Liquid relative permeability through foam-filled porous media: Experiments

For some applications involving liquid foams, such as soil remediation for example, the liquid relative permeability of the foam-filled porous medium is a crucial parameter as it sets the liquid flow rate at which active substances or nutrients (for bacteria) can be delivered deep into the medium. We are interested in the liquid relative permeability of foam-filled porous media, within the range of low liquid saturations, i.e., $\lesssim 20\mathrm{vol}\%$. We fill porous media (packed spherical grains) with different foams made from either alkyl polyglucosides (APG) or saponin, in such a manner that we obtain highly controlled samples in terms of the bubble-to-grain size ratio $r$ and the liquid saturation. The liquid relative permeability of saponin samples exhibits an optimal value as a function of $r$, while it increases significantly for APG foams. The ratio of their relative permeability of APG/saponin reveals two regimes as a function of $r$: for $r\lesssim$ 0.25, the permeability ratio is equal to the ratio corresponding to the bulk foams, while for larger $r$ values, the permeability ratio is increased by one order of magnitude. The foam microstructure changes a lot as the bubble-to-grain size ratio increases up to 0.5, so that a new liquid network is activated, made of surface channels and liquid bridges, the former connecting the latter even at low liquid saturation. These new liquid elements may greatly benefit foams with mobile interfaces such as APGs. One such issue would deserve a dedicated study to be elucidated.

Figure 1

Setup used to prepare liquid foam and fill the porous media. Both bubble size ($D_b$) and liquid volume fraction (${\varepsilon }_f$) of the discharged foam are set during foam preparation—see (b) and (c) respectively. All the details can be found in Appendix pp1.

Figure 2

(a) Reference dimensionless permeability ${\tilde{k}}_{f0}=k_{f0}/D_b^2$, i.e., the bulk (nonconfined) permeability divided by the bubble diameter squared, as a function of the liquid volume fraction for the two foams used in this study, namely saponin foam and APG foam (see Table  2 for details about all the parameters we used). Dotted lines correspond to Eqs. (1a) and (1b). (b) Dimensionless permeability ${\tilde{k}}_f=k_f/D_b^2$ measured as a function of liquid volume fraction (i.e., ${\varepsilon }_f$ or equivalently the liquid saturation $S_w$) for the confined saponin foams, for different bubble and grain sizes, as indicated (in mm). Black symbols correspond to the reference foam. Solid lines correspond to Eq. (2) with coefficient $C\left(r\right)$ presented in Fig. 4. (c) Dimensionless permeability ${\tilde{k}}_f=k_f/D_b^2$ measured as a function of liquid volume fraction (i.e., ${\varepsilon }_f$ or equivalently the liquid saturation $S_w$) for the confined APG foams, for different bubble and grain sizes, as indicated (in mm). Black symbols correspond to the reference foam. Solid lines correspond to Eq. (2) with coefficient $C\left(r\right)$ presented in Fig. 4.

Figure 3

Dimensionless permeability ${\tilde{k}}_f=k_f/D_b^2$ measured as a function of liquid volume fraction (i.e., ${\varepsilon }_f$ or equivalently the liquid saturation $S_w$) for confined APG foams, for different bubble and grain sizes, as indicated (in mm). All the presented size ratios are within the range $0.32\le r\le 0.46$. The thick grey line corresponds to values obtained from Eq. (2) with $C\left(r=0.32\right)=0.029$ and $C\left(r=0.46\right)=0.0175$. As a reasonable collapse is obtained; this suggests that $r$ and liquid fraction can be considered as control parameters.

Figure 4

(a) Coefficient $C\left(r\right)$ introduced in Eq. (2) to describe the decrease in the dimensionless permeability as a function of the bubble-to-grain size ratio, with respect to the reference bulk foam (not confined). Inset: Same data but with log scale for $C\left(r\right)$. Dotted lines correspond to $C\left(r\right)=\exp \left(-r/0.0607\right)$. (b) Functional form $r^{2}C\left(r\right)$ which accounts for the effects of confinement by the grains in the expression of the relative permeability, i.e., permeability of the foam-filled porous sample divided by the permeability of the liquid-saturated porous sample [see Eq. (3)]. The dotted line corresponds to $r^{2}C\left(r\right)=r^2\times \exp \left(-r/0.0607\right)$. (c) Ratio of permeability of APG and saponin foam-filled porous samples as a function of bubble-to-grain size ratio. The point at $r=0$ is related to the corresponding bulk foams.

Figure 5

Picture of a sample as observed through the wall, for high bubble-to-grain size ratio (i.e., $r\approx 0.5$). The liquid content has been made very low (i.e., $2–4\mathrm{vol}\%$) in order to better see the foam microstructure. The liquid bridges form at contact between adjacent grains, and the surface channels (surface plateau borders) connect the liquid bridges. Foam films also connect both surface plateau borders and liquid bridges; that is to say that there are no more bulk pore liquid channels.

Figure 6

(a) The position of the drainage front is detected by light transmission at different positions $z_i$ in the column (${\varepsilon }_{f0}$ is the initial liquid volume fraction, ${\varepsilon }_f$ is the value after the passage of the front). (b) To determine the speed of the front, we plot the normalized gray level $n^{*}$ for the different heights $z_i$ as a function of time. (c) We then plot $z_i-z_1$ as a function of $t_{pi}-t_{p1}$ whose slope directly provides the velocity $v_f$ of the front.

Figure 7

Darcy method. (a) The liquid height is measured at the top of the granular or foam column which allows the liquid flow through the column to be determined, in the situation where the bubbles are not carried away by the flow. (b) Example of measurement preformed with this method: position of the liquid height as a function of time. Bubble size: $490\mu \mathrm{m}$. Grain size: 2 mm. Liquid volume fraction: $7\mathrm{vol}\%$. Permeability value obtained by fitting Eq. (B5): $4.48\times {10}^{-14}{\mathrm{m}}^2$.