Experimental Study of Fingering Flow in Porous Hele-Shaw Cells

2009

We present in this paper an experimental study of the invasion activity during unstable drainage in a 2D random porous medium, when the (wetting) displaced fluid has a high viscosity with respect to that of the (non-wetting) displacing fluid, and for a range of almost two decades in capillary numbers corresponding to the transition between capillary and viscous fingering. We show that the invasion process takes place in an active zone within a characteristic screening length λ from the tip of the most advanced finger. The invasion probability density is found to only depend on the distance z to the latter tip, and to be independent of the value for the capillary number Ca. The mass density along the flow direction is related analytically to the invasion probability density, and the scaling with respect to the capillary number is consistent with a power law. Other quantities characteristic of the displacement process, such as the speed of the most advanced finger tip or the characteristic finger width, are also consistent with power laws of the capillary number. The link between the growth probability and the pressure field is studied analytically and an expression for the pressure in the defending fluid along the cluster is derived. The measured pressure are then compared with the corresponding simulated pressure field using this expression for the boundary condition on the cluster.

2006

Water infiltration into coarse textured dry porous media becomes instable depending on flow conditions characterized through dimensionless quantities, i.e. the Bond number and the Capillary number. Instable infiltration fronts break into flow fingers which we investigate experimentally using Hele-Shaw cells. We further developed a light transmission method to measure the dynamics of water within flow fingers in great detail with high spatial and temporal resolution. The method was calibrated using x-ray absorption and the measured light transmission was corrected for scattering effects through deconvolution with a point spread function. Additionally we applied a dye tracer to visualize the velocity field within flow fingers. We analyzed the dynamics of water within the finger tips, along the finger core behind the tip, and within the fringe of the fingers during radial growth. Our results confirm previous findings of saturation overshoot in the finger tips and revealed a saturation minimum behind the tip as a new feature. The finger development was characterized by a gradual increase in water content within the core of the finger behind this minimum and a gradual widening of the fingers to a quasi-stable state which evolves on time scales that are orders of magnitudes longer than those of fingers' evolution. In this state, a sharp separation into a core with fast convective flow and a fringe with exceedingly slow flow was detected. All observed phenomena could by consistently explained based on the hysteretic behavior of the soil-water characteristic and on the positive pressure induced at the finger tip by the high flow velocity.

Geoderma, 1996

The mechanism for the growth and persistence of gravity-driven fingered flow of water in initially dry porous media is described. A Galerkin finite element solution of the two-dimensional Richards equation with the associated parameter equations for capillary hysteresis in the water retention function is presented. A scheme for upstream weighting of internodal unsaturated hydraulic conductivities is applied to limit smearing of steep wetting fronts. The growth and persistence of a single finger in an initially dry porous media is simulated using this numerical solution scheme. To adequately simulate fingered flow, it was found that the upstream weighting factor had to be negative, meaning that the internodal unsaturated hydraulic conductivities were weighted more by the downstream node. It is shown that the growth and persistence of a finger is sensitive to the character of the porous media water retention functions. For porous media where the water-entry capillary pressure on the main wetting function is less than the air-entry capillary pressure on the main drainage function, a small perturbation will grow into a finger, and during sequential drainage and wetting the finger will persist. In contrast, for porous media where the water-entry capillary pressure on the main wetting function is greater than the air-entry capillary pressure on the main drainage function, the same small perturbation will dissipate by capillary diffusion. The finger widths derived from the numerical simulation are similar to those predicted by analytical theory. 0016-7061/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved SSDI 0016-7061 (95)00086-0

2] To explain the dynamic behavior of the matric potential at the wetting front of gravity driven fingers, we take into account the pressure across the interface that is not continuous and depends on the radius of the meniscus, which is a function of pore size and the dynamic contact angle h d . h d depends on a number of factors including velocity of the water and can be found by the Hoffman-Jiang equation that was modified for gravity effects. By assuming that water at the wetting front imbibes one pore at a time, realistic velocities are obtained that can explain the capillary pressures observed in unstable flow experiments in wettable and water repellent sands.

A series of experiments are performed in a Hele-Shaw cell, consisting of two parallel closely spaced glass plates. A liquid (oil or water, both of viscosity of 1.0 cP) is injected a t a constant volumetric flow rate, q, to radially displace a much more viscous liquid (glycerine, 1050 cP) in the cell. Oil is immiscible with and water is miscible with glycerine. The data presented in this paper are taken mostly a t late stages of the fingering process, when the pattern has multiple generations of splitting. Correlations with time are obtained for the finger length and the overall pattern density. The time-and lengthscales have been found for the immiscible case. At the same dimensionless time, immiscible patterns are similar and have the same generation of splitting. The overall density of each pattern decreases with time. The pattern shows fractal behaviour only after a certain number of generations of splitting. The fractal dimension of the immiscible pattern decreases from 1.9 to 1.82 when the pattern goes from the third to the sixth generation of splitting. The fractal dimension of the miscible pattern reaches a constant value after about ten generations of splitting and the fractal dimension ranges from 1.50 to 1.69 for q/Db = 4.8 x lo5-7.0 x lo6. The miscible patterns are insensitive to dispersion for large q/Db. For immiscible fingers h / b scales with for capillary number Ca ranging from about 8 x to 0.05. For miscible fingers, h / b is insensitive to dispersion and ranges from 5 to 10 for large q/Db. Here D is the molecular diffusion coefficient in glycerine, b the cell gap width and h the splitting wavelength. WITTEN, T. A. & SANDER, L. M. 1983 Diffusion-limited aggregation. Phys. Rev. B 27,5686-5697. Shaw cell containing a more viscous liquid. Proc. R . SOC. Lond. A 245, 312-329. 3086-3088. instability. J . Fluid Mech. 177, 67-82. interface. J . Fluid Mech. 136, 1-30. J . Fluid Mech. 154, 287-301. of flow process in oil reservoirs. Trans. AIME 210, 295-301.

Water Resources Research, 1998

With prolonged rainfall, infiltrating wetting fronts in water repellent soils may become unstable, leading to the formation of high-velocity flow paths, the so-called fingers. Finger formation is generally regarded as a potential cause for the rapid transport of water and contaminants through the unsaturated zone of soils. For the first time, field evidence of the process of finger formation and finger recurrence is given for a water repellent sandy soil. Theoretical analysis and model simulations indicate that finger formation results from hysteresis in the water retention function, and the character of the formation depends on the shape of the main wetting and main drainage branches of that function. Once fingers are established, hysteresis causes fingers to recur along the same pathways during following rain events. Leaching of hydrophobic substances from these fingered pathways makes the soil within the pathways more wettable than the surrounding soil. Thus, in the long-term, instability-driven fingers might become heterogeneity-driven fingers.

Transport in Porous Media, 2011

We report on results from primary drainage experiments on quasi-twodimensional porous models. The models are transparent, allowing the displacement process and structure to be monitored in space and time during primary drainage experiments carried out at various speeds. By combining detailed information on the displacement structure with global measurements of pressure, saturation and the capillary number Ca, we obtain a scaling relation relating pressure, saturation, system size and capillary number. This scaling relation allows pressure-saturation curves for a wide range of capillary numbers to be collapsed on the same master curve. We also show that in the case of primary drainage, the dynamic effect in the capillary pressure-saturation relationship observed on partially water saturated soil samples might be explained by the combined effect of capillary pressure along the invasion front of the gaseous phase, and pressure changes caused by viscous effects in the wetting fluid phase.

Europhysics Letters (epl), 2005

We study viscous fingering during drainage experiments in linear Hele-Shaw cells filled with a random porous medium. The central zone of the cell is found to be statistically more occupied than the average, and to have a lateral width of 40% of the system width, irrespectively of the capillary number $Ca$. A crossover length $w_f \propto Ca^{-1}$ separates lower scales where the invader's fractal dimension $D\simeq1.83$ is identical to capillary fingering, and larger scales where the dimension is found to be $D\simeq1.53$. The lateral width and the large scale dimension are lower than the results for Diffusion Limited Aggregation, but can be explained in terms of Dielectric Breakdown Model. Indeed, we show that when averaging over the quenched disorder in capillary thresholds, an effective law $v\propto (\nabla P)^2$ relates the average interface growth rate and the local pressure gradient.

Geoderma, 2004

Water sometimes infiltrates into a sandy soil along preferential flow paths called fingers, rather than homogeneously. In order to understand the physical fundamentals of fingering flow, we carried out two-dimensional infiltration tests under continuous rainfall. Each finger was composed of two moisture structure zones, a finger core and a finger-swelling zone surrounding the finger core. Each finger can be classified as a low-swell finger (LSF) or a high-swell finger (HSF) based on the swelling velocity. In LSF, water moves laterally across a constant potential boundary where sorptivity is a dominant factor in making the finger swell. In HSF, on the other hand, water moves laterally through a constant flux boundary where pressure gradient is a driving force. In the latter, accordingly, the corresponding water flux can be estimated by the hydraulic conductivity multiplied by the water pressure gradient. Based on these observations, we propose a growth model for a finger moving downward along with subsequent lateral expansion, which makes it possible to estimate the area of soil wetted by fingers. The estimated area is in a fairly good agreement with that observed.

Water Resources Research

Water drainage from porous media is a highly dynamic process often marked by rapid pistonlike air invasion events at the front and other rapid interfacial reconfigurations. Liquid phase entrapped behind the moving front drains at significantly slower rates often via gravity driven flow through corners and crevices. This distribution of slowly draining residual water phase determines the plant available water and biological functioning of soils. The study aims to determine the conditions for the flow regime transition from piston-like invasion at a drainage front to slower corner dominated flow at the pore and sample scale. This transition was observed experimentally for sand and glass beads with fast X-ray tomography, revealing water fragmentation into clusters of full pores interconnected by water corner films. The observed liquid morphology at the transition from piston to corner flow was reproduced by a quasi-static pore network model and predicted by percolation theory. The amount of capillary-retained water at flow transition controlling the subsequent drainage dynamics could be reproduced by an idealized star shaped pore whose geometry is deduced from macroscopic properties of the porous medium. Predictions of water content thresholds at flow transitions were in agreement with other critical saturation values associated with cessation of solute diffusion and of internal drainage (at field capacity) highlighting the criticality of water phase continuity disruption for formation of relatively stable unsaturated conditions controlled by slow corner flow that support life in soil.