Yield stress fluid behavior of foam in porous media

Flow of foams is studied in a model porous medium, in a large range of capillary numbers Ca and relative gas flow rates $f_g$. From pressure measurements, we find that the effective viscosity is a decreasing power-law function of Ca, with the exponent ranging from $-1$ to $-0.75$. Direct observation reveals that the flow is heterogeneous. The fraction of preferential paths increases with both $f_g$ and Ca. In a straight channel of varying cross section, a bubble train behaves as a shear-thinning yield stress fluid. This feature accounts quantitatively for the effective viscosity in the micromodel.

Figure 1

Top: images of the mask used to fabricate the device and a magnified image taken during foam flow. Bottom: Relative foam effective viscosity ${\mu }_{\text{eff}}/\mu$ as a function of Ca for various $f_g$. Lines correspond to the semiempiric predictions (see text): solid ones for the threshold regime, and dashed ones for the Bretherton regime.

Figure 2

Examples of time-averaged image differences highlighting preferential paths. In the inset, the fraction of these paths, $N/N_0$, is plotted as a function of Ca for various relative gas flow rates $f_g$.

Figure 3

Pressure drop induced by a single bubble $\mathrm{\Delta }{P}_b$, normalized by the capillary pressure $〈p_c〉=2\gamma \left(1/h+1/〈w〉\right)$ as a function of Ca, in two geometries: a channel of uniform cross section (red circles) and a channel of sinusoidal width (black crosses). Data correspond to various wavelengths, so that this parameter appears to be irrelevant. Solid lines represent the best fit to the data. For the uniform-cross-section case, we find a Bretherton law $\alpha {\text{Ca}}^{2/3}$, with $\alpha =17$. For the varying-cross-section case, we find $\beta =35$ in the Bretherton regime (for $\text{Ca}>2.3\times {10}^{-4}$) and $\beta =0.13$ in the threshold regime.

Figure 4

Example of normalized instantaneous pressure difference per bubble as a function of time, as determined by the estimation of the menisci curvature on the images. In this experiment, $\text{Ca}={10}^{-5}$. In the six images labeled from A to F, the bubble of interest is highlighted in red. The times corresponding to these six images are displayed on the plot. The dashed line represents the mean value.