Dispersion of inert solutes in spatially periodic, two-dimensional model porous media

Abstract

Taylor dispersion of a passive solute within a fluid flowing through a porous medium is characterized by an effective or Darcy scale, transversely isotropic dispersitivity D*, which depends upon the geometrical microstructure, mean fluid velocity, and physicochemical properties of the system. The longitudinal, D~'t and lateral, D* dispersivity components for two-dimensional, spatially periodic arrays of circular cylinders are here calculated by finite element techniques. The effects of bed voidage, packing arrangement, and microscale Prclet and Reynolds numbers upon these dispersivities are systematically investigated.The longitudinal dispersivity component is found to increase with the microscale Prclet number at a rate less than Pe 2. This accords with previous calculations by Eidsath et al. (1983), although the latter calculations were found to yield significantly lower longitudinal dispersivities than those obtained with the present numerical scheme. With increasing Prclet number, a Pe 2 dependence is, however, approached asymptotically, particularly for square cylindrical arrays -owing to the creation of a linear streamline zone between cylinders.Increasing tortuosity of the intercellular flow pattern reduces the longitudinal dispersivity component and enhances the lateral component. Longitudinal dispersivities for square and hexagonal arrays are found to be quite similar at high porosities; yet they diverge dramatically from one another with decreasing porosity. The longitudinal dispersivity is found to increase markedly with increasing Reynolds number. Comparison of this longitudinal dispersivity with available experimental results shows that/)~'1 experimentally measured for three-dimensional arrays of spheres may be correlated by the present two-dimensional model by an appropriate choice of the array's packing arrangement. In general, the calculated dispersivities were found to be sensitive to the bed packing arrangement and apparently no rationale exists for choosing any one particular geometric microstructure over another for a comparison with existing experimental data. It is thus concluded that existing experimental data pertaining to three-dimensional beds of spherical particles cannot rationally provide a basis for verification of two-dimensional, circular cylindrical dispersion models.The finite-element scheme employed in this work was tested in the purely diffusive, nonflow limit by calculating the composite diffusivities of square cylindrical arrays for different volume fractions and various dispersed solid-continuous phase diffusivity ratios, subsequently comparing these with existing analytical results. An additional test was provided by comparing calculated with analytical axial dispersivities for transport of a dissolved solute in a Poiseuille flow between two parallel plates. O. Notationa B(r), B(r) D Dd, Dc f(r) h il, i2 Ii, 12 !1,12 hi, n2 Pe = 2a V/D 2a12 e Pep-D 1-e lr R Rn Re = 2ap V/# Sp t v -v(r)

Computation of the Longitudinal and Transverse Dispersion Coefficient in an Adsorbing Porous Medium Using Homogenization

“…The Peclet number is defined as Pe = v /D 0 , where v is the average interstitial velocity (the Darcy velocity or specific discharge divided by ϕ), the length of the unit cell and D 0 the molecular diffusion coefficient. Figure 4 also compares our results with experimental (Bear 1972) and numerical data cited in Table IV of Edwards et al (1991). The comparisons involve the dispersion in a 2D periodic medium of circles inside squares for a case with porosity ϕ = 0.37.…”

“…3.13. Edwards et al (1991) and many other authors (Didierjean 1997;Eidsath et al 1983;Souto and Moyne 1997) use this equation for periodic media and present 2D computational results also for higher Peclet numbers. The comparison with experimental values uses a standard plot of the longitudinal dispersion coefficient divided by the molecular diffusion coefficient versus the Peclet number (Arya et al 1988).…”

“…A dissimilar definition of the dispersion tensor by the engineering and mathematics community, differing by a factor equal to the porosity, precluded until now a correct comparison between experimental and theoretical values obtained with homogenization (Tardif d'Hamonville et al 2007). This factor is, however, correctly incorporated in the periodic media literature (Edwards et al 1991).…”

The impact of inertial effects on solute dispersion in a channel with periodically varying aperture

“…When a goes to 0, the channel is smooth and goes to the parallel plate geometry. While this is obviously a simplified model for a real porous medium, Edwards et al 11 illustrated that it is likely relevant for representing flow and transport in a cylindrically packed porous medium. It has also sometimes been considered as an idealized model for the geometry of a geological fracture, 46 although realistic fracture geometries are known to be even more complicated.…”

“…This includes, but is not limited to micro fluidic systems, 2, 3 nutrient transport in bloodflow, 4,5 single and multiphase transport in porous media, [6][7][8][9][10][11][12] and transport in groundwater systems. [13][14][15][16] The basic idea behind Taylor dispersion is simple.…”