Article

# Influence of pore-scale disorder on viscous fingering during drainage

Authors:
To read the full-text of this research, you can request a copy directly from the authors.

## Abstract

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. Comment: 4 pages, 4 figures, submitted to Phys Rev Letters

## No full-text available

... This means that the invading fluid displaces the more viscous one in separated finger-like intrusions, while leaving the fluid inbetween the fingers less or not displaced. Following an increased interest in this phenomenon, two-phase flow have been widely studied in quasi-2-dimensional porous media confined in thin cells with circular and rectangular geometries [2][3][4][5][6][7][8][9][10][11][12][13][14]. In horizontal cells containing rigid disordered porous media, the unstable invasion patterns during drainage are found to be fractal and either form an invasion percolation cluster [15] in the capillary fingering regime [16,17], or long thin fingers resembling DLA patterns in the viscous fingering regime [18,19]. ...
... We do observe in Figures 6, 8 that spatial properties of the structures grow toward limiting curves over time, and in Figure 7 that the local fractal dimensions show a sudden decrease at the outermost radii, i.e., the active growth zone. This confirms that growth dynamics of the patterns follow a behavior where an outer active growth zone surrounds an internally developed and frozen pattern, as described in similar linear systems [3,4]. The fundamental similarity in the growth dynamics of the structures is reflected by their similar global appearance across boundary conditions. ...
... The fundamental similarity in the growth dynamics of the structures is reflected by their similar global appearance across boundary conditions. This is also indicated by the fractal box dimensions, which are all observed in the range from 1.55 to 1.63, consistent with earlier observed values of e.g., 1.53 [4] and 1.62 [19] for viscous fingers in ND porous media. The breakthrough times, shown in Figure 5, are generally observed to decrease with increased deformability and injection pressure, while the growth rates generally seem to have a linear trend with some fluctuations. ...
Article
We study the formation of viscous fingering and fracturing patterns that occur when air at constant overpressure invades a circular Hele-Shaw cell containing a liquid-saturated deformable porous medium—i.e., during the flow of two non-miscible fluids in a confined granular medium at high enough rate to deform it. The resulting patterns are characterized in terms of growth rate, average finger thickness as function of radius and time, and fractal properties. Based on experiments with various injection pressures, we identify and compare typical pattern characteristics when there is no deformation, compaction, and/or decompaction of the porous medium. This is achieved by preparing monolayers of glass beads in cells with various boundary conditions, ranging from a rigid disordered porous medium to a deformable granular medium with either a semi-permeable or a free outer boundary. We show that the patterns formed have characteristic features depending on the boundary conditions. For example, the average finger thickness is found to be constant with radius in the non-deformable (ND) system, while in the deformable ones there is a larger initial thickness decreasing to the ND value. Then, depending on whether the outer boundary is semi-permeable or free there is a further decrease or increase in the average finger thickness. When estimated from the flow patterns, the box-counting fractal dimensions are not found to change significantly with boundary conditions, but by using a method to locally estimate fractal dimensions, we see a transition in behavior with radius for patterns in deformable systems; In the deformable system with a free boundary, it seems to be a transition in universality class as the local fractal dimensions decrease toward the outer rim, where fingers are opening up like fractures in a paste.
... At intermediate Ca where both viscous and capillary forces are important, while the destabilizing effect of disorder is widely accepted 18,21,22,26,27 , quantitative understanding is still lacking. Few works introduced the impact of disorder into Ca via the width of the capillary entry pressures (often termed "capillary disorder"), neglecting its effect on the viscous forces. ...
... Few works introduced the impact of disorder into Ca via the width of the capillary entry pressures (often termed "capillary disorder"), neglecting its effect on the viscous forces. Quantitatively, this assumption is manifested via a decrease in the transitional Ca between CF to VF, leading to more CF-like pattern at a given Ca in more disordered sample 18,21,22,26,27 . We point to a subtle yet crucial contradiction between the decrease in both transitional Ca and stability with increasing disorder, since a more CF-like pattern has actually a much better sweep efficiency (with thicker fingers and less convoluted invasion front), which are considered to indicate higher stability. ...
... For drainage (θ =5 • ), where dominance of bursts increases the importance of the aperture distribution, we use the standard deviation as a measure of the width of the distribution, ω * = σ (ω). Other measures have been previously introduced to represent an effective, disorder-dependent Ca in drainage 21,22,34,35 . For imbibition (θ =120 • ), overlaps and the increase in their occurrence with disorder (Fig. 4d) makes for a more complex λ effect. ...
Presentation
We present a systematic, quantitative assessment of the impact of pore size disorder and its interplay with flow rates and the wettability on immiscible fluid displacement. Pore-scale simulations and micromodel experiments show that increasing disorder destabilizes the displacement, reducing the its efficiency and increasing the fluid-fluid interfacial area, by enhancing trapping at low rates, and fingering at high rates. Lowering disorder enhances the effect of the underlying lattice. Increasing wettability of the invading fluid (contact angle) stabilizes the invasion, smoothing the interface and inhibiting trapping--effects which are suppressed at low disorder and high rates.
... The effect of disorder on the invasion morphology when either capillary or viscous forces are dominant seems to be univocally accepted. For slow drainage, high disorder was shown to promote trapping, leading to a transition from CO to CF [18][19][20][21][22][23][24] . For slow imbibition, quasi-static simulations show a transition from compact to faceted (FA) growth as disorder is decreased 20,25 . ...
... At intermediate Ca where both viscosity and capillarity are important, the qualitative paradigm ("conventional wisdom" 24 ) is that lowering disorder enhances the displacement efficiency 18,[21][22][23][24] . Few works derived a modified capillary number, Ca * , which includes the impact of disorder via the width of the capillary thresholds, neglecting its effect on the viscous forces. ...
... Few works derived a modified capillary number, Ca * , which includes the impact of disorder via the width of the capillary thresholds, neglecting its effect on the viscous forces. Quantitatively, this implies an inverse relationship between Ca * and disorder, namely that increasing disorder (at a given rate, Ca) produces a more CF-like pattern, increasing the transitional Ca between CF to VF, Ca CF/VF 18,[21][22][23][24] . We point to a subtle yet crucial inconsistency in the above: if increasing disorder increases Ca CF/VF , promoting a more CF-like pattern, sweep efficiency should increase (since VF is much less efficient that CF 18,[21][22][23][24]28 ), contrasting the observations stated above. ...
Article
Full-text available
We present a systematic, quantitative assessment of the impact of pore size disorder and its interplay with flow rates and wettability on immiscible displacement of a viscous fluid. Pore-scale simulations and micromodel experiments show that reducing disorder increases the displacement efficiency and compactness, minimizing the fluid-fluid interfacial area, through (i) trapping at low rates and (ii) viscous fingering at high rates. Increasing the wetting angle suppresses both trapping and fingering, hence reducing the sensitivity of the displacement to the underlying disorder. A modified capillary number Ca* that includes the impact of disorder λ on viscous forces (through pore connectivity) is direct related to λ, in par with previous works. Our findings bear important consequences on sweep efficiency and fluid mixing and reactions, which are key in applications such as microfluidics to carbon geosequestration, energy recovery, and soil aeration and remediation.
... In the opposite case, during liquid injection into dry granular media [45], for a given imposed flux, the flow behavior goes from stable invasion toward saturated granular fingers for increasing flow rate and viscosity of the invading fluid. These fingering patterns are thought to form due to the permeability contrast between the channels empty of grains and the granular medium and are in this sense similar to viscous fingering, with a lower viscous pressure drop in empty channels than in the porous medium, as in classical viscous-fingering systems, where the viscosity contrast leads to a similar effect, i.e., where the invading fluid has a lower viscosity than the invaded one [30,42,46]. An important difference with classical viscous fingering is the presence of solid stresses and solid friction in the granular material. ...
... such that the box-counting dimension D B is found as the negative value of the slope of N (s) in a log-log plot [31,46,60,61]. By obtaining box-counting data over a range of sizes s, we estimate D B from the slope of linear fits between an upper cutoff s = 32 cm (cell width) and a lower cutoff s = 1 cm (typical for thinner fingers). ...
... Patterns in categories 3b and 4a have box dimensions within the range of fractal dimensions found for viscous fingers in saturated porous media; i.e., D = 1.53-1.62 [31,46,63]. Figure 10 shows the results of box-counting the air-solid interface of the final structures, averaged per category. ...
Article
We perform experiments where air is injected at a constant overpressure Pin, ranging from 5 to 250 kPa, into a dry granular medium confined within a horizontal linear Hele-Shaw cell. The setup allows us to explore compacted configurations by preventing decompaction at the outer boundary, i.e., the cell outlet has a semipermeable filter such that beads are stopped while air can pass. We study the emerging patterns and dynamic growth of channels in the granular media due to fluid flow, by analyzing images captured with a high speed camera (1000 images/s). We identify four qualitatively different flow regimes, depending on the imposed overpressure, ranging from no channel formation for Pin below 10 kPa, to large thick channels formed by erosion and fingers merging for high Pin around 200 kPa. The flow regimes where channels form are characterized by typical finger thickness, final depth into the medium, and growth dynamics. The shape of the finger tips during growth is studied by looking at the finger width w as function of distance d from the tip. The tip profile is found to follow w(d)∝dβ, where β=0.68 is a typical value for all experiments, also over time. This indicates a singularity in the curvature d2d/dw2∼κ∼d1−2β, but not of the slope dw/dd∼dβ−1, i.e., more rounded tips rather than pointy cusps, as they would be for the case β>1. For increasing Pin, the channels generally grow faster and deeper into the medium. We show that the channel length along the flow direction has a linear growth with time initially, followed by a power-law decay of growth velocity with time as the channel approaches its final length. A closer look reveals that the initial growth velocity v0 is found to scale with injection pressure as v0∝Pin32, while at a critical time tc there is a cross-over to the behavior v(t)∝t−α, where α is close to 2.5 for all experiments. Finally, we explore the fractal dimension of the fully developed patterns. For example, for patterns resulting from intermediate Pin around 100–150 kPa, we find that the box-counting dimensions lie within the range DB∈[1.53,1.62], similar to viscous fingering fractals in porous media.
... However, it is very important to analyze and understand which part of the recording can be labeled an event and which part can not. STA/LTA threshold method is commonly used in seismic data interpretation [55][56][57][58]. If the parameters are selected carefully, it is very easy to use and is very robust [59]. ...
... such that the box-counting dimension D B is found as the negative value of the slope of N (s) in a log-log plot [29,[54][55][56]. By obtaining box-counting data over a range of sizes s, we estimate D B from the slope of linear fits between an upper cutoff s = 32 cm (cell width) and a lower cutoff s = 1 cm (typical for thinner fingers). ...
... where D is the fractal dimension of the pattern, E 1 = 1 is the dimension of the line intersecting it, and E 2 = 2 is the dimension of the image plane containing the sets. [29,56,58]. Figure 12 shows the results of box-counting the airsolid interface of the final structures, averaged per category. ...
Thesis
Fluid induced brittle deformation of porous medium is a phenomenon commonly present in everyday life. From an espresso machine to volcanoes it is possible to see traces of this phenomenon. In a rectangular Hele‐Shaw cell we inject air into a loose porous medium. Then, we monitor this system using optical imaging using a high speed camera (1000 fps) and 4 high frequency resolution accelerometers. Using the numerical and experimental acoustic emissions, different sources of the recorded signal (vibrations due to air, changes in the effective stress due to fluid‐solid interactions) are analyzed. We found that, the peaks in the low frequency range (f < 20 kHz) diminishes while the medium fractures. Furthermore, we propose a new signal localization method based on energy amplitude attenuation and inversed source amplitude comparison. Furthermore, using optical and acoustic datasets and numerical simulations, the mechanics leading Type‐A and Type‐B earthquakes are explained.
... The limiting configurations for low and high capillary numbers are capillary fingering and viscous fingering, respectively; they have been first classified in Lenormand's phase diagram [Lenormand et al., 1988]. More recently, it has been shown that at intermediate regimes, capillary fingering is dominant at smaller scales, while viscous fingering dominates at larger scales [Løvoll et al., 2004; Toussaint et al., 2005]. The experiments described in this paper have been performed in porous micromodels consisting of cylindrical obstacles in Hele-Shaw cells, which we have manufactured by soft lithography [de Anna et al., 2014]. ...
... The typical finger width of the structure created by air through the wetting fluid corresponds to the scale over which the viscous pressure drop in the wetting fluid starts dominating the fluctuations of the capillary pressure thresholds along the interface and dictates the shape of the displacement structure [see, e.g., Toussaint et al., 2005]. When it can be defined without ambiguity, the typical finger width is thus a critical feature of a two-phase flow process. ...
... In this case, the displacement structure can be simply described by invasion percolation [Lenormand and Zarcone, 1985] , which has fractal dimension D c 51:8360:01. At high flow rates the morphology of the front is typical of viscous fingering, which has fractal dimension D v % 1:53 [Toussaint et al., 2005]. As the drainage experiments were performed at intermediate capillary numbers, we expect capillary forces to dominate at a smaller scales, while viscous forces dominate at a larger scales [Løvoll et al., 2004; Toussaint et al., 2005]. ...
Article
Full-text available
... An increased understanding of the basic mechanisms that govern the pore-scale description is of interest for multiple disciplines of science such as soil-science, hydrology, physics, and biology and has shown industrial importance through applications such as enhanced oil recovery, CO 2 sequestration and by mapping and controlling of migrating ground water contaminants (Nsir et al., 2012). Furthermore, it has revealed a variety of pattern-forming processes emerging from the pore scale up to the system scale (Lenormand et al., 1983;Måløy et al., 1985;Méheust et al., 2002;Løvoll et al., 2004;Toussaint et al., 2005), typically governed by the interplay between viscous, capillary, and gravitational forces. The structures have shown to exhibit a complex behavior, characterized by its rich intermittent dynamics (Clotet et al., 2016;Furuberg et al., 1988;Måløy et al., 1992;Moura et al., 2020;Moura, Måløy, Flekkøy, et al., 2017;Planet et al., 2009). ...
... We study two phase flow in a quasi two-dimensional (2-D) porous confinement and look at the simple case of drainage at pore scale, where a nonwetting phase displaces a wetting one. Such experiments have shown to generate displacement structures that depend on the density and viscosity contrast between the fluids, surface tension, and the flow rates at which the system is driven (Lenormand et al., 1983;Løvoll et al., 2004;Toussaint et al., 2005). Furthermore, such structures are assembled by trapped regions of wetting phase, completely surrounded by the invading phase. ...
... Variations of the capillary pressure thresholds promote a small-scale roughness of the front. The separation between stability and instability is set by comparing the Bond and capillary numbers (Toussaint et al., 2005): A stable situation corresponds to Bo > Ca (Méheust et al., 2002), an unstable one to Bo < Ca . ...
Article
Full-text available
We experimentally and numerically study the influence of gravity and finite-size effects on the pressure-saturation relationship in a given porous medium during slow drainage. The effect of gravity is systematically varied by tilting the system relative to the horizontal configuration. The use of a quasi two-dimensional porous media allows for direct spatial monitoring of the saturation. Exploiting the fractal nature of the invasion structure we obtain a relationship between the final saturation and the Bond number SF =Bo^0.097 using percolation theory. Moreover the saturation, pressure and Bond number are functionally related, allowing for pressure-saturation curves to collapse onto a single master curve, parameterized by the Representative Elementary Volume size and by the Bond and Capillary Numbers. This allows to upscale the pressure-saturation curves measured in a laboratory to large representative elementary volumes used in reservoir simulations. The large-scale behavior of these curves follow a simple relationship, depending on Bond and Capillary number, and on the flow direction. The size distribution of trapped defending fluid clusters is also shown to contain information on past fluid flow, and can be used as a marker of past flow speed and direction.
... Immiscible displacement can be classified according to the wettability into drainage or nonwetting invasion, where the displaced fluid preferentially wets the solid (contact angle θ < 90°, measured through the defending fluid), or imbibition of a wetting fluid (θ > 90°). Intensive research has provided a basic understanding of drainage, identifying different invasion behaviors and explaining their dependence on the flow velocity, fluid viscosities, interfacial tension, and the degree of pore-scale disorder (see [4][5][6][7], and references therein). Increasing θ was found to stabilize the displacement and reduce trapping in forced and gravitydriven drainage experiments [8,9]. ...
... For rapid injection, N Ca ≫ N Ca Ã implies dominance of viscous forces leading to viscous fingering. Here, the critical value scales as N Ca Ã ∼ ðL=aÞ −1 ≈ 4 × 10 −3 , suggesting a dependence on the macroscopic characteristic length-the system size [6,7]. For slow injection, N Ca ≪ N Ca Ã , capillary forces govern and invasion becomes strongly dependent on the wettability: for nonwetting invasion, N coop < 0 predicts capillary fingering caused by disorder in capillary (burst) thresholds, whereas for wetting invasion N coop > 0 implies cooperative motion of large parts of the interface (overlaps) and a compact pattern, in agreement with our simulations (Fig. 4) and experiments [15]. ...
Article
Full-text available
We study the impact of the wetting properties on the immiscible displacement of a viscous fluid in disordered porous media. We present a novel pore-scale model that captures wettability and dynamic effects, including the spatiotemporal nonlocality associated with interface readjustments. Our simulations show that increasing the wettability of the invading fluid (the contact angle) promotes cooperative pore filling that stabilizes the invasion, and that this effect is suppressed as the flow rate increases, due to viscous instabilities. We use scaling analysis to derive two dimensionless numbers that predict the mode of displacement. By elucidating the underlying mechanisms, we explain classical yet intriguing experimental observations. These insights could be used to improve technologies such as hydraulic fracturing, CO$_{2}$ geo-sequestration, and microfluidics.
... Although intensive research has focused on the effect of wettability on fluid displacement, less progress has been made in examining how the competition between pore-scale disorder and wettability controls the multiphase flow. In cases where the wettability effect is isolated, previous studies have shown that decreasing disorder stabilizes the displacement for both drainage and imbibition (Chen & Wilkinson 1985;King 1987;Toussaint et al. 2005) and that the disorder also modifies the critical capillary number Ca c corresponding to the cross-over from capillary to viscous fingering (Yortsos, Xu & Salin 1997;Holtzman & Juanes 2010;Xu et al. 2014;Liu, Zhang & Valocchi 2015b). Recently, a systematic study on the impact of disorder and its coupling with wettability was conducted by Holtzman (2016). ...
... For zone I (θ < θ CDM ) and zone IV (θ > θ CFM ), the patterns are respectively compact and unstable, independent of disorder λ. In zone II, increasing the disorder λ destabilizes the displacement front, which has been well investigated in the previous studies (Koiller et al. 1992;Yortsos et al. 1997;Toussaint et al. 2005;Holtzman & Juanes 2010;Xu et al. 2014;Holtzman 2016), and the mechanism is well described by the pore-scale images recorded with microscopy ( figure 4). For zone III, however, increasing the disorder λ stabilizes the displacement front, and the mechanism is given in figure 8(b). ...
Article
Full-text available
Immiscible displacement in porous media is common in many practical applications. Under quasi-static conditions, the process is significantly affected by disorder of the porous media and the wettability of the pore surface. Previous studies have focused on wettability effects, but the impact of the interplay between disorder and contact angle is not well understood. Here, we combine microfluidic experiments and pore-scale simulations with theoretical analysis to study the impact of disorder on the quasi-static displacement from weak imbibition to strong drainage. We define the probability of overlap to link the menisci advancements to displacement patterns, and derive a theoretical model to describe the lower and upper bounds of the cross-over zone between compact displacement and capillary fingering for porous media with arbitrary flow geometry at a given disorder. The phase diagram predicted by the theoretical model shows that the cross-over zone, in terms of contact angle range, expands as the disorder increases. The diagram further identifies four zones to elucidate that the impact of disorder depends on wettability. In zone I, increasing disorder destabilizes the patterns, and in zone II, a stabilizing effect plays a role, which is less significant than that in zone I. In the other two zones, invasion morphologies are compact and fingering, respectively, independent of both contact angle and disorder. We evaluate the proposed diagram using pore-scale simulations, experiments in this work and in the literature, confirming that the diagram can capture the effect of disorder on displacement under different wetting conditions. Our work extends the classical phase diagrams and is also of practical significance for engineering applications.
... Owing to the intimate dependence of such patterns on pore-scale properties and local thermodynamic conditions, displacement processes in the crossover domain are not yet fully understood [4] . Furthermore, the quantitative details of the Ca-M diagram as well as crossover region depend on the scale of the system [27][28][29] , in addition to the pore structure. In larger systems, the boundaries between fluid displacement patterns are not as sharp and the crossover region is typically broader. ...
... In addition, we also need to consider the scaling effect (i.e. scale of rock model) in the characterization of two-phase fluid displacement patterns on the Lenormand diagram [27][28][29] . The fluid behavior would be mixed in the large rock model. ...
Article
To characterize the influence of reservoir conditions upon multiphase flow, we calculated fluid displacements (drainage processes) in 3D pore spaces of Berea sandstone using two-phase lattice Boltzmann (LB) simulations. The results of simulations under various conditions were used to classify the resulting two-phase flow behavior into three typical fluid displacement patterns on the diagram of capillary number (Ca) and viscosity ratio of the two fluids (M). In addition, the saturation of the nonwetting phase was calculated and mapped on the Ca–M diagram. We then characterized dynamic pore-filling events (i.e., Haines jumps) from the pressure variation of the nonwetting phase, and linked this behavior to the occurrence of capillary fingering. The results revealed the onset of capillary fingering in 3D natural rock at a higher Ca than in 2D homogeneous granular models, with the crossover region between typical displacement patterns broader than in the homogeneous granular model. Furthermore, saturation of the nonwetting phase mapped on the Ca-M diagram significantly depends on the rock models. These important differences between two-phase flow in 3D natural rock and in 2D homogeneous models could be due to the heterogeneity of pore geometry in the natural rock and differences in pore connectivity. By quantifying two-phase fluid behavior in the target reservoir rock under various conditions (e.g., saturation mapping on the Ca-M diagram), our approach could provide useful information for investigating suitable reservoir conditions for geo-fluid management (e.g., high CO2 saturation in CO2 storage).
... In previous studies on viscous fingers in porous media [49,68], it was derived that the interface growth should go like v ∝ ∇P − ∇P c , where the thresholds there are due to capillary pressures at the fluid-fluid interface. For moderate capillary numbers and with disorder in the thresholds (i.e., disordered porous media), the authors of [49,68] calculated fractal dimensions with values around 1.5-1.6 for the patterns and derived that the growth in this regime is better described by the dielectric breakdown model (DBM), where v ∝ ( ∇P ) η , with η = 2, rather than DLA, where v ∝ ∇P . ...
... In previous studies on viscous fingers in porous media [49,68], it was derived that the interface growth should go like v ∝ ∇P − ∇P c , where the thresholds there are due to capillary pressures at the fluid-fluid interface. For moderate capillary numbers and with disorder in the thresholds (i.e., disordered porous media), the authors of [49,68] calculated fractal dimensions with values around 1.5-1.6 for the patterns and derived that the growth in this regime is better described by the dielectric breakdown model (DBM), where v ∝ ( ∇P ) η , with η = 2, rather than DLA, where v ∝ ∇P . However, for higher capillary numbers where a disorder in the thresholds is less significant, the growth was found to resemble DLA (the DBM with η = 1). ...
Article
Full-text available
... Extensions to the Lenormand diagram have addressed the effects of gravity [16], pore-scale disorder [17,18], fluid compressibility [3,19], the crossover between regimes including the existence of a Ca-dependent crossover length scale [17,18,[20][21][22], and frictional forces between grains leading to a whole new set of displacement patterns [3,[23][24][25][26][27]]. Yet, this wealth of observations has been restricted to a drainage process. ...
... Extensions to the Lenormand diagram have addressed the effects of gravity [16], pore-scale disorder [17,18], fluid compressibility [3,19], the crossover between regimes including the existence of a Ca-dependent crossover length scale [17,18,[20][21][22], and frictional forces between grains leading to a whole new set of displacement patterns [3,[23][24][25][26][27]]. Yet, this wealth of observations has been restricted to a drainage process. ...
Article
The authors experimentally identify the stabilizing effect of wettability in a porous matrix during the immiscible displacement of a viscous fluid (such as oil) by a much less viscous one (such as water). By altering the wettability of the medium from drainage (contact angle $\ensuremath{\theta}$ = 5\ifmmode^\circ\else\textdegree\fi{}) to imbibition ($\ensuremath{\theta}$ = 120\ifmmode^\circ\else\textdegree\fi{}), the classical viscous-fingering instability is stabilized and even completely suppressed at low capillary numbers (i.e. low injection rates), making the invading fluid better at pushing out the defending fluid. These results have implications for improving oil recovery, CO${}_{2}$ sequestration, and fuel-cell design.
... We thus expect to address flow phenomenologies that are perfectly analogous to viscous fingering at large M in random two-dimensional porous media. The properties of invasion patterns obtained in such configuration, such as the one shown in Figure 5a, are well known, in particular we expect a fractal dimension close to 1.62 (as first measured by Måløy et al., 1985, with a 0.04 uncertainty), and the dynamics of the growth process has been studied in detail by Løvoll et al. (2004) and Toussaint et al. (2005). From eight different porous media of identical statistical properties, we measure the average box-counting curve shown in Figure 5b, which yields a fractal dimension of 1.61 ± 0.03, in excellent agreement with the value from the literature. ...
... From eight different porous media of identical statistical properties, we measure the average box-counting curve shown in Figure 5b, which yields a fractal dimension of 1.61 ± 0.03, in excellent agreement with the value from the literature. From the eight independent numerical runs we also measure the average map of occupancy probabilities (x, y) for the displacing fluid in the reference frame attached to the tip of its most advanced finger (see Toussaint et al., 2005). A cut of the (x, y) topography (see Figure 5d) at its half-maximum value provides the shape of the envelope of the flow pattern, which looks similar to a Saffman-Taylor finger but with a width 0.39 W, where W is the width of the medium, as evidenced from the mean transverse cut of the topography presented in Figure 5d. ...
Article
Full-text available
We develop an efficient computational model for simulating fluid invasion patterns emerging in variable aperture fractures. This two-dimensional model takes into account the effect of capillary force on the fluid-fluid interfaces and viscous pressure drop in both fluid phases. The pressure distribution is solved at each time step based on mass balance and local cubic law, considering an imposed pressure jump condition at the fluid-fluid interface. This pressure jump corresponds to the Laplace pressure which includes both terms related to the out-of-plane (aperture-spanning) curvature and to the in-plane curvature. Simulating a configuration that emulates viscous fingering in two-dimensional random porous media confirms that the model accounts properly for the role of viscous forces. Furthermore, direct comparison with previously obtained experimental results shows that the model reproduces the observed drainage patterns in a rough fracture reasonably well. The evolutions of tip location, the inlet pressures, and the invading phase fractal dimensions are analyzed to characterize the transition from capillary fingering to viscous fingering regimes. A radial injection scenario of immiscible invasion is also studied with varying modified capillary number and viscosity ratio, showing displacement patterns ranging from capillary fingering to viscous fingering to stable displacement. Such simulations using two contact angles show that the invading phase becomes more compact when the wetting condition changes from strong to weak drainage, as already observed in 2-D porous media. The model can be used to bridge the gap in spatial scales of two-phase flow between pore-scale modeling approaches and the continuum Darcy-scale models.
... However, when M 1 for a range of intermediate outflow velocities the flow is dominated by the capillary forces on a smaller scale and the viscous forces on the larger scale. Consequently, the invading patterns created in an intermediate regime demonstrate characteristics of both, capillary and viscous fingering depending on the spatial scale at which they are studied [16]. The critical spatial length-scale between the regimes is defined as l c = a/Ca [16,14]. ...
... Consequently, the invading patterns created in an intermediate regime demonstrate characteristics of both, capillary and viscous fingering depending on the spatial scale at which they are studied [16]. The critical spatial length-scale between the regimes is defined as l c = a/Ca [16,14]. The invading structure patterns on the scales lesser than l c exhibit the characteristics of the capillary fingering while on the scales larger than l c the invading clusters closely resemble a viscous fingering pattern. ...
... Two-phase flow means simultaneous flow of two fluids in the same space. When an immiscible fluid is injected into a porous medium filled with another fluid, different transient flow mechanisms occur depending on the flow conditions, such as capillary fingering (Lenormand and Zarcone 1989), viscous fingering (Toussaint et al. 2005;Måløy et al. 1985;Løvoll et al. 2004) and stable displacement (Frette et al. 1997;Méheust et al. 2002). After the transient flow mechanisms have surpassed, steady state sets in, which is the regime in which the rheology of two-phase flow under different wetting conditions is examined in this work. ...
Preprint
Full-text available
Immiscible two-phase flow in porous media with mixed wet conditions was examined using a capillary fiber bundle model, which is analytically solvable, and a dynamic pore network model. The mixed wettability was implemented in the models by allowing each tube or link to have a different wetting angle chosen randomly from a given distribution. Both models showed that mixed wettability can have significant influence on the rheology in terms of the dependence of the global volumetric flow rate on the global pressure drop. In the capillary fiber bundle model, for small pressure drops when only a small fraction of the tubes were open, it was found that the volumetric flow rate depended on the excess pressure drop as a power law with an exponent equal to 3/2 or 2 depending on the minimum pressure drop necessary for flow. When all the tubes were open due to a high pressure drop, the volumetric flow rate depended linearly on the pressure drop, independent of the wettability. In the transition region in between where most of the tubes opened, the volumetric flow depended more sensitively on the wetting angle distribution function and was in general not a simple power law. The dynamic pore network model results also showed a linear dependence of the flow rate on the pressure drop when the pressure drop is large. However, out of this limit the dynamic pore network model demonstrated a more complicated behaviour that depended on the mixed wettability condition and the saturation. In particular, the exponent relating volumetric flow rate to the excess pressure drop could take on values anywhere between 1.0 and 1.8. The values of the exponent were highest for saturations approaching 0.5, also, the exponent generally increased when the difference in wettability of the two fluids were larger and when this difference was present for a larger fraction of the porous network.
... The frictional fingers develop in the quasistatic limit, where we can neglect the viscous forces. Second, unlike viscous fingers in porous media, which are known to display a fractal interface geometry [8,9,29,30], for frictional fingers we can identify a characteristic length, the finger width. While crossover behavior from frictional to viscous fingers has been observed as the driving rate is increased [19], we focus here on the quasistatic limit where static frictional forces dominate. ...
Article
Experiments on confined two-phase flow systems, involving air and a dense suspension, have revealed a diverse set of flow morphologies. As the air displaces the suspension, the beads that make up the suspension can accumulate along the interface. The dynamics can generate "frictional fingers" of air coated by densely packed grains. We present here a simplified model for the dynamics together with a new numerical strategy for simulating the frictional finger behavior. The model is based on the yield stress criterion of the interface. The discretization scheme allows for simulating a larger range of structures than previous approaches. We further make theoretical predictions for the characteristic width associated with the frictional fingers, based on the yield stress criterion, and compare these to experimental results. The agreement between theory and experiments validates our model and allows us to estimate the unknown parameter in the yield stress criterion, which we use in the simulations.
... 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]. ...
Article
Full-text available
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.
... The use of quasi-2-D porous networks allows for a simplified experimental setup with the benefit of immediate visualization of the flow structures and real-time dynamics. A series of experimental studies has been performed in such systems using Hele-Shaw cells [Hele-Shaw, 1898] and variants to ensure the quasi-2-D geometry of the flow [Måløy et al., 1985;M eheust et al., 2002;Løvoll et al., 2005;Toussaint et al., 2005]. Figure 1 shows a diagram of the experimental setup. ...
Article
In this paper we study the influence of sample geometry on the measurement of pressure-saturation relationships, by analyzing the drainage of a two-phase flow from a quasi-2D random porous medium. The medium is transparent, which allows for the direct visualization of the invasion pattern during flow, and is initially saturated with a viscous liquid (a dyed glycerol-water mix). As the pressure in the liquid is gradually reduced, air penetrates from an open inlet, displacing the liquid which leaves the system from an outlet on the opposite side. Pressure measurements and images of the flow are recorded and the pressure-saturation relationship is computed. We show that this relationship depends on the system size and aspect ratio. The effects of the system's boundaries on this relationship are measured experimentally and compared with simulations produced using an invasion percolation algorithm. The pressure build up at the beginning and end of the invasion process are particularly affected by the boundaries of the system whereas at the central part of the model (when the air front progresses far from these boundaries), the invasion happens at a statistically constant capillary pressure. These observations have led us to propose a much simplified pressure-saturation relationship, valid for systems that are large enough such that the invasion is not influenced by boundary effects. The properties of this relationship depend on the capillary pressure thresholds distribution, sample dimensions and average pore connectivity and its applications may be of particular interest for simulations of two-phase flow in large porous media. This article is protected by copyright. All rights reserved.
... In practice, IP models have mainly been used for slow drainage in porous and fractured media, this is because the use of the IP for imbibition is not well justifiable, due to additional physical mechanisms (e.g., film flow and snap-off) which often accompany imbibition and are not taken into account in IP. I refer the reader to Løvoll et al. (2005) and Toussaint et al. (2005Toussaint et al. ( , 2012 for extensive discussions of various invasion scenarios. From an algorithmic point of view, there is no difference between bond IP and site IP and they can be modeled using a unique IP algorithm on appropriate lattices (e.g. ...
Article
I present a fast algorithm for modeling invasion percolation (IP) with trapping (TIP). IP is a numerical algorithm that models quasi-static (i.e. slow) fluid invasion in porous media. Trapping occurs when the invading fluid (that is injected) forms continuous surfaces surrounding patches of the displaced fluid (that is assumed incompressible and originally saturates the invaded medium). In TIP, the invading fluid is not allowed to enter the trapped patches. I demonstrate that TIP can be modeled in two steps: (1) Run an IP simulation without trapping (NTIP). (2) Identify the sites that invaded trapped regions and remove them from the chronological list of sites invaded in NTIP. Fast algorithms exist for solving NTIP. The focus of this paper is to propose an efficient solution for step (2). I show that it can be solved using a disjoint set data structure and going backward in time, i.e. by un-invading all sites invaded in NTIP in reverse order. Time reversal of the invasion greatly reduces the computational complexity for the identification of trapped sites as one only needs to investigate sites neighbor to the latest invaded/un-invaded site. This differs from traditional approaches where trapping is performed in real time, i.e. as the IP simulation is running, and where it is sometimes necessary to investigate the whole lattice to identify newly trapped regions. With the proposed algorithm, the total computational time for the identification and the removal of trapped sites goes as O(N), where N is the total number of sites in the lattice.
... The well-documented viscous fingering regime [2,3,38,39,42] is seen on the right column, where the fingers get thinner and more pronounced for higher values of injecting air pressure. This is expected, as the higher the pressure, the higher the velocities involved which in its turn translate into higher capillary numbers and smaller finger widths [43,44]. ...
Article
Full-text available
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.
... In practice, IP models have mainly been used for slow drainage in porous and fractured media, this is because the use of the IP for imbibition is not well justifiable, due to additional physical mechanisms (e.g., film flow and snap-off) which often accompany imbibition and are not taken into account in IP. I refer the reader to Løvoll et al. (2005Løvoll et al. ( ) and Toussaint et al. (2005Løvoll et al. ( , 2012) for extensive discussions of various invasion scenarios. From an algorithmic point of view, there is no difference between bond IP and site IP and they can be modeled using a unique IP algorithm on appropriate lattices (e.g. ...
... Two-phase flow means simultaneous flow of two fluids in the same space. When an immiscible fluid is injected into a porous medium filled with another fluid, different transient flow mechanisms occur depending on the flow conditions, such as capillary fingering (Lenormand and Zarcone 1989), viscous fingering (Toussaint et al. 2005;Måløy et al. 1985;Løvoll et al. 2004) and stable displacement (Frette et al. 1997;Méheust et al. 2002). After the transient flow mechanisms have surpassed, steady state sets in, which is the regime in which the rheology of two-phase flow under different wetting conditions is examined in this work. ...
Article
Full-text available
Immiscible two-phase flow in porous media with mixed wet conditions was examined using a capillary fiber bundle model, which is analytically solvable, and a dynamic pore network model. The mixed wettability was implemented in the models by allowing each tube or link to have a different wetting angle chosen randomly from a given distribution. Both models showed that mixed wettability can have significant influence on the rheology in terms of the dependence of the global volumetric flow rate on the global pressure drop. In the capillary fiber bundle model, for small pressure drops when only a small fraction of the tubes were open, it was found that the volumetric flow rate depended on the excess pressure drop as a power law with an exponent equal to 3/2 or 2 depending on the minimum pressure drop necessary for flow. When all the tubes were open due to a high pressure drop, the volumetric flow rate depended linearly on the pressure drop, independent of the wettability. In the transition region in between where most of the tubes opened, the volumetric flow depended more sensitively on the wetting angle distribution function and was in general not a simple power law. The dynamic pore network model results also showed a linear dependence of the flow rate on the pressure drop when the pressure drop is large. However, out of this limit the dynamic pore network model demonstrated a more complicated behavior that depended on the mixed wettability condition and the saturation. In particular, the exponent relating volumetric flow rate to the excess pressure drop could take on values anywhere between 1.0 and 1.8. The values of the exponent were highest for saturations approaching 0.5, also, the exponent generally increased when the difference in wettability of the two fluids were larger and when this difference was present for a larger fraction of the porous network.
... Physical experiments can be effective methods to study the process of oil migration in porous media (e.g., [7,8]). Early experiments were limited by observation techniques; ...
Article
Full-text available
Subsurface migration and accumulation of oil and gas have interested researchers for a long time, but these processes may occur over very long geological periods and are difficult to observe directly, so experimental simulations are warranted. In this study, an experimental method was developed to model hydrocarbon migration in the subsurface structure. Oil migration was simulated in a sandbox model, and industrial CT scanning was used to observe both the internal geometry of the model and the oil migration pathways. In the sandbox model, a NaI solution was used to simulate water, white oil was used to simulate hydrocarbon, and fine quartz sand, glass bead, silica powder, and brown corundum were chosen to represent brittle crust, based on suitable material parameters. A NaI-saturated layered sandbox model was constructed with an along-strike basal discontinuity, which during compression allowed a simple anticline with doubly verging reverse faults to form. Oil was then released continuously at a low rate from an orifice under one limb of the anticline. Initially, the oil migrated vertically through the fault zone until it reached the top of the fault zone; it then migrated laterally along the core of the anticline, saturating a model reservoir by buoyancy and capillary force. This experimental analog helps to explain hydrocarbon migration and accumulation within the Anjihai and Santai anticlines in northwest China.
... fluctuations driven by spatial and temporal disorder on the growth and displacement dynamics. Phase saturation, or occupancy maps [12][13][14], on the other hand, provide macroscopic descriptions of the interface evolution. For (1 + 1)-dimensional interfaces, phase saturation S(z,t) at the longitudinal position z and time t is defined in terms of the interface height, z = H (x,t), as ...
Article
We study the macroscopic representation of noise-driven interfaces in stochastic interface growth models in (1+1) dimensions. The interface is characterized macroscopically by saturation, which represents the fluctuating sharp interface by a smoothly varying phase field with values between 0 and 1. We determine the one-point interface height statistics for the Edwards-Wilkinson (EW) and Kadar-Paris-Zhang (KPZ) models in order to determine explicit deterministic equations for the phase saturation for each of them. While we obtain exact results for the EW model, we develop a Gaussian closure approximation for the KPZ model. We identify an interface compression term, which is related to mass transfer perpendicular to the growth direction, and a diffusion term that tends to increase the interface width. The interface compression rate depends on the mesoscopic mass transfer process along the interface and in this sense provides a relation between meso- and macroscopic interface dynamics. These results shed light on the relation between mesoscale and macroscale interface models, and provide a systematic framework for the upscaling of stochastic interface dynamics.
... The topic of fluid motion inside a porous network has deservedly been subjected to a considerable number of studies over the past decades. Scientists have studied the morphology and dynamics of the flow [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15] and proposed a set of numerical schemes able to reproduce the observed macroscopic patterns [16][17][18][19] and relevant pore-scale mechanisms [20][21][22][23][24][25][26][27]. The topic is also of central importance for the study of groundwater flows and the treatment of soil contaminants [28,29] as well as due to its immediate applications in the energy sector, for example, in hydrocarbon recovery methods [30,31]. ...
Article
Full-text available
The intermittent burst dynamics during the slow drainage of a porous medium is studied experimentally. We have verified a theoretically predicted scaling for the burst size distribution which was previously accessible only via numerical simulations. We show that this system satisfies a set of conditions known to be true for critical systems, such as intermittent activity with bursts extending over several time and length scales, self-similar macroscopic fractal structure and $1/f^\alpha$ power spectrum. The observation of $1/f^\alpha$ power spectra is new for porous media flows and, for specific boundary conditions, we notice the occurrence of a transition from $1/f$ to $1/f^2$ scaling. An analytically integrable mathematical framework was employed to explain this behavior.
... In both cases, gas escapes from a saturated granular matrix-seafloor sediments for pockmarks or dense particle suspension for mud volcanoes. Extensive modeling for gas injection in a saturated, rigid porous medium has been performed in the last 20 years, both with and without the presence of heterogeneities in the system (see for instance [2,[30][31][32][33][34][35], a tentative phase diagram by Geistlinger et al. [36], and references within). When the medium is deformable, however, complex patterns may arise from the interaction between the fluid flow and the grains motion [37,38]. ...
Article
Full-text available
The injection of gas in a liquid-saturated granular bed gives rise to a wide variety of invasion patterns. Many studies have focused on constrained porous media, in which the grains are fixed in the bed and only the interstitial fluid flows when the gas invades the system. With a free upper boundary, however, the grains can be entrained by the ascending gas or fluid motion, and the competition between the upward motion of grains and sedimentation leads to new patterns. We propose a brief review of the experimental investigation of the dynamics of air rising through a water-saturated, unconstrained granular bed, in both two and three dimensions. After describing the invasion pattern at short and long time, a tentative regime-diagram is proposed. We report original results showing a dependence of the fluidized zone shape, at long times, on the injection flow rate and grain size. A method based on image analysis makes it possible to detect not only the fluidized zone profile in the stationary regime, but also to follow the transient dynamics of its formation. Finally, we describe the degassing dynamics inside the fluidized zone, in the stationary regime. Depending on the experimental conditions, regular bubbling, continuous degassing, intermittent regime or even spontaneous flow-to-fracture transition are observed.
... In the case of drainage of a poorly viscous fluid displacing a highly viscous fluid, the viscosity contrast tends to destabilize the initially flat fluid/air interface. At a sufficient flow rate, this can lead to viscous fingering, i.e., localized flow at the scale of the setup (e.g., Saffman and Taylor, 1958;Lenormand, Touboul, and Zarcone, 1988;Lenormand, 1989;Løvoll et al. 2004;Toussaint et al. 2005;Løvoll et al. 2010;Toussaint et al. 2012). This feature can be classified by the capillary number Ca, corresponding to the ratio of the viscous to interfacial forces at the pore scale, which is defined as ...
Article
Streaming-potentials are produced by electrokinetic effects in relation to fluid flow and are used for geophysical prospecting. The aim of this study is to model streaming potential measurements for unsaturated conditions using an empirical approach. A conceptual model is applied to streaming potential measurements obtained from two drainage experiments in sand. The streaming potential data presented here show a non-monotonous behaviour with increasing water saturation, following a pattern that cannot be predicted by existing models. A model involving quasi-static and dynamic components is proposed to reproduce the streaming potential measurements. The dynamic component is based on the first time derivative of the driving pore pressure. The influence of this component is investigated with respect to fluid velocity, which is very different between the two experiments. The results demonstrate that the dynamic component is predominant at the onset of drainage in experiments with the slowest water flow. On the other hand, its influence appears to vanish with increasing drainage velocity. Our results suggest that fluid flow and water distribution at the pore scale have an important influence on the streaming potential response for unsaturated conditions. We propose to explain this specific streaming potential response in terms of the behaviour of both rock/water interface and water/air interfaces created during desaturation processes. The water/air interfaces are negatively charged, as also observed in the case of water/rock interfaces. Both the surface area and the flow velocity across these interfaces are thought to contribute to the non-monotonous behaviour of the streaming potential coefficient as well as the variations in its amplitude. The non-monotonous behaviour of air/water interfaces created during the flow was highlighted as it was measured and modelled by studies published in the literature. The streaming potential coefficient can increase to about 10 to 40 when water saturation decreases. Such an increase is possible if the amount of water/air interfaces is increased in sufficient amount, which can be the case.
... Scientists have studied the morphology and dynamics of porous media flows and proposed a set of numerical schemes able to reproduce the observed macroscopic patterns [2,3,5,7,10,[45][46][47][48][49][50][51][52] and relevant pore-scale mechanisms [6,9,26,27,47,[53][54][55]. These studies have led to a deeper understanding of the pore-scale forces that are ultimately responsible for the macroscopic flow properties and finally to the possible upscaling of the results [6,8,48]. ...
Article
Full-text available
The intermittent dynamics of slow drainage flows in a porous medium is studied experimentally. This kind of two-phase flow is characterized by a rich burst activity and our setup allows us to characterize those bursts directly via images of the flow and pressure measurements. Two different boundary conditions were analyzed: controlled withdrawal rate (CWR) and controlled imposed pressure (CIP). We have characterized geometrical and statistical properties of the bursts from images and pressure measurements. We have shown that in spite of leading to similar final invasion patterns, some dynamical features of the invasion differ considerably between the CWR and CIP boundary conditions. In particular, their pressure signatures are very distinct, which then translates into very distinct features on the power spectrum density of the pressure signals. A fully integrable analytical framework is presented which successfully describes the scaling features of the power spectrum for the CIP case.
... Many studies have focused on the impact of the disorder on fluid displacement, showing that increasing disorder destabilizes the displacement front (J. D. Chen & Wilkinson, 1985;King, 1987;Liu et al., 2015;Toussaint et al. 2005; Z. Wang et al., 2019) and induces the transitions of displacement patterns (Holtzman & Juanes, 2010;Hu et al., 2019). For quasistatic displacement conditions, since multiphase flow is purely controlled by the capillarity, the effect of disorder on viscous force can be neglected. ...
Article
Full-text available
The flow of multiple immiscible fluids in disordered porous media is important in many natural processes and subsurface applications. The pore-scale disorder affects the fluid invasion pathways significantly and induces the transitions of displacement patterns in porous media. Extensive studies focus on pattern transitions affected by disorder under quasistatic or dynamic conditions, but how the disorder controls the pattern transitions from capillary-dominated regime to viscous-dominated regime is not well understood. Here, we combine microfluidic experiments and theoretical analysis to investigate the role of disorder in fluid displacement. We perform drainage experiments with four different disorders under six flow rate conditions and show that increasing disorder destabilizes displacement fronts for all flow rates considered. Based on the scaling analysis of pore-filling events, we propose a theoretical model that describes the pattern transitions from compact displacement to capillary to viscous fingering as functions of disorder and capillary number. The effects of disorder on both capillary and viscous forces are quantified within the theoretical model. The phase diagram predicted by this model agrees well with our experimental results. We further elucidate the role of disorder in fluid displacement via energy conversion and dissipation. We find that increasing disorder enhances the capillary instabilities and induces more energy dissipated in a capillary-dominated regime, with the dissipation ratio increasing from 28.3% to 56.7%. Our work extends the classic phase diagram to consider the effect of disorder and provides a better understanding of the impact of the disorder on flow behaviors by energy dissipation.
... As long as −∆p c is much larger than the characteristic width of the capillary pressure threshold distribution W t , nearly every pore will be invaded. The width of stable invasion fronts in quasi 2D experiments has been found to depend on the generalised fluctuation number F = −∆p c /W t with a transition to unstable fronts when F = 0 [24,[30][31][32][33]. In our experiments F = 0 corresponds to the situation where the invasion structure reaches the critical radius where the local fluctuations are felt, and the wider pores are preferred over narrower ones. ...
Preprint
Full-text available
We present experiments and theory describing the transition from viscosity-stabilized flow to gravitationally unstable fingering for two-phase flow in a 3D synthetic porous medium. Observation is made possible by the use of our newly developed table-top 3D-scanner based on optical index matching and laser-induced fluorescence, which is described in detail. In the experiment, a more dense, more viscous fluid injected at a fixed flow-rate from a point source at the top of the flow cell displaces a less viscous, less dense fluid. We observe a stable invasion zone near the inlet, which increases in size with increasing flow rates, and presents initially a close to hemispherical shape. At later times, the invasion front transits to an unstable mode and a fingering flow regime. The transition occurs at a predicted critical radius, Rc, corresponding to the zero of the combined viscous and gravitational pressure gradient.
... In some classical instabilities, as Saffman-Taylor instability, the finger width as function of distance to tip is one of the most finely studied quantities [56]. The ramification and thickness of the viscous fingering patterns have been characterized in water and miscible non-newtonian fluid system [57] and in drainage in a porous medium [58]. In the fracture dissolution system, the simulation results show that the rougher the aperture, the more ramified dissolution fingers will be Frontiers in Physics | www.frontiersin.org ...
Article
Full-text available
Fractures play an important role as flow paths in porous media. When an undersaturated reactive fluid is flowing in a fracture, the dissolution process will alter the flow paths locally. Dissolution patterns grow slowly, but they may lead to a dramatic reorganization of the flow. Here, we study dissolution in a radial geometry, which is relevant for a number of practical applications, e.g., the acidization of oil reservoirs. Different dissolution patterns are presented in a phase diagram with the Péclet and Damköhler numbers as parameters. We also analyze quantitatively the density of wormholes and the relation between the aperture roughness and the ramified patterns.
... The multiphase interactions combining several factors (including intrinsic topological features, gravity, capillary, and wettability) determine macroscopic drainage properties, such as the residual saturation of wetting phase and the temporal/spatial distributions of saturated zones (Yang et al., 1988;Herring et al., 2016;Li et al., 2017). The majority of previous experiments and numerical models of drainage and injection are focusing on the macroscopic parameters, e.g., porosity, permeability, system size and aspect ratio (Succi et al., 1989;Toussaint et al., 2005;Babadagli et al., 2015;Moura et al., 2015;Rognmo et al., 2017;March et al., 2018), in which the spatial configuration, pore connectivity and their influences on liquid retention are ignored (Prat, 1995;Lin et al., 2018;Yekta et al., 2018). However, for most natural and synthetic porous media, disordered microstructures are dominance (Anguy et al., 2001;Woo et al., 2004). ...
Preprint
Full-text available
Multiphase flow through a porous medium involves complex interactions between gravity, wettability and capillarity during drainage process. In contrast to these factors, the effect of pore distribution on liquid retention is less understood. In particular, the quantitative correlation between the fluid displacement and level of disorder has not yet been established. In this work, we employ direct numerical simulation by solving the Navier-Stokes equations and using volume of fluid method to track the liquid-liquid interface during drainage in disordered porous media. The disorder of pore configuration is characterized by an improved index to capture small microstructural perturbation, which is pivotal for fluid displacement in porous media. Then, we focus on the residual volume and morphological characteristics of saturated zones after drainage and compare the effect of disorder under different wettability (i.e., the contact angle) and gravity (characterized by a modified Bond number) conditions. Pore-scale simulations reveal that the highly-disordered porous medium is favourable to improve liquid retention and provide various morphologies of entrapped saturated zones. Furthermore, the disorder index has a positive correlation to the characteristic curve index (n) in van Genuchten equation, controlling the shape of the retention characteristic curves. It is expected that the findings will benefit to a broad range of industrial applications involving drainage processes in porous media, e.g., drying, carbon sequestration, and underground water remediation.
... The most of the work focused on drainage and inhibitions processes which are generally termed as invasion processes. These were inherently transient processes give different displacement patterns and are classified as capillary fingering, viscous fingering [13,14], and stable front displacement [15]. Soo and Radke [5,16] performed comprehensive studies of emulsion flow through porous media. ...
Article
Flowof oil-in-water (o/w) emulsions throughporousmediawas studied experimentally.Details of the theoretical developments of effective viscosity of emulsions flowing through unconsolidated porous media were presented. The emulsion rheological data were coupledwith the hydraulic radius concept and the Navier-Stokes (NS) equation to account for the pressure gradient dependence of effective viscosity of emulsions in porous media. Thereafter, the emulsion flow through different type of porous media (packed bed with different size of particle diameter)was studied. The rheological behavior of o/wemulsions in porous bedwith oil volumefraction ranging from 10 to 80 % (v/v) was investigated. Emulsions behaved like non-Newtonian shear thinning (n<1) fluids. The effect of emulsion volume fraction, flowrate and number of pore volume (PV) injected on bed pressure drop and flow resistance of different types of porous mediawere studied and discussed. The observed blocking/resistance mechanisms could be attributed in part to the pressure drop-flow rate response of behavior of emulsion flow in porous media. These showed the potential use of emulsions as conformance agents.
... One consequence of an unstable flow at the pore scale is the fingering phenomenon, which is caused by unstable infiltration fronts breaking into flow fingers. In the past, numerous detailed laboratory studies have been conducted by Nsir et al. ([37] and also [21,30,33,46]) to describe the complex immiscible displacement of fluids in a porous medium, driven by buoyancy, viscous and capillary forces. Some common ratios used in viscous cases are usually defined as follows: ...
Article
Full-text available
In this paper, we develop an implementation of gravity effects within a Global Pressure formulation in a numerical scheme based on the implicit pressure explicit saturation (IMPES) approach. We use the Discontinuous Galerkin Finite-Element Method (DGFEM) combined with a generalised Godunov scheme to model an immiscible two-phase flow with predominant gravity effects. The saturation profile of a displacing non-aqueous phase liquid (NAPL) in an initially water-saturated porous medium depends strongly on the ratio between the total specific discharge and the density difference between the NAPL and water. We discuss the solution of the nonlinear Buckley-Leverett equation for the general case in which the flux function is non-monotonic. Using a detailed functional analysis of the characteristics of the given hyperbolic equation, three limit cases are identified as significant for modelling the shock and rarefaction regions. The derived maximum (or entry) and front saturations of NAPL are functions of the viscosity ratio M and the gravity number G. We first test the developed numerical model in the case of a one-dimensional highly gravity-driven flow of NAPL within a homogeneous porous medium. The numerically calculated NAPL entry and front saturations of NAPL agree well with the theoretical values. Furthermore, the numerical diffusion of the shock front is lower than that of the calculated using a first-order Finite Volume method, which is generally used in reservoir engineering because of its robustness. Finally, we apply the developed DGFEM scheme to a 2D heterogeneous porous medium and analyse its capability of modelling the non-uniform saturation field using spatial moment analysis.
... For high Ca, the resulting displacement patterns are reminiscent of diffusion limited aggregation (Witten et al. 1981;Daccord et al. 1986;Meakin et al. 1989;Niemeyer et al. 1984;Conti & Marconi 2010). For low Ca, the displacement dynamics becomes more intricate, and the emerging patterns display a strong dependence on the pore geometry (Lenormand & Zarcone 1985;Lenormand et al. 1983Lenormand et al. , 1988Fernandez et al. 1991;Måløy et al. 1992;Furuberg et al. 1996;Ferer et al. 2004;Toussaint et al. 2005;Holtzman et al. 2012) and the wettability of the medium, that is, the chemical affinity of the solid for each fluid (Stokes et al. 1986;Trojer et al. 2015;Zhao et al. 2016;Odier et al. 2017). In particular, an intermittent injection pressure signal emerges in the limit of low Ca (Furuberg et al. 1996;Måløy et al. 1992). ...
Preprint
We develop a novel moving capacitor' dynamic network model to simulate immiscible fluid-fluid displacement in porous media. Traditional network models approximate the pore geometry as a network of fixed resistors, directly analogous to an electrical circuit. Our model additionally captures the motion of individual fluid-fluid interfaces through the pore geometry by completing this analogy, representing interfaces as a set of moving capacitors. By incorporating pore-scale invasion events, the model reproduces, for the first time, both the displacement pattern and the injection pressure signal under a wide range of capillary numbers and substrate wettabilities. We show that at high capillary numbers the invading patterns advance symmetrically through viscous fingers. In contrast, at low capillary numbers the flow is governed by the wettability-dependent fluid-fluid interactions with the pore structure. The signature of the transition between the two regimes manifests itself in the fluctuations of the injection pressure signal.
... Because of the limitations of experimental model building and observation technique capability, it is difficult to design a proper 3-D model to simulate secondary migration processes. Most published experiments dealing with lateral migration were based on onedimensional or two-dimensional (2-D) models (Emmons, 1921;Lenormand et al., 1988;Catalan et al., 1992;Thomas and Clouse, 1995;Wagner et al., 1995Wagner et al., , 1997Meakin et al., 2000;Tokunaga et al., 2000;Zhang et al., 2003;Hou et al., 2004Hou et al., , 2005Luo et al., 2004;Løvoll et al., 2004Løvoll et al., , 2010Toussaint et al., 2005), which may not reflect the complexity of hydrocarbon lateral migration within the carrier-seal system. Therefore, the main objective in this study is to conduct a 3-D physical experiment using a relatively large box model that should be closer to reality than the previous experiments to investigate the characteristics of oil migration in a 3-D space. ...
Article
Full-text available
A three-dimensional physical experiment was conducted to study secondary oil migration under an impermeable inclined cap. Light-colored oil was released continuously at a slow rate of about 0.1 mL/min from a point at the base of an initially water-saturated porous model. With buoyancy as a primary driving force, a vertical cylindrical shape of an oil migration pathway was observed first, and then a layer-shaped lateral migration pathway was observed beneath the top inclined sealing plate once the oil cluster had reached the top cap. Magnetic resonance imaging was used to observe the migration processes-for example, morphology of the migration pathway, intermittency of oil bubbles, and variation of oil saturation within the migration paths. Results show that the snap-off phenomenon (related to fast local imbibition processes) occurred more commonly during vertical migration than it did during lateral migration. The lateral migration pathway that parallels to the top inclined cap has a typical vertical thickness of 2 to 4 cm (0.8-1.6 in.) (i.e., roughly 40-80 pores). This thickness is consistent with the prediction derived from scaling laws related to pore size and Bond number. Along the lateral migration direction, the sectional area and the horizontal width of the migration pathway fluctuate significantly, although the average oil saturation along the pathway remains almost the same. After stopping the initial oil injection, the sectional area of the migration pathway shrinks significantly. Therefore, we believe that this significant shrinking of the migration pathway is the main reason why only a relatively small volume of oil and gas has been lost during secondary migration.
... For high Ca, the resulting displacement patterns are reminiscent of diffusion limited aggregation (Witten et al. 1981;Daccord et al. 1986;Meakin et al. 1989;Niemeyer et al. 1984;Conti & Marconi 2010). For low Ca, the displacement dynamics becomes more intricate, and the emerging patterns display a strong dependence on the pore geometry (Lenormand & Zarcone 1985;Lenormand et al. 1983Lenormand et al. , 1988Fernandez et al. 1991;Måløy et al. 1992;Furuberg et al. 1996;Ferer et al. 2004;Toussaint et al. 2005;Holtzman et al. 2012) and the wettability of the medium, that is, the chemical affinity of the solid for each fluid (Stokes et al. 1986;Trojer et al. 2015;Zhao et al. 2016;Odier et al. 2017). In particular, an intermittent injection pressure signal emerges in the limit of low Ca (Furuberg et al. 1996;Måløy et al. 1992). ...
Article
We develop a novel ‘moving-capacitor’ dynamic network model to simulate immiscible fluid–fluid displacement in porous media. Traditional network models approximate the pore geometry as a network of fixed resistors, directly analogous to an electrical circuit. Our model additionally captures the motion of individual fluid–fluid interfaces through the pore geometry by completing this analogy, representing interfaces as a set of moving capacitors. By incorporating pore-scale invasion events, the model reproduces, for the first time, both the displacement pattern and the injection-pressure signal under a wide range of capillary numbers and substrate wettabilities. We show that at high capillary numbers the invading patterns advance symmetrically through viscous fingers. In contrast, at low capillary numbers the flow is governed by the wettability-dependent fluid–fluid interactions with the pore structure. The signature of the transition between the two regimes manifests itself in the fluctuations of the injection-pressure signal.
... Petitjeans [12] and Lajeunesse et al. [13] investigated the displacement flow of a high-viscosity fluid by a low-viscosity one in a Hele-Shaw cell experimentally and observed the changes of instability patterns. In addition, many other researchers [14][15][16][17] have also investigated the fingering phenomenon in experimental studies. ...
Article
In this paper, the viscous fingering phenomena of two immiscible fluids with a large viscosity ratio was simulated by the Lattice Boltzmann method. The Rothman–Keller Lattice Boltzmann model was applied to study the viscous fingering phenomena in a microchannel where the high viscosity fluids were displaced by low viscosity fluids. We have investigated the influences of parameters such as viscosity ratio (M), surface wettability, capillary number (Ca), and Reynolds number (Re) on finger structures, breakthrough time (Ts), and areal sweep efficiency (Se). In particular, the effects of surface tension and large viscosity ratio on the phenomenon of fluid accumulation were intensively studied. The simulation results showed that the fluid accumulation became more obvious gradually with the increase of M, which led to more serious displacement effects. Moreover, Se increased as the contact angle increased. Besides, as the viscous fingering phenomenon weakened, the phenomenon of fluid accumulation became more evident. Furthermore, the finger pattern had a tendency to increase as the value of Ca and Re increased, and the phenomenon of fluid accumulation decreased with the decrease of Ts and Se.
... They predicted that the fractal dimension of CF in 3D porous media is about 2.50 and 2.55, respectively, which is similar to our results. Other reports of fractal dimension mainly relate to 2D micromodel systems ( Chen et al., 2017 ;Holtzman and Juanes, 2010 ;Islam et al., 2014 ;Løvoll et al., 2004 ;Toussaint et al., 2005 ), with the fractal dimension of VF reported as being between 1.37 and 1.72 and of CF between 1.61 and 1.89. For comparison, all of the reported fractal dimensions for both 2D and 3D systems are given in Table 3 . ...
Article
Fluid displacement experiments were performed with immiscible fluids in a packed bed of glass spheres. The three-dimensional (3D) structure of the fingering pattern was visualized by computer tomography (CT) for a range of capillary numbers (Ca) between 5.22 × 10⁻⁷ and 1.04 × 10⁻⁴ at the viscosity ratio of log M = −2.203 in the absence of the influence of buoyancy force. Based on the 3D CT images, the characteristics of the fingering pattern at the crossover from viscous fingering (VF) to capillary fingering (CF) were investigated quantitatively. The structure of the fingering pattern changed gradually from streak-like for VF to more compact for CF with a decrease in Ca. Associated with the crossover from VF to CF, saturation of the invading phase (IP) and the fractal number of the structure both increased. The invaded pore size and invaded throat size distributions were also estimated for all values of Ca. The invaded pore size distribution was similar for all Ca values, but a clear shift in the throat size distribution was observed with the crossover from VF to CF.
... As long as − Δp c is negative, with a magnitude much larger than the width of the capillary pressure distribution W t nearly every pore will be invaded and the interface is stable. The width of stable invasion fronts in quasi 2D experiments has been found to depend on the generalised fluctuation number F = −Δp c / W t with a transition to unstable fronts when F = 0 [23,[29][30][31][32]. In our experiments F = 0 corresponds to the situation where the invasion structure reaches the critical radius where the local fluctuations are felt, and the wider pores are preferred over narrower ones. ...
Article
Full-text available
We present experiments and theory describing the transition from viscosity-stabilized flow to gravitationally unstable fingering for two-phase flow in a cubic box, filled with a synthetic porous medium. Observation is made possible by the use of our newly developed table-top 3D scanner based on optical index matching and laser-induced fluorescence, which is described in detail. In the experiment, a more dense, more viscous fluid injected at a fixed flow rate from a point source at the top of the flow cell displaces a less viscous, less dense fluid. We observe a stable invasion zone near the inlet, which increases in size with increasing flow rates, and presents initially a close to hemispherical shape. At later times, the invasion front transits to an unstable mode and a fingering flow regime. The transition occurs at a predicted critical radius, R c , corresponding to the zero of the combined viscous and gravitational pressure gradient.
Article
In terms of the robustness and accuracy of the TAIM stability limits, our analysis and computational results indicate that honoring the divergence of the total-velocity in the linearized system of coupled mass and energy conservation equations is more important than accounting for the rock and fluids compressibility effects. Moreover, we demonstrate through scaling analysis and numerical examples that for most problems of practical interest, a simple stability criterion obtained by assuming incompressible multiphase flow is quite robust. The relationship between the full and simplified stability criteria is analyzed in detail. The methodology is demonstrated using several thermal–compositional examples, including Steam Assisted Gravity Drainage (SAGD).
Thesis
Full-text available
This thesis is an experimental study of flow and transformation of porous media, where we study both fast and slow transformation of the media due to fluid flow. The fast process is mechanical deformation and channel formation due to high fluid pressure, and the slow process is chemical evolution of fractures. In the study of fast deformation, we perform experiments where air is injected at a constant overpressure into a saturated or dry granular medium. From recorded images, we characterize Saffman-Taylor like invasion patterns, surrounding deformation of the medium, flow regimes, and channel growth dynamics. Pore pressure is evaluated numerically, and used to characterize the rheology. In the study of slow transformation, we perform experiments where distilled water is injected at a constant flow rate through a fractured chalk sample. By comparing fracture apertures measured before and after experiments, we study the evolution of fractures for different durations of reactive flow.
Article
Full-text available
The dynamics of solute dispersion and mixing in unsaturated flows is analyzed from photobleaching experiments in two-dimensional porous micromodels. This technique allows producing pulse line (delta-Dirac) injections of a conservative tracer by bleaching a finite volume of fluorescent without disturbing the flow field. The temporal evolution of the concentration field and the spatial distribution of the air and water phases can be monitored at pore scale. We study the dispersion and mixing of a line of tracer under different water saturations. While dispersion in saturated porous media follows an approximately Fickian scaling, a shift to ballistic scaling is observed as soon as saturation is lowered. Hence, at the time scale of observation, dispersion in our unsaturated flows is dominated by the ballistic separation of tracer blobs within the water phase, between trapped clusters and preferential flow paths. While diffusion plays a minor role in the longitudinal dispersion during the time scale of the experiments, its interplay with fluid deformation is apparent in the dynamics of mixing. The scalar dissipation rates show an initial stretching regime, during which mixing is enhanced by fluid deformation, followed by a dissipation regime, during which diffusion overcomes compression induced by stretching. The transition between these two regimes occurs at the mixing time, when concentration gradients are maximum. We propose a predictive analytical model, based on shear-enhanced diffusion, that captures the dynamics of mixing from basic unsaturated porous media parameters, suggesting that this type of model may be a useful framework at larger scales.
Article
We study experimentally the displacement of one fluid by another in a granular pack to uncover relationships between fluid invasion and medium deformation. We develop an experimental setup that allows us to reconstruct the coupled invasion-deformation dynamics in 3D. We simultaneously characterize the fluid invasion pattern and document a transition from fluid-fluid displacement in pores to the formation of conduits by grain motion. We rationalize the findings in terms of a simple poromechanics model that indeed captures this transition as a result of the balance between viscous and frictional forces. These results contribute to elucidating the role of three dimensionality in the timing, mode, and morphology of fluid-fluid displacement and injection-induced deformation in porous media.
Article
Complex fluid flow in porous media is ubiquitous in many natural and industrial processes. Direct visualization of the fluid structure and flow dynamics is critical for understanding and eventually manipulating these processes. However, the opacity of realistic porous media makes such visualization very challenging. Micromodels, microfluidic model porous media systems, have been developed to address this challenge. They provide a transparent interconnected porous network that enables the optical visualization of the complex fluid flow occurring inside at the pore scale. In this Review, the materials and fabrication methods to make micromodels, the main research activities that are conducted with micromodels and their applications in petroleum, geologic, and environmental engineering, as well as in the food and wood industries, are discussed. The potential applications of micromodels in other areas are also discussed and the key issues that should be addressed in the near future are proposed.
Article
We consider the problem of viscous fingering in the presence of quenched disorder, that is both weak and short-range correlated. The two-point correlation function of the harmonic measure is calculated perturbatively, and is used in order to calculate the correction and the box-counting fractal dimension. We show that the disorder increases the fractal dimension, and that its effect decreases logarithmically with the size of the fractal.
Article
CO2 sequestration in saline aquifers is considered to be a promising method for climate mitigation. Studies of the subsurface flow mechanisms of CO2–brine are important for evaluation of sequestration potential and security. However, this process is complicated by the intrinsic heterogeneity of reservoir rocks. The main objective of this study was to identify CO2 distribution and saturation related to changing injection conditions, and to comprehensively compare the characteristics of different drainage processes, under small scale heterogeneous condition. An X-ray computed tomography machine and a micromodel were introduced to investigate the CO2/brine drainage process from pore to core scale in a wide range of injection rates at static and transient states, respectively. Four types of drainage experiments were conducted. Horizontal and vertical (upward) flow directions in a micromodel setup at ambient conditions and vertical (upward and downward) flow directions in a core-flood setup at in-situ conditions. Based on experimental results, it is found that, a higher injection rate gives rise to a higher displacement efficiency but a lower sweep efficiency. The stability of CO2 displacing front become weak with drainage development. The difference in pore structure leads to different CO2 saturation variation with CO2 injection rates change. Even the sand pack is almost homogenous at core scale, the impact of heterogeneous porosity is enhanced with increasing flow rates in upward drainage, while it weakens in downward drainage.
Article
In this perspective we provide a brief overview of the state of knowledge and recent progress in the area of multiphase flow through deformable granular media. We show, with many examples, that the interplay between viscous, capillary, and frictional forces at the pore scale determines the mode of fluid invasion. We pay particular attention to the central role of wettability on the morphology of granular-pack deformation and failure. Beyond their intrinsic interest as processes that give rise to spectacular pattern formation, these coupled phenomena in granular media can control continental-scale fluxes like methane venting from the seafloor and geohazards like earthquakes and landslides. We conclude this perspective by pointing to fundamental knowledge gaps and exciting avenues of research.
Article
Imbibition is the process by which the wetting fluid displaces the non-wetting fluid driven by capillary force. It occurs in many natural and industrial processes, such as enhanced oil recovery and geological carbon sequestration. The imbibition process is highly affected by wettability, viscosity ratio and injection flow rate, and the competition between these factors becomes more complicated when pore-size disorder is involved. In this study, we systematically investigate forced imbibition in four two-dimensional (2D) porous media with different disorders over a broad range of wettability conditions and flow rates. We analyze the relationship between the interfacial length and the invading fluid saturation, and the relationship between displacement front and invading fluid saturation. Results show that disorder and wettability have different impacts on the imbibition process under different capillary numbers. At low capillary number, displacement process is not sensitive to disorder due to the contradicting effects and the imbibition process is governed by the wettability. At intermediate capillary number, only the largest disorder has obvious influence on the displacement process due to the highest entry pressure. The transition from capillary fingering to viscous fingering occurs leading to a higher displacement efficiency. At high capillary number, disorder has a great influence on the front morphology and displacement efficiency while the wettability has relatively weak impact. The pattern transition is also observed at strong imbibition because of the competing effects between the capillary force and the viscous force.
Article
Transient circular pores can open in plasma membrane of cells due to mechanical stress, and failure to repair such pores lead to cell death. Similar pores in the form of defects also exist among smectic membranes, such as in myelin sheaths or mitochondrial membranes. The formation and growth of membrane defects are associated with diseases, for example multiple sclerosis. A deeper understanding of membrane pore dynamics can provide a more refined picture of membrane integrity-related disease development, and possibly also treatment options and strategies. Pore dynamics is also of great importance regarding healthcare applications such as drug delivery, gene or as recently been implied, cancer therapy. The dynamics of pores significantly differ in stacks which are confined in 2D compared to those in cells or vesicles. In this short review, we will summarize the dynamics of different types of pores that can be observed in biological membranes, which include circular transient, fusion and hemi-fusion pores. We will dedicate a section to floral and fractal pores which were discovered a few years ago and have highly peculiar characteristics. Finally, we will discuss the repair mechanisms of large area pores in conjunction with the current cell membrane repair hypotheses.
Article
Full-text available
This review is an expository treatment of the displacement of one fluid by another in a two-dimensional geometry (a Hele-Shaw cell). The Saffman-Taylor equations modeling this system are discussed. They are simulated by random-walk techniques and studied by methods from complex analysis. The stability of the generated patterns (fingers) is studied by a WKB approximation and by complex analytic techniques. The primary conclusions reached are that (a) the fingers are linearly stable even at the highest velocities, (b) they are nonlinearly unstable against noise or an external perturbation, the critical amplitude for the noise being an exponential function of a power of the velocity for high velocities, (c) such exponentials seem to dominate high-velocity behavior, as can be seen from a WKB analysis, and (d) the results of the Saffman-Taylor equations disagree with experiments, apparently because they leave out film-flow phenomena.
Article
Full-text available
We study the viscous fingering or Saffman–Taylor instability in two different dilute or semi-dilute polymer solutions. The different solutions exhibit only one non-Newtonian property, in the sense that other non-Newtonian effects can be neglected. The viscosity of solutions of stiff polymers has a strong shear rate dependence. Relative to Newtonian fluids, narrower fingers are found for rigid polymers. For solutions of flexible polymers, elastic effects such as normal stresses are dominant, whereas the shear viscosity is almost constant. Wider fingers are found in this case. We characterize the non-Newtonian flow properties of these polymer solutions completely, allowing for separate and quantitative investigation of the influence of the two most common non-Newtonian properties on the Saffman–Taylor instability. The effects of the non-Newtonian flow properties on the instability can in all cases be understood quantitatively by redefining the control parameter of the instability.
Article
Full-text available
Diffusion-limited aggregation clusters and the structures observed in viscous fingering experiments at high capillary numbers are tree-like fractals. The different branches may be assigned a branch order in a way that exhibits scaling, and permits a self-similar characterisation in terms of the bifurcation ratio rN and the length ratio rL of branches of different orders. The fractal dimension is given by D=ln(rN)/ln(1/rL). Good agreement between experiments and simulations is found. A crossover function characterises the branch orders.
Article
Full-text available
A wetting fluid is displaced at very low flow rate by a nonwetting fluid in a 250 000-duct transparent etched network. The structure formed by the injected fluid is ramified and the scale invariance is described by a measured fractal dimension 1.80
Article
Full-text available
Conformal mapping models are used to study the competition of noise and anisotropy in Laplacian growth. For this purpose, a family of models is introduced with the noise level and directional anisotropy controlled independently. Fractalization is observed in both anisotropic growth and growth with varying noise. The fractal dimension is determined from the cluster size scaling with cluster area. For isotropic growth d=1.7, at both high and low noise. For anisotropic growth with reduced noise the dimension can be as low as d=1.5 and apparently is not universal. Also, we study the fluctuations of particle areas and observe, in agreement with previous studies, that exceptionally large particles may appear during growth, leading to pathologically irregular clusters. This difficulty is circumvented by using an acceptance window for particle areas.
Article
Full-text available
We show that the fractal growth described by the dielectric breakdown model exhibits a phase transition in the multifractal spectrum of the growth measure. The transition takes place because the tip splitting of branches forms a fixed angle. This angle is eta dependent but it can be rescaled onto an "effectively" universal angle of the diffusion-limited aggregation branching process. We derive an analytic rescaling relation which is in agreement with numerical simulations. The dimension of the clusters decreases linearly with the angle and the growth becomes non-ractal at an angle close to 74 degrees (which corresponds to eta = 4.0+/-0.3).
Article
Full-text available
We performed extensive numerical simulation of diffusion-limited aggregation in two-dimensional channel geometry. Contrary to earlier claims, the measured fractal dimension D=1.712+/-0.002 and its leading correction to scaling are the same as in the radial case. The average cluster, defined as the average conformal map, is similar but not identical to Saffman-Taylor fingers.
Article
Full-text available
We report an algorithm to generate Laplacian growth patterns using iterated conformal maps. The difficulty of growing a complete layer with local width proportional to the gradient of the Laplacian field is overcome. The resulting growth patterns are compared to those obtained by the best algorithms of direct numerical solutions. The fractal dimension of the patterns is discussed.
Article
Full-text available
Diffusion Limited Aggregation (DLA) is a model of fractal growth that had attained a paradigmatic status due to its simplicity and its underlying role for a variety of pattern forming processes. We present a convergent calculation of the fractal dimension D of DLA based on a renormalization scheme for the first Laurent coefficient of the conformal map from the unit circle to the expanding boundary of the fractal cluster. The theory is applicable from very small (2-3 particles) to asymptotically large (n \to \infty) clusters. The computed dimension is D=1.713\pm 0.003.
Article
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.
Article
We consider capillary displacement of immiscible fluids in porous media in the limit of vanishing flow rate. The motion is represented as a stepwise Monte Carlo process on a finite two-dimensional random lattice, where at each step the fluid interface moves through the lattice link where the displacing force is largest. The displacement process exhibits considerable fingering and trapping of displaced phase at all length scales, leading to high residual retention of the displaced phase. Many features of our results are well described by percolation-theory concepts. In particular, we find a residual volume fraction of displaced phase which depends strongly on the sample size, but weakly or not at all on the co-ordination number and microscopic-size distribution of the lattice elements.
Article
An experiment on the Saffman-Taylor instability with wetting fluids is presented that explores a greater range of capillary numbers than did the original experiment of Saffman and Taylor. It turns out that no clear one-half plateau for the finger size is observed, and that the ensemble of experiments cannot be analysed in terms of a single control parameter. The effect of the film of oil left behind the finger is important, and we measure its thickness. A qualitative discussion of the instabilities of the fingers for large capillary numbers is presented, the first instability leading to asymmetrical fingers. Tip splitting appears for larger values of 1/B. The 1/B value for the onset of instabilities is shown to be noise dependent.
Article
The experimental results of Saffman & Taylor (1958) and Pitts (1980) on fingering in a Hele Shaw cell are modelled by two-dimensional potential flow with surface-tension effects included at the interface. Using free streamline techniques, the shape of the free surface is expressed as the solution of a nonlinear integro-differential equation. The equation is solved numerically and the solutions are compared with experimental results. The shapes of the profiles are very well predicted, but the dependence of finger width on surface tension is not quantitatively accurate, although the qualitative behaviour is correct. A conflict between the numerics and a formal singular perturbation analysis is noted but not resolved. The stability of the steady finger to small disturbances is also examined. Linearized stability analysis indicates that the two-dimensional fingers are not stabilized by the surface-tension effect, which disagrees with the experimental observations. A possible reason for the discrepancy between theory and experiment is suggested.
Article
The η model in the linear or radial geometry is investigated numerically. It turns out that the main characteristics of the viscous fingering instability (η=1) at vanishing capillary number such as the λ=1/2 limit are not recovered. For η≠1, the selected finger width decreases with the capillary parameter, indicating the formation of needlelike structures.
Article
The statistical properties of off-lattice diffusion-limited aggregates (DLA) grown in a strip between two reflecting walls are investigated. A large number of independent runs are performed and the cell occupancy distribution is measured and compared with the predictions of a recently proposed mean-field theory (MFT). It is shown that the mean occupancy profile moves at constant speed and has a shape and a selection mechanism similar to that of stable Saffman-Taylor fingers. In particular, there exists a specific contour line of the mean occupancy distribution (rho=0.6rhomax) that has the width and the shape of the Saffman-Taylor finger lambda=0.5. Motivated by the connection to the Saffman-Taylor problem, we extend our study to DLA growth in sector-shaped cells. Again a remarkable agreement is found between the mean occupancy profile and the shape of the selected stable finger in the small surface tension limit. Moreover, whenever the smooth finger is theoretically expected to undergo a tip-splitting instability, one observes, as predicted by the MFT, a qualitative change in the cell occupancy distribution that exhibits `profile crossing'' together with a pronounced flattening of the tip region. We comment on this phenomenon, which was not observed in a previous similar statistical analysis of on-lattice DLA clusters due to the stabilizing effect of lattice anisotropy. The implications of our numerical results to the relevance of the DLA mean-field theory are discussed.
Article
Very unstable viscous fingers moving in a linear channel are investigated as well as diffusion-limited aggregates grown in a strip between two reflecting walls. In both cases a large number of independent runs are performed and the cell occupancy distribution is measured. It is shown that the zone of large occupancy has the width and shape of the Saffman-Taylor finger lambda=0.5. Similarly, in sector-shaped cells, the width of the large occupancy region is the limit width of the stable fingers. In the case of a 90° cell, its shape is that of a predicted analytical solution.
Article
We have measured the coarsening due to surface tension of radially grown fractal viscous fingering patterns. The patterns at late times depend on the structural form at the onset of coarsening, providing information on the age of the fractal. The coarsening process is not dynamically scale invariant, exhibiting two dynamic length scales that grow as L1(t) approximately t(0.22+/-0.02) and L2(t) approximately t(0.31+/-0.02). The measured exponents are in agreement with the results of recent numerical studies of diffusion-controlled coarsening of a diffusion-limited aggregation fractal [Phys. Rev. E 65, 050501 (2002)]].
Article
We present in this paper an experimental study of the invasion activity during unstable drainage in a two-dimensional random porous medium, when the (wetting) displaced fluid has a high viscosity with respect to that of the (nonwetting) 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 lambda 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 is then compared with the corresponding simulated pressure field using this expression for the boundary condition on the cluster.
• R Chandler
• J Koplik
• K Lerman
• J F Willemsen
R. Chandler, J. Koplik, K. Lerman, and J. F. Willemsen, Journal Fluid Mechanics 119, 249 (1982).
• A Levermann
• I Procaccia
A. Levermann and I. Procaccia, Phys. Rev. E 69, 031401 (2004).
• B Davidovitch
• A Levermann
• I Procaccia
B. Davidovitch, A. Levermann, and I. Procaccia, Phys. Rev. E 62, R5919 (2000).
• E W Washburn
E. W. Washburn, Phys. Rev. 17, 273 (1921).
• R Lenormand
• C Zarcone
R. Lenormand and C. Zarcone, Physical Review Letters 54, 2226 (1985).
• A Lindner
• D Bonn
• E Poiré
• M Ben Amar
• J Meunier
A. Lindner, D. Bonn, E. Corvera Poiré, M. Ben Amar, and J. Meunier, J. Fluid Mech. 469, 237 (2002).
• D Bensimon
• S Liang
• B I Shraiman
• C Tang
D. Bensimon, L. P. Kadanoff, S. Liang, B. I. Shraiman, and C. Tang, Rev. Mod. Phys. 58, 977 (1986).
• G Løvoll
• Y Méheust
• R Toussaint
• J Schmittbuhl
• K J Måløy
G. Løvoll, Y. Méheust, R. Toussaint, J. Schmittbuhl, and K. J. Måløy, Phys. Rev. E 70, 026301 (2004).
• J W Mclean
• P G Saffman
• J Fluid Mech
J. W. McLean and P. G. Saffman, J. Fluid Mech. 102, 445 (1981); B. I. Shraiman, Phys. Rev. Lett. 56, 2028 (1986).
• J Mathiesen
• M H Jensen
• E Somfai
J. Mathiesen and M. H. Jensen, Phys. Rev. Lett. 88, 235505 (2002); E. Somfai et al. (2004), cond- mat/0401384; L. Pietronero and E. Tossati, eds., Fractals in Physics (North-Holland, 1986), p. 151.
• E L Hinrichsen
• K J Måløy
• J Feder
• T Jøssang
• J Phys
E. L. Hinrichsen, K. J. Måløy, J. Feder, and T. Jøssang, J. Phys. A 22, L271 (1989); K. J. Måløy, J. Feder, and T. Jøssang, Physical Review Letters 55, 2688 (1985).
• M G Stepanov
• L S Levitov
M. G. Stepanov and L. S. Levitov, Phys. Rev. E 63, 061102 (2001).
• E L Hinrichsen
• K J Måløy
• J Feder
• T Jøssang
E. L. Hinrichsen, K. J. Måløy, J. Feder, and T. Jøssang, J. Phys. A 22, L271 (1989);
• K J Måløy
• J Feder
• T Jøssang
K. J. Måløy, J. Feder, and T. Jøssang, Physical Review Letters 55, 2688 (1985).
• A Arneodo
• J Elezgaray
• M Tabard
• F Tallet
A. Arneodo, J. Elezgaray, M. Tabard, and F. Tallet, Phys. Rev. E 53, 6200 (1996).
• P G Saffman
• G Taylor
P. G. Saffman and G. Taylor, Proc. Soc. London Ser. A 245, 312 (1958).
• P Tabeling
• G Zocchi
• A Libchaber
P. Tabeling, G. Zocchi, and A. Libchaber, J. Fluid Mech. 177, 67 (1987).
• J W Mclean
• P G Saffman
J. W. McLean and P. G. Saffman, J. Fluid Mech. 102, 445 (1981);
• B I Shraiman
B. I. Shraiman, Phys. Rev. Lett. 56, 2028 (1986).
• E Somfai
• R C Ball
• J P Devita
• L M Sander
E. Somfai, R. C. Ball, J. P. DeVita, and L. M. Sander, Phys. Rev. E 68, 020401(R) (2003).
• Ben Amar
M. Ben Amar, Phys. Rev. E 51, R3819 (1995).