Anomalous Roughening of Viscous Fluid Fronts in Spontaneous Imbibition

We report experiments on spontaneous imbibition of a viscous fluid by a model porous medium in the absence of gravity. The average position of the interface satisfies Washburn’s law. Scaling of the interface fluctuations suggests a dynamic exponent $z\simeq 3$, indicative of global dynamics driven by capillary forces. The complete set of exponents clearly shows that interfaces are not self-affine, exhibiting distinct local and global scaling, both for time ($\beta =0.64\pm 0.02$, ${\beta }^{*}=0.33\pm 0.03$) and space ($\alpha =1.94\pm 0.20$, ${\alpha }_{\mathrm{loc}}=0.94\pm 0.10$). These values are compatible with an intrinsic anomalous scaling scenario.

Figure 1

(a) Sketch of the experimental setup. (b) Schematic representation of our Hele-Shaw cell with quenched disorder in the bottom plate. (c) Temporal evolution of the oil-air interface at $H=-9\mathrm{mm}$.

Figure 2

Main plot: Squared average interface position as a function of time, for different oil column heights $H$, showing that the imbibition process follows Washburn’s law, $⟨h⟩=A{t}^{1/2}$. Inset: Dependence of Washburn’s law prefactor $A$ (squared) on column height $H$.

Figure 3

(a) Log-log plot of the interfacial width as a function of time, for different window sizes $l$. The straight lines, with slopes 0.64 and 0.33, result from a data fit in the scaling region. (b) Log-log plot of the interfacial width as a function of window size $l$ for different times. The straight line is a fit to the data at the latest time available, with slope 0.94. The inset shows the correlation length $l_c$ as a function of time. The behavior $t^{1/3}$ is represented by the solid line. The experimental parameters are $b=0.46\mathrm{mm}$ and $H=-9\mathrm{mm}$.

Figure 4

(a) Collapse of the interfacial width of Fig. 3, using the values $\beta =0.64$ and $z=3.0$. The straight line is the result from a data fit. (b) Power spectra of the interfaces analyzed in Fig. 3 at different times. The variable $q=k/(2\pi )$. The solid straight lines, with slope $-2.84$, are fits to the data in the scaling regions. The vertical dashed line indicates the value $q=0.67{\mathrm{mm}}^{-1}$ associated with the lateral size of the unit copper obstacle.