Pore-scale simulation of fluid flow and solute dispersion in three-dimensional porous media

In the present work fluid flow and solute transport through porous media are described by solving the governing equations at the pore scale with finite-volume discretization. Instead of solving the simplified Stokes equation (very often employed in this context) the full Navier-Stokes equation is used here. The realistic three-dimensional porous medium is created in this work by packing together, with standard ballistic physics, irregular and polydisperse objects. Emphasis is placed on numerical issues related to mesh generation and spatial discretization, which play an important role in determining the final accuracy of the finite-volume scheme and are often overlooked. The simulations performed are then analyzed in terms of velocity distributions and dispersion rates in a wider range of operating conditions, when compared with other works carried out by solving the Stokes equation. Results show that dispersion within the analyzed porous medium is adequately described by classical power laws obtained by analytic homogenization. Eventually the validity of Fickian diffusion to treat dispersion in porous media is also assessed.

Figure 1

SEM image of the real sand sample used to extract grain size distribution and approximate the grain shapes.

Figure 2

Grain size distribution of the sand packing under study (filled triangles) and approximation with Weibull distribution (continuous blue line).

Figure 3

Mesh resampling, randomization, and regularization. Original surface mesh (left) and resampled mesh (right).

Figure 4

Details of the mesh refinement process. Stepwise castellation (left), body fitting via vertices moving (center), and precise geometry reconstruction with subsequent mesh refinements (right).

Figure 5

Complete geometry bounding box and location of the two subvolumes chosen for the grid independence analysis.

Figure 6

Visualization of the flow field in the porous medium for $\mathrm{Re}=9.6\times {10}^{-3}$. Velocity magnitude is computed in a central slice, and some flow streamlines are shown. Color coding from blue to red: 0–$2.5\times {10}^{-6}$.

Figure 7

Normalized pressure drops, $\Delta p/\rho V_x^2{\mathcal{L}}_x$, bs normalized superficial velocity, $V_x/\nu$. The slope of the first part of the curve represent the inverse of the permeability of the medium.

Figure 8

Probability density functions (m${}^{-1}$ s) for $\mathrm{Re}=9.6\times {10}^{-3}$ and for, from top to bottom, the $x$-, $y$-, and $z$-velocity components (m s${}^{-1}$) and for, from left to right, the first, fourth, and tenth section. The histogram corresponds to the actual distribution as calculated from the simulations, whereas the red curve is the Gaussian distribution with identical mean and variance.

Figure 9

Probability density functions (m${}^{-1}$ s) for $\mathrm{Re}=66$ and for, from top to bottom, the $x$-, $y$-, and $z$-velocity components (m s${}^{-1}$) and for, from left to right, the first, fourth, and tenth section. The histogram corresponds to the actual distribution as calculated from the simulations, whereas the red curve is the Gaussian distribution with identical mean and variance.

Figure 10

Contour plots of the solute dimensionless concentration in the porous medium at three instants of the simulation for $\mathrm{Pe}=1.9\times {10}^3$.

Figure 11

Comparison between breakthrough curves (normalized dimensionless solute concentration, continuous red line) over dimensionless time $t^{\prime }=t\epsilon /V_x$ obtained by surface averaging over equidistant sections of the porous medium for different Péclet numbers (from top to bottom, $\mathrm{Pe}=1.9\times {10}^{-1}$, $\mathrm{Pe}=1.9\times {10}^1$, $\mathrm{Pe}=1.9\times {10}^3$, $\mathrm{Pe}=1.3\times {10}^5$) with the analytic solutions of the semi-infinite advection-diffusion equation with the fitted parameters (dashed blue line). For the largest $\mathrm{Pe}$ number an insert in log-log scale is included.

Figure 12

Effective transport velocity, as estimated by the method of moments, and its fitting, of the form $a+b/\mathrm{Pe}$, with $a=0.97$ and $b=0.15$.

Figure 13

Dispersion coefficients for different Péclet numbers, as estimated by the method of moments (blue circles) together with two fitted correlation laws; the continuous black line is Eq. (17), whereas the dashed red line is Eq. (18).

Figure 14

Comparison of the dispersion flux $\langle v_x^{\prime }{c}^{\prime }\rangle$ (m s${}^{-1}$) along the flow $x$ coordinate (m) as calculated from the three-dimensional simulations (dotted lines with symbols) and as approximated by Fick's law (computed as the spatial derivative of the analytic solution for semi-infinite media with the fitted parameters, continuous line) for two different Reynolds numbers (top: $\mathrm{Re}=9.6\times {10}^{-3}$; bottom: $\mathrm{Re}=66$) and at two instants (from left to right).