Verification of a dynamic scaling for the pair correlation function during the slow drainage of a porous medium

We give experimental grounding for the remarkable observation made by Furuberg et al. in Ref. [furuberg1988] of an unusual dynamic scaling for the pair correlation function $N(r,t)$ during the slow drainage of a porous medium. The authors of that paper have used an invasion percolation algorithm to show numerically that the probability of invasion of a pore at a distance $r$ away and after a time $t$ from the invasion of another pore, scales as $N(r,t)\propto r^{-1}f\left(r^D\mathrm{/}t\right)$, where $D$ is the fractal dimension of the invading cluster and the function $f(u)\propto u^{1.4}$, for $u\ll 1$ and $f(u)\propto u^{-0.6}$, for $u\gg 1$. Our experimental setup allows us to have full access to the spatiotemporal evolution of the invasion, which was used to directly verify this scaling. Additionally, we have connected two important theoretical contributions from the literature to explain the functional dependency of $N(r,t)$ and the scaling exponent for the short-time regime ($t\ll r^D$). A new theoretical argument was developed to explain the long-time regime exponent ($t\gg r^D$).