Signature of Non-Fickian Solute Transport in Complex Heterogeneous Porous Media

We simulate transport of a solute through three-dimensional images of different rock samples, with resolutions of a few microns, representing geological media of increasing pore-scale complexity: a sandpack, a Berea sandstone, and a Portland limestone. We predict the propagators (concentration as a function of distance) measured on similar cores in nuclear magnetic resonance experiments and the dispersion coefficient as a function of Péclet number and time. The behavior is explained using continuous time random walks with a truncated power-law distribution of travel times: transport is qualitatively different for the complex limestone compared to the sandstone or sandpack, with long tailing, an almost immobile peak concentration, and a very slow approach to asymptotic dispersion.

Figure 1

The probability $\psi ({\tau }_b)$ of traveling between two neighboring voxels for Portland carbonate, for $\mathrm{Pe}=2$, 200, 2000, and $\infty$ (symbols). The probabilities show a power-law trend $\psi \sim {\tau }_b^{-(1+\beta )}$; a line with slope corresponding to $\beta =0.7$ is represented. The dimensionless time is ${\tau }_b=t/t_{b1}$, where $t_{b1}$ is the mean travel time through a grid block $\Delta x/u_{\mathrm{av}}$. Shown in the left inset is the dimensionless velocity field computed in the Portland carbonate image; in the right inset the comparison of $\psi ({\tau }_b)$ for Portland carbonate (black symbols) with $\psi ({\tau }_b)$ of the sandpack (light blue symbols) and Berea sandstone (red symbols) for $\mathrm{Pe}=\infty$ indicates quantitatively different generic behavior characterized by the two parallel solid lines (dark blue and orange color) with slope corresponding to $\beta =1.8$.

Figure 2

Probability of molecular displacement in a micro-CT image of Portland carbonate $P(\zeta )$ as a function of displacement $\zeta$ (solid line), compared with the propagators obtained from NMR experiments from Scheven et al. [3] and Mitchell et al. [4] for the same time $t=1\mathrm{s}$ and flow rate. The coordinates are rescaled by the nominal mean displacement $⟨\zeta {⟩}_0=u_{\mathrm{av}}t$. $u_{\mathrm{av}}=1.3\mathrm{mm}/\mathrm{s}$ and $\mathrm{Pe}=171$ in our simulation.

Figure 3

The preasymptotic behavior of $D_L$ for Portland carbonate computed from the pore-scale model as a function of dimensionless time, $\tau =t/t_1$ where $t_1=L/u_{\mathrm{av}}$, is plotted for different values of Pe (symbols). The solid lines with the slope 0.4 indicate the predicted scaling using Fig. 1 and CTRW: $D_L/D_m\sim \mathrm{Pe}\times {\tau }^{2\beta -1}$, where the slope is $2\beta -1=0.4$ for $\beta =0.7$.

Figure 4

Asymptotic dispersion coefficient $D_L/D_m$ as a function of Pe for the carbonate, sandstone, and sandpack studied (solid lines). The points are experimental data for unconsolidated beadpacks and sandpacks [16, 22]. In the power-law regime for the carbonate we observe the scaling $D_L/D_m\sim {\mathrm{Pe}}^{1.4}$, while for the sandpack and sandstone this scaling is $D_L/D_m\sim {\mathrm{Pe}}^{1.2}$. This corresponds to the values of $\beta =0.7$ for carbonate and $\beta =1.8$ for sandpack and sandstone (as seen in Fig. 1) and agrees with the predictions from CTRW theory.