Capillary bundle models

A CEC column can be considered as a collection of capillary tubes with the average channel size between the particles corresponds to the diameter of each tube. Using such a capillary bundle model, the following relationship between the mean channel diameter and the particle size was derived [37] ... 

Another measure of length is the equivalent hydraulic radius, rh, determined inversely from saturated or unsaturated flow experiments. By comparing Darcy s equation with Poiseuille s law, and invoking a capillary bundle model, one obtains the following definition of rh (Kutflek Nielsen, 1994),... 

The predictions of capillary bundle models have been shown to agree reasonably well with experimental data for packed beds of uniform particles (Carbonell, 1979). However, breakthrough curves computed using this approach appear to underpredict the time of arrival of the peak of the solute pulse in undisturbed soils (Rao ct al., 1976 Bouma Wosten, 1979). This may be due to discrepancies between measured and actual pore size distributions, inappropriate representation of the relationship between and K within individual pores, or the assumed lack of in-kTconneclivily between pore channels (Lindstrom Boersma, 1971 Rao et al., 1976). 

These network models are comprised of different-sized, interconnected elements of uniform shape (e.g., Fig 3-17B). The configuration of elements within the network can be either systematic or random. Marie and Defrenne (1960) were the first to use this type of model to predict solute dispersion. Their network was a modified capillary bundle model with regularly spaced interconnections between parallel tubes of radii r and r2. This model does not consider diffusion. Spreading of a solute in the model is given by ... 

While the above models represent an improvement over classical capillary bundle models, they cannot reproduce the geometrical heterogeneity of natural porous media. As a result, several authors have tried to derive more sophisticated models based on random capillary networks. 

Mathematical Models. Mathematical models of polymer flow in porous rocks are usually based on capillary bundle models of flow in porous media. The rock is visualized to be a bundle of tortuous, non-connected capillaries which have uniform radii, R. This mathematical model represents flow in the tortuous capillaries which have effective length (L ) by equivalent straight capillary tubes with length L. An... 

An equivalent shear rate for flow in porous media can also be derived from the capillary bundle model. For Newtonian fluids, the shear rate in the porous media is given by Equation 6... 

Although many research personnel consider capillary bundle models to be a gross simplification of flow in porous media, they provide a conceptual framework to estimate effects of non-Newtonian properties on flow in porous media. 

Typical results are shown in Tables 3-5 for the Blake-Kozeny model where polymer mobilities are compared at a frontal advance rate of 1 ft/d. For the conditions of this study, the polymer mobility was underestimated by factors ranging from 1.3 to 6.7. Best agreement was observed at 500 ppm for high permeability cores. Poorest agreement occurs at 1500 ppm and 15.5 md cores. The average reduction in polymer mobility was 0.42 for the Blake-Kozeny model and 0.36 for the modified Blake-Kozeny model. Capillary bundle models consistently predict lower polymer mobilities in porous rocks than observed experimentally. 

Capillary bundle models assume the power-law exponent, n, determined from steady shear measurements is identical to the power-law exponent, n, observed during flow in porous rocks. A comparison of n and n is shown in Figure 7. In our experiments, the value of n was greater than n for polymer concentrations above 500 ppm. A review of Chauveteau s data also reveals that the power-law exponent for flow in porous rocks is larger than the power-law index measured from viscometric data. 

The comparison of predicted and experimental polymer mobilities indicates that capillary bundle models which rely on rheological parameters... 

Apparent shear rates estimated from the experimental data using Equation 35 are always higher than those estimated from Equation 11 which are based on the capillary bundle model. The mobility of the polymer at the apparent shear rate is given by Equation 36. 

Polymer mobilities are underestimated by a factor of two or more when predicted using capillary bundle models and rheological parameters obtained from steady shear measurements on the polymer solutions. 

Capillary bundle models are inadequate for prediction of in situ shear rates for polymer flow in porous rocks. 

Figure 6.1. Two variants of the capillary bundle model of a porous medium.

When using the capillary bundle models, the following two-step approach is taken in formulating a macroscopic description of the flow of non-Newtonian fluids in porous media ... 

Based on the capillary bundle model with equivalent radius given by Equation 6.9, several workers (Christopher and Middleman, 1965 Marshall and Mentzner, 1964 Teew and Hesselink, 1980 Greaves and Patel, 1985 Willhite and Uhl, 1986) have found that the equivalent wall shear rate for a power law fluid, 7p j, in a porous medium is given by Equation 6.10 above and that the Darcy velocity is given by the expression ... 

Willhite and Uhl (1986, 1988) studied the flow of xanthan in Berea cores over the concentration range 500-1500 ppm and over a wide range of flow rates. Starting with equations 6.11 and 6.12 as theoretical models, they compared the experimentally measured in-core power law index and effective mobilities of the polymer solution with the bulk values of n and the viscosity prediction based on the power law equations. They found that the power law exponent for the flow of a xanthan biopolymer through Berea sandstone was larger than the bulk value for polymer concentrations above 500 ppm. Like Teew and Hesselink (1980), they found that the effective polymer viscosities are overestimated by the capillary bundle models. Similar results for the flow of xanthan through unconsolidated sandpacks have been found more recently by Hejri et al (1988). 

As a final remark, note that it has been explained why the capillary bundle expressions give a reasonable correlative model for apparent shear rate in porous media. However, the physical basis of the capillary bundle model is not correct and it cannot be used to analyse subtle aspects of the detailed microscopic flows of non-Newtonian (or Newtonian) fluids in porous media. For example, the discussion of shear stress and tortuosity which is given in Teew and Hesselink (1980) in connection with the capillary bundle model must be treated with great caution, as must similar treatments in other works of this type (see, for example, other references in Section 6.3). 

The in-situ viscosity, rj C, v ) referred to above is a function of both polymer concentration and the flow rate (aqueous-phase velocity, v ) in the porous medium. If the polymer solution is purely shear thinning in flow through the porous medium, then one possible model to describe the effective in-situ shear rate within the aqueous phase, is the two-phase generalisation of the capillary bundle model discussed previously (see Chapter 6) ... 

Based on capillary bundle model for non-Newtonian fluid flow ... 

The average pore radius in the pack is estimated from the capillary bundle model of the porous medium. The equation ... 

The second term in eq. 3 is negligible for short time 1 and, consequently, low liquid uptake p. The solution for this limiting case is the Washburn equation (eq. 6). The physicochemical parameters of the liquid and the porous body may be grouped to a system parameter w (eq. 7). This liquid uptake coefficient w is a measure for the rate of imbibition in the system liquid/porous body. It has been shown that the predictions of eq. 7 regarding the dependance of the liquid uptake coefficient w on the physicochemical parameters v and a are experimentally observed (ref. 6). This may be considered as an indication, that the the capillary bundle model is an appropriate model for a porous body. 

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