A random field model for anomalous diffusion in heterogeneous porous media

Heterogeneity, as it occurs in porous media, is characterized in terms of a scaling exponent, or fractal dimension. A feature of primary interest for two-phase flow is the mixing length. This paper determines the relation between the scaling exponent for the heterogeneity and the scaling exponent which governs the mixing length. The analysis assumes a linear transport equation and uses random fields first in the characterization of the heterogeneity and second in the solution of the flow problem, in order to determine the mixing exponents. The scaling behavior changes from long-length-scale dominated to short-length-scale dominated at a critical value of the scaling exponent of the rock heterogeneity. The long-length-scale-dominated diffusion is anomalous.     

Anomalous transport fluctuations in a model of irregular media

We consider a diffusion model with stochastic porosity for which the average solution exhibits an abnormal transport. In this paper we investigate the relation of such an anomalous diffusive property of the mean value with the behavior of the solution corresponding to each realization of the stochastic porosity. Such a solution will correspond to the actual measurements in an experiment made on a particular tube. The most relevant result of our work is that, although the concentration corresponding to each realization diffuses normally for large times, it experiments on large deviations from the mean value during intermediate times.     

A fractional diffusion equation for a marker in porous media

To study the properties of the flow of a marker through an irregular packed bed, porosity is described by stochastic processes with exponentially decaying correlation functions. We show that the ensemble average concentration of the marker satisfies a fractional diffusion equation. (c) 2001 American Institute of Physics.     

Density fluctuations in a model for vibrated granular media

This paper presents the study of density fluctuations in a model for vibrated granular media. Their microscopic origin is shown to be linked to the microscopic disorder in grains packing. Varying vibrations amplitude and duration, several regimes are found for density relaxation. Its power spectrum is well described by power laws.     

Retardation of diffusion in porous media with stagnation zones

The problem of diffusion in a porous medium with stagnation zones, which reduce the rate of transport of particles, is considered. A method based on the concept of entropy barriers associated with the small size of inlets into the stagnation zone is used to solve the diffusion problem. This method made it possible to reduce the diffusion problem in 3D structures to a set of one-dimensional equations that can be solved analytically. The calculations yielded an exact value of the effective diffusion coefficient, which is reduced due to random delays in the stagnation zones, and the transition time required to attain this reduction.     

Quantitative magnetic resonance flow and diffusion imaging in porous media

Quantitative flow and diffusion measurements have been made for water in model porous media, using magnetic resonance micro-imaging methods. The samples consisted of compacted glass beads of various sizes down to 1 mm diameter. Typical flow and diffusion images exhibited a spatial resolution of 117 μm × 117 μm and velocities in the range 1–2 mm/s. Comparison of volume flow rates calculated from the flow velocity maps with values measured directly yielded good agreement in all cases. There was also good agreement between the mean diffusion coefficient of water calculated from the diffusion maps and the bulk diffusion coefficient for pure water at the same temperature. In addition, the mean diffusion coefficient did not depend on the pore sizes in the bead diameter range of 1–3 mm. Our results also show that partial volume effects can be compensated by appropriate thresholding of the images prior to the final Fourier transformation in the flow-encoding dimension.     

2D relaxation/diffusion correlations in porous media

2D correlations between NMR relaxation and/or diffusion have been used to investigate water and oil dynamics in food and micro-emulsion systems. In the case of Mozzarella and Gouda cheese samples, a significant change in D/T2 correlation is appearing with cheese aging. In the case of a water/toluene micro-emulsion, some evidence for coalescence effects is suggested by D/D exchange spectra.     

Probing porous media with gas diffusion NMR.

We show that gas diffusion nuclear magnetic resonance (GD-NMR) provides a powerful technique for probing the structure of porous media. In random packs of glass beads, using both laser-polarized and thermally polarized xenon gas, we find that GD-NMR can accurately measure the pore space surface-area-to-volume ratio, S/V rho, and the tortuosity, alpha (the latter quantity being directly related to the system's transport properties). We also show that GD-NMR provides a good measure of the tortuosity of sandstone and complex carbonate rocks.     

Detection of the high eigenmodes of spin diffusion in porous media

The dynamics of spin diffusion in a fluid is governed by the Torrey-Bloch equations, and the solution is often expressed mathematically in an eigenmode expansion. We report an experimental demonstration of the excitation and detection of a wide range of eigenmodes in porous media by exploring the inhomogeneous internal magnetic field in the pore space. The nodal character of the eigenfunctions of the high eigenmodes was clearly observed. The methodology of excitation and detection of the high eigenmodes may be used to better characterize pore geometry.     

Homogeneous averaging of the stochastic diffusion equation in chaotic porous media

The diffusion of solution components in the pore space of a chaotic porous medium is treated. The problem of homogeneous averaging of the diffusion boundary problem is developed. In the end a system of homogeneous equations is obtained on whose basis one can treat particular transport processes. Russian University of Friendship for the People. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 52–58, September, 1996.     

Macroscopic model of multicomponent fluids in porous media

This article is about a macroscopic model simulating the heat and mass transfer of a fluid containing numerous species in porous media. In order to use it in the modelling of petroleum fields, the physical phenomena studied are gravitational segregation, thermal diffusion, convection and diffusion. The originality of this approach is to simulate the movement of each component in order to deduce the species distribution and mixture flow. The mass flux calculated by an entropic balance is compared with the Darcy equation to evaluate the phenomenological coefficients. After presenting the possible origins of composition variations and the ways that they are usually studied, the article describes the developments that have led to the new model and finally reports the first results obtained.     

The narrow pulse approximation and long length scale determination in xenon gas diffusion NMR studies of model porous media

We report a systematic study of xenon gas diffusion NMR in simple model porous media, random packs of mono-sized glass beads, and focus on three specific areas peculiar to gas-phase diffusion. These topics are: (i) diffusion of spins on the order of the pore dimensions during the application of the diffusion encoding gradient pulses in a PGSE experiment (breakdown of the narrow pulse approximation and imperfect background gradient cancellation), (ii) the ability to derive long length scale structural information, and (iii) effects of finite sample size. We find that the time-dependent diffusion coefficient, D(t), of the imbibed xenon gas at short diffusion times in small beads is significantly affected by the gas pressure. In particular, as expected, we find smaller deviations between measured D(t) and theoretical predictions as the gas pressure is increased, resulting from reduced diffusion during the application of the gradient pulse. The deviations are then completely removed when water D(t) is observed in the same samples. The use of gas also allows us to probe D(t) over a wide range of length scales and observe the long time asymptotic limit which is proportional to the inverse tortuosity of the sample, as well as the diffusion distance where this limit takes effect (approximately 1-1.5 bead diameters). The Padé approximation can be used as a reference for expected xenon D(t) data between the short and the long time limits, allowing us to explore deviations from the expected behavior at intermediate times as a result of finite sample size effects. Finally, the application of the Padé interpolation between the long and the short time asymptotic limits yields a fitted length scale (the Padé length), which is found to be approximately 0.13b for all bead packs, where b is the bead diameter.     

Mean value of an effective refractive index for stratified random media

Abstract

We introduce an effective refractive index of a one-dimensional random medium, which describes a wavefield propagating in this medium. The probability density of the effective index and its mean value are found due to the diffusion approximation.     

Mean value of an effective refractive index for stratified random media

We introduce an effective refractive index of a one-dimensional random medium, which describes a wavefield propagating in this medium. The probability density of the effective index and its mean value are found due to the diffusion approximation.     

Stochastic langevin model for flow and transport in porous media

We present a new model for fluid flow and solute transport in porous media, which employs smoothed particle hydrodynamics to solve a Langevin equation for flow and dispersion in porous media. This allows for effective separation of the advective and diffusive mixing mechanisms, which is absent in the classical dispersion theory that lumps both types of mixing into dispersion coefficient. The classical dispersion theory overestimates both mixing-induced effective reaction rates and the effective fractal dimension of the mixing fronts associated with miscible fluid Rayleigh-Taylor instabilities. We demonstrate that the stochastic (Langevin equation) model overcomes these deficiencies.     

Application of electron spin echo ESR-tomography to study radical diffusion in porous media

ESR-tomography based on electron spin echo was used to study the translational diffusion of stable radical 2,2,6,6-tetramethyl-4-pipyridinoxyl in the solutions filling the pore space of silica gel. The values of the efficient diffusion coefficients were measured for the solution of radicals 2,2,6,6-tetramethyl-4-pipyridinoxyl in methanol and 2-methyl-tetrahydrofuran, depending on the mean size of the silica gel pore. It was shown that the value of the diffusion coefficient decreased as pore size decreased, and furthermore depended on the solvent type. A linear relation between the diffusion coefficient and the spin exchange constant was established within the scope of the quasihomogeneous diffusion model.