Recirculation zones induce non-Fickian transport in three-dimensional periodic porous media

In this work, the influence of pore space geometry on solute transport in porous media is investigated performing computational fluid dynamics pore-scale simulations of fluid flow and solute transport. The three-dimensional periodic domains are obtained from three different pore structure configurations, namely, face-centered-cubic (fcc), body-centered-cubic (bcc), and sphere-in-cube (sic) arrangements of spherical grains. Although transport simulations are performed with media having the same grain size and the same porosity (in fcc and bcc configurations), the resulting breakthrough curves present noteworthy differences, such as enhanced tailing. The cause of such differences is ascribed to the presence of recirculation zones, even at low Reynolds numbers. Various methods to readily identify recirculation zones and quantify their magnitude using pore-scale data are proposed. The information gained from this analysis is then used to define macroscale models able to provide an appropriate description of the observed anomalous transport. A mass transfer model is applied to estimate relevant macroscale parameters (hydrodynamic dispersion above all) and their spatial variation in the medium; a functional relation describing the spatial variation of such macroscale parameters is then proposed.

Figure 1

Elementary cubic cells for (from top to bottom) fcc, bcc, and sic; the cells are presented with the three main planes of symmetry and the corresponding computational domain. The rightmost image in each line represents part of the structure of the whole 32-module computational domain.

Figure 2

Comparison among the BTCs (observed at the outlet of the domains) in the three proposed configurations (fcc, $$ solid black curve; bcc, dashed blue curve; sic, solid red curve) for the operating condition III (${\mathrm{Pe}}_0\simeq 4550, \mathrm{Re}\simeq 2.05\times {10}^{-2}$). Time is scaled with $\tau$.

Figure 3

dBTC curves for fcc, bcc, and sic configurations (from top to bottom) in every operating condition (I to IV from left to right); in each case, three observation points are chosen, at $6\ell$ (solid black curve), $18\ell$ (solid blue curve), and $30\ell$ (dashed red curve), respectively, where $\ell$ is the length of the periodic module; times are normalized by $\tau$ relative to each curve.

Figure 4

Streaklines representation in fcc, bcc, and sic configurations (from top to bottom); from left to right it is possible to appreciate the behavior of streaklines in the core of the fluid domain and in-between two consecutive grains.

Figure 5

Distribution of the angle $\theta$, defined in Eq. (3), between velocity $\mathbf{u}$ and vorticity $\mathbf{\omega }$ vectors in the three proposed configurations (fcc, black circles; bcc, blue crosses; sic, red triangles), for the operating conditions $\mathrm{III}–\mathrm{IV}(\mathrm{Re}\simeq 2.05\times {10}^{-2}$).

Figure 6

PDFs of velocity magnitude, velocity $x$ and $y$ components (from left to right) for the fcc, bcc, and sic configurations (from top to bottom), operating conditions $\mathrm{III}–\mathrm{IV}(\mathrm{Re}\simeq 2.05\times {10}^{-2}$).

Figure 7

Spatial evolution of the cross-section area in fcc, bcc, and sic geometries (from left to right). Top: frontal view of the solid grains within the cubic cell. Middle: slices exemplifying the averaging process along the $x$ axis for the calculation of the cross-section area. Bottom: evolution of the cross-section area along the main flow direction (the $x$ axis). Porosity is that used in the simulation campaign.

Figure 8

Dispersion and exchange coefficient values obtained from the multiple curve fitting. Arrows indicate increasing ${\mathrm{Pe}}_0$, from operating condition I (black, ${\mathrm{Pe}}_0\simeq 46, \mathrm{Re}\simeq 2.04\times {10}^{-4}$) to IV (green, ${\mathrm{Pe}}_0\simeq 18200, \mathrm{Re}\simeq 2.04\times {10}^{-2}$). The exchange coefficient for fcc cases is not presented since this geometry's data are fitted against the ADE model, where no exchange coefficient is needed.