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International audience; The present contribution focuses on the accuracy of reflection-type boundary conditions in the Stokes-Brinkman-Darcy modeling of porous flows solved with the lattice Boltzmann method (LBM), which we operate with the two-relaxation-time (TRT) collision and the Brinkman-force based scheme (BF), called BF-TRT scheme. In parallel, we compare it with the Stokes-Brinkman-Darcy linear finite element method (FEM) where the Dirichlet boundary conditions are enforced on grid vertices. In bulk, both BF-TRT and FEM share the same defect: in their discretization a correction to the modeled Brinkman equation appears, given by the discrete Laplacian of the velocity-proportional resistance force. This correction modifies the effective Brinkman viscosity, playing a crucial role in the triggering of spurious oscillations in the bulk solution. While the exact form of this defect is available in lattice-aligned, straight or diagonal, flows; in arbitrary flow/lattice orientations its approximation is constructed. At boundaries, we verify that such a Brinkman viscosity correction has an even more harmful impact. Already at the first order, it shifts the location of the no-slip wall condition supported by traditional LBM boundary schemes, such as the bounce-back rule. For that reason, this work develops a new class of boundary schemes to prescribe the Dirichlet velocity condition at an arbitrary wall/boundary-node distance and that supports a higher order accuracy in the accommodation of the TRT-Brinkman solutions. For their modeling, we consider the standard BF scheme and its improved version, called IBF; this latter is generalized in this work to suppress or to reduce the viscosity correction in arbitrarily oriented flows. Our framework extends the one- and two-point families of linear and parabolic link-wise boundary schemes, respectively called B-LI and B-MLI, which avoid the interference of the Brinkman viscosity correction in their closure relations. The performance of LBM and FEM is thoroughly evaluated in three benchmark tests, which are run throughout three distinctive permeability regimes. The first configuration is a horizontal porous channel, studied with a symbolic approach, where we construct the exact solutions of FEM and BF/IBF with different boundary schemes. The second problem refers to an inclined porous channel flow, which brings in as new challenge the formation of spurious boundary layers in LBM; that is, numerical artefacts that arise due to a deficient accommodation of the bulk solution by the low-accurate boundary scheme. The third problem considers a porous flow past a periodic square array of solid cylinders, which intensifies the previous two tests with the simulation of a more complex flow pattern. The ensemble of numerical tests provides guidelines on the effect of grid resolution and the TRT free collision parameter over the accuracy and the quality of the velocity field, spanning from Stokes to Darcy permeability regimes. It is shown that, with the use of the high-order accurate boundary schemes, the simple, uniform-mesh-based TRT-LBM formulation can even surpass the accuracy of FEM employing hardworking body-fitted meshes.

We model the flow of a bi-viscous non-Newtonian fluid in a porous medium by a square lattice where the links obey a piece-wise linear constitutive equation. We find numerically that the flow regime where the network transitions from all links behaving according to the first linear part of the constitutive equation to all links behaving according to the second linear part of the constitutive equation, is characterized by a critical point. We measure two critical exponents associated with this critical point, one of the being the correlation length exponent. We find that both critical exponents depend on the parameters of the model., Submitted to Frontiers in Physics

International audience; In this work, we studied experimentally and numerically the pressure–flow rate relationship for yield stress fluids in porous media. We developed and validated 3D numerical simulations of the velocity field via a lattice Boltzmann method based on the TRT scheme, and a specific experimental setup allowing yield stress fluids to flow in a closed-loop system to obtain stable rheological properties over a wide range of flow rates (4 decades). The porous medium studied experimentally is a sandstone. The flow properties were also simulated on a regular sphere packing and a random sphere packing as well as on a 3D geometry of a sandstone obtained by x-ray tomography. These different geometries allow highlighting the role of the heterogeneity of the pore structure on the flow properties. All results are expressed as the flow rate ˜Q versus the difference of the pressure gradient to the critical pressure (Δ˜P−Δ˜Pc); Pc defines the pressure below which there is no flow. We observed both numerically and experimentally three specific scaling regimes, already identified by Talon and Bauer [Eur. Phys. J. E 36, 139 (2013)] for a Bingham fluid in 2D porous media. We also evidenced the existence of two critical pressures: the “true” critical pressure ˜Pc defined as the pressure below which there is no flow and the “pseudo” critical pressure threshold ˜P∞c determined by fitting the data in the high-flow-rate regime. We show that the “true” critical pressure is always lower than the “pseudo” one in heterogeneous porous media and can be equal only in the case of regular porous structures. We explain these observations using an energy minimization principle.

We study non-inertial flows of single-phase yield stress fluids along uneven/rough-walled channels, e.g. approximating a fracture, with two main objectives. First, we re-examine the usual approaches to providing a (nonlinear) Darcy-type flow law and show that significant errors arise due to self-selection of the flowing region/fouling of the walls. This is a new type of non-Darcy effect not previously explored in depth. Second, we study the details of flow as the limiting pressure gradient is approached, deriving approximate expressions for the limiting pressure gradient valid over a range of different geometries. Our approach is computational, solving the two-dimensional Stokes problem along the fracture, then upscaling. The computations also reveal interesting features of the flow for more complex fracture geometries, providing hints about how to extend Darcy-type approaches effectively.

The flow of non-Newtonian fluids is ubiquitous in many applications in the geological and industrial context. We focus here on yield stress fluids (YSF), i.e. a material that requires minimal stress to flow. We study numerically the flow of yield stress fluids in 2D porous media on a macroscopic scale in the presence of local heterogeneities. As with the microscopic problem, heterogeneities are of crucial importance because some regions will flow more easily than others. As a result, the flow is characterized by preferential flow paths with fractal features. These fractal properties are characterized by different scale exponents that will be determined and analyzed. One of the salient features of these results is that these exponents seem to be independent of the amplitude of heterogeneities for a log-normal distribution. In addition, these exponents appear to differ from those at the microscopic level, illustrating the fact that, although similar, the two scales are governed by different sets of equations.

We report on numerical studies of avalanches of an autocatalytic reaction front in a porous medium. The front propagation is controlled by an adverse flow resulting in upstream, static, or downstream regimes. In an earlier study focusing on front shape, we identified three different universality classes associated with this system by following the front dynamics experimentally and numerically. Here, using numerical simulations in the vicinity of the second-order transition, we identify an avalanche dynamics characterized by power-law distributions of avalanche sizes, durations, and lateral extensions. The related exponents agree well with the quenched-Kardar-Parisi-Zhang theory, which describes the front dynamics. However, the geometry of the propagating front differs slightly from that of the theoretical one. We show that this discrepancy can be understood in terms of the nonquasistatic correction induced by the finite front velocity.

International audience; Dispersion of a passive tracer in a tube has been extensively studied in the case of Newtonian fluids since the pioneer work of Taylor (1953). However, the influence of more complex rheological behavior on the transport has only be scarcely investigated. Non-Newtonian fluids are increasingly used in the industry and transport in this type of fluid merits therefore thorough investigations. An example of industrial application is Enhanced Oil Recovery, that is based on the injection of non-Newtonian fluids as polymers or surfactant solutions in porous media, which are then submitted to dispersion phenomena.This work deals with transport of a passive tracer in shear thinning fluids with and without yield stress whose constitutive behaviors are representative of a large number of industrial fluids. We focus on transport in capillary tubes, essential for the understanding of dispersion in porous media. Transport is investigated at different time scales by solving the advection–diffusion equation using a Two-Relaxation-Time Lattice–Boltzmann method. We also derived an analytical expression of the Taylor dispersion coefficient for a large range of fluid rheologies. Dispersion coefficients of all fluids described by the Herschel–Bulkley model can now be determined. Analytical and numerical results are compared and very good accordance is obtained.We discuss the characteristic time scales of the transport before reaching steady state as a function of fluid rheology and Péclet number. We show that the time to reach the dispersive regime is nearly independent of the fluid rheology whereas the effective dispersion coefficient is a function of the rheological parameters.We also present the displacement distribution of the tracer molecules (propagators) as a function of time and show that they are strongly conditioned by the fluid rheology. Indeed, propagators give valuable information on the temporal evolution of the concentration profile towards the stationary Taylor regime.

International audience; Self-sustained reaction fronts in a disordered medium subject to an external flow display self-affine roughening, pinning and depinning transitions. We measure spatial and temporal fluctuations of the front in $1+1$ dimensions, controlled by a single parameter, the mean flow velocity. Three distinct universality classes are observed, consistent with the Kardar-Parisi-Zhang (KPZ) class for fast advancing or receding fronts, the quenched KPZ class (positive-qKPZ) when the mean flow approximately cancels the reaction rate, and the negative-qKPZ class for slowly receding fronts. Both quenched KPZ classes exhibit distinct depinning transitions, in agreement with the theory.

In this paper, we numerically investigate the statistical properties of the nonflowing areas of Bingham fluid in two-dimensional porous media. First, we demonstrate that the size probability distribution of the unyielded clusters follows a power-law decay with a large size cutoff. This cutoff is shown to diverge following a power law as the imposed pressure drop tends to a critical value. In addition, we observe that the exponents are almost identical for two different types of porous media. Finally, those scaling properties allow us to account for the quadratic relationship between the pressure gradient and velocity.

Reaction fronts evolving in a porous medium exhibit a rich dynamical behaviour. In presence of an adverse flow, experiments show that the front slows down and eventually gets pinned, displaying a particular sawtooth shape. Extensive numerical simulations of the hydrodynamic equations confirm the experimental observations. Here we propose a stylized model, predicting two possible outcomes of the experiments for large adverse flow: either the front develops a sawtooth shape, or it acquires a complicated structure with islands and overhangs. A simple criterion allows to distinguish between the two scenarios and its validity is reproduced by direct hydrodynamical simulations. Our model gives a better understanding of the transition and is relevant in a variety of domains, when the pinning regime is strong and only relies on a small number of sites., 5 pages, 4 figures

We analyze experimentally chemical waves propagation in the disordered flow field of a porous medium. The reaction fronts travel at a constant velocity which drastically depends on the mean flow direction and rate. The fronts may propagate either downstream and upstream but, surprisingly, they remain static over a range of flow rate values. Resulting from the competition between the chemical reaction and the disordered flow field, these frozen fronts display a particular sawtooth shape. The frozen regime is likely to be associated with front pinning in low velocity zones, the number of which varies with the ratio of the mean flow and the chemical front velocities. PACS numbers

The extension of a gravity current in a lock-exchange problem, proceeds as square root of time in the viscous-buoyancy phase, where there is a balance between gravitational and viscous forces. In the presence of particles however, this scenario is drastically altered, because sedimentation reduces the motive gravitational force and introduces a finite distance and time at which the gravity current halts. We investigate the spreading of low Reynolds number suspension gravity currents using a novel approach based on the Lattice-Boltzmann (LB) method. The suspension is modeled as a continuous medium with a concentration-dependent viscosity. The settling of particles is simulated using a drift flux function approach that enables us to capture sudden discontinuities in particle concentration that travel as kinematic shock waves. Thereafter a numerical investigation of lock-exchange flows between pure fluids of unequal viscosity, reveals the existence of wall layers which reduce the spreading rate substantially compared to the lubrication theory prediction. In suspension gravity currents, we observe that the settling of particles leads to the formation of two additional fronts: a horizontal front near the top that descends vertically and a sediment layer at the bottom which aggrandises due to deposition of particles. Three phases are identified in the spreading process: the final corresponding to the mutual approach of the two horizontal fronts while the laterally advancing front halts indicating that the suspension current stops even before all the particles have settled. The first two regimes represent a constant and a decreasing spreading rate respectively. Finally we conduct experiments to substantiate the conclusions of our numerical and theoretical investigation.

International audience; Simulating flow of a Bingham fluid in porous media still remains a challenging task as the yield stress may significantly alter the numerical stability and precision. We present a Lattice-Boltzmann TRT scheme that allows the resolution of this type of flow in stochastically reconstructed porous media. LB methods have an intrinsic error associated to the boundary conditions. Depending on the schemes this error might be directly linked to the effective viscosity. As for non-Newtonian fluids viscosity varies in space the error becomes inhomogeneous and very important. In contrast to that, the TRT scheme does not present this deficiency and is therefore adequate to be used for simulations of non-Newtonian fluid flow. We simulated Bingham fluid flow in porous media and determined a generalized Darcy equation depending on the yield stress, the effective viscosity, the pressure drop and a characteristic length of the porous medium. By evaluating the flow in the porous structure, we distinguished three different scaling regimes. Regime I corresponds to the situation where fluid is flowing in only one channel. Here, the relation between flow rate and pressure drop is given by the non-Newtonian Poiseuille law. During Regime II an increase in pressure triggers the opening of new paths and the relation between flow rate and the difference in pressure to the critical yield pressure becomes quadratic: q∝(d~p−d~pc)2 . Finally, Regime III corresponds to the situation where all the fluid is flowing. In this case, q∝(d~p−d~pc) .

Autocatalytic reaction fronts between two reacting species in the absence of fluid flow, propagate as solitary waves. The coupling between autocatalytic reaction front and forced simple hydrodynamic flows leads to stationary fronts whose velocity and shape depend on the underlying flow field. We address the issue of the chemico-hydrodynamic coupling between forced advection in porous media and self-sustained chemical waves. Towards that purpose, we perform experiments over a wide range of flow velocities with the well characterized iodate arsenious acid and chlorite-tetrathionate autocatalytic reactions in transparent packed beads porous media. The characteristics of these porous media such as their porosity, tortuosity, and hydrodynamics dispersion are determined. In a pack of beads, the characteristic pore size and the velocity field correlation length are of the order of the bead size. In order to address these two length scales separately, we perform lattice Boltzmann numerical simulations in a stochastic porous medium, which takes into account the log-normal permeability distribution and the spatial correlation of the permeability field. In both experiments and numerical simulations, we observe stationary fronts propagating at a constant velocity with an almost constant front width. Experiments without flow in packed bead porous media with different bead sizes show that the front propagation depends on the tortuous nature of diffusion in the pore space. We observe microscopic effects when the pores are of the size of the chemical front width. We address both supportive co-current and adverse flows with respect to the direction of propagation of the chemical reaction. For supportive flows, experiments and simulations allow observation of two flow regimes. For adverse flow, we observe upstream and downstream front motion as well as static front behaviors over a wide range of flow rates. In order to understand better these observed static state fronts, flow experiments around a single obstacle were used to delineate the range of steady state behavior. A model using the "eikonal thin front limit" explains the observed steady states.

We demonstrate that the first arrival time in dispersive processes in self-affine fractures are governed by the same length scale characterizing the fractures as that which controls their permeability. In one-dimensional channel flow this length scale is the aperture of the bottle neck, i.e., the region having the smallest aperture. In two dimensions, the concept of a bottle neck is generalized to that of a minimal path normal to the flow. The length scale is then the average aperture along this path. There is a linear relationship between the first arrival time and this length scale, even when there is strong overlap between the fracture surfaces creating areas with zero permeability. We express the first arrival time directly in terms of the permeability., Comment: EPL (2012) 1

Fluid and solute transfer in X-junctions between straight channels is shown to depend critically on the junction angle α in the Stokes flow regime. Experimentally, water and a water-dye solution are injected at equal flow rates in two facing channels of the junction. Planar laser induced fluorescence (PLIF) measurements show that the largest part of each injected fluid "bounces back" preferentially into the outlet channel at the lowest angle to the injection; this is opposite to the inertial case and requires a high curvature of the corresponding streamlines. The proportion of this fluid in the other channel decreases from 50% at α = 90° to 0% at a threshold angle. These counterintuitive features reflect the minimization of energy dissipation for Stokes flows. Finite elements numerical simulations of a 2D Stokes flow of equivalent geometry confirm these results and show that, below the threshold angle αc = 33.8°, recirculation cells are present in the center part of the junction and separate the two injected flows of the two solutions. Reducing further α leads to the appearance of new recirculation cells with lower flow velocities. Fil: Cachile, Mario Andres. Universidad de Buenos Aires. Facultad de Ingeniería. Departamento de Física. Grupo de Medios Porosos; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina Fil: Talon, L.. Universite de Paris Xi. Laboratoire Automatiques et Systeme Thermiques; Francia Fil: Gomba, Juan Manuel. Universidad Nacional del Centro de la Provincia de Buenos Aires. Facultad de Ciencias Exactas. Instituto de Fisica Arroyo Seco; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina Fil: Hulin, J.P.. Universite de Paris Xi. Laboratoire Automatiques et Systeme Thermiques; Francia Fil: Auradou, H.. Universite de Paris Xi. Laboratoire Automatiques et Systeme Thermiques; Francia

In a viscous lock-exchange gravity current, which describes the reciprocal exchange of two fluids of different densities in a horizontal channel, the front between two Newtonian fluids spreads as the square root of time. The resulting diffusion coefficient reflects the competition between the buoyancy driving effect and the viscous damping, and depends on the geometry of the channel. This lock-exchange diffusion coefficient has already been computed for a porous medium, a 2D Stokes flow between two parallel horizontal boundaries separated by a vertical height, H, and, recently, for a cylindrical tube. In the present paper, we calculate it, analytically, for a rectangular channel (horizontal thickness b, vertical height, H) of any aspect ratio (H/b) and compare our results with experiments in horizontal rectangular channels for a wide range of aspect ratios (1/10-10). We also discuss the 2D Stokes-Darcy model for flows in Hele-Shaw cells and show that it leads to a rather good approximation, when an appropriate Brinkman correction is used., Comment: 16 pages,7 figures

International audience; We introduce a model that allows for the prediction of the permeability of self-affine rough channels (one-dimensional fracture) and two-dimensional fractures over a wide range of apertures. In the lubrication approximation, the permeability shows three different scaling regimes. For fractures with a large mean aperture or an aperture small enough to the permeability being close to disappearing, the permeability scales as the cube of the aperture when the zero level of the aperture is set to coincide with the disappearance of the permeability. Between these two regimes, there is a third regime where the scaling is due to the self-affine roughness. For rough channels, the exponent is found to be $3-1/H$ where $H$ is the Hurst exponent. For two-dimensional fractures, it is necessary to introduce a new equivalent aperture $b_c$ to make the scaling regime apparent. $b_c$ is defined as the hydraulic aperture of the most restrective barrier crossing the fracture normal to the flow direction. This regime is characterized by an exponent higher than for the one-dimensional case: it is $2.25$ for $H=0.8$ and $2.16$ for $H=0.3$.

We propose a method for calculating the effective permeability of two-dimensional self-affine permeability fields based on generalizing the one-dimensional concept of a bottleneck. We test the method on fracture faults where the local permeability field is given by the cube of the aperture field. The method remains accurate even when there is substantial mechanical overlap between the two fracture surfaces. The computational efficiency of the method is comparable to calculating a simple average and is more than two orders of magnitude faster than solving the Reynolds equations using a finite-difference scheme.

The convective/absolute nature of the instability of miscible core-annular flow with variable viscosity is investigated via linear stability analysis and nonlinear simulations. From linear analysis, it is found that miscible core-annular flows with the more viscous fluid in the core are at most convectively unstable. On the other hand, flows with the less viscous fluid in the core exhibit absolute instability at high viscosity ratios, over a limited range of core radii. Nonlinear direct numerical simulations in a semi-infinite domain display self-excited intrinsic oscillations if and only if the underlying base flow exhibits absolute instability. This oscillator-type flow behaviour is demonstrated to be associated with the presence of a nonlinear global mode. Both the parameter range of global instability and the intrinsically selected frequency of nonlinear oscillations, as observed in the simulation, are accurately predicted from linear criteria. In convectively unstable situations, the flow is shown to respond to external forcing over an unstable range of frequencies, in quantitative agreement with linear theory. As discussed in part 1 of this study (d'Olce, Martin, Rakotomalala, Salin and Talon,J. Fluid Mech., vol. 618, 2008, pp. 305–322), self-excited synchronized oscillations were also observed experimentally. An interpretation of these experiments is attempted on the basis of the numerical results presented here.

[1] We use the extended Darcy's law, which also accounts for the Brinkman correction, to study macrodispersion in a two-dimensional (2-D) porous medium. The former is necessary when permeability changes fast at a relatively small scale, and in general, it is a more complete description of flow in a heterogeneous medium. Lattice-gas methods are ideally suited to simulate such flows. Simulations using a lattice BGK method and a small-fluctuation approach are described for an isotropic, exponentially decaying correlation function of the permeability field. The analytical results contain the additional parameter Kl/λ2 (where Kl and λ are the typical permeability and velocity variation length, respectively), the sensitivity to which was studied. As expected, the contribution of the Brinkman effect is insignificant for typical field values of this parameter, in which case, the classical results are recovered. At larger values, for example, for heterogeneous media of a small correlation length, and possibly in laboratory applications, the Brinkman correction leads to a decrease in macrodispersivity, reflecting the smoothing effect of the Brinkman correction on the velocity field. Nonetheless, for practical values of the parameters, this reduction is no larger than 50% of the classical expression. The small fluctuation theory was found to be in good agreement with the simulations, provided that it was consistently applied (namely, by not mixing first-order with second-order expansions). The results also show that lattice-gas simulations can be usefully employed to study macrodispersion in heterogeneous porous media.

Predicting the flow of non-Newtonian fluids in porous structure is still a challenging issue due to the interplay betwen the microscopic disorder and the non-linear rheology. In this letter, we study the case of an yield stress fluid in a two-dimensional structure. Thanks to a performant optimization algorithm, we show that the system undergoes a continuous phase transition in the behavior of the flow controlled by the applied pressure drop. In analogy with the studies of the plastic depinning of vortex lattices in high-$T_c$ superconductors we characterize the nonlinearity of the flow curve and relate it to the change in the geometry of the open channels. In particular, close to the transition, an universal scale free distribution of the channel length is observed and explained theoretically via a mapping to the KPZ equation., Comment: 5 pages, 4 figures + 1 Supplementary material

Summarization: We study the steady-state displacement of nonwetting liquid ganglia during immiscible two-phase flows in realistic, stochastically reconstructed porous domains, focusing primarily on the nonlinear Darcian regime that arises when capillary to viscous (or gravity) forces become comparable at the pore scale. During this process, the ganglia undergo a continuous cycle of dynamic coalescence and fragmentation, resulting in two populations (a mobile and a stranded one) with distinct structural and rheological features, that continuously exchange mass between them under “stationary” flow conditions. We use a lattice Boltzmann model for the explicit solution of flow and interfacial dynamics at the pore scale driven by a constant body force field (i.e., gravity), and a periodic clustering algorithm for the identification and classification of mobile and stranded ganglia. Our simulation results reveal that an increase in the applied Bond number (Bo) leads to a gradual mobilization of the initially stranded ganglia population, resulting in a power-law scaling with an exponent being a strong function of the nonwetting-phase saturation. The linear Darcian scaling for the nonwetting phase is progressively restored at high Bo, while the wetting phase appears to maintain a linear Darcian scaling over the entire range of Bo values. We show that the mobilization process is characterized by a critical Bo, which is independent of saturation, above which new flow paths are created, in a similar fashion as a yield-stress fluid flows in a porous medium. Our results also offer a unique insight on the distinct structural characteristics of the mobile and stranded populations (e.g., ganglia size and length), as well as on their velocity and orientation with respect to their size. Presented on: Physical Review Fluids