Boundary Effects on the Tortuosity and Permeability of Idealized Porous Media

Abstract

In this work, the tortuosity and the permeability of a porous medium whose geometry is idealized according to two Euclidean spatial dimensions is studied. Unit porous media solid matrices are taken as iterations toward the 2D Sierpinski carpet fractal geometry. Investigated porous media unit arrangements are either fully periodic (FP), or that of a periodic channel (PC), in which unit porous media geometries are infinitely stacked on both orthogonal directions or only in the flow direction inside a walled channel, respectively. A third case called adjusted fully periodic was also proposed as a counterpoint to the formation of the single preferential flow path of the FP case. The flow regime of interest is the Stokes (or creeping) one, and the numerical approach is done using the lattice Boltzmann method on a porous medium unit domain with a preset pressure drop. Moreover, the scale analysis technique is applied to obtain theoretical correlations for the permeability as a function of porous medium properties for the three cases. Good agreement is found between the correlations obtained herein with results available in the literature. A finding is that whenever the channel walls are removed, the PC case correlation recovers the one for FP. Finally, the results suggest a limit value for the iteration of the carpet beyond which the tortuosity and the permeability of the porous channel and the infinitely periodic (unwalled) porous media are equal.

Appendix 1: Grid Test Methodology

The test seeks to minimize the total error represented by $E_{total}=E_{\mathrm {Ma}}+E_{\varDelta {\varvec{x}}}+E_{\varDelta t}$, where $E_{\mathrm {Ma}}$ represents the error associated with compressibility, $\mathrm {Ma}$ is the Mach number, defined as by $\mathrm {Ma}=|{\varvec{u}}|/c_{s}$, $E_{\varDelta t}$ is the temporal error for a given time step and $E_{\varDelta {\varvec{x}}}$ is the spatial error associated with the mesh. A similar grid test methodology was also employed by Meira et al. (2020) to study the flow of a power-law fluid in a channel partially filled by microscopic porous media.

The error related to compressibility is in the scale $E_{\mathrm {Ma}}\sim {\mathrm {Ma}}^{2}$, for $\mathrm {Ma}<1$. From that, the relationship between the errors is given by:

$$\begin{aligned} E_{\mathrm {Ma}}\propto \frac{E_{\varDelta t}}{E_{\varDelta {\varvec{x}}}}\propto \frac{1}{r^{2}} \end{aligned}$$

(37)

where $E_{\varDelta {\varvec{x}}} \sim \varDelta {\varvec{x}}^{2}$, $E_{\varDelta t} \sim \varDelta t^{2}$. The ratio $r=\varDelta {\varvec{x}}/\varDelta t$ is the lattice spacing divided by the time increment, both defined according to Latt (2008).

To minimize $E_{\mathrm {Ma}}$ and $E_{\varDelta t}$ for an acceptable tolerance, the test consists of keeping the spatial grid size $\varDelta {\varvec{x}}$ fixed and gradually decreasing the time step $\varDelta t$ by increasing the parameter r. This procedure eliminates the temporal errors, leaving the total error $E_{total}$ dominated by $E_{\varDelta {\varvec{x}}}$ only, i.e., $E_{total}\approx E_ {\varDelta {\varvec{x}}}$. In the permeability results of Tables 4 and 5, respectively, for the FP and PC cases, the tolerance was set to $\varDelta t=2\%$.

Table 4 Temporal refinement for the FP case

Table 5 Temporal refinement for the PC case

In the sequence, one refines $\varDelta {\varvec{x}}$ by gradually doubling the number of mesh nodes. Then, as r increases in the same proportion, the quadratic decrease of $\varDelta t$ regarding to $\varDelta {\varvec{x}}$ guarantees that $E_{total}\approx E_{\varDelta {\varvec{x}}}$. Accordingly, for a given tolerance, the successive refinement of $\varDelta {\varvec{x}}$ ensures results independent of $\varDelta {\varvec{x}}$ and $\varDelta t$. For a tolerance of $\varDelta {\varvec{x}}=3\%$ in the variation of the permeability results of the FP and PC arrangements, Table 6 shows the final part of the proposed grid test.

Table 6 Spatial refinement for the FP and PC cases