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

TL;DR: A modified version of the DLA model describes quite well the dynamics of the viscous fingering instabili11 the structure of the displacement fronts is fractal, consistent with results for the diffusionlimited aggregation (DLA) model.

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.