Permeability fractal

A relatively new method to determine the fractal dimension is by using oil permeability measurements. In this method, the permeability coefficient, B, is measured for fat samples containing different SFC. This physical fractal dimension, the permeability fractal dimension. Dp, links the volumetric flow rate of liquid oil penetrating a colloidal fat crystal network with its SFC as (Bremer et al. 1989) ... 

Bremer et al. (1989) described a detailed experimental method used to measure the permeability fractal dimension of a fat crystal network. Similarly to the rheology fractal dimension, the permeability fractal dimension, Dp, could be obtained from the nonlinear regression between Q and O as shown in Figure 17.23 (Tang and Marangoni 2005). 

Figure 17.23. Calculation of permeability fractal dimension for mixtures of the high melting fraction of milkfat and sunflower oil fat.

Several forms of the modified Stake s law have been used in a series of studies concerning aggregates in the liquid phase (Li and Yuan, 2002). In these studies, the particle-liquid density difference has been further modified and adopted for the cases of impermeable bio-logical/microbial aggregates, permeable aggregates, and fractal aggregates. 

Table 17.7 summarizes the effects of the microstructural factors on the microscopy fractal dimensions, Dj, y, and Zlpr- Different fractal dimensions reflect different aspects of the microstructure of the fat crystal networks and thus have different meanings. It is necessary to define which structural characteristic is most closely related to the macroscopic physical property of interest (mechanical strength, permeability, diffusion) and then use the fractal dimension that is most closely related to the particular structural characteristic in the modeling of that physical property. 

We mentioned the possibility of permeability being distributed as a fractal in conjunction with Figure 10. A fractal distribution would show C decreasing moiiotonically at a rate prescribed by its factal dimension. This is not the case with the eolian outcrop which clearly shows two scales of heterogeneity. Given its correspondence with the geologic features, the two-scale interpretation is the only possible consistent interpretation. 

The first one is the Katz and Thompson s model (1986) which interprets transports within pore solids in terms of these percolation ideas [2]. From that, the authors introduced a fractal percolation model to predict the permeability of a disordered porous media. In invasion percolation, a non-wetting fluid can have access to the first connection from one face of the sample to the other only when the driving pressure is sufficient to penetrate the smallest pore-throat of radius rc in the most efficient conducting pathway. So, the permeability of rocks saturated with a single liquid phase is given from the following relationship ... 

Neuman S. P. (1995) On advective transport in fractal permeability and velocity fields. Water Resour. Res. 31, 1455-1460. 

Fig. 12 The variation of apparent permeability coefficient with boundary fractal dimension. (From Ref. " f)...

For carrier-mediated transport of L-lactic acid across human carcinoma cell line, it was found that increasing agitation rate resulted in a larger fractal dimension accompanied by a decrease in the substrate permeability rate. The classical Michaelis-Menten model is known to be only valid for a limited range of glucose concentrations. An alternative model was proposed including convective and non-linear diffusive mechanisms corresponding to the first and second (fractal power function) terms in Eq. (30). 

McNamee, J.E. Fractal character of pulmonary micro-vascular permeability. Ann. Biomed. Eng. 1990, 18,... 

Wheatcraft et al. (1991) considered flow and solute transport in a medium composed of high and low sat distributed according to a Sierpinski carpet fractal, reminiscent of low permeability pebbles distributed in a high permeability matrix. A multigrid solver was used to compute the flow field (Fig. 3 1B) and a particletracking algorithm was used to determine the tracer motion. No diffusion was considered. They found that dispersion increased with the scale of the simulation faster than could be predicted with other models. 

Kemblowski, M.W., and J.-C. Wen. 1993. Contaminant spreading in stratified soils with fractal permeability distribution. Water Resour. Res. 29 419-425. 

Rheological Properties. We have seen that the permeability of a particle gel is related to the fractal structure in a simple way. This is not the case for the rheological properties, because these are affected by several variables, of which the quantitative effect often cannot be readily established. 

The gel structure is determined by the volume fraction of particle material, the size of the building blocks, and the fractal dimensionality. Simple scaling laws are derived for the permeability and for rheological properties as functions of particle concentration. The rheological parameters also depend on those of the particles, especially the extent of the linear range. 

It is interesting to compare eq. (15) with the results obtained on finitely ramified fractals by means of Green function renormalization [9-10]. It has been shown that the fractional uptake curve for a structure possessing fractal dimension dj, walk dimension d, and adsorbing from a reservoir at constant concentration c through an exchange manifold B (which represents the permeable boundary for treuisfer) possessing fractal dimension d scales as... 

T-H-M-C processes are significantly affected by subsurface heterogeneity, which results in scale-dependence of the related parameters. To handle this scale-dependent behavior, we need to characterize this heterogeneity and consider its effects at different scales. In this study, we demonstrate that the measured permeability data from Sellafield site, UK, are very well described by fractional Levy motion (fLm), a stochastic fractal. This finding has important implications for modeling large-scale coupled processes in heterogeneous fractured rocks. 

The conformation of natural organic molecules under various solution conditions is of particular interest in water treatment, as the structure of the molecules may influence their packing and water permeability. The fractal dimension reflects the space occupied by the disordered system. Senesi et al. described a method using a UV/VIS spectrophotometer to measure the fractal dimension of humic aggregates (Senesi (1994), Senesi et al (1994,1996,1997)). At low pH the a egates were of a compact, collapsed, less porous structure, whereas at a higher pH they were more expanded, open, and of fluffy structure. While the method appeared to work well for the published data, the simultaneous variation of absorbance, fluorescence, and scattering casts some doubt on the method. 

Of particular interest in water treatment is the permeability of aggregates. When a regates deposit on a membrane, flux is determined by the resistance of the formed cake. Furthermore, the drag force on the colloid is determined by the aggregate porosity. Veerapaneni and Wiesner (1997) predicted the resistance to fluid flow as a function of fractal dimension. They determined that fluid flow through the aggregates decreased with increasing fractal dimension. 

Strucmre and fractal dimension determine packing and water permeability of a deposit. 

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