Dispersion and Stretching in a 3D Porous media

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HAL Id: insu-02536373

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Submitted on 8 Apr 2020

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Dispersion and Stretching in a 3D Porous media

Mathieu Souzy, Tanguy Le Borgne, Yves Méheust, H. Lhuissier, B. Metzger

To cite this version:

Mathieu Souzy, Tanguy Le Borgne, Yves Méheust, H. Lhuissier, B. Metzger. Dispersion and Stretching in a 3D Porous media. Mixing in Porous Media Conference, Feb 2020, Leiden, Netherlands. 2020. �insu-02536373�

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Dispersion & stretching in 3D porous media

Experimental set-up

Mathieu Souzy

2

_{, T. Leborgne}

2

_{, Y. Meheust}

2

_{, H. Lhuissier}

1

_{& B. Metzger}

1

_{Aix Marseille Universite, CNRS, IUSTI UMR 7343, 13453 Marseille, France}

2

_{Geosciences Rennes, UMR 6118 Universite de Rennes 1, CNRS 35042 Rennes, France}

3D Velocity field

Stretching laws

Dispersion

Mixing of a blob of dye in a porous media

Laser Optical lens Rotating mirror Camera High-pass filter Water bath 12d 12d 100d _{50 slices} x z y Porous media Flexible Container 50 slices d=2 mm x y 5d 5d 9d d=228pixels x y z 0 1 2 3 4 U/ U P ( u U ) 10-4 10-2 1 0 2 4 6 u U P ( v U ) P ( w U ) 0 2 4 -2 -4 10-4 10-2 1 , y/d z/d 0 1 0 1 ₀ 1 2 3 4 5 8 x/d 0 3 1 5 t/τ = 1 0 0 1 y/d z/d 5 0 10 15 20 σ2ln ρ ln ρ 4 0 8 12 P (ln ρ) ln ρ 10 5 15 20 0 0.1 0 0.2 = 0 5 10 15 20 0.47

t/τ

t/τ

0 ,

4.5 ,

9 t/τ =

A blob of dye advected in a 3D porous medium disperses in the surrounding medium while deforming into a set of elongated lamellae.

We measure the 3D velocity field to characterize the dynamics of dispersion and stretching, thus investigating :

σx

σ

y

l(t)

s (t)

What is the distribution of stretching ( ) in a 3D porous medium ?

How is the velocity field distribution related to dispersion ?

ρ =

l(t)

l

0

τ = d

U

Index matching

Random bead pack of solid PMMA spheres

PTV

Uv

w

U

Available online: www.digitalrocksportal.org/

Evolution of a material line advected numerically in the 3D experimental velocity field

ln ρ = λt/τ σ_{ln ρ}2 = µt/τ λ≈ µ ≈ 0.47 ρ = e(λ+µ/2)tτ

Exponential stretching:

Log-normal distribution:

P (ρ)= 1 ρ√2πσln ρ e− (ln ρ ln ρ )2 2σ2 ln ρ dt d/10

Continuous Time Random Walk for a flat velocity distribution leads to anomalous dispersion

Origin of a minimum cut-off velocity :

σx2

d2 ∼

log(t/τ )t

τ

Tracer Finite size:

Umin

U ≈

10dt

d

Tracer Finite diffusion:

Umin U ≈ ( 102 P e) 1/3 100 102 10-2 10-4 1 10 100 0.1 0.01

t/τ

1000 σx2/d2 σ2 y/d2 Experiments CTRW Experiments CTRW with anti-correlation 0.01 0.1 Umin/ U Fickian tF/τ 100 10 Umin U 0 ∞ 1 U U P (U U) 10-2 1 10-2 ₁ 10-4 Umin U Ballistic Transient σ_x2(t = ln U Umin vx 2 U 2 t τ σ2y(t = ln U Umin (1− p) vy 2 U 2 d2 t τ

Fickian behavior beyond:

t

F

= d/U

min

d2 ) )