Direct relations between morphology and transport in Boolean models

We study the relation of permeability and morphology for porous structures composed of randomly placed overlapping circular or elliptical grains, so-called Boolean models. Microfluidic experiments and lattice Boltzmann simulations allow us to evaluate a power-law relation between the Euler characteristic of the conducting phase and its permeability. Moreover, this relation is so far only directly applicable to structures composed of overlapping grains where the grain density is known a priori. We develop a generalization to arbitrary structures modeled by Boolean models and characterized by Minkowski functionals. This generalization works well for the permeability of the void phase in systems with overlapping grains, but systematic deviations are found if the grain phase is transporting the fluid. In the latter case our analysis reveals a significant dependence on the spatial discretization of the porous structure, in particular the occurrence of single isolated pixels. To link the results to percolation theory we performed Monte Carlo simulations of the Euler characteristic of the open cluster, which reveals different regimes of applicability for our permeability-morphology relations close to and far away from the percolation threshold.

Figure 1

Summary of experimental method: (a) A microfluidic sample with a porous structure in the center of the middle channel is created. A dilute suspension of colloidal particles is injected into the device and an external hydrostatic pressure is applied with the help of two reservoirs. The flow rate through the channel is measured from the velocity of the colloidal tracers. Unpatterned reference channels for calibration are added parallel to the structured channel. To account for possible deviations in the channel height two reference channels are used to cross check the results. (b) Equivalent circuit diagram of hydraulic resistances $R_{\mathrm{hyd}}=\frac{\mathrm{\Delta }P}{Q}=\frac{\eta L}{kA}$. (c) Mean particle velocity $\overline{u}$ measured in the two reference and the middle channel containing the porous structure as a function of the applied pressure. The dashed lines are fits to the data points. The average particle velocity $\overline{u}$ is lower in the structured channel due to the lower permeability of the porous part. The fluid velocity in the structure channel is noticeably slower than in the reference channels. Similar fluid velocities are found in the reference channels, however, differences are caused by different length $(\mathrm{\Delta }L\approx 500\mu \text{m})$ and height $(\mathrm{\Delta }h\approx 0.5\mu \text{m})$ of the reference channels and must be taken into account. The error bars are smaller than the symbol sizes.

Figure 2

Boolean models of overlapping grains for (a) circles, (b) ellipses, and (c) rectangles. A set of $N$ points is selected randomly in a plane (total area $L^2$) (Poisson process). At each point a grain with random orientation is placed. Each structure is characterized by its point density $N/L^2$. The white phase corresponds to the conducting phase. (a)–(c) are structures, which we classify as void percolation, whereas the structures (d)–(f) exhibit grain percolation.

Figure 3

Schematic illustration of the Minkowski functionals: (a) The area $V$ of the conducting phase is shown in white, (b) the perimeter $P$ corresponds to the length of the black boundary, and (c) the Euler characteristic $\chi$ is the number difference of the connected components of each phase, which in the case shown would be $2-5=-3$. The open Euler characteristic ${\chi }_o$ (similar to open porosity ${\varphi }_o$ and open perimeter $S_o$) does not count any inclusions, i.e., ${\chi }_o=1-5=-4$, as the inclusion in cluster No. 5 would be neglected.

Figure 4

Cross-property relation between permeability $k$ determined by LB simulations and conductivity $\sigma$ obtained from FEM simulations. The conductivity is normalized by the bulk conductivity ${\sigma }_0$. The permeability is normalized by the square of the critical pore diameter $l_c$ and a constant $c=1/12$, determined from the limit $l_c\to h$ and $\varphi \to 1$, where the system equals the flow problem between infinitely large parallel plates with spacing $l_c$. Inset: Determination of the critical pore diameter $l_c$ from parallel surfaces.

Figure 5

Experimentally (closed symbols) and numerically (open symbols) determined permeability $k$ of void (top) and grain percolation (bottom) vs. different morphological properties: (a), (b) porosity $\varphi$, as expected $k/c{l}_c^2$ vanishes around ${\varphi }_c$ and goes to 1 for $\varphi \to 1$; (c), (d) rescaled porosity, data points collapse with some deviations due to finite-size effects; (e), (f) open porosity, for void percolation ROMC structures have higher permeabilities, for grain percolation ROME structures have higher permeability at equal ${\varphi }_o$; and (g), (h) Euler characteristic: data collapses onto a single curve for void percolation, but for grain percolation deviations are found. The dashed lines in (g) and (h) are fits to Eq. (17) with one free parameter $\alpha$. Error bars are only shown if larger than symbol size.

Figure 6

(a), (b) Current density magnitude from finite-element simulations of the Laplace equation (i.e., conductivity) normalized to the total maximum current. (c), (d) Fluid velocity magnitude from lattice-Boltzmann simulations. For elliptical grains the overlap leads to pronounced stagnant parts, while circular grains form more compact obstacles, as illustrated in (e), (f) respectively. Within such stagnant parts a strong decrease of both the current and the flow velocity is observed. However the decrease of $j$ appears to be faster, as observed in Ref. [50].

Figure 7

Comparison of velocity magnitude distribution $p\left(v\right)$ (solid) and current magnitude distribution $p\left(j\right)$ (dashed) in ROMC (black) and ROME (red) structures shown in Fig. 6. Both distributions, i.e., $p\left(v\right)$ and $p\left(j\right)$, follow similar trends, however, the velocity decays with a significantly faster amplitude.

Figure 8

Distribution for obstacle sizes $p\left(X\right)$ for (a) void percolation and (b) grain percolation. Open bars correspond to ROME and filled bars correspond to ROMC structures. The histograms for grain percolation show a large number of very small obstacles.

Figure 9

Behavior of the Euler characteristic of the percolating cluster ${\chi }_o$ for (a)–(d) void percolation of different ROMR systems (aspect ratios $1:1,1:2,1:4,1:10$) as a function of the porosity $\varphi$ and (e) site percolation on a 2D lattice function of the occupation probability $p$. For the ROMR structures system sizes of $L=10a,20a,50a$, and $100a$ have been simulated. For site percolation $L=2^2,...,2^{14}$. (f) Finite-size scaling analysis of ${\chi }_o$ for site percolation at the percolation threshold for difference linear system size $L$. (g) Finite-size scaling of ${\chi }_o$ in dependence of the linear system size $L$ for ROMR structures.