Prediction of the low-velocity distribution from the pore structure in simple porous media

The macroscopic properties of fluid flow and transport through porous media are a direct consequence of the underlying pore structure. However, precise relations that characterize flow and transport from the statistics of pore-scale disorder have remained elusive. Here we investigate the relationship between pore structure and the resulting fluid flow and asymptotic transport behavior in two-dimensional geometries of nonoverlapping circular posts. We derive an analytical relationship between the pore throat size distribution $f_{\lambda }\sim {\lambda }^{-\beta }$ and the distribution of the low fluid velocities $f_u\sim u^{-\beta /2}$, based on a conceptual model of porelets (the flow established within each pore throat, here a Hagen-Poiseuille flow). Our model allows us to make predictions, within a continuous-time random-walk framework, for the asymptotic statistics of the spreading of fluid particles along their own trajectories. These predictions are confirmed by high-fidelity simulations of Stokes flow and advective transport. The proposed framework can be extended to other configurations which can be represented as a collection of known flow distributions.

Figure 1

(a) Representation of the 2D porous media considered in this study, showing the disordered arrangement of disks (gray circles), and the magnitude of the fluid velocity from a high-fidelity simulation of Stokes flow rescaled by the mean velocity $u/\langle u\rangle$. (b) Zoomed-in view of the red box in (a). (c) Zoomed-in view of the blue box in (b). (d) Schematic of the conceptual model of pipes (cyan squares) associated with pore throats. (e) Same field as in (c), in logarithmic scale. Red color indicates an above-average velocity; blue color indicates a below-average velocity.

Figure 2

(a) Detail of the Delaunay triangulation of the pore geometry. Edges of the triangulation connecting the centers of neighboring disks (red segments) define the pore throats. (b) Proposed conceptual model of porelets, consisting of a collection of pore flows along pore throats (each of given width $\lambda$ and length $c\lambda \sim \lambda$). For this simple geometry each porelet is a Hagen-Poiseuille parabolic profile.

Figure 3

Modulus of the rescaled velocity field $u/\langle u\rangle$ in each of the five geometries considered. On the left the color scale varies linearly, while on the right it changes logarithmically, showing the separation of the flow in channels of high velocity (red areas) and zones of stagnation (blue). The white color is associated with the average velocity.

Figure 4

Zoomed-in views of the magnitude of the velocity field around four typical pore throats with the color map in natural scale (left column) and logarithmic scale (center column) and its interpolation along the segment connecting the grain walls (right column). A parabolic fit (red solid line) is superposed to the interpolation data (blue symbols).

Figure 5

(a) PDF of the rescaled velocity magnitude $u_r=u/\langle u\rangle$ for all five geometries studied, with $\beta =0.25$ (red), $\beta =0.22$ (green), $\beta =0.17$ (blue), $\beta =0.12$ (cyan), and $\beta =0.083$ (black). Symbols correspond to the direct numerical simulations and straight lines are the power-law fits to the symbols. Symbols and curves are shifted vertically for clarity. (b) Fitted exponents plotted against the theoretical exponents $-\beta /2$ [Eq. (9)], where the error bars represent the standard deviation in the least-squares estimate of the time exponent. (c) Symbols represent the same data as in (a), but plotted on semilogarithmic axes to highlight the exponential decay in the distribution of high velocities.

Figure 6

Streamlines for 500 particles initialized at the left end of the channel at four equally spaced time steps. The different colors for different groups of particles help visualize the flow. Shown on the right are the magnifications of small regions (the associated red boxes on the left), illustrating the quality of the streamlines furnished by our spectral-accurate velocity field and high-order time integrator.

Figure 7

Temporal evolution of particle spreading $\sqrt{{\sigma }_s^2}$ as a function of rescaled advective time, for the five geometries considered, with $\beta =0.25$ (red), $\beta =0.22$ (green), $\beta =0.17$ (blue), $\beta =0.12$ (cyan), and $\beta =0.08$ (black). Symbols correspond to the results from direct numerical simulations. Dashed lines are power-law fits to the late-time spreading behavior, which show excellent agreement with the exponents predicted by CTRW theory with the velocity distribution from the proposed porelet model, ${(t/\tau )}^{(1/2+\beta /4)}$ (inset). Solid lines correspond to separate CTRW simulations that account for the correlation structure in the velocity field, which are capable of capturing the transition from the ballistic regime to the superdiffusive regime [19, 41].