Competition ofgravity, capillary and viscous forces during drainage in a two-dimensional porous medium, a pore scale study

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1 Energy 30 (2005) Competition ofgravity, capillary and viscous forces during drainage in a two-dimensional porous medium, a pore scale study Grunde Løvoll a,b,, Yves Méheust a,b,c, Knut Jørgen Måløy a, Eyvind Aker a,c,d, Jean Schmittbuhl b a Department of Physics, University of Oslo, Pb Blindern, N-0316 Oslo, Norway b Laboratoire de Géologie, École Normale Supérieure, 24, rue Lhomond, Paris cedex 5, France c Department of Physics, Norwegian University of Science and Technology, Høgskoleringen 5, N-7491 Trondheim, Norway d WesternGeco, Schlumberger House, Solbråv.23, N-1383 Asker, Norway Abstract We have studied experimentally and numerically the displacement ofa highly viscous wetting fluid by a non-wetting fluid with low viscosity in a random two-dimensional porous medium under stabilizing gravity. In situations where the magnitudes ofthe viscous-, capillary- and gravity forces are comparable, we observe a transition from a capillary fingering behavior to a viscous fingering behavior, when decreasing apparent gravity. In the former configuration, the vertical extension of the displacement front saturates; in the latter, thin branched fingers develop and rapidly reach breakthrough. From pressure measurements and picture analyzes, we experimentally determine the threshold for the instability, a value that we also predict using percolation theory. Percolation theory further allows us to predict that the vertical extension ofthe invasion fronts undergoing stable displacement scales as a power law ofthe generalized Bond number Bo ¼ Bo Ca, where Bo and Ca are the Bond and capillary numbers, respectively. Our experimental findings are compared to the results ofa numerical modeling that takes local viscous forces into account. Theoretical, experimental and numerical approaches appear to be consistent. # 2004 Elsevier Ltd. All rights reserved. Corresponding author. Tel.: ; fax: address: grunde.lovoll@fys.uio.no (G. Løvoll) /$ - see front matter # 2004 Elsevier Ltd. All rights reserved. doi: /j.energy

2 862 G. Løvoll et al. / Energy 30 (2005) Introduction Two-phase flows in porous media are related to many important industrial and geological applications, such as oil recovery or ground water flow modeling [1 4]. For immiscible flows, a wide range ofbehaviors are observed depending on the wetting properties ofthe two fluids, their viscosity ratio, their respective density, and the displacement rate [5]. In this study, we address drainage in a random two-dimensional porous medium. Drainage consists in the immiscible displacement ofa wetting fluid by a non-wetting fluid, with capillary forces acting against the flow. The role of capillary forces is of special importance in a porous medium, where the interface between the two fluids consists of many menisci; capillary forces act at the scale ofthese menisci, that is, at the pore scale (see Fig. 1) and therefore are significant with respect to other forces that govern the displacement. Indeed, the capillary pressure, which is the difference between the pressures in the non-wetting and in the wetting phase at a point ofthe interface, is defined by the well-known Young Laplace law: p c ¼ p nw p w ¼ c 1 R 1 þ 1 R 2 ; Where c is the surface tension between the liquids and R 1 and R 2 are the two principal radii of curvatures for the interface. For a straight tube with perfect wetting, this reduces to p c ¼ 2c=r, where r is the tube radius, while, for a real porous medium the typical capillary pressure results from the typical pore size a. Furthermore, in a random porous medium, the random heterogeneity ofthe capillary pressure distribution greatly affects the local dynamics ofthe interface, with larger pores being more easily invaded by the displacing fluid, along the interface. The study by Saffman and Taylor [6] ofthe displacement ofone fluid by another one in a Hele-Shaw cell has shown how viscous forces can either stabilize or destabilize the interface, depending on whether the displacing fluid is the more viscous or not, respectively, and how the presence ofthe denser fluid below the other one has a stabilizing effect. Destabilization ofthe interface results in a fingering of the displacing fluid into the displaced fluid. The Saffman Taylor theory is not sufficient to describe the displacement in a porous medium, since it does not take into account the capillary fluctuations at the pore scale, nor does it account for the (1) Fig. 1. The difference between the pressure in the non-wetting fluid and that in the wetting fluid is given by the Laplace law; during drainage, capillary forces act against the displacement, and larger pores are more easily invaded.

3 G. Løvoll et al. / Energy 30 (2005) fluctuations in the viscous forces [4,7]. But it still explains the stabilizing or destabilizing tendency of viscous forces and gravity. In what follows, we consider a configuration where a wetting fluid is being displaced by a fluid placed above it, the viscosity and density ofwhich are much smaller than those of the wetting fluid. Hence, viscous forces tend to destabilize the interface against the stabilizing effect ofgravity. The type ofdisplacement observed during drainage in two-dimensional porous medium therefore depends on the relative magnitude of viscous forces and gravity, but also on their relative magnitude with respect to the heterogeneous capillary forces. A set of dimensionless numbers is usually defined to quantify these relative magnitudes. The capillary number Ca is the typical ratio ofthe viscous pressure drop at pore scale to the capillary pressure, while the Bond number quantifies that ofthe typical hydrostatic pressure drop over a pore to the capillary pressure: Ca ¼ Dp visc ¼ l wva 2 (2) Dp cap cj and Bo ¼ Dp grav ¼ Dqga2 ; Dp cap c (3) Where l w is the viscosity ofthe wetting fluid, v is the filtration or Darcy velocity, a is the typical pore size, c is the surface tension, j is the permeability ofthe porous medium, Dq is the density difference in the two fluids, and g is the acceleration due to gravity in the direction offlow. For systems without gravity we expect different flow regimes depending on the capillary number [8]: for very slow (quasi-static) displacements, the displacement is controlled by the heterogeneity ofthe capillary pressures along the interface [5,9]; this capillary fingering regime is well modeled by invasion percolation algorithms [10 12]; for fast displacements, where viscous forces overcome capillary effects, a viscous fingering regime is observed, with a rapid breakthrough of the non-wetting fluid into the wetting fluid [13]. These two flow regimes have been extensively studied, in particular, the fractal properties of the corresponding displacement structures. Experimental studies ofslow displacement under gravity have demonstrated how the interface keeps a finite extension w along the direction ofapparent gravity (see Fig. 2a), and how this width scales as a function of the Bond number [10,12]. Such slow displacement configurations are well modeled by invasion percolation with an invasion probability gradient. Under conditions offast displacements where the effect ofviscous forces dominate, in the presence of gravity, an unstable displacement ofthe interface similar to viscous fingering is observed (see Fig. 2b). In this paper, we present experimental and numerical results obtained in configurations where viscous forces and gravity compete. We study the transition from flow configurations where the interface is stable with respect to viscous instabilities (Fig. 2a) to flow configurations where viscous fingering occurs (Fig. 2b). As mentioned above, though analogous to the Saffman Taylor instability, this transition is not properly described by the Saffman Taylor theory. Three approaches are being used to tackle the problem: experiments, numerical simulations, and percolation theory.

4 864 G. Løvoll et al. / Energy 30 (2005) Fig. 2. Displacement structure ofthe invading wetting fluid observed in our experimental setup for configurations of (a) slow displacement, governed by the competition between capillary forces and gravity, and (b) viscous fingering under gravity. 2. Experiment Our experimental synthetic porous medium consists ofa single layer ofglass beads (diameter 1 mm), which are randomly positioned between two glass plates. It has a porosity / ¼ 0:63 and a permeability j ¼ 0: cm 2 ¼ 1915 darcy. See Refs. [13 15] for an extended description ofthe synthetic porous medium. The boundaries ofthe porous medium are square (35 35 cm); two opposite sides ofthe square are the inlet and outlet for the flow experiment. The medium is initially filled with a 90%/10% mixture in mass ofglycerol and water (viscosity 0.2 Pa s, density 1235 kg m 3 ), dyed with 0.1% Negrosine (black). The temperature ofthe setup is measured to correct for any possible change in the fluid viscosity due to temperature changes. The drainage experiment consists in extracting the mixture from the outlet, letting air invade the porous medium through the inlet. The surface tension c between the two phases is 6: Nm 1. The plane ofthe porous medium can be tilted by an angle h with respect to its horizontal position so as to vary the apparent gravity along the flow direction (see Fig. 3), while varying the extraction speed allows to tune the intensity ofviscous forces.

5 G. Løvoll et al. / Energy 30 (2005) Fig. 3. Sketch ofthe experimental setup. The porous medium is a single random layer ofglass beads. The wetting fluid is extracted from the bottom, allowing air to invade the medium from the top. Tilt angle of the model and extraction speed can be varied. Pressure is recorded at the outlet channel ofthe porous medium. Pictures ofthe displacement structure are taken at regular intervals during the experiment. Various recordings are made during the experiments. Pressure is recorded at the outlet. The displacement structures are analyzed from pictures recorded at regular intervals during the experiment. These pictures contain pixels, which corresponds to a spatial resolution of 2.56 pixels per pore; the color scale in made of256 gray levels. These raw images are filtered in order to clearly separate the wetting from the non-wetting phase and to extract the invasion front. Successive steps of the filtering process are presented in Fig. 4 (see [15] for a Fig. 4. Image filtering process: the raw image (a) is thresholded so as to obtain a black and white image (b) where the two phases are clearly separated. This displacement image is best viewed in reverse video (c) (see also Fig. 2). Removing the trapped wetting fluid clusters, one obtains two domains, the border line ofwhich is the invasion front. The front itself is plotted in black on top of the raw image in (a).

6 866 G. Løvoll et al. / Energy 30 (2005) complete description ofthe filtering process). We refer to displacement structures as the structure shown in black in Fig. 4c, and to invasion front as the line painted in black on top ofthe raw image in Fig. 4a. The invasion front is thus the continuous line separating the air cluster from the glycerol/water mixture. We define the front width w as the root mean square value ofthe front extension in the direction of flow. 3. Numerical model In the simulation, the porous medium is represented by a two-dimensional square network of tubes inclined at a 45 v angle. Each tube has a fixed length l and radius r i that is drawn from a defined distribution. A fragment of the porous network is presented in Fig. 5, with the non-wetting fluid entering the model at constant rate from the top. The network model takes both capillary fluctuations and local viscous pressure field into account when solving the flow field [16 18]. It has previously been used to successfully simulate various aspects ofthe drainage processes in configurations more general than that studied here. Viscous stabilization, viscous fingering and capillary fingering have been addressed [16,17,19]. The model is also able to account for the change in capillary pressure inside the tube, and thus to describe the displacement ofthe meniscus inside the tube. It can therefore be applied to studies oflocal burst dynamics [20]. A detailed description ofthe model can be found in [16,17]. For the present study, the model has been extended to include a tunable gravity field. To allow for simulations on larger system, the following simplification of the model has also been introduced. With respect to the capillary pressure p c ofa meniscus inside a tube, we consider the tube as straight and cylindrical, so that p c ¼ 2c r ; (4) which is independent ofthe position ofthe meniscus in the tube. To ensure numerical stability, the capillary pressure forced linearly to zero over a small region e at each end ofthe tube. By doing so the menisci in the network will either rest at the entrance ofthe tube or move with constant velocity through the tube during a time step. This coarsening ofthe capillary pressure Fig. 5. The pore network considered in the simulations is a regular network oftubes with random radii. 6 9 nodes are drawn here, with the invading non-wetting fluid painted black.

7 G. Løvoll et al. / Energy 30 (2005) Fig. 6. Geometry ofa tube in the simulation. Viscous and capillary pressures are computed using this straight geometry. In order to ensure numerical stability, the capillary pressure goes linearly to zero in a small region e close to the ends ofthe tube. allows for longer time steps. In the simulation, each time step corresponds to the time needed for one more tube in the network to be completely filled. The tube geometry is presented in Fig. 6. The coarsening ofthe model typically reduces the number oftime steps used in one simulation by a factor 10 2 in comparison to the complete model described in [16]. The accuracy ofthe simplified model has been verified by comparing its results to those provided by the complete model. For sufficient high displacement rates, the consistency between the two models is satisfactory. In the simplified model, the flow q in each tube is written as [21] q ¼ pr2 k ðdp p c Þ; (5) l eff where k ¼ r 2 =8 is the permeability ofthe tube, l eff ¼ l nw x þ l w ðl xþ (see Fig. 6 for the signification of l and x) is the effective viscosity, and Dp ¼ p j p i is the pressure difference between the two nodes at the opposite ends ofthe tube. Since the geometry ofthe porous network in the simulation is different from that ofthe experimental porous medium, the Bond and capillary numbers have to be defined carefully in order for the comparison to the experimental values to be valid. The Bond number is defined as Bo ¼ Dp grav ¼ Dqglr ; (6) Dp cap 2c where l is the tube length and r is the average tube radius over the system (excluding e and l e regions). The capillary number is estimated as Ca ¼ Dp visc ¼ DP visclr Dp cap 2cL ; (7) where hp visc is the total viscous pressure drop over the whole model when it is completely filled with wetting fluid (initial configuration), and L is the length ofthe system.

8 868 G. Løvoll et al. / Energy 30 (2005) In what follows we present results from a series of numerical experiments on a system of nodes where we keep the capillary number constant and vary the apparent gravity. The length ofthe tubes in the network is l ¼ 1:0 mm; the radii r i are drawn from a flat distribution in the interval [0.1, 0.4 mm]. The liquids are incompressible and immiscible with a viscosity contrast (l nw ¼ 0:0014 Pa s, l w ¼ 0:140 Pa s), a surface tension (c ¼ 6:4 Nm 1 ) and a density difference (q nw ¼ 1:3 kg=m 3, q w ¼ 1230 kg=m 3 ) chosen to match the liquid pair used in the experiments. In the simulations, the invasion front is defined as the position of the interface in the tubes separating the continuous wetting phase from the non-wetting phase. We define the front width in the same way as in the experiment. 4. Results For drainage with high viscosity contrast (l nw 5 l w ), where gravity stabilizes the invasion front, theoretical arguments based on percolation theory [10,22] in a stabilizing gradient could be developed [12,15,23]. The theory predicts the following scaling law for the displacement from which w under stable displacement [15] w ðbo CaÞ a ¼ðBo CaÞ ðv=1þvþ (8) where m is the correlation length exponent in percolation theory, for two-dimensional systems m ¼ 4=3 [22], and thus a 0:57. For convenience, we introduce the generalized Bond number Bo Bo Ca. Since the front width w in Eq. (8) diverges when Bo! 0, we expect a criterion for interface stabilization in the form Bo Ca ¼ Bo > 0 (9) This criterion is analogous to that found by Saffman and Taylor [6] for the displacement in an Hele-Shaw cell (qg ðvlþ=j > 0). But it should be noted that the dynamics ofthe instability is radically different from what is observed for Hele-Shaw cell experiments; this is obvious when considering the finite width ofthe invasion front or the trapping ofwetting fluid behind the front Stabilization of the invasion front In order to check ifstabilization occurs and under which circumstances, we investigate the evolution ofthe width ofthe displacement front in the flow direction as a function oftime. In Fig. 7, the front width is plotted as a function of time normalized by the time at breakthrough, t b, for both the experiments and the numerical simulations. From these curves, it is apparent that the invasion fronts reach saturation (stabilize) if they correspond to a generalized Bond number Bo > 0. The value Bo ¼ 0 is that for which the average hydrostatic pressure drop and viscous pressure drop over a pore are equal. As a consequence, for this value of the Bond number, the overall pressure difference over the model is expected to be constant. We have checked this by plotting the evolution ofthe total pressure difference over the model as a function ofnormalized

9 G. Løvoll et al. / Energy 30 (2005) Fig. 7. Evolution of the front vertical extension as a function of time in (a) the experiments and (b) the simulations. time. The pressure difference over the model, P, consists ofa hydrostatic contribution DP hyd and a viscous contribution DP visc : P ¼ P outlet P front ¼ DP hyd þ DP visc : (10) Experimental pressure measurements are plotted in Fig. 8a, and numerical results are presented in Fig. 8b. Both curves demonstrate that pressure reaches a constant level for Bo Scaling of the front width Fig. 9 presents the measured front width w as a function of the generalized Bond number Bo, using a log log scale, for configurations of stable displacement. The results from both the experiments (Fig. 9a) and numerical simulations (Fig. 9b) are consistent with the predicted scaling: w Bo 0:57 (11) The vertical shift between the data in Fig. 9 is due to a different width ofthe capillary threshold distribution in the experiments and the simulations [15,23]. 5. Discussion and conclusion When gravity is stabilizing and larger than the viscous forces, the invasion of a non-wetting liquid into a wetting liquid is stable. This problem could be understood by using a combination

10 870 G. Løvoll et al. / Energy 30 (2005) Fig. 8. Evolution ofthe pressure difference between the outlet and the invasion front, P ¼ P outlet P front, as a function oftime, (a) in the experiments and (b) in the simulations. P consists oftwo contributions, a hydrostatic pressure drop (positive) over the model and a viscous pressure drop (negative) over the model: P ¼ DP hyd þ DP visc. If DP hyd > DP visc, pressure decreases with time. The pressure difference over the model at breakthrough is used as the reference pressure: Pðt b Þ0. ofdarcy s law and a mapping to percolation theory, which allowed us to predict the scaling of the front width w (Eq. (8)) and the stabilization criterion (Eq. (9)) [15]. It is perhaps not surprising that our criterion for stabilization is indeed the same as the Saffman Taylor criterion [6]. Despite this, the geometry in our system is radically different from the geometry in the Hele-Shaw cell. Our system is a real porous medium where the relevant length scale is the pore scale; in contrast, the relevant size for Hele-Shaw experiments is the overall width ofthe channel. The differences between the two systems are clearly seen ifwe look at two effects. First, in a real porous medium we have a finite width w ofthe invasion front even for stable displacement, while in the Hele-Shaw geometry the front is flat as long as the displacement is stable. Second, the finite front gives rise to another characteristic feature of drainage in a porous media, which is the trapping ofclusters ofwetting liquid behind the front. This

11 G. Løvoll et al. / Energy 30 (2005) Fig. 9. Scaling of the front width as a function of the generalized Bond number Bo, for the experiments and numerical simulations. The two scaling are consistent, and in good agreement with the law predicted by the percolation theory, Eq. (11). phenomenon is absent in the Hele-Shaw geometry but prominent in our experiments (see Fig. 2a). The process oftrapping might even lead to a violation ofthe Saffman Taylor stabilization criterion for systems with a lower viscous contrast since trapping decreases the permeability for the non-wetting fluid [14]. We currently working on extending our results to situations where the displacements is unstable. Preliminary results seem to indicate that the mapping to percolation theory in a gradient is no longer valid, as characteristic length scales exhibit scaling laws that differ from what we observed in configurations ofstable displacement. Acknowledgments The work was supported by NFR, the Norwegian Research Council, VISTA, the Norwegian academy ofscience and letters research program with Statoil and the French/Norwegian collaboration PICS contract number References [1] Dullien FAL. Porous media fluid transport and pore structure, 2nd ed. San Diego: Academic Press, Inc; [2] Bear J. Dynamics offluids in porous media. New York: American Elsevier Publishing Company; [3] Sahimi M. Flow phenomena in rocks: from continuum models to fractals, percolation cellular automata, and simulated annealing. Review ofmodern Physics 1993;65(4): [4] Homsy GM. Viscous fingering in porous media. Annual Review offluid Mechanics 1987;19: [5] Lenormand R, Zarcone C. Invasion percolation in an etched network: measurement ofa fractal dimension. Physical Review Letters 1985;54(20):

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