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# Viscous Fingering Fractals in Porous Media

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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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Viscous fingering is the occurrence of narrow fingers of an invading less viscous fluid such as water in a porous medium filled with a more viscous fluid such as oil, and its occurrence dramatically affects enhanced oil recovery by water flooding. We conduct 2D simulations using the lattice Boltzmann method for two-phase flow through a porous medium initially saturated with a fluid of a given viscosity in which a fluid of another viscosity is injected from the left side of the model. We conduct suites of simulations over viscosity ratios ($$\sim 1/(\mathrm {mobility\ ratio})$$) from M = 0.01 through M = 100 and for wetting angles from non-wetting to fully wetting. We plot the phase space of saturation (= Recovery Factor) versus wetting angle and viscosity ratio. We remove the dominant viscosity ratio effect to study the effect of wetting angle and find that while there is some tendency for the saturation to be higher with increasing wettability, the saturation landscape is complex with hills and valleys in which optimal wetting angles exist that maximize saturation. Furthermore, the phase space landscape is found to depend on the porous matrix geometry. We also plot saturation post-breakthrough and find that the saturation continues to increase albeit at an ever decreasing rate. This research demonstrates the potential of the lattice Boltzmann method for two-phase flow to reveal unexpected behavior and phenomena with both scientific and practical significance such as optimization of recovery factors in enhanced oil recovery (EOR).
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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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When a viscous fluid filling the voids in a porous medium is driven forwards by the pressure of another driving fluid, the interface between them is liable to be unstable if the driving fluid is the less viscous of the two. This condition occurs in oil fields. To describe the normal modes of small disturbances from a plane interface and their rate of growth, it is necessary to know, or to assume one knows, the conditions which must be satisfied at the interface. The simplest assumption, that the fluids remain completely separated along a definite interface, leads to formulae which are analogous to known expressions developed by scientists working in the oil industry, and also analogous to expressions representing the instability of accelerated interfaces between fluids of different densities. In the latter case the instability develops into round-ended fingers of less dense fluid penetrating into the more dense one. Experiments in which a viscous fluid confined between closely spaced parallel sheets of glass, a Hele-Shaw cell, is driven out by a less viscous one reveal a similar state. The motion in a Hele-Shaw cell is mathematically analogous to two-dimensional flow in a porous medium. Analysis which assumes continuity of pressure through the interface shows that a flow is possible in which equally spaced fingers advance steadily. The ratio lambda = (width of finger)/ (spacing of fingers) appears as the parameter in a singly infinite set of such motions, all of which appear equally possible. Experiments in which various fluids were forced into a narrow Hele-Shaw cell showed that single fingers can be produced, and that unless the flow is very slow lambda = (width of finger)/(width of channel) is close to 1/2, so that behind the tips of the advancing fingers the widths of the two columns of fluid are equal. When lambda =1/2 the calculated form of the fingers is very close to that which is registered photographically in the Hele-Shaw cell, but at very slow speeds where the measured value of lambda increased from 1/2 to the limit 1\cdot 0 as the speed decreased to zero, there were considerable differences. Assuming that these might be due to surface tension, experiments were made in which a fluid of small viscosity, air or water, displaced a much more viscous oil. It is to be expected in that case that lambda would be a function of mu U/T only, where mu is the viscosity, U the speed of advance and T the interfacial tension. This was verified using air as the less viscous fluid penetrating two oils of viscosities 0\cdot 30 and 4\cdot 5 poises.
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Zinc metal leaves are grown two-dimensionally by electrodeposition. The structures clearly remind us of the random patterns simulated by computer according to the Witten-Sander diffusion-limited-aggregation model. The scale invariance is tested by computing the density-density correlation function for the digitized patterns of the photographs. The Hausdorff dimension averaged over many examples is D=1.66+/-0.03, which is in excellent agreement with that of the two-dimensional diffusion-limited-aggregation model (D≅53).
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Remarkable parallels in the behavior of diffusion-limited aggregation and two-fluid displacements in porous media exist; hence, the former can be used to simulate the latter. Both processes can be described by the application of Laplace's equation with similar boundary conditions. Displacements can be stabilized by reversing the flow direction and interfacial tension can be incorporated to broaden dendrites or fingers. Furthermore, diffusion-limited aggregation can be used to simulate flow in anisotropic or inhomogeneous porous media.
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Published in Petroleum Transactions, AIME, Vol. 219, 1960, pages 264–272. Paper presented at 35th Annual Fall Meeting of SPE, Oct. 2–5, 1960 in Denver, and at the California Regional Fall Meeting of SPE, Oct. 20–21, 1960 in Pasadena, Calif. Abstract Artificially consolidated sand models, representing one-quarter of a five-spot, have been developed and used to study factors affecting miscible displacement. Sweep efficiency at breakthrough, size of the mixing zone between two miscible liquids and per cent are contacted by drive after breakthrough were determined for the high mobility ratios encountered in actual reservoirs. Quantitative relationships between the degree of viscous fingering and mobility ratio were obtained by measuring the length of the fluid interface. Scaled miscible-slug experiments, supported by field evidence reported in the literature, have shown that when the slug is followed by dry gas the process is less efficient than expected. In addition to low areal sweep efficiencies encountered for high mobility ratio displacements, the effectiveness of a miscible slug is greatly reduced. This is caused by an accelerated growth of the mixing zone between the driving and the displacing fluids. Introduction Considerable interest in miscible displacement as a secondary recovery process has been shown by the oil industry in recent years. In addition to extensive laboratory investigations, numerous field operations have been initiated so that 39 miscible displacement projects had been reported in the United States by early 1959. By far the most popular approach in these projects is the use of an LPG slug followed by dry gas.
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Diffusion-controlled cluster formation has been simulated on lattices of dimensionality 2-6. For the case of a sticking probability of 1.0 at nearest-neighbor sites, we find that the radius of gyration (Rg) of the cluster is related to the number of particles (N) by Rg∼Nβ (for large N). The exponent β is given by β∼6/5d, where d is the classical (Euclidean) dimensionality of the lattice. These results indicate that the Hausdorff (fractal) dimensionality (D) is related to the Euclidean dimensionality (d) by D≈5d/6 (d=2-6). Similar results can be obtained from the density-density correlation function in two-dimensional simulations. Nonlattice simulations have also been carried out in two- and three-dimensional space. The radius-of-gyration exponents (β) obtained from these simulations are essentially equal to those obtained in the lattice model simulations. We have also investigated the effects of sticking probabilities (S) less than 1.0 on diffusion-limited cluster formation on two- and three-dimensional lattices. While smaller sticking probabilities do lead to the formation of denser clusters, the radius-of-gyration exponents are insensitive to sticking coefficients over the range 0.1≤S≤1.0.
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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.