Permeability of Porous Materials Determined from the Euler Characteristic

We study the permeability of quasi-two-dimensional porous structures of randomly placed overlapping monodisperse circular and elliptical grains. Measurements in microfluidic devices and lattice Boltzmann simulations demonstrate that the permeability is determined by the Euler characteristic of the conducting phase. We obtain an expression for the permeability that is independent of the percolation threshold and shows agreement with experimental and simulated data over a wide range of porosities. Our approach suggests that the permeability explicitly depends on the overlapping probability of grains rather than their shape.

Figure 1

(a) Sketch of the experimental setup to measure the permeability through a porous material. (b) Illustration of the morphological quantities to describe a porous sample that consists, in this example, of $N=9$ grains (circles) leading to five connected solid clusters. The porosity $\varphi$ corresponds to the area fraction of the entire liquid phase ($\mathrm{\text{hatched area}}+\mathrm{\text{white inclusion}}$), while the open porosity ${\varphi }_o$ is only defined by the conducting phase (hatched area). The perimeter is the length of the dashed line, and the Euler characteristic is $\chi =-3$ (2 liquid phase components $-5$ solid clusters) and ${\chi }_o=-4$ (the latter neglecting the liquid phase inside the cavity formed in one cluster). Exemplarily, we show one realization for the Boolean process (c) for ROMC $\varphi =0.406$, ${\varphi }_o=0.375$, $N=200$, ${\chi }_o=-18$ and (d) for ROME $\varphi =0.805$, ${\varphi }_o=0.78$, $N=80$, and ${\chi }_o=-17$. The arrows point to the pore throats corresponding to the critical pore diameter $D_c$.

Figure 2

Experimentally (open symbols) and numerically (closed symbols) measured permeability $k$ of ROMC (closed downward triangle, open downward triangle) and ROME (closed upward triangle, open upward triangle) structures as a function of the porosity $\varphi$. All data points are normalized by $cl_c{}^2$. The dashed lines correspond to the interpolation formula given in Ref. 28. Inset: $k$ vs the rescaled porosity, according to Eq. (2). The increasing deviations between experimental and numerical data at small permeabilities are a result of discretization errors and numerical artifacts that increase close to ${\varphi }_c$. Due to finite size effects, some $\varphi$ are shifted to negative values.

Figure 3

Experimentally (open symbols) and numerically (closed symbols) measured permeability of ROMC structures (closed downward triangle, open downward triangle) and ROME structures (closed upward triangle, open upward triangle) vs $(1-{\chi }_o)/N$. The error bars are smaller than the symbol size. The solid line is a fit of Eq. (4) with $\alpha =1.27$. Inset: $k$ in dependence of the open porosity ${\varphi }_o$.

Figure 4

Contour plots of the simulated velocity field for two realizations of each process with equal open porosities (a),(b) ${\varphi }_o=0.4$ and (c),(d) ${\varphi }_o=0.85$. The plots show the local average velocity magnitude. A logarithmic color bar, which is identical for (a),(b) and (c),(d), respectively, is used for visualization. The unit is $\mu \mathrm{m}/\mathrm{s}$. As illustrated in the insets, elliptical grains form larger but less compact interconnected obstacles, which leads to more tortuous streamlines and stagnant zones where the velocity goes to zero.