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

## TL;DR

This paper experimentally verifies a previously observed dynamic scaling law for the pair correlation function during slow drainage in porous media, connecting experimental data with theoretical models for different temporal regimes.

## Contribution

It provides the first experimental validation of the dynamic scaling law for the pair correlation function in porous media drainage, linking theory and experiment.

## Key findings

- Confirmed the scaling law $N(r,t) \,\propto\, r^{-1}f(r^{D}/t)$ experimentally.
- Connected theoretical models to explain short-time and long-time behavior of $N(r,t)$.
- Developed a new theoretical argument for the long-time regime exponent.

## Abstract

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 $N(r,t)\propto r^{-1}f\left(r^{D}/t\right)$, 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 ($t \gg r^{D}$).