Growth activity during fingering in a porous Hele Shaw cell

tial resolution measurements. After fingers had fully developed, we added a dye tracer in order to distinguish mobile and immobile water fractions. Fully developed fingers consist of a tip, a core with mobile water, and a hull with immobile water. We analyzed the dynamics of water saturation within the finger tip, 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 at time scales that are orders of magnitude 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, with the exception of saturation overshoot, could be consistently explained based on the hysteretic behavior of the soil-water characteristic.

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.

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.

International Journal of Multiphase Flow, 2014

We experimentally studied the displacement of a viscous wetting fluid (water) by an inviscid non-wetting fluid (air) injected at the bottom of a vertical Hele-Shaw cell filled with glass microbeads. In order to cover a wide parameter space, the permeability of the porous medium was varied by using different bead size ranges and diverse air flow rates were generated by means of a syringe pump. A LED light table was used to back illuminate the experimental cell, allowing a high speed camera to capture images of the drainage process at equal time intervals. The invasion occurred in intermittent bursts. Image processing of the bursts and fractal analysis showed successive transitions from capillary invasion to viscous fingering to fracturing during the same experiment, dependent on the medium permeability, the air injection flow rate, and the vertical position in the cell. The interplay between the capillary, viscous and gravity forces determines the nature of the invasion pattern and the transitions, from capillary invasion to viscous fingering with decreasing fluid pressure on one hand and from viscous fingering to fracturing with decreasing effective overburden pressure on the other hand.

Physical Review E, 2017

Energy, 2005

We have studied experimentally and numerically the displacement of a highly viscous wetting fluid by a non-wetting fluid with low viscosity in a random two-dimensional porous medium under stabilizing gravity. In situations where the magnitudes of the viscous-, capillary-and gravity forces are comparable, we observe a transition from a capillary fingering behavior to a viscous fingering behavior, when decreasing apparent gravity. In the former configuration, the vertical extension of the displacement front saturates; in the latter, thin branched fingers develop and rapidly reach breakthrough. From pressure measurements and picture analyzes, we experimentally determine the threshold for the instability, a value that we also predict using percolation theory. Percolation theory further allows us to predict that the vertical extension of the invasion fronts undergoing stable displacement scales as a power law of the generalized Bond number Bo à ¼ Bo À Ca, where Bo and Ca are the Bond and capillary numbers, respectively. Our experimental findings are compared to the results of a numerical modeling that takes local viscous forces into account. Theoretical, experimental and numerical approaches appear to be consistent. #

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

Physical Review Fluids

Lifted Hele-Shaw cells typically display viscous fingering of liquids, which in turn leads to branched fractal patterns in the absence of any anisotropies. Recently, experiments involving parallely lifted Hele-Shaw cells with holes in the cell plates, also termed as "multiport lifted Hele-Shaw cells," have been used to generate more regular meshlike patterns in the liquid film. Although such patterns promise usefulness in several applications, their spatiotemporal evolution needs to be theoretically understood for better synthesis. As a first step, therefore, we examine the stability of fingers evolving from a single hole by focusing on flow of an annular film of liquid placed in a lifted Hele-Shaw cell. We use linear stability analysis to find the growth rate of azimuthally periodic perturbations of the inner and outer interfaces around the evolving base state of the liquid film. To validate the results of our stability analysis, we also perform resolved numerical simulations of the setup via an in-house solver based on lubrication theory, which uses front-tracking method to evolve the interface in time and space. For a wide range of parameters and wave numbers, we find excellent agreement in the growth rates predicted by the linear stability analysis with the numerical simulation. The results of the stability analysis are expressed in terms of the capillary number, initial nondimensional plate separation, and initial ratio of the interface radii. Furthermore, using the results from our linear stability analysis, we generate a phase map to demarcate the flow regimes corresponding to unstable and stable states for the interfaces. Numerical simulations of the interface evolution over finite times are consistent with the results predicted by the proposed analysis. These finite-time simulations successfully capture the presence of shielding of the fingers at both the inner and outer interface. The proposed theoretical analysis and insights obtained through numerical simulations thus provide a framework for accurately predicting and experimentally realizing stable fluid patterns in a multiport Hele-Shaw cell.

The phenomenon of interfacial motion between two immiscible viscous fluids in the narrow gap between two parallel plates (Hele-Shaw cell) is considered. This flow is currently of interest because of its relation to pattern selection mechanisms and the formation of fractal structures in a number of physical applications. Attention is concentrated on the fingers that result from the instability when a less-viscous fluid drives a more-viscous one. The status of the problem is reviewed and progress with the thirty-year-old problem of explaining the shape and stability of the fingers is described. The paradoxes and controversies are both mathematical and physical. Theoretical results on the structure and stability of steady shapes are presented for a particular formulation of the boundary conditions at the interface and compared with the experimental phenomenon. Alternative boundary conditions and future approaches are discussed.

Physical Review E, 2010

This paper reports some experimental results on two-phase flows in model two-dimensional porous media. Standard microfluidic techniques are used to fabricate networks of straight microchannels and to control the throat size distribution. We analyze both the invasion mechanism of the medium by a nonwetting fluid and the drainage after the percolation for capillary numbers lying between 10 −7 and 10 −2. We propose a crude model allowing a description of the observed capillary fingering that captures its scaling properties. This model is supported by numerical simulations based on a pore-network model. Numerical simulations and experiments agree quantitatively.