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Multiphase and fingered flow in a Hele-Shaw cell

Giulia Spina

Relatrice: Prof.ssa Marina Serio

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Purpose of the experimentTo study in detail the finger phenomena and the behavior of the finger at

different flow rates.

Fields of interestCONTAMINANTS• Unstable wetting front leads to much faster percolation of pollutants• The MIM model explains the persistence of contaminants in ground

OIL ENGINEERING• In order to avoid infiltration water-oil, gas-oil

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Porous media

Porous soils are characterized by a strong irregular solid matrix, whose constituents got dimensions that vary in a range of many orders of magnitude.

The complement of the solid matrix can be occupied by a liquid or a gas, or both.

volumetric phase densities

porosity a=air, w=water

0/VVw

0/VVa

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Hele-Shaw cell

Hele-Shaw cell enables to 2D analysis of infiltration phenomena.

water light reference

Fine sand

homogeneous sand

Heterogeneous sand 160x60x0.3cm

LIGHT SOURCE

Air outflow

Non ponding water, two different flow rates (4.8 ml/min, 1.2 ml/min)

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Water potential

In rigid, unsaturated soil

Where:

is the gravitational potential

is the matric potential, which accounts for surface effects. This is the Young-Laplace equation

• surface tension• principal radii of curvature of the ellipsoid that represents the interface• p pressure • water density

r>0 if it lies within the water phase

Often the equivalent height of a water column is used instead of the potential.

1 cmWC 1hPa=1mBar

mgw

)( 0zzgwg

)/1/1( 21 rrpp waawm

wa

21, rr

w

gh www /

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meniscus

Example: in a capillary ( )

The rise (or fall) in the capillary follows from surface forces.

Distribution of different phases (water: grey, a air: black) within a 2D section through a

porous media at different matric potentials.

According to the Young-Laplace equation,

one can note different radii of curvature of the

interfaces.

rzzg waw /2)( 0

0zrrr 21

z

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Fluid-dynamics laws

• Conservation of mass• volumetric water content • water flux

Actually a conservation of volume since water may be considered as incompressible• Darcy-Buckingham law

The water flux is proportional to the pressure gradient. The proportionality constant is the conductivity .

This linear relation was found by Darcy in 1856; later improved by Buckingham by substituting with

In a porous media p is substituted by , the water potential• Richards equation

Obtained by inserting the flux law in the conservation equation

0 wt j

pKjw )(

wK )(K

0))(( wt K

K

wj

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Matric potential and hydraulic conductivity

Hydraulic functions for various soil textures in the Mualem-van Genuchten (thick lines) and in the Burdine-Brooks-Corey parametrization

• Non linear relations due to pore structure• Hysteretic behaviour of the potential

The between the two lines is the same;

but a further small increase leads to saturation

of the pore.

p

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Light Transmission Method• After Hoa (Water Resources Research, vol17, Feb 1981)In the hypothesis of normal incidence the factor of light transmission through a

diopter is given by the Fresnel’s law:

n=n1/n2; n1,n2 are the refractive indices of the two mediaThe closer n is to 1, the bigger isWith

the transmission of light is favored by the presence of water.

As the sand refractive index always changes, this method leads to qualitative results. In order to obtain the absolute water content, a calibration was performed, by using X-Ray.

This is a very appropriate method in order to achieve a great spatial and temporal resolution

2)1/(4/ nnII inout

1,33.1,6.1 airwatersand nnn

991.0,98.0,946.0 ,,, watersandairwaterairsand

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Pressure measurements• 12 tensiometers were employed to measure the water potential at a fixed point in the

cell.

• the presence of a air bubble in the tensiometer makes it less useful, because air can grow and plug the porous membrane.

• every tensiometer has to be calibrated.• Finger exactly reaching the sensor is a lucky case; the chance grows by increasing

the number of sensors, but this was not possible with our device (max tens. number was 12).

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Results: water contentusing LTM

A C++ program calculates the mean light intensity in an area of about 15x15 pixels, corresponding to the central part of the finger.

Flow rate 4.8 ml/min Flow rate 1.2 ml/min

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Results: pressure datausing Tensiometers

Results are more variegated, because the finger can reach the sensor in many different ways, and what we obtain is always a mean. Here are showed only the best results.

Flow rate 4.8 ml/min Flow rate 1.2 ml/min

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Finger occurrence

After Saffman & Taylor (1958) for two fluids of different density and viscosity, driven by a pressure gradient, one observes unstable displacement if:

permeability porosity

density of replacing (1) and replaced (2) fluid viscosity

interfacial velocity

Because both the viscosity and the density of water are much greater than that of air, the equation simplifies to:

Where is the saturated hydraulic conductivity. The first term is simply the flux through the system.

So we obtain:

This condition is achieved in our experiment by superimposing fine sand, with lower saturated hydraulic conductivity, to the coarse sand.

0)()( 2121 Ukg

k

U

sKkgU 11 / sK

sw Kj

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Hysteresis and finger stabilityGlass proposed in 1989 an explanation of the finger phenomena. Hysteresis in the

water retention curve has a great effect.

drainage curve fringe core

wetting curve

finger tip; high saturation

Glass recognizes a finger core and a finger fringe. The water content in the two parts is different, but the water potential is the same.

This prevents horizontal water flow and widening of the finger.

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Horizontal saturationIn order to check the explanation of Glass, we perform a horizontal analysis with the LTM.

As supposed, the water content is higher in the center of the finger. One can notice also the presence of the peak due to finger tip.

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Solute transportAfter reaching a stable structure, we let a colored liquid, blue dye, flow in the cell.

One cannot see clearly the dye tip, because it mixes with water during transport from pump to the top of the cell (this is an effect of convection and dispersion).

2,5 min

40 min

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Convection-dispersion model (mention)

• breakthrough of solute step• In a heterogeneous velocity field we expect that the variance of transport distances

increases with time.• Each particle can change its velocity by moving from one streamline to another

through molecular diffusion.• According to CLT, the distribution of travel distances approaches a Gaussian

distribution.

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Convection-dispersion model (mention)

• One may consider two terms:

convective solute flux

dispersive component; described in analogy to a diffusion process; z is the axis parallel to water flux.

is the solute concentration in the water phase, is the effective diffusion

coefficient.• The total solute flux law, according to conservation of mass, leads:

swconvs cjj

zcDj seffdisps /

wsc , effD

0/// zcDcjzct seffsws

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mobile-immobile model (mention)

• In many experiments the shape of the distribution is not Gaussian.• This phenomena can be explained by separating the available into two

components, , which is mobile, and , which is not.

The total concentration is decomposed into the two parts of the water phase as:

• As only the mobile part is flowing, the convective dispersive solute flux states:

• Inserting these two equations into the mass balance equation, and assuming the fluid is well mixed, i.e. , leads to:

Where is the retardation factor. The solute is retarded with respect to a hypothetical substance that is present in the mobile phase only.

In the experiment one can clearly see and , but the well mixed condition can be achieved only on very longer time scales.

m im

imsimmsms ccc ,,

zcDcjj mseffmsws /,,

imm cc

0// zjtcR sm

mimR /1

m im

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Future developments• Numerical simulation: starting from the Richards equation one may add terms to copy

the behavior of fingers. For example Eliassi & Glass (Water Resources Research, vol 38, 2002) proposed a hold-back-pile-up term, related to a Nhd, hypodiffusion number, that accounts for the material non-linearity

pile-up

hold-back

• Microscopic resolution: in the university of Heidelberg a microscope will be mounted to see the meniscus modifications in the different phases of the finger (tip, drainage, stable phase).

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Acknowledgments • Prof. Marina Serio, University of Turin• Prof. Kurt Roth, Institute of environmental physics, University of Heidelberg• D Sc Fereidoun Rezanezhad, Institute of environmental physics, University of

Heidelberg• D Sc Marco Maccarini, Institute of physics and chemistry, University of Heidelberg

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references

• R.J.Glass “Mechanism of finger persistence in homogeneous, unsaturated, porous media: theory and verification” Soil Science, vol 148, July 1989

• N.T.Hoa “A new method allowing the measurements of rapid variations of the water content in sandy porous media” Water Resources Research, vol17, Feb 1981

• M. Eliassi & R. J. Glass “On the porous continuum modeling of gravity driven fingers in unsaturated materials: extension of standard theory with a hold-back-pile-up effect” Water Resources Research, vol 38, 2002