Pressure-dependent scaling scenarios in experiments of spontaneous imbibition

The scaling properties of the rough liquid-air interface formed in the spontaneous imbibition of a viscous liquid by a model porous medium are found to be very sensitive to the magnitude of the pressure difference applied at the liquid inlet. Interface fluctuations change from obeying intrinsic anomalous scaling at large negative pressure differences, to being super-rough with the same dynamic exponent $z\simeq 3$ at less negative pressure differences, to finally obeying ordinary Family-Vicsek scaling with $z\simeq 2$ at large positive pressure differences. This rich scenario reflects the relative importance on different length scales of capillary and permeability disorder, and the role of surface tension and viscous pressure in damping interface fluctuations.

Figure 1

Sequence of interfaces for experiments with oil-column heights $H=-5$ (top) and $15\text{mm}$ (bottom). It can be seen that higher pressure differences lead to smoother interfaces. In each experiment, as time goes on, the interface slows down and gets rougher. The two experiments were carried out with the same disorder configuration.

Figure 2

Squared average interface position $⟨h⟩$ as a function of time $t$ for the oil-column heights $H$ used in this paper, showing that the interfaces obey Washburn’s law $⟨h⟩=A{t}^{1∕2}$.

Figure 3

Log-log plot of the interfacial width as a function of time for different window sizes. The straight lines are linear fits in the scaling region. $H=-5\text{mm}$.

Figure 4

Power spectrum of the interfacial fluctuations, for $H=-5\text{mm}$. The power law provides the value of the spectral roughness exponent ${\alpha }_s$. The vertical dashed line indicates the value $q=0.67{\text{mm}}^{-1}$ associated with the lateral size of the unit’s copper obstacle.

Figure 5

Data collapse of the interfacial width of Fig. 3 using $\beta =0.41$ and $z=3$. The experiments correspond to $H=-5\text{mm}$.

Figure 6

Log-log plot of the interfacial width as a function of time for different window sizes. The straight lines are linear fits in the scaling region. $H=15\text{mm}$.

Figure 7

Log-log plot of the power spectrum at different times, for $H=15\text{mm}$. The linear fit in the power law region gives the value of ${\alpha }_s$. The vertical dashed line indicates the value $q=0.67{\text{mm}}^{-1}$ associated with the lateral size of the unit’s copper obstacle.

Figure 8

Data collapse of the interfacial width of Fig. 6 using $\beta =0.36$ and $z=2$. The experiments correspond to $H=15\text{mm}$.

Figure 9

Crossover length and correlation length versus time for a negative applied pressure difference $\left(H=-5\text{mm}\right)$.

Figure 10

Crossover length and correlation length versus time for medium applied pressure differences $\left(H=-1\text{mm}\right)$.

Figure 11

Different length scales present in spontaneous imbibition for high applied pressure difference $\left(H=15\text{mm}\right)$.