Channelization of buoyant nonwetting fluids in saturated porous media

Transcription

1 WATER RESOURCES RESEARCH, VOL. 49, , doi: /wrcr.20514, 2013 Channelization of buoyant noetting fluids in saturated porous media C. Huber, 1 Andrea Parmigiani, 1 Jonas Latt, 2 and J. Dufek 1 Received 17 December 2012; revised 6 August 2013; accepted 28 August 2013; published 7 October [1] We study the development of capillary instabilities during the invasion of a buoyant noetting phase in a saturated porous media. Capillary instabilities are generally attributed to heterogeneities in the porous medium resulting in the existence of fluid pathways opposing different resistance to the flow ( passive control ). We use a simple macroscale theoretical model based on the postulate that the noetting fluid will be distributed in the porous medium to minimize the resistance to transport. This theoretical argument is used to show that after their formation, some capillary instabilities can grow at the expense of others. The competitive growth between capillary channels arises because of pore-scale fluid interactions that occur even in a porous medium offering identical pathways at the pore scale. The evolution of the pore volume fraction of noetting fluid in capillary fingers is therefore dynamically controlled by fluctuations in the noetting phase saturation and its effect on the relative permeability ( active control ). The theoretical model predicts (1) the growth of heterogeneities in noetting fluid saturation among competing capillary channels if the second derivative of the invading phase relative permeability with respect to its saturation is positive, and (2) that the amplitude of the perturbation in noetting fluid content between competing fingers increases with the interfacial tension. We use a porescale multiphase flow numerical model to test the validity of the postulate for optimal transport of noetting fluids and the two ensuing predictions. We observe that the numerical calculations are in excellent agreement with the theoretical predictions. Citation: Huber, C., A. Parmigiani, J. Latt, and J. Dufek (2013), Channelization of buoyant noetting fluids in saturated porous media, Water Resour. Res., 49, , doi: /wrcr Introduction [2] Multiphase flows in porous media have a wide array of engineering, geological, and environmental applications. The contamination of water reservoirs during the doward migration of dense nonaqueous phase liquids (DNAPL) has, for example, been subject to numerous studies [Ewing and Berkowitz, 1998, 2001; Berglund, 1997]. The transport and storage ability of porous reservoirs hosting two immiscible fluid phases are controlled by their respective spatial distributions. Fingering instabilities, the process by which the distribution of each fluid phases becomes heterogeneous during the injection of the invading fluid, are expected to arise as a consequence of pore-scale heterogeneities (capillary fingering) or of a difference in viscosity between a low viscosity invading fluid and a more viscous defending fluid [Lenormand et al., 1988; Lïvoll et al., 2004, 2011]. The development of fingering 1 School of Earth and Atmospheric Sciences, Georgia Institute of Technology, Atlanta, Georgia, USA. 2 Computer Sciences Department, University of Geneva, Geneva, Switzerland. Corresponding author: C. Huber, School of Earth and Atmospheric Sciences, Georgia Institute of Technology, 311 Ferst Dr., Atlanta, GA 30332, USA. (christian.huber@eas.gatech.edu) American Geophysical Union. All Rights Reserved /13/ /wrcr instabilities at the invading front affects greatly the storage and transport of each fluid phase and is therefore a big concern for remediation processes or carbon sequestration strategies. In this study, we focus on the buoyant migration of a noetting invading phase in a saturated porous medium. Several experimental and numerical studies have shown that, at low injection rate, a buoyant invading fluid (either a dense NAPL migrating doward or a light NAPL migrating upward) forms fingers because of a capillary instability [Ewing and Berkowitz, 1998; Berglund, 1997; Yortsos et al., 1997; Nsir et al., 2012]. In this study, we focus on the development of and evolution of capillary instabilities during the migration of the invading fluid, using a macroscale scaling argument based on energy optimization and numerical calculations that solve for the multiphase dynamics at the pore scale. We show cases where the capillary coupling between the two fluids, rather than the pore geometry, controls the redistribution of invading fluid among competing capillary channels. [3] Numerical studies can be generally divided in two groups: (1) macroscale models derived from the multiphase Darcy equation [Nieber et al., 2005; Cueto-Felgueroso and Juanes, 2009] and (2) pore-scale approaches such as pore-network and stochastic models [Ewing and Berkowitz, 1998, 2001; Lenormand et al., 1988; Yortsos et al., 1997; Blunt, 2001]. Continuum models such as those based on variants of the Richards equation [Nieber et al., 2005; Cueto-Felgueroso and Juanes, 2009] solve for the 6371

2 migration of the invading fluid at a scale much greater than the pore. The multiphase permeability and capillary effects are therefore introduced as constitutive equations in these macroscopic models. The Richards equation was shown to be intrinsically stable and the generation of capillary fingering required the introduction of higher order terms in the governing equation [Cueto-Felgueroso and Juanes, 2009; Ewing and Berkowitz, 2001]. These macroscopic models do not solve for the force balance between capillary, buoyancy, and viscous forces at the pore scale, but rather compute an average over a large number of pores from empirically derived relationships between the capillary forces and the invading fluid saturation S. Alternatively, pore-scale models are often limited to the study of a small number of pores or require, in the case of pore-network models, some geometrical simplifications to model the flow over volumes where continuous properties computed from averages become meaningful. In the present study, we use a highly parallel multiphase lattice Boltzmann model to benefit from a realistic treatment of the pore-scale force balance while solving the dynamics in a volume large enough to relate our results to macroscale continuous properties such as porosity, fluid saturation, and discharge. 2. Macroscopical Model [4] As we aim to study the evolution of developed capillary fingers in a porous medium, we first present a macroscale scaling argument that shows the importance of an accurate description of capillary forces. Figure 1 shows the setup of an ideal porous medium, where low permeability regions alternate with high permeability channels. The saturated porous medium is subjected to the injection of a buoyant noetting phase at a constant mass input rate _M. The flow of noetting fluid is assumed to be buoyancy driven and therefore mostly vertical. [5] The overall hydraulic conductivity for the noetting phase K tot, which is equivalent to the inverse of the hydraulic resistivity, is X K tot ¼ X K i i i ; ð1þ where the K i are the individual noetting phase conductivities for the different channels and i their relative width. Using the simple geometry illustrated in Figure 1 for n high permeability channels of similar width, assuming that each high porosity channel hosts the same amount (volume fraction) of noetting fluid, i.e., gas saturation S c (volume fraction of the pore space occupied by the noetting phase) and porosity c, we obtain i i k r0; ðs 0 Þk 0 K tot ð c Þ ¼ 1 n L þ n L k r1; ðs c Þk 1 ; ð2þ where k 0, k 1, k r0; ðs 0 Þ, and k r1; ðs c Þ are, respectively, the permeability of the two different types of channels in the porous medium and the relative permeabilities as function of gas saturation for the low and high conductivity regions. In equation (2), is the specific gravity of the noetting fluids, and S 0 represents the steady saturation of noetting Figure 1. Heterogeneous porous media. The even numbered channels represent the low permeability k 0 (low hydraulic conductivity K 0 ) regions and separated by high permeability channels k 1 (with odd numbers). Each of the five low or high permeability channels is identical to the pore scale and should therefore grow identical capillary channels under the same injection conditions. Colors are just added to better visualize the structure of the porous medium. fluid in the channels of permeability k 0 and S c in the high permeability channels. c refers to the configuration with uniform noetting fluid saturation S c in high porosity channels. [6] We postulate that the existence of an optimization principle similar to minimization of entropy production controls the distribution of noetting fluid and favors transport for the buoyant invasion of noetting fluid in a saturated porous medium. One of the aims of this study is to use such an assumption to obtain theoretical predictions about the organization of the mass transport of noetting fluid in the porous medium and assess the validity of the postulate by testing these predictions with pore-scale numerical calculations. One can argue that Saffman-Taylor instabilities associated with viscous fingering obey a similar type of principle, as the instability of the front reflects a configuration that minimizes the dissipation of energy in comparison to a stable invasion front. [7] Constructal theory [Bejan and Lorente, 2008] has been developed over the last two decades as a novel approach to derive asymptotical solutions to dynamical processes according to simple optimization principles. It has been applied with success to a large range of applications, modeling natural systems [Bejan and Lorente, 2010] as well as socioeconomic patterns [Bejan, 2012]. From constructal theory, we assume that the configuration with the lowest resistance to the flow of the noetting fluid, under the given constraints of the system studied (porous media 6372

3 Figure 2. Invasion of the noetting buoyant fluid (orange) in a single channel of high porosity medium. The formation of capillary instabilities, here the viscosity ratio between the two fluids is 1, is controlled here by the least -resistant pathways for invasion, i.e., the distribution of pore throats along the path of the invading fluid. geometry and fixed mass flux of buoyant noetting phase), will emerge naturally. We can compute the overall hydraulic conductivity for the noetting phase for a perturbed case where the saturation for two of the n channels is perturbed by 1 S 1 ¼ S c þ ; S n ¼ S c S i ¼ S c ; for i ¼ 2;...; n 1: ð3þ [8] Here the numbering of channels does not matter, and channels can be renumbered without influencing our ensuing conclusions. This perturbed configuration will be referred as. The overall hydraulic conductivity for configuration is K tot ð Þ ¼ k r0;ðs 0 Þk 0 1 n L þ ðn 2Þ k r1; ðs c Þk 1 þ k r1; ðs c þ Þk 1 L L þ k r1; ðs c Þk 1 : L [9] We can compare the two results for K tot and find conditions under which the perturbed configuration offers less resistance to the noetting fluid transport, i.e., K tot ð Þ > K tot ð c Þ. Subtracting the overall hydraulic conductivities for these two configurations leads to K tot ð Þ K tot ð c Þ ¼ k 1 L ð4þ k r1; ðs c þ Þ 2 k r1;ðs c Þ þ k r1;ðs c Þ : [10] Considering here small perturbations 1, K tot ð Þ K tot ð c Þ ¼ k 1 2 k 2 ð5þ 2 þo 3 ; ð6þ S¼S c where Oð 3 Þ represents all the contributions with order greater or equal to 3 in terms of the perturbation. We find that the flux of noetting phase is greater for the perturbed configuration (i.e., greater overall hydraulic conductivity) 2 k 2 > 0: ð7þ [11] Because the relative permeability for the noetting invading fluid as function of the saturation S is sensitive to capillary forces at the pore scale, this condition depends on the interfacial tension between the fluid phases and tension forces between each fluid and the solid matrix. If equation (7) is satisfied, the flow of noetting phase through the porous media is more efficient for a heterogeneous distribution of noetting phase saturation among high porosity channels. This is true even when the channels are identical, as long as small perturbations associated with random noise in the invasion process are present. From equation (7), the degree of heterogeneity in the gas saturation between the different channels depends also only on the curvature of the relative permeability with respect to the gas saturation. Because interfacial tension exerts a control on the curvature of k r; ðsþ [Asar and Handy, 1988; Shen et al., 2010], it suggests that the growth of perturbations in noetting fluid saturation is expected to be stronger for greater interfacial tensions [Asar and Handy, 1988; Shen et al., 2010]. This macroscale scaling argument emphasizes the importance of resolving for the appropriate balance of capillary, buoyancy, and viscous force at the pore scale instead of using standard empirical correlations between the capillary pressure and the pore saturation. 3. Numerical Approach [12] Pore-scale approaches based on pore network models are based on idealized porous media geometry where the medium consists of a succession of conduits (pore throats) and pores with generally cylindrical and spherical shapes and using a random distribution of radii. The pores are assumed to be fully saturated by either the invading or the defending fluid. The invadibility of a given pore is obtained directly from the position of the pore with respect to the water table, the density contrast between the fluids, and the pore radius [Ewing and Berkowitz, 1998, 2001]. In that sense, the migration of the invading phase is purely 6373

4 deterministic as it is controlled by the porous medium. The control exerted by the pore size distribution in the development of a capillary instability will be referred to as passive control, and it is a deterministic process. We argue that the effect of a nonlinear relative permeability for the noetting fluid leads to a stochastic reorganization of the distribution of the invading phase between identical channels. This feedback between transport and invasion is dynamic as opposed to the passive control defined above, and the goal of the numerical investigation is to test the existence and importance of this process on welldeveloped capillary channels. [13] Because we want to observe the dynamics of capillary channels and the emerging behavior of these channels (distribution of noetting fluid at steady state), we use a numerical approach that solves for the multiphase dynamics at the pore scale. It is different from macroscale continuum approaches in the sense that it does not require any assumption, i.e., empirical closure equations, for the treatment of capillary effects between the fluid phases. Moreover, our model solves for the force balance between the two fluid phases within each pore, unlike pore-network models. The numerical model we use is based on a multiphase lattice Boltzmann method, and it allows us to calculate the migration of a buoyant noetting phase in a porous medium with complex pore geometries. We use the parallel open source library Palabos ( org/) for all flow calculations. The multiphase flow is solved with the Shan-Chen model for multicomponents [Shan and Chen, 1993; Shan and Doolen, 1995]. Bubbles of noetting fluid are injected periodically at the bottom inlet and ascend buoyantly to the porous medium where they form a capillary layer. The injection process can be parameterized in terms of a capillary number Ca ¼ Q =R ð 2, where Q is the volumetric rate of noetting fluid injected in the porous medium, its dynamic viscosity, is the interfacial tension, and R is the radius of injected bubbles of noetting fluid. The values of Ca used in our calculations range from 0.28 to 0.54 from the highest value of interfacial tension to the lowest. The contact angle between the two fluids and the solid substrate is set to 135 degrees and fixed for all calculations. The viscosity ratio between the two fluids was set to unity. Once the pressure in the noetting fluid layer at the inlet reaches the capillary entry pressure, invasion starts (see Figure 2). For more details about the numerical model, the reader is referred to section 3.1. [14] We use a synthetic porous medium grown numerically that consists of a succession of identical channels at the pore scale (see Figure 1). The porous medium is constructed numerically with a crystal nucleation and growth algorithm derived from the model of Avrami [1940] and mostly similar to Hersum and Marsh [2006]. Each channel is nodes large and contains more than 15,000 crystals with various shapes and an average size on the order of 10 grid nodes. The nucleation and growth model allows us to crystallize a single texture and use it for both the low (36 %) and high (64%) porosity channels by removing more or less of the latest crystallized phases. To test the channelization instability (heterogenous distribution of noetting phase in identical channels), we build a large domain ( ) by alternating high (odd numbers) and low porosity (even numbers) channels. As a result, all high porosity channels are identical to the pore scale (Figure 1). We computed the permeability for the low and high porosity channels with a single phase pore-scale flow model and obtained 0.35 and 4.13, respectively, with these values being normalized by the grid spacing used in the calculations. It means that, in this particular study, high porosity channels are about an order of magnitude more permeable than the low porosity ones. [15] As the high porosity channels are identical, the passive control exerted by the porous media on the invading phase is identical for each channel. The deterministic nature of pore-network models would therefore predict that identical capillary fingers grow in each high porosity channel. We show, however, that capillary fingers grow at the expense of others in high porosity channels. The difference in invading fluid saturation among identical high porosity channels is not controlled by the pore structure (passive control), but rather by the capillary coupling between the two fluid phases (dynamic control). This effect is not deterministic as it is initiated by random fluctuations (noise) in the injection rate of the invading phase and controlled by the capillary forces between the invading and defending phases at the pore scale. Therefore, in this ideal porous medium, the growth of certain capillary channels at the expense of others can not be predicted by pore-network models, and, because of the scale at which it operates, cannot be solved explicitly by macroscale approaches Shan-Chen Lattice Boltzmann Scheme for Immiscible Fluid [16] The lattice Boltzmann method (LBM) is a computational fluid dynamics method that offers an efficient and convenient alternative to traditional solvers for a large variety of multiphysics fluid dynamic problems. Instead of discretizing a set of partial differential equations that describe the evolution of pressure and velocity, the LBM solves a discretized version of the Boltzmann equation, the analog of the Navier- Stokes equation at the molecular level. The Boltzmann equation describes the space-time evolution of the probability distribution function f ðx; v; tþ, that describes the probability of finding a particle at the position x with a velocity v at time t. The discrete version of particle distribution function, f i must be represented in a discretized 6-dimensional phase space. This implies that both space and velocities must be discretized; this is generally done on a cubic lattice. The discrete set of lattice velocities v i connect each grid point to its neighbors. We use a D3Q19 lattice where 8 < ð0; 0; 0Þ i ¼ 0; v i ¼ ð61; 0; 0Þ; ð0; 61; 0Þ; ð0; 0; 61Þ; i ¼ 1 6; : ð61; 61; 0Þ; ð61; 0; 61Þ; ð0; 61; 61Þ; i ¼ 7 18 [17] This means that for each grid point x of a 3- dimensional space, we define a set of f i composed of 19 components that connect x to its 19 (x þ v i ) neighbors. The weights for each distribution f i are defined with: ð8þ w 0 ¼ 1=3; w 1 6 ¼ 1=18; w 7 18 ¼ 1=36: ð9þ [18] Commonly, the collision operator is simplified and the single relaxation time LB scheme based on the 6374

5 Figure 3. Calculations of the relative permeabilitysaturation relationship for high porosity channels with a fixed Bond number B¼0:26. The error bars represent the magnitude of the fluctuations around steady state values for the saturation of the noetting phase and its discharge. The calculated data, computed over the range of saturations of interest, are fitted by two power-law relationship, one with fixed exponent (set to 4) and one where the exponent is also a fitting parameter. Bhatnagar-Gross-Krook approximation (BGK) [Bhatnagar et al., 1954] is used. With the BGK approximation, the LB scheme is split into two steps, a collision step, where the f i s exchange information between each other f out i ðx; t þ 1Þ ¼ f i ðx; tþþ 1 ð f eq i ðx; tþ f ðx; tþþ; ð10þ and a streaming step, where the fi out s stream and transport information to neighbor lattice grid points: f i ðx þ v i ; t þ 1Þ ¼ fi out ðx; t þ 1Þ: ð11þ [19] In the BGK model, the collision step represents a relaxation of the particle distribution functions f i toward the a local equilibrium f eq i with a rate 1/, where the f eq i s is defined by " # f eq i ðx; tþ ¼ w i 1 þ v i u ð c 2 þ v i uþ s 2c 4 u2 s 2c 2 s : ð12þ where q c s ffiffi is a constant characteristic of the lattice topology (c s ¼ for the D3Q19 lattice), is the density of the 1 3 fluid, and u its macroscopic velocity at the position x. The density and momentum j ¼ u of the fluid are obtained from the moments of the distributions: ¼ X i f i ; u ¼ X i v i f i : ð13þ [20] The choice of relaxation time controls the kinematic viscosity of the fluid: ¼ c 2 s 1 : ð14þ 2 [21] The lattice Boltzmann community has developed and used a variety of models for multiphase and multicomponent fluid flow applications [Gunstensen et al., 1991; Shan and Doolen, 1995; Swift et al., 1996; He et al., 1999; Aursjï et al., 2011]. Although these models differ depending on their approach to model nonideal fluid behaviors, all of them belong to the same class of diffuse-interface methods [Anderson et al., 1998]. Diffuse-interface methods allow to study immiscible fluids flow with interfacial tension without the need to track the deformable interface at the cost of a diffuse interface region of finite thickness. [22] We selected the multicomponent Shan-Chen (SC) method for immiscible fluid flow [Shan and Chen, 1993; Shan and Doolen, 1995]. The implementation of no-slip surfaces between the immiscible fluids and the solid porous medium is accomplished by a reflection of the incoming particle distribution function on solid nodes (bounce-back method). We introduce two particle distribution functions fi and fi w to represent the noetting and wetting fluids, respectively. Both distributions follow the same evolution described by equations (10) and (11). The wetting and noetting fluids are coupled by repulsion forces that are proportional to their densities u eq X u! ¼ X! þ F! : ð15þ [23] In equation (12),! ¼ 1= and F ¼ F coh þ F ads þ Fb is the sum of the fluid-fluid and fluid-solid interaction forces. F coh are the cohesion forces responsible for phase separation and interfacial tension, while F ads represent adhesion forces between the solid boundaries and the fluids. Finally, buoyancy can be introduced in F b as an external body force. We refer the reader to Huang et al. [2007] and Parmigiani et al. [2011] for a complete description of implementation of cohesion F coh and adhesion F ads forces in the Shan-Chen LB scheme. 4. Relative Permeability [24] Our theoretical model predicts that the dependence of the relative permeability on the volume fraction S for the noetting phase controls the existence of instabilities in the distribution of the buoyant noetting fluid in the pore space. We focus on obtaining the k r;nm ðsþ relationship in high porosity channels (1) to make sure that the condition of equation (7) is satisfied and that large-scale calculations lie in the heterogenous noetting phase distribution regime and (2) to estimate the noetting phase volume flux variability between the different high-porosity channels. [25] Relative permeabilities are generally empirically defined as an extension of Darcian flow to immiscible multiphase transport in porous media. As such, the relative permeability for a noetting fluid depends on several factors such as the viscosity of the two fluids, the structure and topology of the porous medium, interfacial tension, and the contact angle between the three phases. Because, we use the same porous medium and fix the values of the fluids viscosity as well as the contact angle, the only factor that can affect the relative permeability for the noetting fluid 6375

6 Figure 4. (a, b) Distribution of the buoyant noetting phase in the porous media for runs shown respectively in Figures 4a and 4c. (c) A close-up of the bottom part of the porous media around channel 5. The lower numbers on each channel represent the volume fraction of pore space occupied by the noetting phase. We use red for channels with excess noetting phase and blue for depleted channels. between different calculations is the interfacial tension. Figure 3 shows the relative permeability-saturation relationship for the buoyant noetting phase for high porosity channels. The curvature of this relationship satisfies the 2 k r; =@S 2 > 0. The numerical results are fitted with two different power laws, one where the exponent is fixed to 4, following a Corey-type model, and the other where the exponent is a free parameter. We expect that the curvature of k r; ðsþ is controlled by the interfacial tension and increases with increasing interfacial tension or decreasing Bond number B B¼ Dgr2 ; ð16þ where r is the radius of injected bubbles [Asar and Handy, 1988; Shen et al., 2010]. We revisit our assumption about the effect of the interfacial tension on the curvature of k r; ðsþ in the discussion section. 5. Results [26] We conducted five calculations with different values of interfacial tension between the two fluids ð0:25 B 0:5Þ to test the two predictions obtained from the simple scaling argument: (1) that channelization instabilities will occur 2 k r; =@S 2 > 0 and will lead to heterogenous distribution of noetting phase in the competing capillary fingers and (2) that the amplitude of the heterogeneities is expected to increase with interfacial tension. A snapshot of the distribution of the noetting fluid for two different interfacial tensions is shown in Figure 4. All runs lead to an heterogenous distribution of the noetting phase among the five high porosity channels as predicted by our theoretical model (see Figure 5). Figure 5 illustrates the typical results we obtain. First, a transient phase where the capillary pressure builds up in the inlet region as a result of the constant injection of buoyant noetting fluid. This stage is followed by a steady state (constant volume of 6376

7 Figure 5. Results of three large scale calculation showing the evolution of the volume fraction of buoyant noetting phase in the porous media as function of time. Time is made dimensionless by using the injection rate of noetting fluid t ¼ Q t=v b, where Q is the volume of noetting fluid injected at the inlet per unit time, and V b is the volume of an individual bubble of noetting fluid as they are injected. (a, b) Run conducted at intermediate B¼0:32. (c) For a lower interfacial tension and higher Bond number B¼0:48. (d) Same results for a higher interfacial tension B¼0:26. In each calculation, the saturation of noetting fluid is normalized by the corresponding average steady state value. noetting phase) perturbed by the onset of the instability where S grows in certain high porosity channels at the expense of others. [27] Figures 5c and 5d show the pore volume fraction occupied by the noetting phase for two other calculations, (c) for a lower interfacial tension (equivalent to B¼0:48), and (d) for a higher interfacial tension (leading to B¼0:26) than those in Figures 5a and 5b. We observe a qualitatively similar behavior; however, the steady-state average S value and the amplitude of the instability (i.e., variability in S in the high porosity channels) increases with interfacial tension (or decrease with B) as expected from our theoretical model. Each calculation was run on 4096 Intel cores of the supercomputer Ranger at the TACC (NSF facility) and was limited to the usage of 100,000 CPU-hours. Within these computing resources, we observe that, as the Bond number B decreases, or in other words as the interfacial tension increases, the relative variability in S increases from 20 to 60% (see Table 1). 6. Discussion [28] In section 2, we showed mathematically that if one does postulate that the distribution of an invading noetting buoyant fluid can be determined by an optimization of the resistance to its flow, the curvature of the relative permeability k r; ðsþ is then expected to control the distribution of the two fluids in the porous medium. In other words, in light of the optimization argument, the invading fluid is expected to be distributed so as to minimize the resistance to the flow, in the limits of the constraints of the problem. These constraints include, here, for example, a fixed injection flux at the inlet, the geometrical setup of the porous medium, and finally the ability of the invading fluid to travel laterally from one capillary channel to another (here granted by the low resistance to lateral flow at the inlet). Limitation for the invading fluid to find an optimal configuration for transport is discussed in greater details below. [29] All five of our calculations show convincingly that, under the conditions tested here, after a steady overall Table 1. Amplitude of Instability (in Terms of Saturation of the Noetting Phase) Versus Bond Number B B¼0:26 B¼0:32 B¼0:48 DS ¼ ðs max S min Þ=2S avg 60% 40% 2 k r =@Sg 2at S g ¼ S avg

8 Figure 6. Relative permeability of the noetting invading phase as function of saturation for a single high porosity channel. (a-c) The relationship shows a more nonlinear relationship as the interfacial tension increases (decreasing Bond number). The error bars are estimated from the output of the numerical substitution by estimating the maximum amplitude of fluctuations around the steady state values for the relative permeability and the noetting phase saturation. (d) The dependence of the amplitude of the differential growth of initially similar capillary channels and the curvature of the relative permeabilitysaturation curve at the steady state value for S. saturation is reached, the invading phase distribution grows heterogeneously between identical channels in agreement to what is expected from the optimization principle discussed above. The latter predicted that (1) if the invasion process is able to reach an optimal or pseudo-optimal transport configuration, the resulting distribution of noetting fluid between the different capillary fingers is heterogeneous, and (2) that the amplitude of this noetting fluid saturation heterogeneity is controlled by the curvature of k r; ðsþ. In Figures 6a and 6c, we show the relative permeability for the noetting fluid as function of its saturation S for the three different interfacial tension values (different Bond number). The blue band represents the average saturation at steady state S avg ¼ S c for each calculation and therefore indicates where the curvature of k r; needs to be estimated to test the condition of equation (7). Each point on these plots was obtained from the steady state discharge of noetting fluid through a single high porosity channel. The large error bars represent very conservative estimates of the transient behavior associated with the noise in S and noetting phase discharge for these multiphase flows. The errors are estimated directly from the amplitude of the temporal fluctuations observed around the steady state solution. [30] In our calculations, the only factor that is varied from a run to the other is the strength of the interfacial tension between the two fluids, with the contact angle, porous medium topology, and viscosities remaining identical. The interfacial tension will therefore control the changes in curvature in the relationship between the relative permeability for the noetting fluid and its saturation. From Figures 6a and 6c, we clearly see an increase in curvature d 2 k r; =ds 2 at S avg with increasing interfacial tension or decreasing B around the average steady saturation S avg. Our theoretical model predicts that the amplitude of the perturbation for the distribution of noetting fluid in high porosity channels increases with the curvature of k r; ðsþ, which is consistent with our observed numerical results in Figure 6d. From the limited number of calculations we have conducted, we observe thatthe amplitude of the perturbation scales like d 2 k r; =ds 2 0:03. [31] Multiphase invasion instabilities in homogeneous porous media are generally attributed to either a difference in viscosity between the invading and defending fluids (viscous fingering) or the control of capillary forces that allow only the largest pores to be invaded (capillary channels) [Lenormand et al., 1988]. These fingering instabilities have deep implications for engineering applications and natural flows, with a growing interest in particular related to NAPL remediation and geological CO 2 sequestration. The process we focus on in this paper belongs to the class of fingering instabilities but differs from standard capillary and viscous 6378

9 fingering instabilities. We studied the evolution of capillary fingering instabilities established in identical strips of porous media that offer an identical spatial distribution of pore sizes and shapes. The random noise generated by the injection process leads to small perturbations in local noetting phase saturation in the capillary fingers. Because of the nonlinear feedback between relative permeability (which accounts for capillary coupling of the two fluids at the pore scale) and the respective volume fraction of the two fluids, these perturbations affect the overall resistance to the ascent of the invading buoyant fluid, and therefore perturbation can grow and become quite significant (up to 60% relative changes in saturation). [32] In our calculations, the evolution of the capillary fingering instability (the growth of fingers at the expense of others) is controlled by a dynamical process (perturbation in local invading fluid pore volume fraction) rather than by the variation of capillary entry pressure due to the heterogeneous distribution of pore sizes. The redistribution of invading fluid among competing fingering instabilities is controlled by the nonlinear relative permeability. There are, however, potential limitations for this redistribution process to occur. First and foremost, the growth of noetting fluid perturbations requires some lateral transport of noetting fluid from the shrinking to the growing channel. If the lateral transport of noetting fluid is limited by a low lateral permeability, the redistribution and optimization of transport process can only affect narrow regions around the capillary instability. In this particular set of calculations, the lateral transport of noetting fluid was favored by the existence of the inlet and therefore did not limit the redistribution of noetting fluid among the different channels. The lateral transport of noetting fluid between shrinking and growing fingers requires lateral pressure gradients. In our calculations, these lateral pressure gradients result from lateral variations of capillary entry pressure at the level of the inlet. Lower resistance pathways for the invading fluid are characterized by lower capillary entry pressures and can therefore channelize more invading fluid than the more resistant pathways. 7. Conclusions [33] The distribution of different immiscible fluid phases in porous media controls their respective mobility and their storage potential. We present a theoretical model based on an optimization argument (minimization of the resistance to the flow of the invading fluid) and postulate that if such principle is valid, the distribution of noetting fluid in competing and initially identical capillary fingers evolves toward a heterogeneous state where some of these fingers grow at the expense of others. The assumption for an optimal transport, if valid, leads to another significant outcome: the amplitude of noetting fluid saturation perturbations is expected to depend on the curvature of the relative permeability with respect to the saturation of noetting fluid. Because all the factors that contribute to the relative permeability-saturation relationship for the noetting fluid are fixed in our calculations at the exception of the interfacial tension, we expect that the existence of preferential pathways is controlled by the interfacial tension between the two fluids. Independent pore-scale numerical calculations of multiphase flow in a porous medium are in excellent agreement with these predictions and provide support for the theoretical model. The calculations show that even in portions of a porous media that are identical at the pore scale and are subjected to equivalent injection conditions, the buoyant injection of a noetting phase results in a heterogeneous saturation. The redistribution of noetting fluid among capillary instabilities is controlled by the dynamic coupling between the invading and defending fluid phases at the pore scale and the ability of the noetting fluid to travel laterally from one finger to another. Future studies considering homogeneous porous media will be needed to establish the natural horizontal wavelength associated with this process and quantify its dependence on the injection rate of invading fluid and the horizontal permeability of the medium. 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