Permeability length scale

When the permeability length scale is the shortest length of the problem,... 

When the permeability length scale is the shortest length of the problem, <3 and L, the layer-induced shift, A/i, is proportional to the density of the liquid and does not depend on the viscosity. It has the form of the Sauerbrey equation for mass loading. This effect results from the inertial motion of the liquid trapped by the inhomogeneities in the interfacial layer ... 

In contrast, diffusion of MeOH measured via permeabilify measurements (assuming a partition coefficient of 1) was lower (1.3 x 10 and 6.4 x 10 cm s for Nafion 117 and BPSH 40, respectively) and showed no concentration dependence. The differences observed between the two techniques are related to the length scale over which diffusion is monitored and the partition coefficient, or solubility, of MeOH in the membranes as a function of concentration. For the permeability measurements, this length is equal to the thickness of the membrane (178 and 132 pm for Nafion 117 and BPSH 40, respectively), whereas the NMR method observes diffusion over a lengfh of approximately 4-8 pm. 

The interfacial capacitance can also provide a significant insight into the permeability of interfacial supramolecular assemblies. While information of this kind complements studies using redox-active probes in solution, it also provides information on a significantly shorter length scale, i.e. that of electrolyte ions and solvent molecules. For example, for dense, defect-free monolayers, the limiting capacitance is very much lower (5-10 pF cm-2) than that found for an unmodified interface (20-60 pF cm 2). 

Two fundamentally different regimes can exist 1) those characterized by transport-controlled reaction and 2) those characterized by kinetic rate-controlled reaction (2.)- In the case of transport-controlled reaction, the reaction rate constant is much faster than any of the transport processes involved so that the length scale over which a moving fluid comes to equilibrium is small. In this regime, therefore, the walls of a dissolution channel are essentially discontinuities in permeability while in the kinetic rate-controlled case, where equilibrium between the fluid and the reacting mineral occurs over some distance, the boundaries of a channel are blurred by a more gradual permeability change. 

Figure 5. Contours of permeability at the same point in the time evolution of a channel for a variety of Damkohler numbers. Flow is from bottom to top and the shaded region contains the reactive cement. The initial perturbation is a 1 meter high by 0.3 meter wide zone. Permeability contours are equally spaced at increments of 2 millidarcies. True velocity of 1 x 10 m/s and a dispersion coefficient of 1 x 10 m /s. The length scale is taken as 10 meters, the width of the aquifer.

Figure 4. Pressure waveforms illustrating the length scale effect on fast and slow P-wave events. Waveforms correspond to vug sizes of I, 1.5, 2, and 3 cm, in a matrix of 3 Darcy permeability.

We correlate the effect of randomness, porosity, and fluid parameters with permeability fields and probability distributions predicted by our model, thus improving our understanding of heterogeneous media for applications to natural systems. Normally, at core scales for reservoirs and aquifers among others, the unknown permeability distribution in the subsurface on all length scales is much needed for practical goals (Sitar et al., 1987 Cooke et al., 1995). 

Fig. 2.1. Upper. Cross-section of a consolidated heap of sand, impregnated with a resin. Lower. A model that could be used to study, for example, the permeability of this material. The real material and the model have in common a well-defined characteristic length scale (the period for the model and the correlation length for the real material), from which a representative elementary volume can be specified...

The mean pore size decreases when the length scale decreases and we find that the permeability inside the fractal aggregate varies as... 

PC and the diffusion and permeability coefficients of Helium have been calculated. The transport coefficients in these 50 A microstructures agreed with those of 33 A micro-structures to within 30% it seems, therefore, that a length scale of several dozen angstroms is sufficient for evaluating the solute s transport coefficients. Similar conclusions can also be drawn for the MD simulations Changing from a 20 A to a 30 A simulation cell did not significantly alter diffusion coefficients of O2 in PIB [49]. 

Pillai, K.M. Advani, S.G. Numerical and analytical study to estimate the effect of two length scales upon the permeability of a fibrous porous medium. Transp. Porous Media 1995, 21, 1-17. 

It is almost impossible to cover the entire range of models in Figure 25.1, and in this chapter we will limit ourselves to the different modeling approaches at the continuum level (micro-macroscopic and system-level simulations). In summary, there are computational models that are developed primarily for the lower-length scales (atomistic and mesoscopic) which do not scale to the system-level. The existing models at the macroscopic or system-level are primarily based on electrical circuit models or simple lD/pseudo-2D models [17-24]. The ID models are limited in their ability to capture spatial variations in permeability or conductivity or to handle the multidimensional structure of recent electrode and solid electrolyte materials. There have been some recent extensions to 2D [29-31], and this is still an active area of development As mentioned in a recent Materials Research Society (MRS) bulletin [6], errors arising from over-simplified macroscopic models are corrected for when the parameters in the model are fitted to real experimental data, and these models have to be improved if they are to be integrated with atomistic... 

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