collapse for the residual saturation at breakthrough, with above

1-

Conclusions:

• At finite speed, dynamic effects are observed on the retention (pressure-saturation) curves: the relationship between the pressure drop from one phase to the other is function of the flow speed

• These effects can be attributed to viscous effects added to the capillary pressure

• It is possible to account for them with simple arguments and obtain a general relationship, in simple geometries and controlled invasion speed

Lovoll et al., TIPM 2011