Diffusion Coefficients in Porous Media

Altenberger and Tirrell [11] utilized the Langevin equation for particle motion coupled with hydrodynamics described by the Navier-S takes equation to determine particle diffusion coefficients in porous media given by... 

Weissberg, HL, Effective Diffusion Coefficient in Porous Media, Journal of Applied Physics 34, 2636, 1963. 

Diffusion is the process of molecular transport associated with the stochastic movement of the individual diffusants. Diffusion coefficients in porous media may be defined on the basis of the generalized Fick s first law... 

In an aquifer, the total Fickian transport coefficient of a chemical is the sum of the dispersion coefficient and the effective molecular diffusion coefficient. For use in the groundwater regime, the molecular diffusion coefficient of a chemical in free water must be corrected to account for tortuosity and porosity. Commonly, the free-water molecular diffusion coefficient is divided by an estimate of tortuosity (sometimes taken as the square root of two) and multiplied by porosity to estimate an effective molecular diffusion coefficient in groundwater. Millington (1959) and Millington and Quirk (1961) provide a review of several approaches to the estimation of effective molecular diffusion coefficients in porous media. Note that mixing by molecular diffusion of chemicals dissolved in pore waters always occurs, even if mechanical dispersion becomes zero as a consequence of no seepage velocity. 

This is beyond the resolution of the NMR imaging techniques that are discussed in Section 5. In this section, we shall consider the theory behind the application of PFG methods to measure diffusion coefficients, and then go on to consider how diffusion coefficients in porous media reveal structural information. Some PFG pulse sequences that overcome particular problems experienced for porous media are discussed. PFG methods on samples in which there is bulk flow of fluid will be considered in Section 6. 

Literature Relations Predicting Effective Diffusion Coefficients in Porous Media... 

Many investigators have studied diffusion in systems composed of a stationary porous solid phase and a continuous fluid phase in which the solute diffuses. The effective transport coefficients in porous media have often been estimated using the following expression ... 

The result (9.30) is the macroscale transport equation of the diffusion problem in porous media, and Djj is the homogenized diffusion coefficient. 

Diffusion coefficients for porous media are generally referred to as effective diffusivities, since the actual molecular diffusion process occurs in the fluid phase and interactions with the porous medium inhibits the chemical movement. There are both physical and chemical factors that go into estimating effective diffusivities. The physical effects are twofold. First, some fraction of the porous media is solid, limiting the volume through which fluid phase diffusion can occur. This is quantified by the porosity, which is defined as the ratio of the volume of void space to the total volume. Second, the connectivity between pore spaces in soil and sediment grain packs (as well as other porous media) are circuitous and lengthen the distance a molecule must travel to traverse the material. This lengthening of the diffusion path is quantified... 

The objective of most of the theories of transport in porous media is to derive analytical or numerical functions for the effective diffusion coefficient to use in the preceed-ing averaged species continuity equations based on the structure of the media and, more recently, the structure of the solute. 

MC simulations and semianalytical theories for diffusion of flexible polymers in random porous media, which have been summarized [35], indicate that the diffusion coefficient in random three-dimensional media follows the Rouse behavior (D N dependence) at short times, and approaches the reptation limit (D dependence) for long times. By contrast, the diffusion coefficient follows the reptation limit for a highly ordered media made from infinitely long rectangular rods connected at right angles in three-dimensional space (Uke a 3D grid). 

The effective diffusivity depends on the statistical distribution of the pore transport coefficients W j. The derivation shows that the semi-empirical volume-averaging method can only be regarded as an approximation to a more complex dynamic behavior which depends non-locally on the history of the system. Under certain circumstances the long-time (t —> oo) diffusivity will not depend on t (for further details, see [191]). In such a case, the usual Pick diffusion scenario applies. The derivation presented above can, with minor revisions, be applied to the problem of flow in porous media. When considering the heat conduction problem, however, some new aspects have to be taken into accoimt, as heat is transported not only inside the pore space, but also inside the solid phase. 

Dispersivity is a property that depends on the nature of the sediment or rock in question, as well as the scale on which dispersion is observed. There is no typical value a dispersivity of 1 cm might be observed in a laboratory experiment, whereas a value of 100 m (10 000 cm) might be found to apply in a field study. Dispersion is generally more rapid along the direction of flow than across it, so oil > t. Typical values of the diffusion coefficient D in porous media are in the range 10-7 to 10-6 cm2 s-1. 

Then, the effective diffusivity of the macro-porous granular material is evaluated. The transport is again governed by the Fick s equation (23), but diffusion also takes place in the solid phase, which formally represents the nano-porous material, cf. Fig. 15. The diffusion coefficient in the solid phase is Dsohd = i//nanoD — 0.112D, where D is the bulk diffusivity. The concentration field in the macro-porous media is the solution of Eq. (23) with a diffusivity... 

In addition to elucidation of molecular structures, NMR can also extract valuable information about physicochemical parameters. Because of the omnipresence of protonated solvents in CE/CEC, mobile-phase events can be monitored with NMR. Early studies using E-NMR involved the calculation of diffusion coefficients, electrophoretic mobilities, and viscosity [27]. Stagnant mobile-phase mass transfer kinetics and diffusion effects [60] and fluid mass transfer resistance in porous media-related chromatographic stationary phases [61] have been studied with NMR spectroscopy. NMR imaging of the chromatographic process [62] and NMR microscopy of chromatographic columns [63] have also been reported. Several applications of NMR to on-line studies of CE/ and CEC/ NMR are highlighted. 

The ordinary diffusion equations have been presented for the case of a gas in absence of porous medium. However, in a porous medium, whose pores are all wide compared to the mean free path and provided the total pressure gradient is negligible, it is assumed that the fluxes will still satisfy the relationships of Stefan-Maxwell, since intermolecular collisions still dominate over molecule-wall collisions [19]. In the case of diffusion in porous media, the binary diffusivities are usually replaced by effective diffusion coefficients, to yield... 

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