Viscous flow parameter

Viscous flow This mode of transport is due to a total pressure gradient of a continuum fluid mixture (Figure 7.3-2). Hence, there is no separation of species due to the viscous flow. The driving force is the total pressure gradient and the parameter characterizing the transport is the mixture viscosity, p, and the viscous flow parameter, B, which is a function of solid properties only. The flow inside the pore is assumed laminar, hence the velocity profile is parabolic in shape. 

If we define the viscous flow parameter Bq as Bq = F/S (which is the parameter characterising the capillary), the viscous flux equation then becomes ... 

For a porous medium, the viscous flow parameter characterizes the properties of the porous medium, such as radius, its distribution, orientation, and overlapping, etc. Eq. (7.5-4a) when written in terms of pressure becomes ... 

First, we calculate the viscous flow parameter for the capillary... 

We have obtained equations for the viscous flow in a capillary. For a porous medium, the viscous flow parameter Bq can be treated as a fitting structural parameter or it can be calculated from the assumption of a model for the solid structure as it is a function of only solid properties. Like before, if we assume that the solid can be idealised as a bundle of parallel capillaries of equal diameter and length Lc running through the medium in the direction of flow, the average velocity in the capillary is (from eq. 7.5-2) ... 

If we now introduce the specific permeability coefficient Bq (viscous flow parameter) of a porous medium as follows (Carman, 1956) ... 

The proper starting equation to derive the reduced equation for this limiting case is eq. (8.8-19a). The dependence of the Knudsen diffusivity, the binary diffusion coefficient and the viscous flow parameter Bq on the pore size and pressure is ... 

The parameter Bq is the viscous flow parameter and is a function of solid structure only. Since a porous solid usually exhibits a pore size distribution, this parameter must be determined experimentally. 

Figure 12.4-1 shows the effect of the viscous flow parameter aCo on the reduced time lag (scaled against the time lags corresponding to no viscous flow). We see that the viscous flow reduces the time lag as we would expect physically. 

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