Biphasic flow: structure, upscaling, and

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Oct 3, 2011 - Holtzman and Juanes, 2010; Chevalier et al., 2009; Johnsen et al.,. 73. 2006; Varas et al., ...... C. R. Mecanique 331, 835–. 410. 842. 411. 19 ...

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Biphasic flow: structure, upscaling, and

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consequences for macroscopic transport properties

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R. Toussaint (1), K.J. M˚ aløy (2), Y. M´eheust (3), G. Løvoll (2, 4), M. Jankov (2), G. Sch¨afer (4), J. Schmittbuhl (1) (1) IPGS, CNRS, University of Strasbourg, Strasbourg (France) (2) Physics Department, University of Oslo, Oslo (Norway) (3) G´eosciences Rennes, University of Rennes 1, Rennes (France) (4) Det Norske Veritas AS, Research and Innovation, Høvik (Norway) (5) LHYGES, CNRS, University of Strasbourg, Strasbourg (France)

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October 3, 2011

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Abstract

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In disordered porous media, biphasic immiscible fluid flows orga-

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nize in patterns that sometimes exhibit fractal geometries over a range

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of length scales, depending on the capillary, gravitational and viscous

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forces at play. These forces, as well as the boundary conditions, also

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determine whether the flow leads to the appearance of fingering path-

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ways, i.e., unstable flow, or not. We present here a short review of

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these aspects, summarizing when these flows are expected to be stable

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or not, what fractal dimensions can be expected, and in which scale

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ranges. We base our review on experimental studies performed in two-

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dimensional Hele-Shaw cells, or addressing three dimensional porous

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media by use of several imaging techniques. We first present configu-

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rations in which solely capillary forces and gravity play a role. Next,

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we review configurations in which capillarity and viscosity are the main

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forces at play. Eventually, we examine how this microscopic geome-

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try affects the macroscopic transport properties: its fractal structure

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and the way in which the scale range depends on the flux have for

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instance an impact on the relationship between saturation and the

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pressure difference between the two phases, at the macroscopic scale.

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An example of such an upscaling is illustrated in details.

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1

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The physics of two phase flows in porous media is a complex and

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rich topic, with obvious applications to the hydraulics of the vadoze

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zone, be it water infiltration, its evaporation, or the transport of Dense

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Non-aqueous Phase Liquids (DNAPL) down to the aquifers (Dridi,

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2009). Hydrogeologists and soil scientists aim at relating volumetric

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flow, pressure head, and water content at the Darcy scale, that is,

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a meso-scale above which the medium and the flow are described by

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continuous mathematical fields. The basic laws of multiphase flows

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treated at mesoscopic scale as a continuum require a closure of partial-

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flow Darcy relations. The key point of this closure is a functional

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relation between the capillary pressure (in a water-air system) and

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(water) saturation in the form of retention curves.

Introduction

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The physics community has been mostly concerned with character-

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izing and understanding flow structures/patterns from the pore scale

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and up. These structures and processes have a ajor impact on the

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retention curves. Notably, viscous fingering may have strong influence

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on retention curves resulting in dynamic saturationpressure curves in

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porous media.

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These flow structures can vary from the compact to the ramified

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and fractal (Lenormand , 1989; Lenormand et al. , 1989; M˚ aløy et al.,

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1985; M´eheust et al., 2002; Sandnes et al., 2011; Holtzman and Juanes,

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2010). One major issue is to simplify this complexity by keeping just

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enough information to describe the relevant physics at the relevant

2

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scale for the flow considered, without discarding important informa-

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tion. The porous body of a piece of soil or rock consists of pores and

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fracture networks of different length scales and shapes, whose perme-

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ability is often anisotropic and presents large spatial variations (see e.g.

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Zimmerman and Bodvarsson (1996); M´eheust and Schmittbuhl (2000,

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2001); Neuville et al. (2010a,b); Neuville et al. (2011a); Neuville et al.

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(2011b,c)). In general the soil/rock is a dynamic medium where the

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porosity can be modified by the fluids involved due to chemical reac-

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tions and desorption/adsorption mechanisms, in addition to the fluid

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pressure and the mechanical stress acting on the porous medium. The

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chemical composition and nano/micro structure of the rock further

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decides the wetting properties of the fluids which is crucial for the cap-

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illary front advancement in two-phase flow. For example, when a fluid

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with high viscosity is displaced in a porous medium by a fluid with a

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lower viscosity, the displacing fluid tends to channel through the paths

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of lower flow resistance, thereby forming pronounced fingers. The in-

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fluence of the physical properties of the fluids plays important practical

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role on natural flows: e.g., in soil and groundwaters, the identification

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of pollution sources is difficult due to the fact that organic pollutants

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can rapidly migrate down to the bottom of the aquifer and/or along

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paths different from the water (Benremita et al., 2003).

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In addition, when the porous medium is deformable, branching

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structures can be observed with transitions to fracturing of the porous

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medium (Lemaire et al., 1993; Cheng et al., 2008; Sandnes et al., 2011;

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Holtzman and Juanes, 2010; Chevalier et al., 2009; Johnsen et al.,

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2006; Varas et al., 2011) or formation of fingers, channels or bubbles

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in it (Johnsen et al., 2007, 2008; Kong et al., 2011; Vinningland et al.,

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2007a,b, 2010, 2011; Niebling et al., 2010a,b).

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In this review we will address the detailed structure and dynam-

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ics of two phase flow in fixed and disordered porous media, based on

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pore scale experiments. We will limit the discussion to drainage, i.e.

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to situations where a non wetting fluid displaces a wetting one – even

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though imbibition, where a wetting fluid invades a non wetting one, is

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of equal practical importance. The discussion will be limited to me-

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dia that are isotropic and homogeneous at large scales, and to cases

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where chemical reactions and adsorption or desorption between the

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fluids and the porous medium can be neglected. The structures of

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clusters of moving fluid and the dynamics of drainage in porous media

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depend on several parameters like the density difference, the surface

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tension, the wetting properties, the viscosities and the flow rates of the

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fluids involved. These different parameters correspond to forces. The

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forces that dominate on different length scales and their interplay give

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rise to separate scaling regimes. Up-scaling, which consists in relating

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the pore scale description to properties defined at the Darcy scale or

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even at the macroscopic scale, is a central topic within hydrology and

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oil recovery. Only by understanding the scaling of the structures and

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dynamics within each regime, and what controls the crossover lengths

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involved, is it possible to perform up-scaling. The structures involved

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are typically fractal within some scaling range; their fractal dimension

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depends on length scales, and often result from fluctuations occur-

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ring at smaller scales. At the end of this short review we provide an

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example of upscaling of recent experimental data; the experiments in

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question were aimed at studying the crossover between capillary and

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viscous fingering in a quasi two-dimensional porous medium.

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2

Capillary and gravitational effects

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When drainage is performed in the limit of infinitely slow displacement

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velocities, the pressure drop accross the porous medium is controlled

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by the capillary pressure drop accross the interface between the two

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fluids. The criterium for advancement of the interface into a given pore

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is that the capillary pressure drop be larger than the capillary pressure

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threshold needed to invade the pore neck that separates that pore from

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the already-invaded adjacent pore. The value of the capillary pressure

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threshold fluctuates from pore neck to pore neck, with a distribution

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function determined by the geometry of the porous medium. In the

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case of zero gravity or for a horizontal 2D porous medium, the next

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pore throat/neck to be invaded will be, among the pores that touch

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the interface, the one whose pore throat has the smallest capillary

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pressure threshold. This idea is the basis of the invasion percolation

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algorithm (de Gennes and Guyon, 1978; Chandler et al., 1982; Wilkin-

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son and Willemsen, 1983), where random numbers representing the

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capillary pressure threshold values are distributed on a lattice and

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where the front is moved at each time step at the location along the

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interface corresponding to the smallest threshold value. The fact that

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the fluid front always moves at the most easily invaded pore neck and

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nowhere else is actually not always true in real flows, even if it is a

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good approximation. What drives the advancement of the front is the

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capillary pressure build-up in the fluid. The capillary pressure will not

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relax immediately after invasion of a new pore but is controlled by a

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back contraction of the fluid interface. This is the reason for the so

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called Haines jumps which may lead to invasion of several pores in one

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jump (Haines, 1930; Furuberg et al., 1996; M˚ aløy et al., 1985)

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When the displaced fluid is incompressible (or lowly compressible),

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trapping takes place. Trapping is very important in two dimensions

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(2D) (Wilkinson and Willemsen, 1983) but much less significant in

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three dimensions (3D). Experiments addressing this so-called capil-

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lary fingering in 2D model systems were first performed by Lenormand

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(1989)who found a fractal dimension of the invaded structures equal

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to D = 1.83, in consistency with the results of numerical simulations

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based on invasion percolation with trapping (Wilkinson and Willem-

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sen, 1983). In 3D, several experiments have been performed (Chuoke et

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al., 1959; Paterson et al., 1984a,b; Chen et al., 1992; Hou et al., 2009;

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Mandava et al., 1990; Nsir et al., 2011; Jianzhao et al., 2011). The

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fractal dimension found at small scale, between 2 and 2.6 is compat-

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ible with the dimension D = 2.5 found in three dimensional invasion

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percolation models (Wilkinson and Willemsen, 1983).

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Even though the capillary fingering structure is fractal, in practice

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it is well described by a fractal dimension only within a window of

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length scales ranging from the pore size up to a crossover length on

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a larger scale. In the case where the density difference between the

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two fluids is different from zero, but where viscous forces are small

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compared to the others, this crossover length corresponds to a scale at

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which the capillary threshold fluctuations become of the same order of

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magnitude as the difference in hydrostatic pressure drop between the

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two fluids. This means that the crossover will always occur when the

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fluid structures become large enough. When a lighter fluid is displacing

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a heavier one from above at a slow flow rate resulting in low viscous

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forces, a stable displacement is observed , that is, the displacing fluid

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does not finger its way trough the displaced fluid; the crossover length

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sets the width of the rough interface between the two fluids (Birovljev

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et al., 1991; M´eheust et al., 2002; Løvoll et al., 2005), parallel to the

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average flow direction. Gravitational effects can easily be accounted for in the invasion percolation model by mapping the system onto a problem where the capillary threshold values are modified linearly by the hydrostatic pressure difference between the two fluids (Wilkinson, 1984; Birovljev et al., 1991; Auradou et al., 1999). By using this theory of percolation in a gradient , Wilkinson (1984) predicted theoretically the scaling of the front width ξ observed in a gravitational field as −ν

ξ/a ∝ B0ν+1 ,

(1)

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where the Bond number B0 = ∆ρga2 /γ is the ratio of the difference

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of the hydrostatic pressure drops in the two fluids on the length scale

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of a single pore to the capillary pressure drop. Here, ∆ρ is the density

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difference between the fluids, g the gravitational acceleration, a the

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characteristic pore size and γ the surface tension between the two flu-

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ids. Gravitationally-stabilized fluid fronts occurring during very slow

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two-dimensional drainage have been studied both experimentally and

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by computer simulations (Birovljev et al., 1991). The results were

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found consistent with the theoretical prediction of Wilkinson (1984).

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When a lighter fluid is injected into a heavier fluid from below (Frette

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et al., 1992; Birovljev et al., 1995; Wagner et al., 1997), gravitational

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fingering of the displacing fluid through the displaced fluid occurs; a

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scaling behavior consistent with Eq. (1) has has also been found in

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this unstable case: the characteristic length scale ξ then corresponds

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to the width of the unstable gravitational fingers, perpendicularly to

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the average flow direction. The same simple mapping to invasion per-

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colation as described above can also be performed in the case of slow

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displacement in a rough fracture joint filled with particles (Auradou et

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al., 1999).

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When comparing systems with different capillary pressure threshold

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distributions, Eq.(1) needs to be modified. From the phenomenology of

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the invasion process that we have explained above, it is quite intuitive

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that a gravity-stabilized front in a porous medium presenting a narrow

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capillary noise (i.e. a narrow distribution of the capillary thresholds)

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will give a flatter front than a porous medium with a wide capillary

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noise. Instead of equation Eq. (1), it has therefore been suggested to

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take into account a dimensionless fluctuation number F = ∆ρga/Wt ,

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in which capillary fluctuations are accounted for in terms of the width

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Wt of the capillary pressure distribution (Auradou et al., 1999; M´eheust

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et al., 2002). Experiments to check the dependence of the displace-

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ment process on the capillary noise are difficult, because controlling

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the capillary noise in a systematic way is not straightforward. These

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experiments therefore remain to be done.

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3

Capillary and viscous effects

The crossover between capillary fingering and regimes for which viscous effects are dominant was first studied in the pioneering work of Lenormand (1989). He classified the different flow structures in a phase diagram depending on the viscosity contrast M = µi /µd between the fluids and the capillary number Ca = ∆Pvisc /∆Pcap , which is the ratio of the characteristic viscous pressure drop at the pore scale to the capillary pressure drop. Here µi and µd are the viscosities of the injected and displaced fluid, respectively. From the Darcy equation, the capillary number can be evaluated as

Ca =

∆Pvisc µa2 v a∇Pvisc = . = ∆Pcap γ/a κγ

(2)

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where γ is the surface tension, a the characteristic pore size, κ the in-

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trinsic permeability of the medium, v the seepage velocity associated to

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the imposed flow rate of the displaced fluid, µ the viscosity of the most

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viscous fluid, and γ/a the typical capillary pressure drop accross the

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interface. Lenormand identified three flow regimes: (i) stable displace-

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ment, for which the interface roughness is not larger than one linear

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pore size, (ii) capillary fingering, which we have discussed in section 2,

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and (iii) viscous fingering, which occurs when large scale fingers of the

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displacing fluid develop inside a more viscous defending fluid, resulting

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in a much faster breakthrough of the displacing fluid. It is important

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to keep in mind that the observed structures will depend on the length

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scale considered. For large systems it is therefore not meaningful to

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talk about a sharp transition in a phase diagram between capillary

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and viscous fingering, because one will always have both structures

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present, i.e. capillary fingering on small length scales, and either vis-

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cous fingering or stable displacement on large length scales. When the

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two fluids involved have different viscosities, the viscous pressure drop

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between two points along the fluid interface will typically be differ-

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ent in the two fluids. This viscosity contrast will produce a change

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in the capillary pressure along the fluid interface, therefore playing a

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role similar to that of density contrasts in the presence of a grav-

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itational field (see section 2). At sufficiently large length scales, the

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difference in viscous pressure drop between the two sides of the in-

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terface will become larger than the typical fluctuations in capillary

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pressure threshold. This means that at sufficiently large length scales,

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and thus for a sufficient large system , viscous pressure drops , rather

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than capillary forces associated to random capillary thresholds, de-

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termine the most likely invaded pores; consequently, viscous fingering

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will always dominate at sufficiently large scales when a viscous fluid

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is injected into another more viscous fluid. At these large scales, and

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in the absence of a stabilizing gravitational effects, two-dimensional

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flows exhibit tree-like branched displacement structures with a fractal

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dimension Dv = 1.6 (M˚ aløy et al., 1985). M´eheust et al. (2002) have introduced an generalized fluctuation number F =

∆ρga − Wt

aµv κ

(3)

which is the ratio of the typical total pressure drop in the fluids over one pore , including both viscous and gravitational pressure drops, to the width of the capillary pressure threshold distribution Wt . The experiments by M´eheust et al. (2002), identical to those by Birovljev et al. (1991) but performed at larger flow rates and therefore under significant viscous effects, showed that the characteristic width of the rough interface parallel to the macroscopic flow could be characterized with the fluctuation number according to an equation analog to Eq.(1): −ν

ξ/a ∝ F ν+1

(4)

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where the exponent ν/(1 + ν) = 4/7 is consistent with percolation the-

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ory (ν = 4/3) in 2D. Since the viscous pressure field is not homogeneous

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like the gravitational field, this result is not obvious. Note that in this

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case ξ can be interpreted as the length scale at which the sum of the

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viscous and gravitational pressure drop becomes of the same order of

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magnitude as the spatial fluctuations of the capillary pressure thresh-

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old. In terms of fluid-fluid interface, ξ corresponds to the crossover

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scale between capillary fingering structures at small scale and the sta-

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bilized structure, which is linear (dimension 1) at large scales. When

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the displacement is large enough for viscous forces to play a role, the

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fractal dimension typical of viscous fingering structures is also seen at

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intermediate scales (M´eheust et al., 2002). Even in the case where the

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two fluids involved have the same viscosity, the width of the front was

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found to be consistent with Eq.(4) (Frette et al., 1992). As observed in

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the experiments of Frette et al. (1992) the effect of trapping turns out,

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at least in two dimensions, to be of central importance. The trapped

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islands result in a decrease in the relative permeability of the invaded

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fluid, which is equivalent to having a fluid with a higher viscosity. This

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is the effect responsible for the well-known decreasing dependence of

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a soil’s matric potential on its water content. This result is consistent

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with the scaling relation Eq.(4), which is expected from theoretical

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arguments for percolation in a stabilizing gradient (Xu et al., 1998;

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Wilkinson, 1984; Lenormand , 1989). Note thate other scaling rela-

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tions have been derived theoretically and observed experimentally by

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other authors, as reported by Xu et al. (1998). As mentioned previously, when a viscous fluid is injected into a 2D medium filled with a more viscous fluid, viscous fingers occur. The scaling of the finger width was studied experimentally by Løvoll et al. (2004) and Toussaint et al. (2005). The measurements were found to be consistent with a scaling law in the form ξ/a ∝ Ca−1

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(5)

This result is different from the scaling laws that can be explained

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from the theory of percolation in a gradient, as observed with stabi-

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lizing viscous forces.

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Toussaint et al. (2005), the destabilizing field is highly inhomogeneous

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and screened around the fingers, which may explain why the behaviour

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expected from percolation in a gradient is not observed, but a rather

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simpler one instead. Note however that scaling laws based on the the-

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ory of percolation in a gradient are still expected for some types of

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unstables flows (Xu et al., 1998).

In the experiments of Løvoll et al. (2004) and

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The scaling law (5) can be explained from a simple mean field ar-

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gument: consider an approximation for the pressure field for which

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the pressure gradient is homogeneous around the mobile sites at the

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boundary between the two fluids; the pressure difference between two

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points at a distance l is l∇P ∼ lCaγ/a2 . This relation holds not only

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when the viscosity of one fluid can be neglected with respect to that

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of the other one (for example, for air and water), in which case the

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definition of the capillary number if given by Eq. (2); it also holds in

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the general case of two viscous fluids, in which the capillary number

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can be defined from Eq. (2) by replacing the viscous pressure drop by

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a the difference in viscous pressure drop between the two fluids. If the

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viscous pressure drop between these two points exceeds the character-

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istic random fluctuations of the capillary pressure threshold from one

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pore throat to another along the interface, then the viscous pressure

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field is dominant in determining at which of the two points considered

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new pores are going to be invaded. On the contrary, if the random

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fluctuations of capillary threshold exceed the viscous pressure drop be-

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tween the two points, then this random pressure difference component

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dominates. Its magnitude is of the same order as the average capillary

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pressure value, γ/a. Hence, capillary effects are expected to dominate

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for scales l such that lCaγ/a2 < γ/a, whereas viscous effects will dom-

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inate for larger scales, such that lCaγ/a2 > γ/a: this explains the

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observed cross over scale ξ = a/Ca between the structures character-

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istic of capillary fingering and those characteristic of viscous fingering

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.

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to viscous fingering

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In many situations of non-miscible biphasic flow in porous media, the

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flow gets organized in fingering structures (preferential paths); in these

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unstable configurations, the fluids arrange in fractal geometries with

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nontrivial fractal dimensions depending on the observation scale, and

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the scale range over which each dimension is observed depends on the

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imposed boundary conditions (such as the macroscopic fluxes). This

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microscopic structuration has far reaching consequences for the up-

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scaled relationships between, for example, saturation and the pressure

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difference between the two phases.

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An example of upscaling from capillary

An example of such a situation has been mentioned briefly earlier

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in this review:

if gravity is negligible, only capillarity and viscosity

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play a role on the flow ; when a lowly viscous fluid displaces a more

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viscous one, the fluid-fluid interface is unstable due to viscous pressure

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gradients increasing at the most advanced parts of the invader, so that

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fingers naturally arise. The development of such an interface instabil-

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ity occuring during drainage was studied optically in Hele Shaw cells,

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at several controlled injection rates (Løvoll et al., 2004; Toussaint et

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al., 2005). Air was injected into an artifical porous medium composed

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of a monolayer of immobile glass beads sandwiched between two glass

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plates, and initially filled with a wetting dyed glycerol-water solution.

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The Hele-Shaw cell dimensions were denoted L × W × H, H = 1 mm

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being the cell thickness as well as the typical glass bead diameter; flow

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was imposed along the length L, with impermeable lateralboundaries

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defining a channel of width W (see Fig. 1). It was shown that the path-

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way of the air, defined as the locations where the occupancy probability

12

Ca = 0.022

inlet

outlet

x

L-x Ca = 0.033

λW

outlet

inlet

a/Ca

x

W

λW

a/Ca

x

outlet

inlet

Ca = 0.062

L Figure 1: Invasion structure of a lowly viscous fluid (white) into a much more viscous one (dark grey) during drainage in an artificial 2D porous medium of width W and extent L, at three different extraction speeds. The position of the invasion tip is denoted x. Characteristic crossover scales between fractal regimes, λW and a/Ca, separate a straight finger structure, a viscous fingering geometry, and a capillary fingering geometry, down to the pore scale a. 314

exceeds half its maximum value, was a finger of width λW positioned

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in the centre of the channel, with λ = 0.4. This was attributed to a

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similarity between the process of selection of the the pore throats to be

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invaded and a Dielectric Breakdown Model with η = 2, as was shown

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by computing the average invasion speed taking into account the width

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of the capillary threshold distribution (Toussaint et al., 2005). This

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structure is illustrated in Fig. 1.

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From approximations on the shape of the pressure around this fin-

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ger, mostly controlled by the viscous pressure drop, one can derive an

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upscaled pressure-saturation relation (Løvoll et al., 2010).

13

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Indeed, the pressure presents to first order a linear viscous pressure

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drop from the tip of the invasion cluster, at position x, to the outlet

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of the system, at position L. Over the rest of the system, the pressure

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gradient is screened by the finger, rendering the pressure in the wetting

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fluid essentially constant at a value close to the sum of the air pressure

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and the entrance pressure γ/a. Hence, the pressure difference between

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the two phases, with a pressure in the wetting fluid Pw measured at

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the outlet, the one in the non wetting phase equal to the atmospheric

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pressure Pn.w. , and a correction to this entrance pressure, writes as ∆P ∗

=

Pw. − Pn.w. − γ/a = (L − x)∇P

(6)

=

(L − x)∆Pvisc /a

(7)

=

(L − x)∆Pcap Ca/a

(8)

=

(L − x)

γCa a2

(9)

333

where ∆Pvisc and ∆Pcap are considered at the pore scale. Thus, there

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is a linear relationship between the viscous pressure drop accross the

335

cell and the distance between the finger tip and the outlet. Besides, the relationship between saturation and capillary number can be inferred from the fractal structure of the non-wetting invading fluid. At scales above the width λW , the finger is a linear structure of dimension 1. Between the scale λW and a/Ca, the structure has a viscous fingering geometry of fractal dimension Dv ∼ 1.5 (M˚ aløy et al., 1985; Løvoll et al., 2004). Between the crossover scale a/Ca and the pore scale a, the structure has a capillary fingering geometry of fractal dimension Dc = 1.83. Hence, the total number of pores invaded by the non viscous fluid can be evaluated as a function of these fractal dimensions, the ratio of the finger length to its width, x/(λW ), and the ratios of the latter length to the two others lengths, the crossover

14

length and the pore size. This leads to:

Nn.w.

x = λW



λW aCa−1

Dv 

aCa−1 a

Dc (10)

Together with the relationship between the total number of pores and the characteristic model dimensions,

Ntot =

LW a2

(11)

and the relationship between the wetting phase and non-wetting phase saturations, Sn.w. = 1 − Sw. = 336

Nn.w. , Ntot

(12)

Eq. (10) leads to

1 − Sw. = Sn.w. = λ

Dv −1

   a 2−Dv a2 ∆P ∗ Dv −Dc 1− Ca W γLCa

(13)

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This relationship allows to collapse all the pressure difference

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curves measured as a function of saturation in the set of experiments

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performed by Løvoll et al. (2010) onto a unique master curve, for

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capillary numbers ranging from around 0.008 to 0.12. Since P ∗ =

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a2 ∆P ∗ /(γLCa) is nothing else than the capillary pressure measured

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at the scale L of the experimental model, Eq. (13) in effect relates that

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capillary pressure to the imposed seepage velocity. In other words, it

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defines the dependence on Darcy flow velocity of what is commonly de-

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noted dynamic capillary pressure, measured at scale L. This example

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shows how both viscous and capillary effects play a role in constrain-

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ing the geometry of the invasion structures, resulting in a dynamic

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capillary pressure, as it is traditionally called (Hassanizadeh, 2002),

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that is simply due to the upscaling of the invasion structure, with only

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capillary and viscous effects seen at the REV scale, and without any

351

dynamic capillary/wetting effects occuring at the pore scale.

15

Ca = 0.009 Ca = 0.022 Ca = 0.029 Ca = 0.033 Ca = 0.054 Ca = 0.058 Ca = 0.060 Ca = 0.062 Ca = 0.079 Ca = 0.110

4000

Δ P*(Pa)

3000 2000 1000 0 0

0.05

0.1

S n.w.

0.15

0.2

0.25

Figure 2: Dependence of the pressure difference between the two phases and the saturation of the invading fluid, at different injection speeds. Figures 2 and 3 illustrate respectively the raw measurements at several injection speeds and how Eq. (13) allows to collapse these curves of saturation versus pressure The viscous pressure drop accross the cell drops linearly as the finger progresses into the cell, from a maximum value at the beginning of the invasion of γLCa/a2 , down to 0 at breakthrough of the invasion finger. In the previous equation, the wetting saturation is indeed initially 1 as it should be at initial total saturation, but we also obtain the final and maximum value of the residual wetting saturation as 1 − Sw.r. = Sn.w.r. = λDv −1

 a 2−Dv CaDv −Dc . W

(14)

352

This relation between the residual saturation and the capillary number

353

is indeed consistent with the observed residual saturations, as shown

16

Snw(tbt)

1.25

1

0.2 0.1 0

0.75

0

0.1

0.2

P

*

Ca

0.5

Ca = 0.009 Ca = 0.022 Ca = 0.029 Ca = 0.033 Ca = 0.054 Ca = 0.058 Ca = 0.060 Ca = 0.062 Ca = 0.079 Ca = 0.110 * * P =1-S

0.25

0 0

0.2

0.4

0.8

0.6

1

1.2

*

S

Figure 3: The collapse of the relationship between the reduced pressure difference (between the two phases), P ∗ = a2 ∆P ∗ /(γLCa), and the reduced  a −2+Dv saturation of the invading fluid, S∗ = λ1−Dv W CaDc −Dv Sn.w. , at different injection speeds, shows the influence of the structure on the upscaling. Dotted curve: prediction. Inset: Residual saturation at breakthrough.

17

354

in the inset of Fig.3.

355

Besides the relation between saturation and the macroscopic pres-

356

sure difference between the phases, other macroscopic relations can

357

be obtained via upscaling, as e.g., in some situations, the relative per-

358

meability. For example, in other experiments where both fluids were

359

injected at the same time, the trapped structures of wetting fluids

360

were observed to be fractal up to a certain cutoff depending on the

361

imposed flux (Tallakstad et al., 2009). The upscaling explaining the

362

cutoff and the structures allowed to explain the measured scaling law

363

of the relative permeability of the viscous phase as a function of the

364

imposed flux, κrel ∼ Ca−1/2 .

365

5

366

We have discussed the local flow structures that are observed ex-

367

perimentally during drainage in a disordered porous medium. They

368

are fractal, with a fractal dimension that depends on the observation

369

scale. At small scales, capillary fingering exhibits a fractal dimension

370

of 1.8 for two-dimensional media, and between 2 and 2.6 for three-

371

dimensional media. At larger scales a branched structure characteris-

372

tic of viscous fingering is seen, with a fractal dimension 1.6 for two-

373

dimensional systems. The crossover between the two behaviors occurs

374

at a length scale for which the differential viscous pressure drop equals

375

the typical capillary pressure threshold in the medium. This means

376

that for horizontal flow, unstable viscous fingering is always seen at

377

large enough scales, even if the medium exhibits no permeability het-

378

erogeneities at the Darcy scale. From the definition of the crossover

379

length, it follows that it scales as the inverse of the capillary number,

380

which explains why experiments performed at a given experimental

381

scale and at very slow flow rates have evidenced capillary fingering,

382

while those performed at very large flow rates have evidenced viscous

383

fingering. As for the effect of gravity, it can be to either destabilize

Conclusion

18

384

or stabilize the interface, depending on which fluid is the densest. In

385

the latter case, it acts against capillary effects and, when the displac-

386

ing fluid is the most viscous, against the destabilizing viscous forces,

387

resulting in an amplitude of the interface roughness that scales as a

388

power law of the generalized fluctuation number (or generalized Bond

389

number). In horizontal two-dimensional flows, viscous fingering is ob-

390

served to occur up to another characteristic length that is a fixed frac-

391

tion of the width of the medium. Upscaling of the local flow structures

392

is possible once one knows the fractal dimensions typical of the flow

393

regimes, and the relevant length scale range for each of them. We have

394

given an example of how the measured capillary pressure can be re-

395

lated theoretically to water saturation, a relation that is confirmed by

396

measurements. In that example, the capillary pressure measured at

397

the scale of the experimental setup exhibits dynamic features, i. e., a

398

dependence on the flow rate, that is fully explained by the geometry

399

of the upscaling, without any dynamic effects in the physical capillary

400

pressure as defined at the pore/interface scale.

401

6

402

This work was supported by the CNRS through a french-norwegian

403

PICS grant, the Alsace region through the REALISE program, and

404

the Norwegian NFR.

405

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