1. 1 Multiphase and fingered flow in a Hele-Shaw cell Giulia Spina Relatrice: Prof.ssa Marina Serio

2. 2 Purpose of the experiment To study in detail the finger phenomena and the behavior of the finger at different flow rates. Fields of interest CONTAMINANTS 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

3. 3 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

4. 4 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)‏

5. 5

6. 6 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

7. 7 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.

8. 8 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

9. 9 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.

10. 10 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 media The closer n is to 1, the bigger is With 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

11. 11 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).

12. 12 Results: water content using 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

13. 13 Results: pressure data using 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

14. 14 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.

15. 15 Hysteresis and finger stability Glass 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.

16. 16 Horizontal saturation In 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.

17. 17 Solute transport After 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

18. 18 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.

19. 19 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:

20. 20 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.

21. 21 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).

22. 22 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

23. 23 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