Drainage in a Rough Gouge-Filled Fracture

Transcription

1 Transport in Porous Media 50: , Kluwer Academic Publishers. Printed in the Netherlands. 267 Drainage in a Rough Gouge-Filled Fracture H. AURADOU 1,K.J.MÅLØY 2, J. SCHMITTBUHL 3 and ALEX HANSEN 4 1 F.A.S.T. (UMR 7608), Université, Paris XI, Bat. 502, Orsay Cedex, France. auradou@fast.u-psud.fr 2 Fysisk institutt, Universitetet i Oslo, Postboks 1048, Blindern, N 0316 Oslo, Norway. maloy@fys.uio.no 3 Laboratoire de Géologie (UMR 8538), École Normale Supérieure, 24 rue Lhomond, F Paris Cedex 05, France. schmittb@geophy.ens.fr 4 Institutt for fysikk, Norges teknisk-naturvitenskapelige Universitet, N 7491 Trondheim, Norway. Alex.Hansen@phys.ntnu.no (Received: 14 November 2000; in final form: 20 November 2001) Abstract. We study experimentally and numerically slow drainage of a non-wetting fluid in a saturated fracture. Facing surfaces of the experimental fracture (25 cm 25cm) are obtained from the casting of a brittle fracture of a granite block. They are rough but mated leading to a constant aperture where glass beads (1 mm) are spread in a single layer to mimic the influence of gouge particles trapped within the fracture. During injection, snap off might appear owing to the buoyancy difference between fluids, which splits the non-wetting invader into bubbles. Geometry and connectivity of this new fluid structure are characterized and shown to be significantly controlled by the long range spatial correlations of the fracture topography. Two coexisting types of sub structures emerge along percolating clusters: string-like links and compact blobs. Saturation and trapping are shown to be significantly influenced by the buoyancy effect. A numerical model to describe the experiments is introduced. It is based on an invasion percolation algorithm but includes spatially correlated contributions resulting from the roughness of the crack surfaces and gravity. Numerical results are shown to be consistent with experimental observations. The model allows us to extend the analysis to regimes where gravity forces are completely dominating and to obtain statistical results exploring numerous different fractures with the same properties (roughness statistics, pore size distribution, size). A description of the injection pressure evolution is proposed. Key words: drainage, percolation, fracture roughness. 1. Introduction Two phase flow in fractured rocks is of interest for several important implications: oil and gas recovery, exploitation of geothermal energy, storage of radioactive waste, soil pollution, etc. Most of fluid flows occur along a network of fractures or faults. However, the study of flow within a single fracture appears as the foundation for understanding complex fracture networks. Drainage within fractured rock has been widely studied (see Bear et al., 1993 for a review). Less work has been done about the drainage within a single fracture. Two phase flow between glass plate roughened by glass beads has been studied by Fourar et al. (1993) and Fourar and Bories (1995). Glass and Nicholl (1995) ob-

2 268 H. AURADOU ET AL. served the trapping of air into water between two roughened glass plates. A parametric description of two-phase flow along a single fracture in a volcanic tufa was proposed by Rasmussen (1995). Amundsen et al. (1999) has studied experimentally the drainage in an artificially constructed self-affine aperture. Experiments of drainage in natural fractures have been performed by Reitsma and Kueper (1994) and Longino and Kueper (1999). Flow in a single fracture is highly controlled by the geometry of the aperture and subsequently by the topography of the fracture. Despite numerous geological processes (erosion, dissolution, precipitation, complex slip history, etc), geometry of natural joints are shown to exhibit the same scaling invariance as fresh artificial fractures in rocks (Brown and Scholz, 1985; Power et al., 1987; Cox and Wang, 1993; Schmittbuhl et al., 1993a, 1995). As a consequence, artificial selfaffine surfaces are good descriptions of natural fractures. The roughness of crack surfaces is self-affine with a robust roughness exponent close to The estimate of the roughness exponent seems independent of the material (Power et al., 1987; Måløy et al., 1992; Schmittbuhl et al., 1995) and of the rupture mode (Bouchaud et al., 1990; Bouchaud, 1997). The scaling is observed over a large range of scales: from micro-meters to several meters (Brown and Scholz, 1985; Power et al., 1987; Schmittbuhl et al., 1993a). Reliability of self-affine analysis have been extensively checked (Schmittbuhl et al., 1994; Simonsen et al., 1998). A classical approach to describe an open fracture, is to reduce it to two parallel facing planes (Bear et al., 1993). Experimentally such a configuration for two phase flow corresponds to a Hele-Shaw cell (Hele-Shaw, 1898; Vicsek, 1992). However, open fractures are often filled with particles, that is, gouge particles. These particles are shown to be broadly distributed (power law distribution) from micro-meter to centimeter because of the fragmentation process (Chester et al., 1993; Sammis and Steacy, 1995). A simple extension of a Hele-Shaw configuration to incorporate porosity effects, is to fill it with mono-disperse particles (Måløy et al., 1985). In this case, the gouge particles form a two-dimensional porous medium. The reduction of a broad particle distribution to a peaked distribution is a strong simplification since it transforms a three-dimensional porous medium to a two-dimensional layer. However, this configuration allows a direct observation of the fluid interface motion, and hence is justified. Numerous experiments have been performed in this framework in the limit of low injection speed, that is, negligible viscous effect (Lenormand and Zarcone, 1985; Lenormand et al., 1988). Influence of gravity in a two-dimensional porous medium has been addressed (Birovljev et al., 1991, 1995). When a lighter non-wetting fluid displaces, at low capillary number and along an inclined topography, a wetting denser fluid, the different relative buoyancy of the fluids monitors the migration of the fluid interface. For instance, a stable invasion front is observed when the light fluid propagates downwards into the heavier fluid. When the lighter fluid propagates upwards into the heavier fluid, the propagation is unstable (Frette et al., 1992). Finger-like structures that often fragment are then observed.

3 DRAINAGE IN A ROUGH GOUGE-FILLED FRACTURE 269 Extension of gravity effects to three-dimensional porous media has also been performed (Clement et al., 1985; Frette et al., 1992; Meakin et al., 1992; Glass et al., 2000). The present paper focuses on the comparison between experimental and numerical results on drainage in a single rough fracture filled with a single layer of mono-disperse particles. It can be seen as a rough porous Hele-Shaw cell. We emphasize that the simplification to a mono layer is very important for a clear comparison between experiments and simulations. Moreover, as discussed later in the paper, it captures most of the physical processes and can be extended to a complete three-dimensional layer. The experimental study is based on recently published results by us (Auradou et al., 1999) and complementary analyses. The foundations of the numerical model are described in Schmittbuhl et al. (2000). Experimentally, we used fracture walls that were obtained from the fracture of a granite block. Facing walls are rough but exactly mated and the aperture is constant all over the fracture. Beads are spread along the aperture in a random single layer. The average fracture planes are oriented horizontally but the fracture roughness with local hills and valleys is responsible for both stabilized and fingerlike structures to be present. The fracture topography is analysed and shown to exhibit spatially long-range correlations in contrast to the fluctuations of the pore size. Pores exist between beads as a result of the spherical shape of the beads and the non-compact but random positions of them along the aperture. Because of the bead layer, the interface between fluids that propagates within the fracture aperture is pinned. Trapping of both injected and displaced fluids appears at slow injection rate. Modeling of two-phase flow in rocks concerns mainly flow in porous media. For low injection speed, the invasion percolation model has been very successful (Wilkinson and Willemsen, 1983; Lenormand et al., 1988; Sahimi, 1993; Berkowitz and Ewing, 1998; Ewing and Berkowitz, 1998). Less numerous works describe drainage or imbibition along single fractures, especially when it includes rough fracture walls. Different numerical approaches have been developed to study the influence of wall roughness. Murphy and Thomson (1993) proposed a model of two-phase flow in a variable aperture fracture dividing the global aperture in sub-region of parallel plate description and including viscous effects. A graph theory algorithm has been introduced by Yaug et al. (1995) to describe drainage and imbibition in a single fracture including fractal rough walls. Schmittbuhl et al. (1993b) considered the percolation without any dynamics along self-affine surfaces. They showed the strong influence of spatial correlations along the surface. For realistic roughness exponents, clusters at the percolation threshold are compact and very different from percolation under uncorrelated noise (Stauffer, 1992). Invasion percolation in correlated media has been recently reviewed by Sahimi (1998). Wagner et al. (1997b) developed an invasion percolation model along a self-affine noise including migration. However, these approaches describe the invasion within an empty

4 270 H. AURADOU ET AL. fracture with an aperture fluctuation like a self-affine surface. In this case, the aperture might be large and accordingly the interface between fluids might present with a complex three-dimensional shape. In this case, we emphasize that the general assumption of invasion percolation algorithm which is a direct and simple link between the geometry of the pore and the geometry of the fluid front, is not clear. Numerical models that include viscous and gravity effect in addition to capillarity have recently been studied by Glass et al. (2001). The paper is structured as follows. The second section give details on the experimental setup. Section 3 introduces the dimensionless numbers that are required to fully describe the problem. Special attention is paid on the fluctuation number which is the tuning parameter of this experimental study. Section 4.1 presents the experimental results when different pairs of fluids are used to explore low fluctuation numbers. Section 4.2, on the other hand, describes experiments for larger fluctuation numbers when the effective gravity is changed by the use of a centrifuge. In Section 5 we describe the numerical model. Section 6 presents the quantitative comparisons between experiments and simulations through the analyses of the cluster dimension, the fracture saturation and the wetting fluid trapping. In Section 7, we discuss the injection pressure evolution obtained from simulations. The last section is devoted to a summary of the results and a conclusion. 2. Experimental Setup 2.1. THE FRACTURE CELL AND FLUID INTERFACE IMAGING In case of hydraulic fracturing, fractures are opened in mode I without shear slip and fracture surfaces in contact are exactly mated. Such an open fracture might be filled with circulating gouge particles and partly sealed. We propose to simplify the aperture of such fracture to be the volume between two identical fresh artificial fracture surfaces translated apart in the direction normal to the mean fracture plane by a constant distance (see Figure 1). Different fracture walls were used. Facing fracture walls of a given experiment have the same geometry since they both come from the cast of the same granite fracture surface. The latter is obtained from a fracture test of a granite block (Lanhelin, Brittany, France) of size 25 cm 25 cm 40 cm. The test is a mode I dynamical fracture propagation from two facing notches initially scratched on the block which was then loaded in compression. Three mother granite fractures were produced and used. From a mother granite fracture containing two facing surfaces, one epoxy cast of each side of the fracture were obtained using a silicon replica technique. Finally three different pairs of mated epoxy casts were made. For each experiment, only one of the epoxy cast is used. The facing surface is a clay replica of the epoxy cast in order to squeeze a porous medium made of a mono layer of beads, between the fracture surfaces. The porosity of the monolayer was measured to φ = When making the experimental fracture cell,

5 DRAINAGE IN A ROUGH GOUGE-FILLED FRACTURE 271 Figure 1. Sketch of the experimental setup. Two zooms of the fracture cell are included: a top view (T )andasideview(s). For both, gray disks represent the glass beads. the clay is first forced directly onto the rough epoxy plate to shape it as a mated fracture surface. The clay is then separated from the epoxy cast. A porous monolayer is made by spreading glass beads of diameter b = 1 mm onto a horizontal sheet (30 cm 30 cm) of sticking contact paper. After having removed the excess beads that do not stick to the contact paper, the porous mono-layer attached to the contact paper, is placed between the rough epoxy plate and the clay. The contact paper lies on the clay and prevent a direct contact between glass beads and clay. The gouge-filled fracture is sealed by a 2 cm thick glass plate placed on the top of the flat side of the epoxy plate and a 1 cm thick aluminum plate placed on the bottom of the clay (see Figure 1). Both plates are squeezed together with clamps at the four corners. The external borders of the experimental cell are sealed with a silicon glue. Four holes at each corner of the cell serve as outlets and are connected to a pump. A small hole (0.5 mm) in the middle of the plate forms the inlet (see Figure 1). In a fault gouge a distribution of particles sizes is generally found and it is not typically a single layer (Chester et al., 1993; Sammis and Steacy, 1995). This experiment is an approximation in which we choose a narrow distribution of particle

6 272 H. AURADOU ET AL. sizes. It is, however, important to keep in mind that the simulations described below is more general and can also be applied in a three-dimensional system. One of the main reason for choosing this experimental model system with a mono-layer is that it makes a direct comparison between the observed structures in the experiment and the simulations optically possible and subsequently simple. Glass and epoxy are transparent materials. They allow a direct observation through the upper side of the fracture, of the porous medium located in the fracture aperture. Fluid displacement within the aperture is visualized and pictures are taken using a Kodak DCS 420 CCD camera placed above the fracture (see Figure 1). Four halogen lamps are set along the sides of the cell to improve the lighting. The camera provided gray images on 8 bits with a resolution of pixels. The spatial resolution of the images is 60 µm per pixel, or 16 pixels per glass bead. The camera is controlled by a computer which sets a constant time step between each image. The fracture cell is first filled with a colored wetting fluid. A syringe pump (YA-12) withdraws the colored liquid at a low flow rate from the sides inducing an invasion of the non-wetting fluid from the center. Experiments are stopped when the invader fluid reaches the external border of the model. The fluid interface appears as a color contrast between the colored wetting fluid and the white areas of the (white) clay covered by the transparent non-wetting invader. After each experiments the model is either rebuilt or refilled. To check that the refilling procedure does not change the fluid migration significantly because of the trapping of small air bubbles or clay deformation, experiments were repeated under the same conditions after refilling. No significant changes in the structures were observed. Before each experiment, an initial image is taken. To reduce the image noise, all of the images taken during the experiment are subtracted from the initial image. The gray level distribution of the image presents two peaks which corresponds, respectively, to the white and the dark parts of the image. The image is clipped with a threshold set at the minimum between the peaks. Some small spots can be present on the thresholded image but these are deleted by extracting from the image the biggest clusters. Table I summarizes the description of all the experiments performed for this study SELF-AFFINE SCALING OF FRACTURE SURFACES Figure 2 shows a gray level representation of the roughness of one of the fractures. The topography h(x,y) of the epoxy replica is measured by a mechanical profiler. The rough interface was placed on a (x, y) translation table and translated by 0.25 mm between each height measurement. A total of 720 profiles containing 920 height estimates were obtained. The accuracy of the profiler measurement is 5 µm for x and y positions and 3 µm for height estimates (Lopez and Schmittbuhl, 1998).

7 DRAINAGE IN A ROUGH GOUGE-FILLED FRACTURE 273 Table I. List of all the experiments performed W. F. N. W. F. g/g 0 B F F L F a Fract. R/N C a (10 7 ) 1 W S R 9 2 W A Flat cell N 18 3 G A Flat cell N 9 4 W A N R N 18 7 G A N N R R R R R 8 14 G A R R R G A N N N N 20 The columns respectively give the name of the wetting fluid (W for water and G for the glycerol), the name of the non-wetting fluid (A for air and S for silicon oil), the acceleration, the bond number given by B = ρga 2 /4σ, the fluctuation number and the fluctuation number at the system scale, F L = F 2w/a, the fluctuation number at the pore scale F a = F 2w a /a with w a = w(a/l) ξ,the name of the fracture (2 is the same fracture than 2 excepted that the injection point is located at the highest point of the fracture), the refilling condition (N when it the first time the cell is used and R when the cell is refilled) and the capillary number. The rough interface was analysed by the Fourier spectrum and average wavelet coefficient (AWC) methods. The former method is based on the estimate of the Fourier spectrum of each profile. The Fourier spectrum is expected for a self-affine topography to scale as: P(f) f 1 2ζ where ζ is the roughness exponent (Feder, 1988; Schmittbuhl et al., 1995). The latter method is based on the computation of the wavelet transform of the profiles (Mehrabi et al., 1997; Simonsen et al., 1998). In our case we used the Daubechies-12 wavelet filter. For a self-affine function h(x,y), it can be shown that the averaged wavelet coefficient verifies W[h](λ) λ 1/2+ζ for one dimension analysis, where λ is the spatial scale parameter of the wavelet. Figure 2 shows the result of the Fourier Spectrum and the A.W.C. techniques applied on the topography of the fracture. For both techniques, results have been averaged over all parallel profiles. Both techniques show that

8 274 H. AURADOU ET AL. Figure 2. The upper plot show the topography of an epoxy cast of a granite fracture. The roughness has been measured by use of a mechanical profiler long 720 profiles of 920 data points each. It covers an area of 18 cm 23 cm. The lower left part of the figure show the average of Fourier spectrum of each profile. The power law behavior is consistent with a self-affine scaling with a roughness exponent of ζ = The result is confirmed with the Average Wavelet Coefficient method plot on the lower right of the figure. the cast is self affine with a roughness exponent ζ 0.8. This result is in good agreement with recent experimental studies of brittle fractures (Brown and Scholz, 1986; Schmittbuhl et al., 1995; Bouchaud, 1997; Sahimi, 1998). Since the fracture surface is self-affine, the height difference h( x) between two pores increases statistically with the distance between the pores x: h x ζ,whereζ is the roughness exponent. The spatial correlation function is also apowerlaw: h(x)h(x + δ) δ 2ζ showing the long range spatial correlations that exist along the fracture topography. Cutoffs exist for both small and large scales but they are respectively smaller than the bead size b and of the order of the system

9 DRAINAGE IN A ROUGH GOUGE-FILLED FRACTURE 275 size L. From measurements of heights h(x,y), we calculated the width w of the height distribution defined as w = h 2 h 2 where the average is over the entire data set, finding w 2.5b. 3. Dimensionless Parameters 3.1. THE CAPILLARY NUMBER We consider the case of a central drainage, that is to say injection from the center of the fracture, of a non-wetting fluid assuming that the fracture is perfectly saturated by a wetting fluid. Fluids are immiscible. The relative importance of viscous and capillary forces at the pore level is expressed by the capillary number C a, C a = µu γ, (1) where γ is the surface tension between fluids, µ is the viscosity of the wetting phase, and U is a characteristic flow velocity estimated here as U = L/T,where T is the duration of the experiment and L the distance between the injection point and the side of the model which is of the order of 100 b. The physical properties of the fluids are summarized in Table II. Experiments were performed with a flow rate of the order of few microlitres per minute. The number of pores invaded per minute is in the range of To check if the injection rate is sufficiently slow to neglect viscous effects, experiments were performed at two different capillary numbers C a and C a These experiments were done with the same model (same fracture and porous medium) that was refilled. The final structures were not significantly changed from one experiment to another, indicating that the injection speed was sufficiently slow. Table II. Viscosity µ, density ρ and the fluid fluid surface tension γ of the fluids used Silicon oil Water Air Glycerol µ [cp] ρ [g cm 3 ] γ [dyn cm 1 ] The glycerol has been diluted with 20% water and was originally 98% purified in mass percent THE FLUCTUATION NUMBER The capillary pressure drop P c across the fluid interface is given by the Young Laplace equation

10 276 H. AURADOU ET AL. ( 1 P c = γ + 1 ), (2) R 1 R 2 where R 1 and R 2 are the principal radii of curvature or the vertical and horizontal radii of curvature, and γ is the surface tension. In the limit of slow injection, capillary forces dominate viscous forces (low capillary number). The condition for the drainage invasion of throat located at (x, y) is that the capillary pressure P c (x, y) exceeds the capillary threshold P t (x, y), P c (x, y) > P t (x, y). (3) This condition is also used in the invasion percolation simulation (de Gennes and Guyon, 1978; Wilkinson and Willemsen, 1993) for modeling of slow twophase flow. In the presence of density contrast and gravity, the capillary pressure P c (x, y) will depend on the height level owing to the hydrostatic pressure. Roughness of the fracture is significant and introduces height fluctuations. Because of the self-affine property of the roughness, the height fluctuations are highly spatially correlated over large distances (up to the sample size). In the following we will define a reference capillary pressure P r (t) at the height h = 0. P r (t) is the pressure difference between the invader fluid and the displaced fluid for an interface sitting at a height h = 0. It is a function only of time since it is the driving force (monitored by the injection pump in experiment). The capillary pressure at a given throat along the fluid interface is then related to P r through the formula P c (x, y) = P r (t) + ρgh(x, y), (4) where ρ is defined as the density of the displaced non-wetting fluid minus the density of the wetting invading fluid. Without any density contrast between fluids, the topography of the roughness will have no influence on the invasion and results from invasion percolation in a two-dimensional flat horizontal porous media are recovered. With a significant density contrast between fluids, spatial correlations included in the fracture topography will monitor the invasion. Influences of the buoyancy and of the capillary forces have to be compared. Because of the pore size fluctuations, capillary thresholds also fluctuate. Figure 3 is a sketch of the distributions of the capillary thresholds for both drainage and imbibition. These distributions are characterized by their means, respectively P t D and P t I and their widths respectively W D and W I. All these quantities are characteristic pressures in the problem. Another typical pressure is the hydrostatic pressure across a pore: P h = ρga where a is the typical pore size which is of the same order as the bead size: a b. To compare gravity and capillary effects during a drainage step, one has to compare the fluctuations of the capillary threshold W D and the fluctuations of the hydrostatic pressure P h through the dimensionless fluctuation number F, F = ρga W D. (5)

11 DRAINAGE IN A ROUGH GOUGE-FILLED FRACTURE 277 Figure 3. Schematic distributions N(P t ) of capillary pressure thresholds P t for drainage (D) and for imbibition (I). The average thresholds are P t D and P t I for imbibition. Widths of the distributions are respectively W D and W I. When F is large, gravity controls the drainage and subsequently long range correlations of the roughness will monitor the shape of the invader cluster. On the contrary, for low fluctuation number F, gravity effects are negligible and geometry of invasion percolation in a 2D porous medium should be recovered. Using Equation (3) and introducing dimensionless pressures p r and p t by dividing Equation (4) by the characteristic pressure W D, the local criterion for a drainage process becomes (x, y) p r (t) > p t (x, y) Fh. (6) a When gravity and capillary effects are compared at the system scale L, the typical magnitude of the roughness at the system scale has to be included. We propose to estimate this magnitude by computing the double of the root mean square of the fracture topography over the whole fracture: 2w = 2 h 2 L h 2 L. The width 2w has to be estimated in unit of the pore size a. The fluctuation number at the system scale becomes F L = 2Fw a. (7) When F L is large (F L 1) gravity effects are dominant up to the system scale. This number F L will be used for comparison with the numerical results. Note that the fluctuation number at the system scale F L is for a rough surface larger than the local fluctuation number F : F L >F. When increasing the gravity effects, first

12 278 H. AURADOU ET AL. large scales will be influenced: F L > 1. It is only when gravity effects are sensitive at the pore scale (2Fw a )/a > 1, where w a = w(a/l) ζ that the entire system will be gravity controlled EXPERIMENTAL TUNING OF F There are various manners to experimentally tune F L = 2Fw/a. The first one is to change the model size. Since w L ζ an increase of the system size will expand w. However, if the system is increased by a factor 2, w is only increased by a factor 2 ζ = 1.74 (ζ = 0.80). Another way is to change the pair of fluids. This will modify ρ and γ.we use this technique to explore low fluctuation numbers F. By density matching the fluids, there is no gravity influence: F L = 0. Another possibility is to tune the acceleration of gravity by running the experiment in a centrifuge. We use this second approach for reaching high values of the fluctuation number F. The problem in estimating F is that it depends on both the pore size distribution and the wetting properties of the fluids. However, F can be measured at 1g 0 by performing gravitationally stabilized experiments similar to the experiments described in Birovljev et al. (1991). Here g 0 is the standard acceleration of gravity g 0 = 9.8m/s 2. For the measurement of F at 1g 0, we used a flat plate made of the same material as the rough fracture plates (see Table I). The epoxy is casted using a flat plate and the clay squeezes in between a flat two-dimensional porous medium. The experiment in this case is a side injection (not a central injection) letting one side wall of the cell open (not sealed with a silicon glue). The flat fracture cell is filled by the wetting fluid and then tilted at an angle α from horizontal. The wetting fluid is slowly pumped from the lower edge. Air is free to enter the cell through the open upper side. A stable front is observed and recorded by the CCD camera. The width of the front, W, is defined as the standard deviation of the front position in the direction of propagation, W = y 2 y 2. (8) Here the average is over all points belonging to the front. The front width W is estimated from an average at different times of five different fronts. Figure 4 shows the evolution with time of the width W of fronts for experiments at different angles α using air and glycerol. The uncertainty when computing the front width, W, is estimated as the maximum deviation of the set of W-measurements from W. The fluctuation number F is estimated in a way similar to that used by Birovljev et al. (1990) for estimating the bond number (see Equation (11)) F sin(α) = ( ) (ν+1)/ν W, (9) a

13 DRAINAGE IN A ROUGH GOUGE-FILLED FRACTURE 279 Figure 4. Evolution of the front width W with time for different tilt angles. Circles and squares were obtained when air drained water for a tilt angle α = 1.6degandα = 4.8 deg respectively. Filled diamonds correspond to the case where air drained the Glycerol Water mixture and with α = 1.6 deg. Horizontal lines indicate for each experiment the average front width W. The vertical lines correspond the estimated deviation W. where ν = 4/3 is the universal exponent that describes the scaling of the correlation length with the percolation threshold (Stauffer, 1985) THE BOND NUMBER Because of gravity, snap off might appear along slopes of the topography and create bubbles, that is, splitting of the invading non-wetting fluid in several independent clusters. Subsequently, imbibition might exist locally during bubble motion. Such snap off occurs typically when the hydrostatic pressure difference along the cluster becomes larger than the capillary pressure threshold difference P t D P t I (see Figure 3), ρgh(l) > P t D P t I, (10) where l is the size of the cluster and h(l) its height difference. The bigger the cluster, the more probable the snap off will be triggered. Because of the bead shape, a local imbibition is supposed to happen close to throat outlet where the radius of the pore is at maximum. Subsequently, imbibition thresholds are close to zero: P t I 0, and the distribution is a sharp peak: W I 0 (see Figure 3). The trigger of snap off is a function only of the drainage properties of the system and might be characterized by use of a second dimensionless number: the Bond number B = ρga P t D, (11)

14 280 H. AURADOU ET AL. which compares capillary and hydrostatic pressures at the pore scale. As we introduced a fluctuation number F L at the system scale, we define a bond number B L when the cluster extends over the system size L, B L = ρgh(l) P t D = Bh(L). (12) a Accordingly, when the cluster extends sufficiently the criterion for snap-off is B L > 1or h(l)>a/b. Here also large scales will be responsible for snap-off before pore scales since: B L >B. It should be noted that usually the bond number B and the fluctuation number F are independent since they are sensitive to two independent properties of the pore size distribution: the first moment of the distribution for the bond number and the second moment of the distribution for the fluctuation number. However, for some specific pore size distributions, both moments of the distribution are linked and subsequently F and B are related (Auradou et al., 1999). 4. Phenomenological Description from Experiments 4.1. EXPERIMENTS WITH DIFFERENT FLUID PAIRS Three different wetting/non-wetting pairs of fluids were used: water and air, glycerol and air, and silicon oil (DC 1107 from KeboLab) and water. The glycerol (purified at 98%) has been diluted with 20% of water by weight. Properties of the fluids are reported in Table II. The fluids containing water were colored by 0.2 g per liter nigrosine and de-gassed to remove air bubbles. The silicon oil has been chosen since its density matches water. Experiments were performed to check the wetting properties of the fluid pairs by using droplets of one of the fluids immersed in the other on substrates of glass, contact paper, and epoxy. From these observations we concluded that the pairs air/water, air/glycerol, and silicon oil/water are respectively non-wetting/wetting pairs. By performing experiments with various pair of fluids, we achieved three different fluctuation numbers F L : 0, 0.65, 1.02 (see Table II). The zero fluctuation number F L = 0 corresponds to the pair silicon oil water that are density matched. F L = 0.65 is obtained when the pair of fluid is air water. F L = 1.02 is achieved for the pair air glycerol Density Matched Fluids Using the pair of fluids: silicon oil/water that are density matched, we explored a zero fluctuation number: F L = 0. The interface migration in this case is expected to be insensitive to gravity. Figure 5 shows the non-wetting invader cluster in black superimposed on the fracture topography h(x,y) drawn with various gray levels. The invaded structure is adjusted to the underlying topography h(x,y) by using

15 DRAINAGE IN A ROUGH GOUGE-FILLED FRACTURE 281 Figure 5. Picture of the invading non-wetting fluid (black cluster) during a drainage experiment. Fluids are density matched (F = 0): water and silicon oil. The image is taken from above the fracture cell. Fracture roughness is shown by the gray level background of the image. High points are in light gray and low ones are in dark gray. The size of the picture is 230 mm 180 mm. four marks on the corners of the model as reference points. These marks are nail heads that are also observed by the mechanical profiler. The image is rotated and re-scaled until marks exactly match. In Figure 5, the invader cluster has grown independently of the fracture topography. Hills in light gray or valleys in dark gray are both invaded. The structure of the cluster is similar to that of invasion percolation (Wilkinson and Willemsen, 1983; Lenormand and Zarcone, 1985) Fluids with Density Contrast We measured as described previously the fluctuation number F for two different pairs of fluids: air/water and air/glycerol (see Table III). At g = 1g 0, the former pair is characterized by a fluctuation number: F = 0.13 lower than the latter: F = Note that the fluctuation number at the system scale F L is closer to one, respectively (F L = 0.65 and 1.02) indicating that gravity should modify the structure at the largest scales. However, the fluctuation number at the pore scale

16 282 H. AURADOU ET AL. Table III. The widthof the front, W normalized by the bead size a measured for different pair of fluids and tiltangles α measured in degrees Non wetting-wetting fluids α W/a W/a F F Air water Air water Air glycerol The fluctuation number F is determined using Equation (9). Fw a /a is respectively and such that gravity effects at the pore scale is very small. For these two fluids the system will be in a cross-over regime between capillary dominated and gravity dominated flow, and it is difficult to visually see the gravitational effect. However, influence of the gravity is observable when computing the evolution of the height histogram for the non-wetting invader that is the lighter. Figure 6 shows the comparison of the histogram evolution for silicon oil/water (F = 0) and air/glycerol (F = 0.20). In Figure 6(a) (F = 0) all histograms are roughly symmetric around the injection height. Growth exists independently of the local height of the non-wetting invader. On the contrary, when fluids are not density matched (F = 0.2), histograms are not symmetric. The preferential growth of the light fluid within high pores of the model is observed as a peak on the right side of the histogram (i.e. large heights) EXPERIMENTS WITH VARYING GRAVITATIONAL ACCELERATION As mentioned previously and fully described in Auradou et al. (1999), significant influences of the gravity buoyancy (i.e. large fluctuation number) were observed when increasing the acceleration of gravity g and keeping the fluid properties: air/glycerol. Several experiments were performed in a large centrifuge in the L.C.P.C. in Nantes, France (see Table I). The centrifuge rotates with a 5.5 m arm, a basket that is 1.40 m long, 1.15 m wide and 1.50 m high. Because of the large size of the basket, we were able to install the complete experiment within the basket. During rotation of the centrifuge, the basket is free to tilt and rest in a position where acceleration is normal to the bottom of it and subsequently to the fracture cell. Changing the rotation speed allows to adjust the magnitude of the acceleration of gravity from: g = 1g 0 to g = 7g 0. The maximum acceleration was chosen to get a sufficient high fluctuation number F and preserve the experimental setup specially the CCD camera. The image subtraction procedure was not used for the experiments in the centrifuge because the setup was subject to small vibrations. For these pictures the threshold was adjusted by comparing visually the initial and the thresholded image.

17 DRAINAGE IN A ROUGH GOUGE-FILLED FRACTURE 283 Figure 6. Histogram of the invaded heights for different volumes V in unit of bead volume a 3 for two experiments done in the same cell: same porous media and same fracture but with different pairs of fluids. Figure (a) shows the histogram evolution for silicon/water (F = 0), while Figure (b) corresponds to the histogram evolution for air/water (F = 0.13). By performing experiments with various accelerations, we achieved a set of three different fluctuation numbers F L : 1.02, 3.07, and 6.15 using the pair air glycerol respectively at 1g 0,3g 0,and6g 0 (see Table I) Fragmentation, Migration, and Coalescence The most explicit observation of the influence of gravity is the fragmentation of the non-wetting invader during injection. Bubbles of the invader are created; they migrate and merge. These phenomena can only be observed when local imbibition takes place. This is achieved when the height difference between pores of the front

18 284 H. AURADOU ET AL. Figure 7. Three steps of an experiment at 6g 0 (F L = 6.1 andca = )areshown: (a) describes the snap off of the non-wetting invader after an injected volume of 3578 in pore volume unit. The image size is 85mm 205 mm. Arrow 1 shows the moving bubble. Figures (b) and (c) illustrate the coalescence of two clusters for an injected volume of 5725 in pore volume unit. Images (b) and (c) are separated by 5 min and their size is 44 mm 149 mm beads. (From Auradou et al., 1999). becomes of the order of a/b, thatisb L 1. To increase this height difference, the injection point was located at the lowest point of the fracture (not at the center). These effects are illustrated in Figure 7. Figure 7(a) shows the snap off of the injected non-wetting fluid. A bubble is created and migrates along the topography of the fracture aperture searching for local hills. The bubble stops when a summit larger than the bubble size is found. If the summit is already occupied by an invader cluster, both merge as shown in Figure 7(b) and (c). An important consequence of this phenomenon is that the injected fluid might be disconnected because of the roughness topography and not continuously connected in particular to the inlet, at a given time of the injection. We checked that bubble migrations were always happening at low speed. If we computed the local capillary number C a at the bubble scale during these events, it is found to be always smaller than C a = during snap off and C a = during coalescence Blobs and String-Like Links The bond number B is estimated from the fluid properties (see Table II) and the average pore size: a = 1 mm. For all experiments, the estimate of the bond number

19 DRAINAGE IN A ROUGH GOUGE-FILLED FRACTURE 285 Figure 8. Influence of the gravity on the structure of the non-wetting fluid (in black). These experiments were done on the same cell (same fracture and porous media) with the same pair of fluids air versus glycerol water mixture but under, respectively, 1g 0,3g 0,and6g 0 (F L = 2Fw/a = 1.02, F L = 3.07, and F L = 6.15). The inlet of the fracture cell is visible in the lower right corner of the pictures. The image background represents the porous medium along the fracture aperture. The dimensions of the pictures are all 97 mm 85 mm. (From Auradou et al., 1999). B is reported in Table I. The bond number B is always small: from B = 0for density matched fluid to B = 0.27 for air/glycerol at 6 g 0. As developed previously, the bond number describes the stability of the cluster and the probability of snap off. For low bond numbers, snap off are rare. The lighter fluid migrates along slopes of the roughness topography but without any splitting of the cluster. The migration process appears as worm-like (or string-like) structures. Figure 8 shows the non-wetting invader cluster for three different accelerations 1g 0,3g 0,and6g 0 corresponding to F L = 1.02, 3.07, and When gravity is increased, the invader cluster is reduced. Trapping of non-wetting fluid are less important and string-like structures are more visible. Parts of the cluster remain massive and do not shrink. They will be called blobs in the following. 5. A Buoyancy-Driven Invasion Percolation Model 5.1. ALGORITHM The numerical model we developed is based on the invasion percolation algorithm (Wilkinson and Willemsen, 1983). The model is inherently discrete and the discretization size corresponds to the pore size a. The porous medium is described as a regular grid where nodes are pores and bonds are throats. Invasion percolation algorithm relies on local thresholds. In our model they are attached to each bond to describe the capillary pressure threshold for a fluid interface to pass a given throat. Three important extensions of the model with respect of a classical invasion

20 286 H. AURADOU ET AL. percolation model have been implemented. Details of the modeling can be founded in the Appendix and in Schmittbuhl et al. (2000). First, we included the effect of the fracture topography even if the model is purely two dimensional and flat. The roughness of the fracture is incorporated in the model as an hydrostatic contribution to local thresholds. More explicitly, local capillary pressure thresholds are modified according to the local height of the throat to which they are attached. Summits have a reduced threshold contrary to valleys. The local hydrostatic contribution, that is, the fracture roughness, is computed as an artificial self-affine surface with a prescribed roughness exponent. The second main feature of the model is the simultaneous description of drainage and imbibition. Indeed both processes might appear either in a sequence when fingers develop up to snap-off or simultaneously during a bubble move. The main assumption of the model (as in most invasion percolation models) is that viscous effects during local interface motion are negligible even if they happen quickly. Capillary and gravity effects are supposed to be dominant for all time and space scales. The last specificity of the model is the boundary conditions. We included a central injection condition to mimic the experiment. The main difficulty with this choice is the management of an inlet that is not located on a local summit. Indeed, any droplet that is injected is unstable and immediately move along the topography. We always assume that the injection rate is significantly slower than the move of the fluid PARAMETER SETTINGS Five independent parameters are required to run a simulation: the system size L, the roughness exponent of the fracture surfaces ζ, the seed of the random generator, the fluctuation number F, and the bond number B. Fracture surface: In order to model fracture surfaces, the roughness exponent is set to 0.8. No influence of this parameter is studied. Each self-affine geometry is computed from a seed. Changing the seed provide independent realizations. Statistics are done over at least 100 realizations for each set of the physical parameters (except for L = 1024 where statistics were obtained for 30 independent surfaces). System size: The system size L is defined as the number of nodes (or beads) along a grid size (the grid is square). Simulations are done for: L = 16, 32, 64, 128, 256, 512, The unit of size is the pore size a. Experimentally, the number of pores along the model side is of the order of 200. We study numerically the scaling of the height distribution width w with the system size L using 100 synthetic surfaces for each size with a roughness exponent ζ = 0.8. Numerical data are nicely fit by: w = 0.074L 0.8. The typical width at the pore scale is w(a) = Fluctuation number: The fluctuation number F is the ratio of the gravity forces over the magnitude of the capillary threshold fluctuations (see Equation (5)). When similar beads are spread along a regular lattice, pores have all the same size and

21 DRAINAGE IN A ROUGH GOUGE-FILLED FRACTURE 287 fluctuations of the capillary thresholds disappear. The fluctuation number F becomes large and drainage is only gravity controlled. On the contrary, usual invasion percolation regime is reached when no gravity influence exists, that is, for F = 0. Influence of buoyancy on the fluid flow is tuned by this F parameter. Bond number: The model describes fluid migration at scales larger than the pore scale. If gravity effects are so important that they are predominant even at the pore scale, then bubbles may appear within throats owing to snap off. This effect is not included in the model since the discretization size corresponds to the throat length. To be consistent, the bond number B has to be adjusted to have B 1 everywhere in the model all along the simulation ILLUSTRATION OF THE DRAINAGE SIMULATION This section is devoted to graphical illustration of the simulated drainage process. Figure 9 shows the influence of the fluctuation number. The fracture is horizontal and discretized as a grid. The topography is shown as the picture background: light gray pixels corresponds to high pores (large h) and dark gray pixels corresponds to low pores (small h). Black dots represent pores filled with the non-wetting light fluid. For this set of figures the migration process has been suppressed by decreasing significantly the bond number B = Illustration of the simulated snap off effect may be found in Schmittbuhl et al. (2000). For case (a) (F = 10 6 ) the cluster grows independently of the fracture topography. Both deep valleys and high hills are invaded. The general shape of the cluster is very comparable to that of a classical invasion percolation cluster. When the fluctuation number is increased, sensitivity to the topography increases. The invading cluster develops mainly on local hills and becomes more compact, that is, less trapping of the wetting fluid. At the transition (see Figure 9(b)) F 1, the cluster follows the spatial correlations of the topography at large scales and becomes anisotropic. At small scales, the cluster still has a structure close to that of classical invasion percolation including large trapping. When comparing figures for a fluctuation number larger than one (see Figure 9(c) and (d)), very little difference of the cluster geometry is detectable. Experiments were performed for a limited range of fluctuation number (between F L = 0andF L = 6.2) compared to the simulations (between F L = and F L = ). The experimental structures seen in Figure 8 with F L = 1.02, 3.07, 6.15, respectively corresponds to the simulated structures between Figure 9(b) (F L = 0.125) and Figure 9(c) (F L = 12.5). A very good visual agreement is found between the simulated structures and the experiments. For F L of the order of 10 a structure with strings and blobs is seen in both cases. When F L is reduced the clusters becomes larger and with more trapping for both the simulations and the experiments. It has to be noted that an exact simulation of experiments is not accessible. Indeed, though the roughness of the fracture is measurable, the description of the

22 288 H. AURADOU ET AL. Figure 9. Percolating cluster for four fluctuation number: F = 10 6, 10 2, 1, 10 2 (F L = 2Fw/a = , 0.125, 12.5, ). The system size is geometry of each pore, that is, the capillary pressure threshold of each throat, is not measurable. Subsequently, a complete description of the experimental fracture cell is not possible. Therefore, simulations and experiments can only be compared statistically. 6. Quantitative Comparison between Experiments and Simulations 6.1. INVADER CLUSTER DIMENSION We propose to separate both populations of sub-structures of the invader cluster by use of a specific technique: the Inside Mass Method (Auradou et al., 1999). The Inside Mass Method computes the mass M(r) inside disks that are located inside

23 DRAINAGE IN A ROUGH GOUGE-FILLED FRACTURE 289 the external perimeter of the cluster. All possible disks inside the external perimeter are considered and the average mass M(r) for each radius r is calculated. On the contrary to classical average mass methods, this method has an upper limit R s of r which is the largest disk which can be inscribed inside the external perimeter of the cluster. The Inside Mass Method was used to perform a direct quantitative comparison between the experimental and the simulated structures. Figure 10(a) shows log M(r) for the experiments plotted as function of log(r). M(r) is the average of M(r) over clusters obtained at the breakthrough for a given F L.Curvesare fitted by a power law and the dimension D of the cluster is obtained as: M(R) R D. Table IV shows the experimental increase of the cluster dimension D from 1.84 to 2.0 with the fluctuation number F L. This measurement illustrates the compactness increase of blobs when buoyancy effects become dominant. We also performed the method on the simulated clusters. As shown in Figure 10(b), the dimensions of blobs of non-wetting invader fluid is increasing with the fluctuation number from D = 1.85 ± 0.01 for F L = to D = 1.97 ± 0.01 for F L = At low fluctuation number, results from classical invasion percolation are recovered. The dimension for high fluctuation number is close to compact (D = 2). Numerical estimates (see Table IV) are computed from the analysis of 100 independent fractures of similar size L = 512a as in the experiments (different fracture topography with the same roughness exponent ζ = 0.8 and different capillary pressure thresholds but using the same distribution). Average and r.m.s. of the mass dimension D num for the same fluctuation numbers as for the experiments are shown in Table IV. The numerical estimates are clearly consistent with the experiments (see Table IV). This might be compared to the scaling behavior obtained for percolation with self-affine noise where there is no history of the injection and no trapping (Schmittbuhl et al., 1993b). In this study, it was shown that for low roughness exponent (i.e. weak spatial correlation of the self-affine noise), the cluster mass behaves as M L 2 β/ν, (13) where the ratio of the critical exponents: β/ν is not different from that of percolation with spatially uncorrelated noise, β/ν Schmittbuhl et al. (1993b) analysed the influence of the increase of the roughness exponent up to realistic values for natural fracture surfaces (ζ 0.8). It was shown that the critical exponent ν is increased and diverges (ν )forζ = 0.8. The invader dimension is then: D = 2 and clusters appear as fully compact. Our simulation is close to the latter study if the bond number is sufficiently high (Schmittbuhl et al., 2000) SATURATION Saturation is often used to characterize the invasion of a porous medium. Since the porous medium is two dimensional the saturation of the fracture is directly

24 290 H. AURADOU ET AL. Figure 10. log 10 ( M(r) ) as a function of log 10 (r) where M(r) is the average of the inside mass M(r) at breakthrough for a given F L. (a) Experiments for five fluctuation numbers F L = 0; 0.65; 1.02; 3.07; Solid lines show linear fits that provide estimates of the fractal dimension D. (b) Simulations for four fluctuation numbers: F L = ; 0.218; 21.8; The system size is Results are averaged over all possible position of the percolating clusters and for 100 independent surfaces. Linear fits give estimates of the fractal dimension D. accessible from the pictures that are taken during experiments. We define the saturation S as the ratio between the number of black pixels to cover the non-wetting invader cluster and the total number of pixels representing the porous medium. For a given fluctuation number at the system scale F L, we have computed the mean saturation S at break through. An average is performed on all experiments

25 DRAINAGE IN A ROUGH GOUGE-FILLED FRACTURE 291 Table IV. Fractal dimension of the structures at the breakthrough for various F L F L D fit D fit D num rms(d) num D fit is the slope of the linear regression of the log 10 ( M(r) ) within a window starting at the pore level and ending at r/r s = 0.5. D fit is the maximal deviation of the slope of a single set from D fit. Experimental results are compared to numerical estimates obtained from the analysis of 100 independent fractures: the average of the fractal dimension D num and the root mean square rms(d) num. having the same fluctuation number F L. The result is shown in Figure 11. The saturation S decreases when buoyancy increases. The decrease is significant: up to a factor 3. The measured saturation is compared and found to be consistent with the saturation obtained from simulations (see Figure 11). The simulation results are for a Figure 11. Variation of the saturation S with F L = 2Fw/a for both experiments and simulations. Experimental results are shown as ( ) and correspond to the average over all experiments, at the breakthrough, done for a given F L. Vertical error bars indicate the maximal deviation of S from S. Results from computer simulations are shown as ( ) with a line. The error bars reflects the r.m.s. fluctuations for 100 realizations in the simulations.

26 292 H. AURADOU ET AL. system with the same size as in the experiments (L = 200a). This was obtained by a linear interpolation of the results from the L = 256a and L = 128a system sizes. Stars in Figure 11 shows the average of the saturation estimate when 100 independent surfaces are analysed. The large error bars apart simulation averages indicates the r.m.s. of the estimates. The error bars for the experimental results are estimated from the maximal deviation of S from the average S for a small number of experiments. Figure 12 shows the evolution of the saturation in the simulations with respect to the fluctuation number F L for different system sizes L. For each fluctuation number, the saturation is averaged over 200 independent surfaces for size L = 32, and 64, 100 independent surfaces for L = 128, 256, 512. A significant variation of the fluctuation numbers has been explored: F = 10 6 to F = 10 2 which covers eight orders of magnitudes. Two domains have to be separated. At low fluctuation numbers the saturation is constant for a given system size and higher than for large fluctuation numbers. Most of the transition occurs for a range of fluctuation numbers between 0.01 and 1. Finite size effects are significant at low fluctuation numbers. These are reduced at high fluctuation numbers. Size effects are consistent with the sensitivity of the invader mass to the system size as described previously. We expect the following size effect of the saturation: S L D 2, (14) Figure 12. The saturation defined as the ratio of the total mass of non-wetting injected fluid and the system surface (L 2 ). Evolution of the saturation in the simulations is shown as function of 2Fw/a for different system sizes, where F is the fluctuation number, w the width of the topography distribution and a the pore size.

27 DRAINAGE IN A ROUGH GOUGE-FILLED FRACTURE 293 where D is the dimension of the cluster. Then for low fluctuation numbers (F 1), the saturation reduces with an increase of the system or fracture size: S L On the contrary, for high fluctuation numbers (F 1), saturation is size independent. The cross-over regime in Figure 12 suggests that it exists a characteristic length ξ c (Auradou et al., 1999; Schmittbuhl et al., 2000) that can be estimated by searching for the scale at which capillary effects balance gravity contributions, that is, the scale at which the fluctuation number F L is one, 2Fw(ξ c ) = 1, (15) a where w(ξ c ) is the width of the height distribution at scale ξ c. From the self-affine behavior of the fracture geometry, we know that the height width scale as ( ) ξ ζ w(ξ) = w(a), (16) a where w(a) is the width of the height distribution at the pore scale. As mentioned previously, w(a) has been estimated for numerical synthetic surface: w(a) = Finally, the cross-over length ξ c measured in units of pore size a is ( ) 2Fw(a) 1/ζ ξ c /a =. (17) a When ξ c is close to the pore size ξ c /a 1, buoyancy dominates at any scales. On the contrary, when ξ c reaches the system size ξ c /a L, the system is driven by capillary pressures. No effect of the fracture topography exists. A qualitative implication of the cross-over length estimate is to show that when buoyancy effects exist, they influence first large scales of the invader cluster. Possible fluctuations of the saturation for given fracture size and fluctuation number are also of interest. Figure 13 presents the relative fluctuations of the saturation as function of the fluctuation number. Fluctuations are obtained from the statistics over different independent surfaces by computing the r.m.s. of the mass of the non-wetting fluid of each surface. Fluctuations are relative, that is, divided by the mean saturation. For all computed system sizes, the relative fluctuations of the saturation significantly increase with the fluctuation number, that is, with the long range correlations existing along the fracture surfaces. This results from the very different configurations that might exist when correlations along the surfaces are developed. To illustrate this influence, different clusters obtained for the same system size and the same fluctuation number are presented in Figure 14. Cluster shapes vary from very compact and massive (Figure 14(a)) to very thin stringy objects (Figure 14(d)). Intermediate structures includes various combinations of both populations: compact regions (i.e. blobs ) and string-like links. This dual structure is very different from that for standard non-correlated invasion percolation.

28 294 H. AURADOU ET AL. Figure 13. Simulations of relative fluctuations of the saturation of the non-wetting fluid as function of 2Fw/a for different system sizes, where F is the fluctuation number, w the width of the topography distribution and a the pore size. As indicated from the numerical results, broad fluctuations exist. These fluctuations are illustrated for the experiments in Figure 15 where two clusters obtained by using two different fractures for F L = The fluctuations in saturation S increase with F L both for the experiments and the simulations TRAPPING The effects of trapping have been studied by extracting by image analysis the trapped clusters of wetting fluid within the invading non-wetting fluid. We measured the ratio ϱ of the number of pixels describing the trapped wetting fluid and the number of pixels belonging to the non-wetting cluster. In Figure 16 we show the experimental average ϱ at break through for different fluctuation number F L. We see that the ratio ϱ decreases while the hydrostatic effect is increased, which means that the trapping process becomes less efficient. In the same figure we also show the results of the numerical simulations described above. A system size comparable to the experiments has been obtained by linear interpolation of the results from the system sizes L = 256a and L = 128a since the experimental system size is L = 200a. Consistent results are obtained between the experiments and the simulations (see Figure 16). As for the saturation plot, error bars for numerical results describe the r.m.s.of the trapping ratio ϱ. The error bars for the experimental results are estimated from the maximum deviation of ϱ from ϱ for a small number of experiments.

29 DRAINAGE IN A ROUGH GOUGE-FILLED FRACTURE 295 Figure 14. Four samples of the simulated percolating cluster for a high fluctuation number: F = 10 2 (2Fw/a = ). The system size is Figure 17 illustrates the evolution of the wetting fluid trapping during the drainage of the fracture for different F numbers in the simulations. Trapping significantly decreases when long range correlations drive the drainage process. Accordingly, trapping is reduced for high fluctuation numbers. As mentioned previously, simulations allow an exploration to very high fluctuation numbers and clearly exhibit a limit for the trapping decrease (i.e. regime of high fluctuation numbers) AVERAGE HEIGHT Another way of illustrating the role of the gravity buoyancy can be obtained by following the average height of the non-wetting invader fluid during injection. The average height h NW is defined as h NW = 1 h(i, j), (18) N NW i,j where N NW is the number of pixels that describe the non-wetting cluster.

30 296 H. AURADOU ET AL. Figure 15. Two experimental clusters obtained at the breakthrough in two different fractures for F L = Both pictures are at the same scale. The line is 100 mm long. Figure 18 shows the evolution of the average height as function of the fluctuation number at breakthrough for the simulations. For low fluctuation numbers, drainage happens independently of the height position of the pore. Since fractures have zero mean height, the average height <h> NW is zero. When long range correlations of the topography are monitoring the invasion, the non-wetting invader fluid which is lighter searches for hills. Subsequently, average height of the non-wetting fluid increases significantly. This effect has been clearly shown experimentally (see Figure 6). The link between the average height of the invader cluster and the fluctuation number, that is, the fluid properties, might be used as a tool for characterizing the roughness of an aperture. Assuming that there is for this fracture a possibility to estimate the average height of the invader, by changing the pair of fluids, different regions of the height distribution will be explored. For density matched fluid, for instance, the average height of the aperture will be estimated. Using fluids with a high fluctuation number (high density contrast or low surface tension contrast), extrema of the roughness will be explored.