Influence of pore-scale disorder

on viscous fingering during drainage.

Renaud Toussaint, Jean Schmittbuhl Institute of Globe Physics Strasbourg (IPGS/ EOST), University Louis Pasteur

Grunde Løvoll, Knut Jørgen Måløy Complex, Physics Departement, University of Oslo,

Yves Méheust, Complex, Physics Departement, NTNU, University of Trondheim,

Alain Steyer, Institute of Globe Physics Strasbourg (IPGS/ EOST), University Louis Pasteur,

Gerhard Schäfer, Institute of Mechanics of FLuids and Solids, Strasbourg, University Louis Pasteur.

Contact: http://eost.u-strasbg.fr/recherche/renaud

Experimental setup:

Horizontal Hele-Shaw cell filled with monolayer of glass beads, saturated with water glycerol.

References: 1. Influence of pore scale disorder on viscous fingering during drainage, Toussaint, Løvoll, Méheust, Måløy and Schmittbuhl, Europhysics Letters 2005, in press, http://eost.u-strasbg.fr/recherche/renaud

2. Growth activity during fingering in a porous Hele Shaw cell, Løvoll, Méheust, Toussaint, Schmittbuhl, Måløy, Phys Rev E, 70, art 026301, 2004 .

Drainage and viscous fingering: Air injection at various speeds.

Intermediate regime between capillary and viscous fingering:

3 Sizes L x W x a:

L=815 mm

W=110, 215,430 mm

a=1 mm

,/2 avCa f

5.001.0 Ca

fv : viscosity : average Filtration velocity : interfacial tension : cell permeabilitywith a : cell thickness

Ca 0.22Ca 0.06Ca 0.03

Optical top view:

Extracting the invasion

cluster: thresholding.

Graymap: simulated

pressure field (Laplace)

Two lengths scales:

Maximum size of isolated viscous clusters,

Zone more occupied than the average i nthe

center of the cell

fw

W

Tip speed

(in z-direction) roughly constant

Extracting information about statistically stationary process, in reference frame attached to propagating tip

and lateral cell walls:

invasion rate, number of cells invaded per unit length and unit time, in strips at z behind the tip,

normalized:

)/)/(( 2tipfCa vvaWn

)(zn

)(z )1)(( dzz

independence of Ca:

number of cells invaded per unit length behind the tip, Spatial -integral of , normalized by ,

with reduced spatial coordinates z/W

)(z

Can

independence of Ca, and of system size:

Longitudinal projection (z/W),

Lateral projection, invaded density function of (x/W):

number of cells invaded per unit lateral width: same collapse.

Determination of zone more occupied than the average in the longitudinal direction,

with .Note: for Saffman Taylor in empty HS cell,

before tip splitting,

W 4.0

5.0

)(x

Extracting the Saffman-Taylor glove around the fingers: in the co-moving reference frame, attached to

propagating tip and lateral cell walls:

an invasion probability is determined: for each snapshot, at (x/W,y/W), 1 counts presence of invader, 0 counts invaded fluid. Average over all times and exps:

Experimental estimate of occupancy probability

, also studied by Arneodo et al in DLA)/,/( WzWx

Arneodo et al, PRE 1996, Phys Rev Lett. 1989, Somfai and Ball, Phys Rev E 2003:

Note: in off lattice Diffusion Limited Aggreagtion , 62.0

Thresholding of the occupancy probability at half maximum , Comparison with Saffman

Taylor mathematical solution for :4.02/max

Other characteristic: Fractal dimension by box-couting.

Two scaling regimes, D=1.83 (capillary fingering) below ,

D=1.53 (viscous fingering, <1.71 DLA) above .

Scaling law for transition scale:

fw

)( fws

Caaw f /~

Snapshot: is the maximum size of isolated pockets of invaded fluid left behind.

fw

Experimental determination of : after removal of trapped islands, average length of lateral slices across the structure

fw

Scaling of , together with small and large scale saturations, rule the final proportion of emptied pores - efficiency of the drainage.

)(Caw f

1. Theoretically, transition scale as can be explained by assuming a roughly flat distribution of capillary thresholds , of width :

For a given shape of invasion cluster, scales s such as are dominated by the weakest threshold, and behave as capillary fingering (invasion percolation): D=1.83

For larger scales s such as , pores are invaded simultaneously, at a rate dependent of the local pressure gradient in the pore throat: viscous fingering scale.

Caaw f /~

tW

tboundary WPs .

tboundary WPs .

2. At large scale, a/ fractal dimension (1.53) lower than DLA (1.71), and b/ width fraction covered lower than DLA ( ) or Saffman Taylor in empty Hele-Shaw cell ( ).

Key for these features: relationship between average invasion growth rate and pore scale pressure drop across interface, relates this problem to DBM, , which have such large scale characteristics.

Perspectives:

Influence of pore scale capillary threshold distribution on Representative Volume Element continuous laws describgin the influence of Capillary threshold: beyond phenomenological relative premeability relations (Van Genuchten) ?

1. Numerical hybrid models, viscous flow + quenched capillary thresholds for the internal Boundray Condition.

2. Experimental, laboratory (dm-m scale): Artificial 2D porous medium, few types of pore throat size distribution probed.

3. Large scale: Influence on 3D drainage, measure of capillary threshold distribution (porosimetry). Tests on large scale drainage, pollutant transport, on SCERES (dam scale), IFARE, IMFStrasbourg.

62.04.0 5.04.0

HeavisidetP

Pressure in first pore along the interface: in air, in water glycerol. 0P 1P

Washburn equation, speed across pore throat, average over quenched disorder:

At low speeds, : 2)4/( PWav t

At large speeds, :

Dielectric Breakdown Model (DBM), ,

2 Pv

Pv / Diffusion Limited Aggregation (DLA), DBM with (Paterson 84)

1

Incompressibility plus Darcy law: Laplace equation, 0P

:influence of disorder in capillary thresholds on boundary condition.

2