Porous media interstitial velocity

Diffusion, convection, and dispersion all contribute to the spread of a front. Let us see how much each mechanism contributes to the spread. First, let us see when the diffusion transport is important as compared to the convective transport. We use v2Dot to calculate the spreading distance from a point source 68% of the injected source is within this distance. Table 2.2 shows the results for different time periods compared with the traveled distances during the same time periods by a convective flow of 1 m/day. A typical flow rate in petroleum reservoirs is 1 m/day (interstitial velocity). A typical value of diffusion coefficient of 4 X 10 mVs in a porous medium is used. In the first 5 seconds, the diffusive transport is more important than the convective transport. Soon after, the convective flow becomes the dominant mechanism. 

The characteristic interstitial velocity ue can be related to the superficial velocity by equating the time required for a fluid element to travel with the superficial velocity q for a fixed apparent length L within the porous medium and the time for a fluid element to travel with the characteristic interstitial velocity ue in the pore for a fixed passage length of L/t. Hence, we obtain... 

Withdrawal speed of plate in dip coating Mean interstitial or effective pore velocity in porous medium,... 

To the extent that dispersion in an inertia free porous medium flow arises from a nonuniform velocity distribution, its physical basis is the same as that of Taylor dispersion within a capillary. Data on solute dispersions in such flows show the long-time behavior to be Gaussian, as in capillaries. The Taylor dispersion equation for circular capillaries (Eq. 4.6.30) has therefore been applied empirically as a model equation to characterize the dispersion process in chromatographic separations in packed beds and porous media, with the mean velocity identified with the interstitial velocity. In so doing it is implicitly assumed that the mean interstitial velocity and flow pattern is independent of the flow rate, a condition that would, for example, not prevail when inertial effects become important. 

If Eq. (6.6.1) is applied to a porous medium by a simple capillary model, then the superficial velocity is given by U = eu, where e is the porosity and is the convection velocity in Eq. (6.6.1). Locally, the interstitial convection velocity is made up of hydraulic and electroosmotic contributions and is given by... 

When bulk fluid flow is present (v 0), concentration profiles can be predicted from Equation 10-17, subject to the same boundary and initial conditions. This set of equations has been used to describe concentration profiles during micro-infusion of drugs into the brain [33]. In addition to Equation 10-17, conservation equations for water are needed to determine the variation of fluid velocity in the radial direction. Relative concentrations are predicted by assuming that the brain behaves as a porous medium (i.e., velocity is related to pressure gradient by Darcy s law, see Equation 6-9). Water introduced into the brain can expand the interstitial space this effect is balanced by the flow of water in the radial direction away from the infusion source and, to a lesser extent, by the movement of water across the capillary wall. 

If the interstitial velocity in the porous medium is denoted as v, the residence time of the fluid in the porous medium is (L/v), which must be the same as the residence time that the fluid spent in the capillary, that is ... 

Eq. (7.5-8b) suggests that the interstitial velocity in the porous medium, v, is less than the velocity in the capillary as the medium length L is smaller than the capillary length. 

Superficial velocity of fluid flowing in a porous medium is less than the interstitial velocity according to the Dupuit relation ... 

When the porous medium saturated with the electrolyte is embodied in an external electric field E, there appears a nonzero volumetric body force within the Debye layer, which sets the ions in that region into motion. Far from the particle surfaces, this volumetric force is zero, since the solute there is neutral. However, the electrolyte is brought into motion also in the latter region as a result of the solute s viscosity. These processes lead to the appearance of an interstitial flow velocity field u(R). This velocity field, when integrated over a representative volume of the porous medium, yields a nonzero seepage velocity U in the absence of any macroscopic pressure gradient applied to the porous medium. This process is called electro-osmosis, and the velocity U is called the electro-osmotic velocity. 

Northrup, M.A., et al.. Direct measurement of interstitial velocity field variations in a porous medium using fluorescent-particle image velocimetry, Chem. Eng. Sci.. 48( I), 13-22 (1993). 

The above discussion shows how the presence of inaccessible pore volume causes salt peaks to move through a porous medium more slowly than polymer peaks. Only one final point remains to be made — the connected pore volume of a core, measured by saturation starting from an evacuated condition, is not just the polymer pore volume it is the total pore volume occupied by water, including that fraction inaccessible to polymer. With this as the pore volume, then the salt front velocity is the true interstitial velocity and the polymer moves faster its velocity is greater because it does not enter the inaccessible pore volume. 

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