Diffusion-limited-aggregation-like displacement structures in a three-dimensional porous medium

Abstract

We have studied the structure that arises when a low-viscosity fluid displaces a high-viscosity fluid in a transparent three-dimensional porous medium. The viscosity ratio m is not far from 1 (m\ensuremath{\simeq}14) in this system. The fluids we use have equal densities so that there are no gravity effects. The injected fluid forms a ramified structure when an intermediate displacement rate is used (${\mathit{N}}_{\mathrm{Ca}}$\ensuremath{\simeq}${10}^{\mathrm{\ensuremath{-}}4}$). We have measured the projected area A of these growing structures. The cluster mass M is found to scale with a characteristic length ${\mathit{R}}_{\mathit{A}}$ obtained from this area, as M\ensuremath{\sim}${\mathit{R}}_{\mathit{A}\mathit{A}}^{\mathit{D}}$. We find the exponent to be ${\mathit{D}}_{\mathit{A}}$=2.5\ifmmode\pm\else\textpm\fi{}0.1. The physical model was simulated on the computer using the three-dimensional off-lattice diffusion-limited aggregation (DLA) algorithm. Projections of these clusters show features in common with the experimentally generated patterns. Quantitatively we find reasonably good agreement in terms of the exponent ${\mathit{D}}_{\mathit{A}}$, although the experiments are outside the DLA regime of viscous fingering.