Fluctuations in Fluid Invasion into Disordered Media

Interfaces moving in a disordered medium exhibit stochastic velocity fluctuations obeying universal scaling relations related to the presence or absence of conservation laws. For fluid invasion of porous media, we show that the fluctuations of the velocity are governed by a geometry-dependent length scale arising from fluid conservation. This result is compared to the statistics resulting from a nonequilibrium (depinning) transition between a moving interface and a stationary, pinned one.

Figure 1

Left: Interfaces $h(x,t)$ in a system of width $L=500$ and average separation $\overline{h}=320$ from the reservoir. The configurations are separated by time intervals $\Delta t=500$ at 240 different times. Interface propagation is made visible by adding the constant shift velocity $\overline{v}$. The sequence of 20 black interfaces shows pinned and moving regions. Right: Time series of $v(t)$, the velocity for $L=\overline{h}=40,80,160,\dots ,1280$ from top to bottom, logarithmic scale in $v(t)$.

Figure 2

Activity pattern in space (vertical) and time (horizontal). $L=256$, $\overline{h}=160$, lateral periodic boundary conditions. High velocities are dark, low velocities are light, and gray scale is nonlinear for better visibility. At any given time, activity is restricted to $n=L/\overline{h}\approx 2$ narrow avalanches of width ${\xi }_c$.

Figure 3

Left: Rescaled distributions $\mathcal{P}(\frac{v(t)-\overline{v}}{\overline{v}})$ for various values of $1/\overline{v}\propto \overline{h}=L=40,\dots ,1280$. Right: Power spectrum $S(\omega )=⟨|\widehat{v}(\omega ){|}^2⟩\propto {\omega }^{-4/3}$ for $\overline{h}=L=1280$. The same exponent relates average avalanche size $s$ and duration $\tau$.

Figure 4

Left: Fluctuations $\Delta v$ vs mean $\overline{v}$ for constant width $L\equiv 500$ (open circles) and $L=\overline{h}$ (solid circles) follows $\Delta v\sim {\overline{v}}^a$ with $a=1/2$ and 1, respectively; cf. Eq. (8). Lines with slope $1/2$ and 1, respectively, are to guide the eye. Right: The same for the local model of Ref. 21, with system size $L={10}^4$. The line has slope $1/3$.