Inertial Effects on Flow and Transport in Heterogeneous Porous Media

We investigate the effects of high fluid velocities on flow and tracer transport in heterogeneous porous media. We simulate fluid flow and advective transport through two-dimensional pore-scale matrices with varying structural complexity. As the Reynolds number increases, the flow regime transitions from linear to nonlinear; this behavior is controlled by the medium structure, where higher complexity amplifies inertial effects. The result is, nonintuitively, increased homogenization of the flow field, which leads in the context of conservative chemical transport to less anomalous behavior. We quantify the transport patterns via a continuous time random walk, using the spatial distribution of the kinetic energy within the fluid as a characteristic measure.

Figure 1

Spatial structure (top row) and normalized velocity field (bottom row) for the different systems (from left to right): homogeneous system (HS), randomly heterogeneous system (RHS), and structured heterogeneous system (SHS). HS and SHS are each composed of ($4\times 3$) 12 units, and the RHS contains ($16\times 12$) 192 units. The color bar is in logarithmic scale $\log (\mathbf{U}/\overline{U})$.

Figure 2

Dependence of the normalized participation number (${\pi }^{*}$) on Reynolds number (Re), for HS (blue circles), RHS (gray squares), and SHS (red diamonds), averages of 10 realizations for each system. In each case, $\pi$ was normalized by the initial value (at the smallest Re), so that the first value is equal to 1; thus, while $0\le \pi \le 1$, here ${\pi }^{*}\ge 1$. Because the spatial distribution of the velocity field may vary significantly among realizations, particularly for the SHS, we use ${\pi }^{*}$ to examine the standard deviation of the realizations (error bars). Where not visible, the error bars lie within the symbols.

Figure 3

Left column: Probability density function ($\mathrm{\Omega }$) of the normalized particle velocities $({\mathbf{U}}_p/\overline{\mathbf{U}})$, averaged over 10 realizations, for $\mathrm{Re}=0.4$ (blue) and $\mathrm{Re}=4$ (red); HS (upper), RHS (middle), and SHS (bottom). Right column: The slope ($1+\beta$) of the $\mathrm{\Omega }$ power-law region for each Re (each averaged over 10 realizations). The two colored markers in the plots represent the Re shown in the left column. Inset: Comparison between the velocity field density functions (for a single realization) of the fluid throughout the system (purple) and the particles that are transported within it (green) with $\mathrm{Re}=0.4$.

Figure 4

Normalized cumulative breakthrough curves (${\mathcal{N}}_p$) for two Re, using the streamline-based particle tracking simulations (markers) and the associated CTRW framework results (lines), $\mathrm{Re}=0.4$ (blue), and $\mathrm{Re}=4$ (red).