NMR flow velocity mapping in random percolation model objects: Evidence for a power-law dependence of the volume-averaged velocity on the probe-volume radius

Abstract

Two- or three-dimensional lacunar objects were fabricated using computer-simulated random site-percolation networks as templates. The flow of water pumped through the pore space was studied with the aid of nuclear-magnetic-resonance- (NMR) microscopy modified for mapping of the velocity vector field. The percolation backbones of the objects were determined by exclusion of all pixels or voxels of the spin-density images with velocities below the noise level. An evaluation procedure was established which reliably renders the fractal dimensions of the whole cluster and of its backbone. The volume-averaged velocity magnitude as a function of the probe-volume radius r was found to obey a power law ${\mathit{v}\mathit{¯}}_{\mathit{V}}$∼${\mathit{r}}^{\mathrm{-}\lambda }$ in the range a<r<${\xi }_{\mathit{v}}$, where a is the voxel edge length and ${\xi }_{\mathit{v}}$ the velocity correlation length. The exponents turned out to be λ=0.32±0.04 and λ=0.82±0.03 for the two- and three-dimensional objects, respectively. In order to test the time dependence of the mean-squared displacement on fractals, 〈${\mathit{r}}^2$〉=α${\mathit{t}}^{2/{\mathit{d}}_{\mathit{w}}}$, expected for random walks of a fractal dimension ${\mathit{d}}_{\mathit{w}}$, self-diffusion of gaseous methane was examined in the pore spaces of the same objects with the aid of field-gradient NMR diffusometry. The results are in accordance with the theoretical predictions for anomalous diffusion on percolation clusters. This finding is supported by studies of incoherent water flow in the percolation network. © 1996 The American Physical Society.