Tracing diffusion in porous media with fractal properties

This work is concerned with conditional averaging methods which can be used for modeling of transport in porous media with volume reactions in the fluid phase and surface reactions at the fluid/solid interface. The model under consideration takes into account convection, diffusion within the pores and on larger scales, and homogeneous and heterogeneous reactions. Near the interface with fractal properties, the fluid flow is slow, and diffusion, as a transport mechanism, dominates over convection. Following the conditional moment closure paradigm, we employ a diffusion tracer as a reference scalar field that makes the conditional averaging sensitive to the proximity of a point to the interface. The resulting conditionally averaged reactive transport equations are governed by the probability density function (PDF) of the diffusion tracer, and this makes the study of its behavior an important problem. We consider a hitting time stochastic interpretation of the diffusion tracer, establish integral equations relating it to a subsidiary distance tracer, and obtain distance-diffusion inequalities. Assuming that the fluid/solid interface and pores themselves possess fractal properties which are quantified, in particular, by a variant of the Minkowski-Bouligand fractal dimension, we investigate the interplay between the interface and network scenarios of fractality in the scaling laws of the diffusion tracer PDF. We also discuss and employ several hypotheses, including a lognormal cascade hypothesis on the behavior of the diffusion tracer at different length scales

Development of a novel ultra-high vacuum diffusion apparatus for investigating Knudsen diffusion in complex pore channels

Disordered porous materials with rough surfaces are widely used in several industrial applications, such as fuel cells, heterogeneous catalysis, molecular separation, and oil and gas recovery. Many of these processes are diffusion-controlled, due to narrow pore size. In porous catalysts, molecules diffuse through the pore network and react on the active sites of the pore walls. This indicates that the shape and surface morphology of the pores may affect catalytic performance. Knudsen diffusion plays a crucial role in catalysts where gaseous molecular transport is dominated by molecule–wall collisions. Consequently, the pore geometry of disordered porous materials, including the pore shapes and pore wall roughness, has attracted significant research interest. However, the high cost and limited availability of techniques for building a system that can simulate the Knudsen regime in complex pores with rough inner surface representative of porous catalysts have hindered the practical investigation of the effect of the pore morphology on Knudsen diffusion. In this study, to address these limitations and accurately measure the Knudsen diffusivity in complex pore channels, a unique vacuum diffusion apparatus was designed and employed. The inherent characteristics of the novel apparatus enabled the system to rapidly simulate the Knudsen regime, emulating the complexity of actual porous media, and to accurately measure the relevant data for calculations. In addition, this high-vacuum diffusion apparatus was used to investigate Knudsen diffusion in complex pore channels with various lengths, pore shapes, and surface roughness. The measurement of the Knudsen diffusivity in the channels with various lengths and shapes was guided by computer simulations and theoretical calculations to validate the functioning and the accuracy of the apparatus. Three fractal surface channels (with N= 1, 2, and 3 generations) were fabricated using a three-dimensional printer via selective laser sintering, and the effect of the surface roughness of the fabricated channels on the Knudsen diffusion can be analysed in the unique apparatus. The developed ultra-high vacuum diffusion apparatus enables us to experimentally probe Knudsen diffusion in complex geometries more directly than has ever been achieved

Non-Fickian dispersion in porous media : 1. Multiscale measurements using single-well injection withdrawal tracer tests

International audienceWe present a set of single-well injection withdrawal tracer tests in a paleoreef porous reservoir displaying important small-scale heterogeneity. An improved dual-packer probe was designed to perform dirac-like tracer injection and accurate downhole automatic measurements of the tracer concentration during the recovery phase. By flushing the tracer, at constant flow rate, for increasing time duration, we can probe distinctly different reservoir volumes and test the multiscale predictability of the (non-Fickian) dispersion models. First we describe the characteristics, from microscale to meter scale, of the reservoir rock. Second, the specificity of the tracer test setup and the results obtained using two different tracers and measurement methods (salinity-conductivity and fluorescent dye­optical measurement, respectively) are presented. All the tracer tests display strongly tailed breakthrough curves (BTC) consistent with diffusion in immobile regions. Conductivity results, measured over 3 orders of magnitude only, could have been easily interpreted by the conventional mobile-immobile (MIM) diffusive mass transfer model of asymptotic log-log slope of 2. However, the fluorescent dye sensor, which allows exploring much lower concentration values, shows that a change in the log-log slope occurs at larger time with an asymptotic value of 1.5, corresponding to the double-porosity model. These results suggest that the conventional, one-slope MIM transfer rate model is too simplistic to account for the real multiscale heterogeneity of the diffusion-dominant fraction of the reservoir

Opacity of fluffy dust aggregates

Context. Dust grains coagulate to form dust aggregates in protoplanetary disks. Their porosity can be extremely high in the disks. Although disk emission may come from fluffy dust aggregates, the emission has been modeled with compact grains. Aims. We aim to reveal the mass opacity of fluffy aggregates from infrared to millimeter wavelengths with the filling factor ranging from 1 down to $10^{-4}$. Methods. We use Mie calculations with an effective medium theory. The monomers are assumed to be 0.1 ${\rm \mu m}$ sized grains, which is much shorter than the wavelengths that we focus on. Results. We find that the absorption mass opacity of fluffy aggregates are characterized by the product $a\times f$, where $a$ is the dust radius and $f$ is the filling factor, except for the interference structure. The scattering mass opacity is also characterized by $af$ at short wavelengths while it is higher in more fluffy aggregates at long wavelengths. We also derive the analytic formula of the mass opacity and find that it reproduces the Mie calculations. We also calculate the expected difference of the emission between compact and fluffy aggregates in protoplanetary disks with a simple dust growth and drift model. We find that compact grains and fluffy aggregates can be distinguished by the radial distribution of the opacity index $\beta$. The previous observation of the radial distribution of $\beta$ is consistent with the fluffy case, but more observations are required to distinguish between fluffy or compact. In addition, we find that the scattered light would be another way to distinguish between compact grains and fluffy aggregates.Comment: 16 pages, 17 figures, published in A&A, 568, A4

Transport Phenomena Modelled on Pore-Space Images

Fluid flow and dispersion of solute particles are modelled directly on three-dimensional pore-space images of rock samples. To simulate flow, the finite-difference method combined with a standard predictor-corrector procedure to decouple pressure and velocity is applied. We study the permeability and the size of representative elementary volume (REV) of a range of consolidated and unconsolidated porous media. We demonstrate that the flow-based REV is larger than for geometry-based properties such as porosity and specific surface area, since it needs to account for the tortuosity and connectedness of the flow paths. For solute transport we apply a novel streamline-based algorithm that is similar to the Pollock algorithm common in field-scale reservoir simulation, but which employs a semi-analytic formulation near solid boundaries to capture, with sub-grid resolution, the variation in velocity near the grains. A random walk method is used to account for mixing by molecular diffusion. The algorithm is validated by comparison with published results for Taylor-Aris dispersion in a single capillary with a square cross-section. We then accurately predict experimental data available in the literature for longitudinal dispersion coefficient as a function of Peclet number. We study a number of sandpack, sandstone and carbonate samples for which we have good quality three-dimensional images. There is a power-law dependence of dispersion coefficient as a function of Peclet number, with an exponent that is a function of pore-space heterogeneity: the carbonates we study have a distinctly different behaviour than sandstones and sandpacks. This is related to the differences in transit time probabilities of solute particles travelling between two neighbouring voxels. We then study the non-Fickian behaviour of solute transport in porous media by modelling the NMR propagators and the time-dependent dispersion coefficients of different rock types. The behaviour is explained using Continuous Time Random Walk (CTRW) theory: transport is qualitatively different for the complex porous media such as carbonates compared to the sandstone or sandpack, with long tailing and an almost immobile peak concentration. We discuss extensions of the work to reactive transport and the simulation of transport in finely-resolved images with billions of voxels

First-passage statistics of colloids on fractals: Theory and experimental realization

In nature and technology, particle dynamics frequently occur in complex environments, for example in restricted geometries or crowded media. These dynamics have often been modeled invoking a fractal structure of the medium although the fractal structure was only indirectly inferred through the dynamics. Moreover, systematic studies have not yet been performed. Here, colloidal particles moving in a laser speckle pattern are used as a model system. In this case, the experimental observations can be reliably traced to the fractal structure of the underlying medium with an adjustable fractal dimension. First-passage time statistics reveal that the particles explore the speckle in a self-similar, fractal manner at least over four decades in time and on length scales up to 20 times the particle radius. The requirements for fractal diffusion to be applicable are laid out, and methods to extract the fractal dimension are established

Non-Fickian dispersion in porous media explained by heterogeneous microscale matrix diffusion

International audienceMobile-immobile mass transfer is widely used to model non-Fickian dispersion in porous media. Nevertheless, the memory function, implemented in the sink/source term of the transport equation to characterize diffusion in the matrix (i.e., the immobile domain), is rarely measured directly. Therefore, the question can be posed as to whether the memory function is just a practical way of increasing the degrees of freedom for fitting tracer test breakthrough curves or whether it actually models the physics of tracer transport. In this paper we first present a technique to measure the memory function of aquifer samples and then compare the results with the memory function fitted from a set of field-scale tracer tests performed in the same aquifer. The memory function is computed by solving the matrix diffusion equation using a random walk approach. The properties that control diffusion (i.e., mobile-immobile interface and immobile domain cluster shapes, porosity, and tortuosity) are investigated by X-ray microtomography. Once the geometry of the matrix clusters is measured, the shape of the memory function is controlled by the value of the porosity at the percolation threshold and of the tortuosity of the diffusion path. These parameters can be evaluated from microtomographic images. The computed memory function compares well with the memory function deduced from the field-scale tracer tests. We conclude that for the reservoir rock studied here, the atypical non-Fickian dispersion measured from the tracer test is well explained by microscale diffusion processes in the immobile domain. A diffusion-controlled mobileimmobile mass transfer model therefore appears to be valid for this specific case