Search

Transport processes in the subsurface are coupled with the heterogeneity of the porous structure. Obtaining an accurate description of mass transport in such media at all time scales remains a crucial task, as the subsurface is strongly impacted by human activity. Spreading of pollutants in the transient but also in the asymptotic regime, as well as the time to reach a critical location (e.g. aquifer, well) significantly depend on the underlying heterogeneity of the permeability field. Moreover, in the case of solute retention, transport is also characterized by local exchange kinetics that depend on the local aquifer properties. Consequently, exchange (retention) times are expected to be spatially heterogeneous. In this work we focus on the influence of spatially heterogeneous permeability and exchange times on the transient and asymptotic transport regimes and provide a parametric study. To this goal, we simulate in a first part the transport in a two-dimensional heterogeneous medium under spatially varying permeability and mobile–immobile mass transfer parameters. Equations are solved using a Lattice-Boltzmann two-relaxation-time (TRT) algorithm. We assume the following relation between the local permeability K and the local exchange time: τ∝Kγ. Taking into account this relation, we investigate the impact of the Damköhler number (Da, ratio of the advection and exchange time scales), the disorder of the permeability field and the value of the exponent of the coupling function (γ) on the spatial evolution of the concentration field and breakthrough curve. We show that, depending on the parameters (Da, γ, etc.), we can observe transient, non-Fickian dispersion, which is characterized by non-exponential tails of the solute breakthrough curves and a non-linear evolution of the spatial variance of the solute distribution. In a second part, we present a new continuous time random walk (CTRW) model to upscale these transport behaviors. The model is based on a spatial Markov model for particle velocities that couples advective–dispersive transport and heterogeneous mass transfer through a compound Poisson process. The upscaled model can be fully parameterized by the statistics of permeability and the hydraulic gradient (with no fitting parameter), that is, in terms of medium and flow characteristics. The results of the CTRW model fully capture the non-Fickian transient transport regimes both for the breakthrough curves and spatial concentration variance. In the longtime limit, as expected from the central limit theorem, the CTRW model predicts normal, Fickian behavior. Finally we show, that the time to reach the asymptotic regimes in heterogeneous media (e.g. heterogeneous exchange times) is parameter dependent and in average is two orders of magnitude larger than for the respective homogeneous case. To summarize, the coupling between the heterogeneous permeability field and the local mass transfer properties can strongly influence transient and asymptotic transport regimes and potentially explain experimentally observed non-Gaussian behaviors., M.D. acknowledges the support of the Spanish Research Agency (10.13039/501100011033) and the Spanish Ministry of Science and Innovation through the grants CEX2018-000794-S and PID2019-106887GB-C31 (HydroPore).

The behavior of shear-oscillated amorphous materials is studied using a coarse-grained model. Samples are prepared at different degrees of annealing and then subject to athermal and quasistatic oscillatory deformations at various fixed amplitudes. The steady-state reached after several oscillations is fully determined by the initial preparation and the oscillation amplitude, as seen from stroboscopic stress and energy measurements. Under small oscillations, poorly annealed materials display shear-annealing, while ultra-stabilized materials are insensitive to them. Yet, beyond a critical oscillation amplitude, both kind of materials display a discontinuous transition to the same mixed state composed by a fluid shear-band embedded in a marginal solid. Quantitative relations between uniform shear and the steady-state reached with this protocol are established. The transient regime characterizing the growth and the motion of the shear band is also studied., 10 pages, 11 figures. Accepted for publication in The Journal of Chemical Physics

The flow through a capillary tube with non-constant radius and where bubbles of yield stress fluid are injected is strongly non-linear. In particular below a finite yield pressure drop, $P_y$, flow is absent, while a singular behaviour is expected above it. In this paper we compute the yield pressure drop statistics and the mean flow rate in two cases: (i) when a single bubble is injected, (ii) when many bubbles are randomly injected in the fluid., 12 pages, 8 figures

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.

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 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.

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.

A computational investigation of variable density and viscosity, miscible displacements in horizontal Hele-Shaw cells is presented. As a first step, two-dimensional base states are obtained by means of simulations of the Stokes equations, which are nonlinear due to the dependence of the viscosity on the local concentration. Here, the vertical position of the displacement front is seen to reach a quasisteady equilibrium value, reflecting a balance between viscous and gravitational forces. These base states allow for two instability modes: first, there is the familiar tip instability driven by the unfavourable viscosity contrast of the displacement, which is modulated by the presence of density variations in the gravitational field; second, a gravitational instability occurs at the unstably stratified horizontal interface along the side of the finger. Both of these instability modes are investigated by means of a linear stability analysis. The gravitational mode along the side of the finger is characterized by a wavelength of about one half to one full gap width. It becomes more unstable as the gravity parameter increases, even though the interface is shifted closer to the wall. The growth rate is largest far behind the finger tip, where the interface is both thicker, and located closer to the wall, than near the finger tip. The competing influences of interface thickness and wall proximity are clarified by means of a parametric stability analysis. The tip instability mode represents a gravity-modulated version of the neutrally buoyant mode. The analysis shows that in the presence of density stratification its growth rate increases, while the dominant wavelength decreases. This overall destabilizing effect of gravity is due to the additional terms appearing in the stability equations, which outweigh the stabilizing effects of gravity onto the base state.

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.

We investigate the linear stability of miscible, viscosity-layered Poiseuille flow. In the Stokes flow regime, diffusion is observed to have a destabilizing effect very similar to that of inertia in finite-Reynolds-number flows. For two-layer flows, four types of instability can dominate, depending on the interface location. Two interfacial modes exhibit large growth rates, while two additional bulk modes grow more slowly. Three-layer Stokes flows give rise to diffusive modes for each interface. These two diffusive interface modes can be in resonance, thereby enhancing the growth rate. Furthermore, modes without inertia and diffusion are also observed, consistent with a previous long-wave analysis for sharp interfaces. In contrast to that earlier investigation, the present analysis demonstrates that instability can also occur when the more viscous layer is in the centre, at larger wavenumbers.

We address the issue of the convective or absolute nature of the instability of core-annular pipe flows, in experiments using two miscible fluids of equal density but different viscosities, the core fluid being much less viscous than the wall one. We use a concentric co-current injection of the two fluids. An axisymmetric parallel base state is obtained downstream the injector. The core radiusRIand the Reynolds numberReof the so-obtained base state are varied independently due to the control of the flow rate of each fluid. However, a downstream destabilization of this base state was observed within the explored range of the two control parametersRIandRe. Moreover, the fixed location of this destabilization, observed for some particular parameters, suggests an absolute nature of the instability. We present a tentative delineation of the nature (convective or absolute) of the instability and discuss the accessible measurements to experimentally address this issue.

Flow in discrete networks can be observed in biological, geological, or technical systems. Often, the number of individual segments is very large. Therefore, discrete pressure and flow calculation in each single segment becomes very time consuming, and a continuum model becomes attractive. We apply a global upscaling method based on a spatial average to investigate a porous media network model with heterogeneous pore length distribution. For this purpose, the porous media was modeled by triangular networks. For such networks, we characterize the representative elementary volume (REV) size and show that using window sizes smaller than the REV requires an heterogeneous Darcian description. We find that the permeability has to be distributed according to a log-Gaussian distribution with variance and correlation length depending on the window size and the porous microstructure. Finally, we apply the procedure to an anisotropic regular network for which an analytical solution can be found and show that using this method leads to the correct analytical behavior.

In this article, we propose a simple model to understand the critical behavior of path opening during flow of a yield stress fluid in porous media as numerically observed by Chevalier and Talon (2015). This model can be mapped to the problem of a contact line moving in an heterogeneous field. Close to the critical point, this line presents an avalanche dynamic where the front advances by a succession of waiting time and large burst events. These burst events are then related to the non-flowing (i.e. unyielded) areas. Remarkably, the statistics of these areas reproduce the same properties as in the direct numerical simulations. Furthermore, even if our exponents seem to be close to the mean field universal exponents, we report an unusual bump in the distribution which depends on the disorder. Finally, we identify a scaling invariance of the cluster spatial shape that is well fit, to first order, by a self-affine parabola.

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.

This work focuses on the numerical solution of the Stokes-Brinkman equation for a voxel-type porous-media grid, resolved by one to eight spacings per permeability contrast of 1 to 10 orders in magnitude. It is first analytically demonstrated that the lattice Boltzmann method (LBM) and the linear-finite-element method (FEM) both suffer from the viscosity correction induced by the linear variation of the resistance with the velocity. This numerical artefact may lead to an apparent negative viscosity in low-permeable blocks, inducing spurious velocity oscillations. The two-relaxation-times (TRT) LBM may control this effect thanks to free-tunable two-rates combination Λ. Moreover, the Brinkman-force-based BF-TRT schemes may maintain the nondimensional Darcy group and produce viscosity-independent permeability provided that the spatial distribution of Λ is fixed independently of the kinematic viscosity. Such a property is lost not only in the BF-BGK scheme but also by "partial bounce-back" TRT gray models, as shown in this work. Further, we propose a consistent and improved IBF-TRT model which vanishes viscosity correction via simple specific adjusting of the viscous-mode relaxation rate to local permeability value. This prevents the model from velocity fluctuations and, in parallel, improves for effective permeability measurements, from porous channel to multidimensions. The framework of our exact analysis employs a symbolic approach developed for both LBM and FEM in single and stratified, unconfined, and bounded channels. It shows that even with similar bulk discretization, BF, IBF, and FEM may manifest quite different velocity profiles on the coarse grids due to their intrinsic contrasts in the setting of interface continuity and no-slip conditions. While FEM enforces them on the grid vertexes, the LBM prescribes them implicitly. We derive effective LBM continuity conditions and show that the heterogeneous viscosity correction impacts them, a property also shared by FEM for shear stress. But, in contrast with FEM, effective velocity conditions in LBM give rise to slip velocity jumps which depend on (i) neighbor permeability values, (ii) resolution, and (iii) control parameter Λ, ranging its reliable values from Poiseuille bounce-back solution in open flow to zero in Darcy's limit. We suggest an "upscaling" algorithm for Λ, from multilayers to multidimensions in random extremely dispersive samples. Finally, on the positive side for LBM besides its overall versatility, the implicit boundary layers allow for smooth accommodation of the flat discontinuous Darcy profiles, quite deficient in FEM.

We analyze the displacement of a viscous fluid by a miscible more viscous one in heterogeneous porous media. We performed lattice Bhatnagar–Gross–Krook simulations, which were previously successfully applied to the study of the dispersion of a passive tracer in a stochastic heterogeneous porous medium. In the present situation, the flow is stable (no viscous fingering) and leads to an overall Gaussian dispersion, the coefficient of which decreases as the viscosity ratio increases. The results are in reasonable agreement with the stochastic approach of Welty and Gelhar.

We consider miscible displacement between parallel plates in the absence of diffusion, with a concentration-dependent viscosity. By selecting a piecewise viscosity function, this can also be considered as 'three-fluid' flow in the same geometry. Assuming symmetry across the gap and based on the lubrication ('equilibrium') approximation, a description in terms of two quasi-linear hyperbolic equations is obtained. We find that the system is hyperbolic and can be solved analytically, when the mobility profile is monotonic, or when the mobility of the middle phase is smaller than its neighbours. When the mobility of the middle phase is larger, a change of type is displayed, an elliptic region developing in the composition space. Numerical solutions of Riemann problems of the hyperbolic system spanning the elliptic region, with small diffusion added, show good agreement with the analytical outside, but an unstable behaviour inside the elliptic region. In these problems, the elliptic region arises precisely at the displacement front. Crossing the elliptic region requires the solution of essentially an eigenvalue problem of the full higher-dimensional model, obtained here using lattice BGK simulations

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

We study the dynamics of nonwetting liquid blobs during immiscible two-phase flows in stochastically reconstructed porous domains predominantly saturated by a wetting fluid. The flow problem is solved explicitly using a Lattice-Boltzmann model that captures both the bulk phase and interfacial dynamics of the process. We show that the nonwetting blobs undergo a continuous life cycle of dynamic breaking up and coalescence producing two populations of blobs, a mobile and a stranded one, that exchange continuously mass between them. The process reaches a “steady state” when the rates of coalescence and breaking up become equal, and the macroscopic flow variables remain practically constant with time. At steady state, mass partitioning between mobile and immobile populations depends strongly on the applied Bond number Bo and the initial nonwetting phase distributions. Three flow regimes are identified: a single-phase flow Darcy-type regime at low Bo numbers, a non-Darcy two-phaseflow regime at intermediate values of Bo, where the capillary number scales as Ca ∝ Bo 2 , and a Darcy-type two-phase flow regime at higher values of Bo. Our numerical results are found to be in good agreement with recent experimental and theoretical works.

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

A viscous lock-exchange gravity current corresponds to the reciprocal exchange of two fluids of different densities in a horizontal channel. The resulting front between the two fluids spreads as the square root of time, with a diffusion coefficient reflecting the buoyancy, viscosity, and geometrical configuration of the current. On the other hand, an autocatalytic reaction front between a reactant and a product may propagate as a solitary wave, namely, at a constant velocity and with a stationary concentration profile, resulting from the balance between molecular diffusion and chemical reaction. In most systems, the fluid left behind the front has a different density leading to a lock-exchange configuration. We revisit, with a chemical reaction, the classical situation of lock-exchange. We present an experimental analysis of buoyancy effects on the shape and the velocity of the iodate arsenous acid autocatalytic reaction fronts, propagating in horizontal rectangular channels and for a wide range of aspect ratios (1/3 to 20) and cylindrical tubes. We do observe stationary-shaped fronts, spanning the height of the cell and propagating along the cell axis. Our data support the contention that the front velocity and its extension are linked to each other and that their variations scale with a single variable involving the diffusion coefficient of the lock-exchange in the absence of chemical reaction. This analysis is supported by results obtained with lattice Bathnagar-Gross-Krook (BGK) simulations Jarrige et al. [Phys. Rev. E 81, 06631 (2010)], in other geometries (like in 2D simulations by Rongy et al. [J. Chem. Phys. 127, 114710 (2007)] and experiments in cylindrical tubes by Pojman et al. [J. Phys. Chem. 95, 1299 (1991)]), and for another chemical reaction Schuszter et al. [Phys. Rev. E 79, 016216 (2009)].

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.

We present a numerical analysis of solutal buoyancy effects on the shape and the velocity of autocatalytic reaction fronts, propagating in thin tilted rectangular channels. We use two-dimensional (2D) lattice Bathnagar-Gross-Krook (BGK) numerical simulations of gap-averaged equations for the flow and the concentration, namely a Stokes-Darcy equation coupled with an advection-diffusion-reaction equation. We do observe stationary-shaped fronts, spanning the width of the cell and propagating along the cell axis. We show that the model accounts rather well for experiments we performed using an Iodate Arsenous Acid reaction propagating in tilted Hele-Shaw cells, hence validating our 2D modelization of a three-dimensional problem. This modelization is also able to account for results found for another chemical reaction (chlorite tetrathionate) in a horizontal cell. In particular, we show that the shape and the traveling velocity of such fronts are linked with an eikonal equation. Moreover, we show that the front velocity varies nonmonotonically with the tilt of the cell, and nonlinearly with the width of the cell.

Autocatalytic reactions may propagate as solitary waves, namely, at a constant front velocity and with a stationary concentration profile, resulting from a balance between molecular diffusion and chemical reaction. When the reaction is exothermic, a thermal wave is linked to the chemical front. As the thermal diffusivity is nearly two orders of magnitude larger than the molecular one, the temperature profile spreads over length scales (mm) two orders of magnitude larger than the concentration one. Using an infrared camera, we measure the temperature profiles for a chlorite-tetrathionate autocatalytic reaction. The profiles are compared quantitatively to lattice Bhatnagar-Gross-Krook (BGK) numerical simulations. Our analysis also accounts for the lack of observation of the thermal wave for the iodate arsenous acid reaction.

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] The miscible displacement of a Newtonian or shear-thinning fluid by another one of same rheological properties has been studied optically in a flat transparent model fracture with a random distribution of identical cylindrical obstacles on one of the walls. At the local scale, the concentration variation on individual pixels satisfies a Gaussian convection-dispersion relation with local transit time (x, y) and dispersivity ld(x, y). The variation of ld with the Peclet number Pe shows that it results from a combination of geometrical and Taylor dispersion, respectively dominant at low and high Pe values. Using shear-thinning solutions instead of a Newtonian fluid enhances the velocity contrasts (and therefore geometrical dispersion) and reduces Taylor dispersion. At the global scale, the front geometry is studied from the isoconcentration lines c = 0.5 (equivalent to lines of constant (x, y) value): beyond a transition travel time, their width in the direction parallel to the flow reaches a constant limit varying linearly with Log(Pe) with a slope increasing with the shear-thinning character of the fluid. These characteristics are compared to previous observations on other model fractures with a self-affine roughness displaying channelization effects.

We investigate the transition from unbanded to banded spherulitic growth in mixtures of ethylene carbonate with polyacrylonitrile. By carefully considering systematic errors, we show that the band spacing diverges with a power-law form showing scaling over nearly two decades. We also observe that the bands disorder as the transition point is approached. The critical exponent is nonclassical. One possible explanation is that the nonequilibrium transition is actually weakly first order (subcritical).

Change-of-type behavior from hyperbolic to elliptic is common to quasilinear hyperbolic systems. This issue is addressed here for the particular case of miscible flow of three fluids between two parallel plates. Change of type occurs at the leading edge of the displacement front and reflects the failing of the equilibrium assumption, necessary for the quasilinear hyperbolic formalism, at the front. To cross the elliptic region requires the solution of the full, higher-dimensionality problem, obtained here using lattice gas simulations. For the specific example, it is found that the system self-selects a front structure independent of injection conditions.

[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.

We investigate the hierarchy of optimal paths in a disordered landscape, based on the best path, the second best path and so on in terms of an energy. By plotting each path at a height according to its energy above some zero level, a landscape appears. This landscape is self-affine and controlled by two Hurst exponents: the one controlling the height fluctuations is 1/3 and the one controlling the fluctuations of the equipotential lines in the landscape is 2/3. These two exponents correspond to the exponents controlling energy and shape fluctations in the directed polymer problem. We furthermore find that the density of spanning optimal paths scale as the length of the paths to −2/3 and the histogram of energy differences between consecutive paths scale as a power law in the difference size with exponent −2.5.

Using lattice Boltzmann simulations, we analyze the different regimes of propagation of an autocatalytic reaction front in heterogenous porous media. The heterogeneities of the porous medium are characterized by the standard deviation of its log-normal distribution of permeability and its correlation length. We focus on the situation where chemical reaction and flow field act in opposite directions. In agreement with previous experiments we observe upstream, downstream fronts as well as static, frozen ones over a range of flow velocity which depends drastically on the heterogeneities of the flow field. The transition between the static regime and the downstream one account for large enough low-velocity zones, whereas the transition from static to upstream regime is found to be given by a kind of percolation path.

To address the issue of miscible viscous fingering instability in buoyancy free conditions, experiments have been performed under microgravity conditions in parabolic flights. A Hele-Shaw cell, two parallel plates separated by a small gap, has been used with two miscible fluids of viscosity ratio 100 (the injected fluid is the less viscous). The influence of the initial thickness of the pseudointerface between the two fluids has been studied, using flow rates large enough to prevent further mixing during displacement. The selected wavelength, measured on the observed fingering pattern, does not depend on the initial front thickness: It is around three times the gap of the cell, i.e., significantly lower than the value of five, observed on earth. However, the initial thickness does control the displacement length required for the instability to occur. Our results are in reasonable agreement with existing and new numerical simulations.

Autocatalytic reaction fronts between reacted and unreacted species may propagate as solitary waves, that is, at a constant front velocity and with a stationary concentration profile, which result from a balance between molecular diffusion and chemical reaction. A velocity field in the supporting medium may affect the propagation of such fronts through different phenomena: advection, diffusion enhancement, front shape changes, etc. Here, we report on an experimental study and lattice Bhatnagar–Gross–Krook numerical simulations of the effect of an oscillating flow on the autocatalytic reaction between iodate and arsenous acid in a Hele–Shaw cell. In the low frequency range covered by the experiments, the front behavior is controlled by the flow across the gap and is reproduced with two-dimensional numerical simulations. It is shown that the front velocity oscillates at the frequency of the flow, whereas the front width oscillates at twice that frequency. Moreover, the Taylor regime in the presence of a Poi...

We report on experiments with two miscible fluids of equal density but different viscosities. The fluids were injected co-currently and concentrically into a cylindrical pipe. The resulting base state is an axisymmetric parallel flow. The ratio of the two fluid flow rates determines the relative amount of the fluids, thus the radius of the inner core fluid. Depending on this radius and the total flow rate, two different and unstable axisymmetric patterns, denoted by mushrooms and pearls, were observed. We delineate the diagram of occurrence of the two patterns as a function of the various parameters.

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

A non-newtonian fluid is a fluid which relation between it's shear rate and the stress under which it is put, is not linear. In a porous medium, the stress imposed to the fluid depends on the imposed pressure, but also on the pores size, and therefore on the macroscopic scale permeability. Some fluids have a rheology such that the fluid show a change of behaviour reaching a yield stress. If the pore size is random, then the fluid will present heterogeneous regime changes in the medium. The flow will then show a first regime where the whole fluid will be under the threshold, a regime where the whole fluid will be far above the threshold, and an intermediate regime for which both rheologies coexists. We are interested in intermediate regime for the flow of non-newtonian fluids in macroscopic porous media, and study it with numerical simulations. More particularly, we look at the flow of a Bingham fluid and that of a Carreau fluid. The Bingham fluid doesn't flow under a yield stress. Under the threshold, it behaves as a solid. Beyond, it's shear-rate/stress relation is an affine law. Carreau fluids have a shear-rate/stress relation that change regime between that of a newtonian fluid, and a power law. The macroscopic scale study is done simulating a Darcy-Brinkman law in a heterogeneous permeability field. We use for our simulations the lattice-Boltzmann method, on a regular node grid, and more specifically Irina Ginzburg two relaxation-time scheme. For each fluid, we study the flow-pressure relationship, as well as the geometric properties and the multi-scale properties in the fluid regions in the same flow regime (clusters), properties such as their size and shape. We also link these geometrical properties to the percolation theory, which studies the behaviour of randomly opening node maps and predicts fractal properties., Un fluide non-newtonien est un fluide dont la relation entre son taux de déformation et la contrainte qui lui est imposé n'est pas linéaire. Dans un milieu poreux, la contrainte imposée au fluide dépend de la pression imposée mais aussi de la taille des pores, et donc de la perméabilité à l'échelle macroscopique. Certains fluides peuvent avoir une rhéologie qui présente un changement à partir d'un seuil en contrainte. Si la taille des pores est aléatoire, le fluide va alors changer de régime de façon hétérogène dans le milieu. L'écoulement pourra alors présenter un premier régime où tout le fluide est en dessous du seuil, un régime où tout le fluide est au-dessus du seuil, et un régime intermédiaire pour lequel les deux types de rhéologie coexistent. Nous nous intéressons à ce régime intermédiaire pour des écoulements de fluide non-newtonien en milieu poreux macroscopique, étudiés par des simulations. Plus particulièrement, nous regardons l'écoulement d'un fluide de Bingham et celui d'un fluide de Carreau. Le fluide de Bingham ne s'écoule qu'à partir d'une contrainte seuil. En dessous du seuil, il se comporte comme un solide. Au-delà, sa relation taux de déformation/contrainte suis une loi affine. Les fluides de Carreau ont une relation taux de déformation/contrainte qui change de régime entre une loi linéaire et une loi de puissance. L'étude à l'échelle macroscopique se fait en simulant une loi de Darcy-Brinkman dans un champ de perméabilité hétérogène. Nous utilisons pour nos simulations la méthode de Boltzmann sur réseau, avec une grille de nœuds régulière, plus particulièrement le schéma à deux temps de relaxation d'Irina Ginzburg. Pour chaque fluide, nous regardons la relation débit-pression, ainsi que les propriétés géométriques des différents régimes d'écoulement. Nous caractérisons plus particulièrement les propriétés multi-échelles des régions dans un le même régime d'écoulement (clusters), telles que leur taille ou leur forme. Nous faisons aussi le lien entre ces propriétés géométriques et la théorie de la percolation, qui étudie le comportement de cartes de nœuds s'ouvrant aléatoirement et qui prédit des propriétés fractales.