Verification of a Dynamic Scaling for the Pair Correlation Function during the Slow Drainage of a Porous Medium

In this Letter we give experimental grounding for the remarkable observation made by Furuberg et al. [Phys. Rev. Lett. 61, 2117 (1988)PRLTAO0031-900710.1103/PhysRevLett.61.2117] of an unusual dynamic scaling for the pair correlation function N(r,t) during the slow drainage of a porous medium. Those authors use 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)∝r^{-1}f(r^{D}/t), where D is the fractal dimension of the invading cluster and the function f(u)∝u^{1.4}, for u≪1 and f(u)∝u^{-0.6}, for u≫1. Our experimental setup allows us to have full access to the spatiotemporal evolution of the invasion, which is used to directly verify this scaling. Additionally, we connect 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≪r^{D}). A new theoretical argument is developed to explain the long-time regime exponent (t≫r^{D}).