Enhanced reaction kinetics and reactive mixing scale dynamics in mixing fronts under shear flow for arbitrary Damk\"ohler numbers

Mixing fronts, where fluids of different chemical compositions mix with each other, are typically subjected to velocity gradients, ranging from the pore scale to the catchment scale due to permeability variations and flow line geometries. A common trait of these processes is that the mixing interface is strained by shear. Depending on the P\'eclet number $Pe$, which represents the ratio of the characteristic diffusion time to the characteristic advection time, and the Damk\"ohler number $Da$, which represents the ratio of the characteristic diffusion time to the characteristic reaction time, the local reaction rates can be strongly impacted by the dynamics of the mixing interface. This impact has been characterized mostly either in kinetics-limited or in mixing-limited conditions, that is, for either very low or very high $Da$. Here the coupling of shear flow and chemical reactivity is investigated for arbitrary Damk\"ohler numbers, for a bimolecular reaction and an initial interface with separated reactants. Approximate analytical expressions for the global production rate and reactive mixing scale are derived based on a reactive lamella approach that allows for a general coupling between stretching enhanced mixing and chemical reactions. While for $Pe<Da$, reaction kinetics and stretching effects are decoupled, a scenario which we name "weak stretching", for $Pe>Da$, we uncover a "strong stretching" scenario where new scaling laws emerge from the interplay between reaction kinetics, diffusion, and stretching. The analytical results are validated against numerical simulations. These findings shed light on the effect of flow heterogeneity on the enhancement of chemical reaction and the creation of spatially localized hotspots of reactivity for a broad range of systems ranging from kinetic limited to mixing limited situations

The impact of inertial effects on solute dispersion in a channel with periodically varying aperture

International audienceWe investigate solute transport in channels with a periodically varying aperture, when the flow is still laminar but sufficiently fast for inertial effects to be nonnegligible. The flow field is computed for a two-dimensional setup using a finite element analysis, while transport is modeled using a random walk particle tracking method. Recirculation zones are observed when the aspect ratio of the unit cell and the relative aperture fluctuations are sufficiently large; under non-Stokes flow conditions, the flow in non-reversible, which is clearly noticeable by the horizontal asymmetry in the recirculation zones. After characterizing the size and position of the recirculation zones as a function of the geometry and Reynolds number, we investigate the corresponding behavior of the longitudinal effective diffusion coefficient. We characterize its dependence on the molecular diffusion coefficient Dm, the P'eclet number, the Reynolds number, and the geometry. The proposed relation is a generalization of the well-known Taylor-Aris relationship relating the longitudinal dispersion coefficient to Dm and the P'eclet number for a channel of constant aperture at sufficiently low Reynolds number. Inertial effects impact the exponent of the P'eclet number in this relationship; the exponent is controlled by the relative amplitude of aperture fluctuations. For the range of parameters investigated, the measured dispersion coefficient always exceeds that corresponding to the parallel plate geometry under Stokes conditions; in otherwords, boundary fluctuations always result in increased dispersion. The transient approach to the asymptotic regime is also studied and characterized quantitatively. We show that the measured characteristic time to attain asymptotic conditions is controlled by two competing effects: (i) the trapping of particles in the near-immobile zone and, (ii) the enhanced mixing in the central zone where most of the flow takes place (mainstream), due to its thinning

Influence of fracture scale heterogeneity on the flow properties of three-dimensional discrete fracture networks (DFN)

International audienceWhile permeability scaling of fractured media has been so far studied independently at the fracture- and network- scales, we propose a numerical analysis of the combined effect of fracture-scale heterogeneities and the network-scale topology. The analysis is based on 2 106 discrete fracture network (DFNs) simulations performed with highly robust numerical methods. Fracture local apertures are distributed according to a truncated Gaussian law, and exhibit self-affine spatial correlations up to a cutoff scale Lc. Network structures range widely over sparse and dense systems of short, long or widely distributed fracture sizes and display a large variety of fracture interconnections, flow bottlenecks and dead-ends. At the fracture scale, accounting for aperture heterogeneities leads to a reduction of the equivalent fracture transmissivity of up to a factor of 6 as compared to the parallel plate of identical mean aperture. At the network scale, a significant coupling is observed in most cases between flow heterogeneities at the fracture and at the network scale. The upscaling from the fracture to the network scale modifies the impact of fracture roughness on the measured permeability. This change can be quantified by the measure a2, which is analogous to the more classical power-averaging exponent used with heterogeneous porous media, and whose magnitude results from the competition of two effects: (i) the permeability is enhanced by the highly transmissive zones within the fractures that can bridge fracture intersections within a fracture plane; (ii) it is reduced by the closed and low transmissive areas that break up connectivity and flow paths. Citation: de Dreuzy, J.-R., Y. Méheust, and G. Pichot (2012), Influence of fracture scale heterogeneit

Convective dissolution of CO2_2 in 2D and 3D porous media: the impact of hydrodynamic dispersion

Convective dissolution is the process by which CO$_2$ injected in deep geological formations dissolves into the aqueous phase, which allows storing it perennially by gravity. The process results from buoyancy-coupled Darcy flow and solute transport. Proper theoretical modeling of the process should consider in the transport equation a diffusive term accounting for hydrodynamics (or, mechanical) dispersion, with an effective diffusion coefficient that is proportional to the local interstitial velocity. A few two-dimensional (2D) numerical studies, and three-dimensional (3D) experimental investigations, have investigated the impact of hydrodynamic dispersion on convection dynamics, with contradictory conclusions. Here, we investigate systematically the impact of the dispersion strength $S$ (relative to molecular diffusion), and of the anisotropy $\alpha$ of its tensor, on convective dissolution in 2D and 3D geometries. We use a new numerical model and analyze the solute fingers' number density (FND), penetration depth and maximum velocity; the onset time of convection; the dissolution flux in the quasi-constant flux regime; the mean concentration of the dissolved CO2; and the scalar dissipation rate. The efficiency of convective dissolution over long times is observed to be mostly controlled by the onset time of convection. For most natural porous media ($\alpha = 0.1$), the onset time is found to increase as a function of $S$, in agreement with previous experimental findings and in stark contrast to previous numerical findings. However, if $\alpha$ is sufficiently large this behavior is reversed. Furthermore, results in 3D are fully consistent with the 2D results on all accounts, except that in 3D the onset time is slightly smaller, the dissolution flux in the quasi-constant flux regime is slightly larger, and the dependence of the FND on the dispersion parameters is impacted by $Ra$.Comment: 30 pages, 18 figure

Mesoscopic structure of dry-pressed clay samples from small-angle X-ray scattering measurements . In : Proceedings of the XIIIth International Conference on Small-Angle Scattering

Weakly hydrated samples of platelet-shaped nano-particles obtained by dry-pressing suspensions of the synthetic Na fluorohectorite clay are studied. The particles consist of stacks of several tens of 1 nm-thick nanosilicate platelets. They form a compound of quasi-two-dimensional particles whose average director is aligned with the direction of the uniaxial stress applied at dehydration. Small-angle X-ray scattering images from these samples are either isotropic or anisotropic, depending on the sample orientation with respect to the X-ray beam. From anisotropic images, changes in the scattering objects' orientation distribution probability (ODP) function are investigated as the temperature is lowered, thus triggering swelling of the individual particles by water intercalation. This is done, on the one hand, by inferring the width of the ODP function from the eccentricity of quasi-elliptic iso-intensity cuts of the small-angle scattering images, and, on the other hand, by obtaining the ODP function from azimuthal profiles of the images. The decays of the scattering intensity as a function of momentum transfer along the two principal directions of the images exhibit power law behaviors. A crossover scale between two power law regimes is observed on the profiles recorded along the horizontal axis; it corresponds to the typical pore size along the direction of the initially applied load. These results are compared with a previous study of similar systems

Fluid trapping during capillary displacement in fractures

International audienceThe spatial distribution of fluid phases and the geometry of fluid-fluid interfaces resulting from immiscible displacement in fractures cast decisive influence on a range of macroscopic flow parameters. Most importantly, these are the relative permeabilities of the fluids as well as the macroscopic irreducible saturations. They also influence parameters for component (solute) transport, as it usually occurs through one of the fluid phase only. Here, we present a numerical investigation on the critical role of aperture variation and spatial correlation on fluid trapping and the morphology of fluid phase distributions in a geological fracture. We consider drainage in the capillary dominated regime. The correlation scale, that is, the scale over which the two facing fracture walls are matched, varies among the investigated geometries between L/256 and L (self-affine fields), L being the domain/fracture length. The aperture variability is quantified by the coefficient of variation (δ), ranging among the various geometries from 0.05 to 0.25. We use an invasion percolation based model which has been shown to properly reproduce displacement patterns observed in previous experiments. We present a quantitative analysis of the size distribution of trapped fluid clusters. We show that when the in-plane curvature is considered, the amount of trapped fluid mass first increases with increasing correlation scale Lc and then decreases as Lc further increases from some intermediate scale towards the domain length scale L. The in-plane curvature contributes to smoothening the invasion front and to dampening the entrapment of fluid clusters of a certain size range that depends on the combination of random aperture standard deviation and spatial correlation

The flow of a foam in a two-dimensional porous medium

International audienceFoams have been used for decades as displacing fluids for enhanced oil recovery and aquifer remediation, and more recently, for remediation of the vadose zone, in which case foams carry chemical amendments. Foams are better injection fluids than aqueous solutions due to their low sensitivity to gravity and because they are less sensitive to permeability heterogeneities, thus allowing a more uniform sweep. The latter aspect results from their peculiar rheology, whose understanding motivates the present study. We investigate foam flow through a two-dimensional porous medium consisting of circular obstacles positioned randomly in a horizontal transparent Hele-Shaw cell. The local foam structure is recorded in situ, which provides a measure of the spatial distribution of bubble velocities and sizes at regular time intervals. The flow exhibits a rich phenomenology including preferential flow paths and local flow nonstationarity (intermittency) despite the imposed permanent global flow rate. Moreover, the medium selects the bubble size distribution through lamella division-triggered bubble fragmentation. Varying the mean bubble size of the injected foam, its water content, and mean velocity, we characterize those processes systematically. In particular, we measure the spatial evolution of the distribution of bubble areas, and infer the efficiency of bubble fragmentation depending on the various control parameters. We furthermore show that the distributions of bubble sizes and velocities are correlated. This study sheds new light on the local rheology of foams in porous media and opens the way toward quantitative characterization of the relationship between medium geometry and foam flow properties. It also suggests that large-scale models of foam flows in the subsurface should account for the correlation between bubble sizes and velocities

Spontaneous imbibition dynamics in two-dimensional porous media : a generalized interacting multi-capillary model

The capillary bundle model, wherein the flow dynamics of a porous-medium is predicted from that of a bundle of independent cylindrical tubes/capillaries whose radii are distributed according to the medium's pore size distribution, has been used extensively. The model lacks interaction between the flow channels, thus fails at predicting complex flow configuration, including those involving two-phase flow. We propose here to predict spontaneous imbibition in quasi-two-dimensional (quasi-2D) porous-media from a model based on a planar bundle of interacting capillaries. The imbibition flow dynamics, particularly, breakthrough time, global wetting fluid saturation at breakthrough, and capillary carrying the leading meniscus are governed by the distribution of the capillaries' radii and their spatial arrangement. For an 20 interacting capillary system, the breakthrough time can be 39% smaller than that predicted by the classic, non-interacting, capillary-bundle-model of identical capillary radii distribution. We propose a stochastic approach to use this model of interacting capillaries for quantitative predictions. Using the capillary diameter distribution as that of the pore sizes in the target porous medium, and computing the average behavior of a randomly-chosen samples of such interacting-capillary-bundles with different spatial arrangements, we obtain predictions of the position in time of the bulk saturating front, and of that of the visible leading front, that agree well with measurements taken from the literature. This semi-analytical model is quick to run and provides fast predictions on one-dimensional spontaneous imbibition in porous-media whose porosity structure can reasonably be considered two-dimensional, e.g., paper, thin porous-media in general, or layered aquifers