Porous media relaxation times

Generally, the BTE with single relaxation time approximation can be written as Equation (15). In order to simulate flow in the porous media, we consider the drag effect of the medium and present the LBM equation by the following form of the statistical average ... 

No exact general criterion is available when it is necessary to include the relaxation terms in the equations of change however, relaxation terms are necessary for viscoelastic fluids, dispersed systems, rarefied gases, capillary porous mediums, and helium, in which the frequency of the fast variable transients may be comparable to the reciprocal of the longest relaxation time. 

It appears, then, that the mechanical degradation process is intimately connected with the molecular structure of the macromolecule and the resulting fluid rheology that arises from this structure. For a flexible coil macromolecule, such as HPAM or polyethylene oxide, the polymer solutions are known to display viscoelastic behaviour (see Chapter 3) and thus a liquid relaxation time, may be defined as the time for the fluid to respond to the changing flow field in the porous medium. It may be computed from several possible models (Rouse, 1953 Warner, 1972 Durst et al, 1982 Haas and Durst, 1982 Bird et al. 1987). The finite extendible non-linear elastic (FENE) (Warner, 1972 Bird et al, 1987a Haas and Durst, 1982 Durst et al, 1982) dumbbell model of the polymer molecule may be used to find the relaxation time, tg, as it is known that this model provides a good description of HPAM flow in porous media (Durst et al, 1982 Haas and Durst, 1982) the expression for fe is ... 

In an early paper, Sadowski and Bird (1965) recognised that using a bulk viscosity function for the fluid together with a capillary-hydraulic radius model for the porous media in the manner described above did not take into account any time-dependent elastic phenomena. They suggested that, in a tortuous channel of a porous medium, elastic effects would not be seen provided that the fluid s relaxation time was small compared with the transit time through the contraction/expansion. The fluid would have enough time to readjust to the changing flow conditions. However, if the transit time is small compared with the fluid s relaxation time, then the elasticity of the fluid would have an effect. 

The general concensus Is that the steep Increase In pressure beyond a critical flow rate Is a direct consequence of viscoelasticity In relation to the unsteadiness of the flow field In a porous medium. During flow the molecules can become extended, and this Is particularly the case In converglng/dlverging extenslonal flows, such as the fluid experiences In pore throats In a reservoir rock. If the fluid relaxation time Is small with respect to the time It takes to pass through a contraction or expansion, the fluid will accommodate quickly and no elastic effects are observed. If, on the other hand, relaxation time Is relatively large, elastic effects become dominant, resulting In excessive pressure Increase. 

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