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Abstract

Gas displacing a high-viscosity fluid in a two-dimensional porous disk intrudes in the form of ramified fingers similar to the structures obtained in diffusion-limited aggregation. We find that the resulting finger structures are described by a fractal dimension $D=1.62\ifmmode\pm\else\textpm\fi{}0.04$ consistent with $D$ for diffusion-limited aggregation clusters. This result confirms the analogy between diffusion-limited aggregation and two-fluid displacement in porous media introduced by Paterson.
... It is clearly a multidisciplinary subject and its complex nature has been studied by hydrologists, physicists, chemists, geoscientists, biologists, and engineers for decades. The structures observed are controlled by the characteristic forces involved like viscous [1][2][3][4], capillary [4][5][6], and gravitational forces [7] in addition to wetting properties [6,8] and fluctuations in the local geometry of the porous medium. These structures range from compact to the ramified and fractal [9,10]. ...
... At high injection rates, however, the patterns strongly depend on the viscosities of the two fluids. When the viscosity of the displaced fluid μ d is higher than the viscosity of the invading fluid μ i , a viscous fingering instability is observed [17], creating a fractal ramified structure [2,3]. However, when the injected fluid has a higher viscosity than the displaced fluid a compact stable structure is formed. ...
... The set of experiments with different defending phase viscosities and invading phase pressures have yielded a wide range of dynamical features. The observed invasion dynamics included the standard viscous fingering behavior in porous media [2,3,38,39], particularly in the regime of higher defending fluid viscosity, but we have also observed many additional features that are unaccounted for in the literature. In order to better visualize and understand these features, we will introduce two matricial concepts: the invading matrix and flipping matrix. ...
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We describe a crossover from the viscous fingering instability to a compact invasion regime during viscously unstable drainage of porous media, and we investigate the underlying mechanisms of this compact fluid displacement. The study is based on a series of drainage experiments in a radial porous Hele-Shaw cell where we systematically vary the viscosity of the defending (wetting) fluid and the overpressure of the invading (nonwetting) fluid to map out the resulting invasion structures as a function of viscosity ratio and injection pressure. We show that above a threshold of injection pressure and viscosity ratio a more stable and compact invasion structure emerges within the viscous fingering patterns, i.e., a roughly circular displacement with viscous fingers on the outside. The onset of the stable displacement is found to begin at a rather low viscosity ratio M between the invading and defending fluids, i.e., when M>10^−3 for injection pressures of 3-5 kPa. We find that the ratio between the length of the outer fingers and the size of the compact invasion scales with the viscosity ratio and approaches a more or less constant value during growth, resulting in structures with proportionate growth and larger compact invasions for higher viscosity ratios. As opposed to the viscous fingering instability, we describe rich ganglion dynamics within the compact invasion structures and show that the pressure gradient is not screened by the outer fingers.
... It is a multidisciplinary subject that has been investigated for decades by hydrologists, physicists, chemists, geoscientists, biologists, and engineers due to its practical importance and complexity. The structures observed are controlled by the forces involved, such as viscous [15][16][17][18][19][20][21], capillary [18][19][20][21][22][23][24][25], and gravitational forces [26][27][28][29][30][31], as well as wetting properties [32][33][34][35][36][37][38] and changes in the local geometry of the porous medium [39,40]. The structures vary in shape and complexity [15,16,18,32,37,[41][42][43][44][45][46][47], from compact to ramified and fractal [48,49]. ...
... The structures observed are controlled by the forces involved, such as viscous [15][16][17][18][19][20][21], capillary [18][19][20][21][22][23][24][25], and gravitational forces [26][27][28][29][30][31], as well as wetting properties [32][33][34][35][36][37][38] and changes in the local geometry of the porous medium [39,40]. The structures vary in shape and complexity [15,16,18,32,37,[41][42][43][44][45][46][47], from compact to ramified and fractal [48,49]. The fractal nature of porous media is itself important for a number of applications [50,51], such as electrolyte diffusion through charged media [52] a topic of relevance for the development of modern battery technology [53]. ...
... where Eqs. (20) and (17) give the cutoff pressure ∆p * Fig. 7 shows the scaling function H(∆p/∆p * ) for various values of κ from invasion percolation simulations [16] which confirms (22) and (23). We will in the following use F(∆p) to calculate the work. ...
... It is a multidisciplinary subject that has been investigated for decades by hydrologists, physicists, chemists, geoscientists, biologists, and engineers due to its practical importance and complexity. The structures observed are controlled by the forces involved, such as viscous [15][16][17][18][19][20][21], capillary [18][19][20][21][22][23][24][25], and gravitational forces [26][27][28][29][30][31], as well as wetting properties [32][33][34][35][36][37][38] and changes in the local geometry of the porous medium [39,40]. The structures vary in shape and complexity [15,16,18,32,37,[41][42][43][44][45][46][47], from compact to ramified and fractal [48,49]. ...
... The structures observed are controlled by the forces involved, such as viscous [15][16][17][18][19][20][21], capillary [18][19][20][21][22][23][24][25], and gravitational forces [26][27][28][29][30][31], as well as wetting properties [32][33][34][35][36][37][38] and changes in the local geometry of the porous medium [39,40]. The structures vary in shape and complexity [15,16,18,32,37,[41][42][43][44][45][46][47], from compact to ramified and fractal [48,49]. The fractal nature of porous media is itself important for a number of applications [50,51], such as electrolyte diffusion through charged media [52] a topic of relevance for the development of modern battery technology [53]. ...
... (23) Figure 7 shows the scaling function H (Δp/Δpp) for various values of κ from invasion percolation simulations [16] which confirms (22) and (23). ...
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We present a theoretical and experimental investigation of drainage in porous media. The study is limited to stabilized fluid fronts at moderate injection rates, but it takes into account capillary, viscous, and gravitational forces. In the theoretical framework presented, the work applied on the system, the energy dissipation, the final saturation and the width of the stabilized fluid front can all be calculated if we know the dimensionless fluctuation number, the wetting properties, the surface tension between the fluids, the fractal dimensions of the invading structure and its boundary, and the exponent describing the divergence of the correlation length in percolation. Furthermore, our theoretical description explains how the Haines jumps’ local activity and dissipation relate to dissipation on larger scales.
... Once the meniscus reaches its maximal possible curvature compatible with the geometrical constraints and the contact angles of water on PDMS and glass, the situation becomes unstable, because the Laplace pressure can no longer equilibrate the elastic deformation of the channel, and the meniscus advances rapidly until the channel is fully relaxed. The onset of the jump is therefore an occurrence of a capillary threshold, as encountered in the drainage of Newtonian fluids in porous media ( 1983; Lenormand & Zarcone 1985;Måløy et al. 1985;Méheust et al. 2002), or in the flow of foams in porous media (Géraud et al. 2016). Hence, the maximal deformation of the channel is correlated, through Laplace pressure, to the maximal curvature of the meniscus. ...
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We explore air invasion in an initially water-filled dead-end compliant microchannel containing a constriction. The phenomenon is driven by the pervaporation of the liquid present in the channel through the surrounding medium. The penetration is intermittent, jerky, and characterised by a stop-and-go dynamics as the bubble escapes the constriction. We demonstrate that this sequence of arrest and jump of the bubble is due to an elas-tocapillary coupling between the air-liquid interface and the elastic medium. When the interface enters the constriction, its curvature strongly increases, leading to a depression within the liquid-filled channel which drives a compression of the channel. As the interface is forced to leave the constriction at a given threshold, due to the ongoing loss of liquid content by pervaporation, the pressure is suddenly released, which gives rise to a rapid propagation of the air bubbles away from the constriction and a restoration of the rest shape of the channel. Combining macroscopic observations and confocal imaging, we present a comprehensive experimental study of this phenomenon. In particular, the effect of the channel geometry on the time of arrest in the constriction and the jump length is investigated. Our novel microfluidic design succeeds in mimicking the role of inter-vessel pits in plants, which transiently stop the propagation of air embolism during long and severe droughts. It is expected to serve as a building block for further biomimetic studies in more complex leaf-like architectures, in order to recover this universal phenomena of intermittent propagation reported in real leaves.
... This diagram, with the identified flow regimes and transition regions, is represented in Figure 1. Using similar Hele-Shaw type experimental set-ups (in 2D), subsequent experimental campaigns have also identified the mechanisms behind the unstable flow regimes in idealized porous media [5,6], comparisons with Saffman-Taylor theoretical solutions [7], fractal dimension analysis [3,8], as well as pore-continuum upscaling relations [9]. ...
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This paper presents the development of a laboratory scale apparatus and first experimental results on the characterization of fingering patterns of immiscible fluids in a porous rock (Fontainebleau sandstone), using three dimensional full-field measurements from x-ray tomography. The few existing studies that have extended experimental investigation of immiscible fluid flow from 2D to 3D have been primarily interested in the pore scale or performed on idealized porous media. While the heterogeneities inherent to natural rocks are known to play an important role on subsurface fluid flow regimes, a limited number of studies have approached the problem of characterizing the time resolved 3D multiphase flow in these material, at the mesoscale. The series of experiments reported in this paper has been performed at a low viscosity ratio, water invasion into oil as the defending fluid, and different capillary numbers (1.8 orders of magnitude). The results illustrate the qualitative transition in the flow regime, from capillary fingering to viscous fingering. While a full quantitative characterization of geometrical features of fluid fingers will require further technical refinements, a qualitative understanding can be already gathered from the results presented herein.
... A velocity distribution in Poiseuille flow and in a steady Couette flow has been found. Fractal properties of viscous fingering which arise when a low viscosity fluid is injected from a point source into the thin space between two parallel plates originally flooded with a more viscous, immiscible fluid have been studied in [116,117]. This problem is connected to the Diffusion-Limited Aggregation (DLA) [118] and to the viscous flows in porous media [119,120]. ...
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In this article, we construct fluid equations in fractal dimensions based on the concept of the product-like fractal measure introduced by Li and Ostoja-Starzewski in their formulation of anisotropic media. Three main problems were discussed and analyzed: the Rayleigh problem, the steady Burger's vortex and the Kelvin–Helmholtz instability. This study confirms the importance of fractal dimensions in fluid mechanics where several mechanisms were revealed. In particular, long-tails took place in Rayleigh problem analogous to those arising in several superdiffusion processes. Besides, the analysis of the steady Burger's vortex has proved that the flux rate of dissipation of energy per unit length of vortex depends on the viscosity of the fluid and it is finite for insignificant viscous effects, a scenario which is detected in small-scale turbulent fluid flows. Moreover, it was proved that both the Rayleigh–Taylor and the Kelvin–Helmholtz instabilities are affected by the fractal dimensions and that, for a particular value of the characteristic length as a function of the wavelength, the Kelvin–Helmholtz instability may be suppressed, a particular scenario which is observed in compressible fluid flows and other physical engineering processes.
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Viscoelastic (Maxwellian) slow, immiscible liquid-liquid displacement in a permeable medium is considered. The necessary and sufficient criteria for stability are that the displacing fluid is denser and less mobile than the displaced fluid. The instability criteria and critical wave length are found to be the same as those for ordinary viscous liquid-liquid displacements in permeable media.
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We investigate the impact of wettability distribution, pore size distribution and pore geometry on the statistical behaviour of trapping in pore-throat networks during capillary displacement. Through theoretical analyses and numerical simulations, we propose and prove that the trapping patterns, defined as the percentage and distribution of trapped elements, are determined by four dimensionless control parameters. The range of all possible trapping patterns and how the patterns are dependent on the four parameters are obtained. The results help us to understand the impact of wettability and structure on trapping behaviour in disordered media.
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What happens when one attempts to push water through a fluid of higher viscosity? Under appropriate experimental conditions, the water breaks through in the form of highly branched patterns called viscous fingers. Water was used to push a more viscous but miscible, non-newtonian fluid in a Hele-Shaw cell. The resulting viscous finger instability was found to be a fractal growth phenomenon. Reproducible values of the fractal dimension df were found and were interpreted using a modification of the diffusion limited aggregation model.
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Transport properties such as conductivity on scale-invariant structures such as random aggregates are governed by a spectral dimension exponent d̃. We investigate the behavior of d̃ for branched structures without loops using geometric constraints on their current-carrying backbone. The requirement that the structure fit into d-dimensional space imposes a new constraint on d̃. The distance Δ between branches of the backbone must be of the order of the span N measured along the backbone. Under mild scaling assumptions this distance Δ controls the conductance. We express d̃ in terms of the fractal dimension D and the scaling power δ relating N to the geometric size L (N∼Lδ): δ=D(2/d̃-1). This implies that d̃<4/3 for such branched structures with D<2.
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The results of experiments involving instability, known as fingering, in a circular Hele Shaw cell with inward and outward flow are presented. The width of fingers in this situation is examined, and an approximate equation for the growth of fingers is proposed. The equation rα = cos (nθ) is shown to fit the shape of long fingers.