Pore-Scale Mixing and the Evolution of Hydrodynamic Dispersion in Porous Media

We study the interplay of pore-scale mixing and network-scale advection through heterogeneous porous media, and its role for the evolution and asymptotic behavior of hydrodynamic dispersion. In a Lagrangian framework, we identify three fundamental mechanisms of pore-scale mixing that determine large scale particle motion, namely, the smoothing of intrapore velocity contrasts, the increase of the tortuosity of particle paths, and the setting of a maximum time for particle transitions. Based on these mechanisms, we derive a theory that predicts anomalous and normal hydrodynamic dispersion in terms of the characteristic pore length, Eulerian velocity distribution, and Péclet number.

Figure 1

Three-dimensional structure of a Berea sandstone sample. The top panel highlights the void space in shades of blue. The lines in the bottom panel show particle paths. The color scheme indicates particle speeds from (white-blue-black-orange) low to high.

Figure 2

(a) PDFs of (circles) Eulerian flow speed $p_e(v)$, (crosses) mean flow speed $p_m(v)$ and (squares) flux weighted mean speed distribution $p_s(v)$ normalized by the Eulerian mean speed $⟨v_e⟩$ for the Berea sample of Fig. 1. The solid line denotes the power-law $v^{\alpha -1}$ with $\alpha =0.35$. (b) First passage time distribution ${\psi }_0(t|v)$ for local Péclet numbers of (crosses) ${\mathrm{Pe}}_v={10}^2$ (circles) 10, (squares) 1 and (diamonds) ${10}^{-1}$. (c) Tortuosity $\chi$ versus Pe for Gamma distributed $p_e(v)$ with exponents (solid line) $\alpha =0.35$, (dotted) 0.5, (dash-dotted) 0.8, and (dashed) 1. The horizontal line indicates the advective tortuosity ${\chi }_a$.

Figure 3

(a) First arrival time PDFs at distance $x\approx 32{\ell }_0$ and (b) ${\sigma }^2(t)$ for (crosses) $Pe={10}^4$, (circles) ${10}^3$, (squares) ${10}^2$, and (rhombs) 10. (c) Streamwise hydrodynamic dispersion coefficients versus Péclet number from (full circles) direct numerical simulations, and (empty circles) the model predictions. The gray symbols show the predictions for Gamma-distributed Eulerian speeds with exponents (top to bottom) $\alpha =0.5$, 0.8, 1, the corresponding lines show the theoretical scalings.