Kozeny-Carman formula

The geometrical parameters of the porous medium are the grain size d and the number of grains per unit volume p. The geometrical parameters of the network of capillary tubes are the diameter D of the tubes and the number of tubes passing through a unit surface in the Oxy plane. Tortuosity i is a common parameter of the porous medium and the capillary-tube network model. The Kozeny-Carman formula expresses, based on [14.27], the dependence of k on d, t, and e, establishing a link between and D, on the one hand, and between p and d, on the other. These relations are obtained under the hypothesis that the porous medium and the network of capillary tubes have the same porosity e and the same specific area Os 

For a granular medium consisting of spherical balls, the porosity and specific areas are given by  

For the capillary-tube network medium, the porosity and specific area are  

The latter relation is more often used in the form  

T able 14.1. Permeability values calculated using the Kozeny-Carman formula (equation [14.35]) for a porous medium consisting of spherical balls of diameter d. 

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All terms of 0(1), 0(e), and O(e ) cancel identically to yield the above relation. The Kozeny-Carman formula, Eq. (4.7.16), can be similarly expanded to give... 

This formula for a low-permeable porous medium is known as the Kozeny-Carman formula. The Kozeny constant K is approximately equal to 5. In particular, if the porous medium consists of identical spherical particles of diameter d,... 

This is known as the Kozeny-Carman formula, and the Kozeny constant k has been experimentally estimated as k 5. The Kozeny-Carman formula can be simplified to... 

The best way to use the Kozeny-Carman model and other permeability models (e.g. the anisotropic model by Gebart) [18], is to use them as interpolation formulas for intermediate volume fractions between known values. Extrapolation should be done with extreme caution because the models are developed for idealized reinforcements. Typical values for the permeability of different types of reinforcement are given in Table 12.1. 

Here p is the density of the infiltrant,, > is the acceleration due to gravity (9.8 m/s2) and Patm is the pressure of the surrounding space above the liquid infiltrant. This model is useful for predicting the influence of pressure on the rate of infiltration. Another formula is the Kozeny-Carman equation [Carman, 1956] ... 

A porous medium is modeled as made up of uniformly distributed straight circular capillaries of the same diameter. The flow through each capillary is an inertia free Poiseuille flow. By comparing the Poiseuille pressure drop and the Darcy pressure drop formulas, deduce an expression for the permeability. Discuss the difference between the result obtained and the Kozeny-Carman permeability. 

The Navier-Stokes (NS) equations can be used to describe problems of fluid flow. Since these equations are scale-independent, flow in the microscale structure of a porous medium can also be described by a NS field. If the velocity on a solid surface is assumed to be null, the velocity field of a porous medium problem with a small pore size rapidly decreases (see Sect. 5.3.2). We describe this flow field by omitting the convective term v Vv, which gives rise to the classical Stokes equation We recall that Darcy s theory is usually applied to describe seepage in a porous medium, where the scale of the solid skeleton does not enter the formulation as an explicit parameter. The scale effect of a solid phase is implicitly included in the permeability coefficient, which is specified through experiments. It should be noted that Kozeny-Carman s formula (5.88) involves a parameter of the solid particle however, it is not applicable to a geometrical structure at the local pore scale. 

The most important parameters that control the permeability are the grain size distribution and the density of the fill. A first estimate of the permeability can be obtained from the particle size distribution by empirical formulae proposed by Hazen (1911), Kozeny-Carman (Carman, 1956) and others. In particular the fines content has a strong influence on the permeability. 

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