Permeability of solid foam: Effect of pore connections

In this paper, we study how the permeability of solid foam is modified by the presence of membranes that close partially or totally the cell windows connecting neighboring pores. The finite element method (FEM) simulations computing the Stokes problem are performed at both pore and macroscopic scales. For foam with fully interconnected pores, we obtain a robust power-law relationship between permeability and aperture size. This result is due to the local pressure drop mechanism through the aperture as described by Sampson for fluid flow through a circular orifice in a thin plate. Based on this local law, pore-network simulation of simple flow is used and is shown to reproduce FEM results. Then this low computational cost method is used to study in detail the effect of an open window fraction on the percolation properties of the foam pore space. The results clarify the effect of membranes on foam permeability. Finally, Kirkpatrick's model is adapted to provide analytical expressions that allow for our simulation results to be successfully reproduced.

Figure 1

PUC with fully open windows (a), with partially closed windows characterized by the same aperture size (b), identical aperture rate (c), definitions of the aperture size $t_o$, and the full window size $t_w$ (d).

Figure 2

FEM macroscale samples: skeleton mesh (a) and pore-space mesh (b). For sake of visibility, a mesh of size $2\sqrt{2}\times 2\sqrt{2}\times 2$ ($D_b$ units) is depicted.

Figure 3

Network structures used in pore-network simulations.

Figure 4

(a) FEM results at identical aperture size with $\varphi =0.98$, (b) FEM results at identical aperture rate for various $\varphi$, (c) permeability as a function of the mean aperture: FEM results (blue dot for identical aperture, green dot for identical rate, red cross for “without membrane” foam with $\varphi$ varying from 0.8 to 0.99), pore-network model with Sampson local permeabilities and $N_v=14$ (blue line for identical aperture, green line for identical rate). Note that the mean aperture is calculated without including the four square windows, which are parallel to the macroscopic flow direction $〈t_{o,\mathrm{eq}}〉/D_b=(2t_{o,\mathrm{sq}}+8t_{o,\mathrm{hex}})/10D_b.$

Figure 5

Dimensionless permeability $K\left(x_{\mathrm{ow}}\right)/K\left(1\right)$ as a function of open window fraction $x_{\mathrm{ow}}$ for FEM simulations (black square) and pore-network simulations (blue cross and red dot) on samples mixing two local permeabilities with various ratios $k_{hex}/k_{sq}$ and having a Kelvin structure ($N_v=14$). Error bars are calculated using (maximal value–minimal value)/2.

Figure 6

Pore-network simulations: Height of the largest cluster of interconnected pores (black diamond) and fraction of open porosity (green diamond) as a function of the open window fraction $x_{\mathrm{ow}}$ for $N_v=14$ and large samples $N_p\sim {10}^6$. Green line corresponds to fraction of open porosity calculated by using Eq. (6a).

Figure 7

Pore-network simulations: (a) dimensionless permeability $K\left(x_{\mathrm{ow}}\right)/K\left(1\right)$ as a function of open window fraction $x_{\mathrm{ow}}$ for various neighbor pores number $N_v$ (arrows point to the abscissa $x_{\mathrm{ow}}=x_{\mathrm{ow}}^{★}=2/N_v$); (b) same data with another abscissa $(x_{\mathrm{ow}}-x_{\mathrm{ow}}^{★})/(1-x_{\mathrm{ow}}^{★})$.

Figure 8

Fraction of closed porosity, $1-R_{op}$ (a) and excess fraction of open windows within the open pore space (b) as a function of the reduced fraction of open windows for various neighbor numbers $N_v$ and large samples ($N_p\sim {10}^6$). Dashed lines correspond to theoretical curves calculated by considering the first simplest closed clusters [Eqs. (5a) and (5b)]. Full lines correspond to curves calculated by using approximate formulas [Eqs. (6a) and (6b)].

Figure 9

Pore-network simulations: (a) $(x_{\mathrm{ow}}^{\prime }-x_{\mathrm{ow}}^{★})/(1-x_{\mathrm{ow}}^{★})$ as a function of open window fraction $x_{\mathrm{ow}}$ for various neighbor pores number $N_v$, (b) Open porosity fraction (dashed line) and open porosity fraction without dead ends (full line) as a function of the open window fraction $x_{\mathrm{ow}}$ for $N_v=6$ (black) and $N_v=14$ (green).

Figure 10

Comparison between EM model and network simulations (black diamond) for various neighbor pore numbers. “EM0” (dashed blue line) is based on the global open window fraction [Eq. (9)], and “EM1” (red line) is based on the open window fraction within the open pore space [Eq. (10)].

Figure 11

(a) Cross section of foam; (b) geometry of a half pore representative of pores contained inside the foam cross section. Note that we have to consider $n+1$ configurations for the position $p$ of the window associated with the local permeability $k_i$. Figure depicts the case $p=2$.

Figure 12

For each lattice, bonds per unit cross section (gray area) considered for the calculation of ${\sigma }_w$. A couple (${\alpha }_p$, fraction of bonds $p_b$ included within unit cross section) are associated to each bond. ${\sigma }_w$ is calculated by $\sum {\alpha }_p{p}_b{D}_b^2/\text{area}$.

Figure 13

Comparison of EM model predictions (full line) to network simulations (cross) with $N_v=14$.