Flow, Diffusion, Dispersion, and Thermal Convection in Percolation Clusters: NMR Experiments and Numerical FEM/FVM Simul
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Flow, Diffusion, Dispersion, and Thermal Convection in Percolation Clusters: NMR Experiments and Numerical FEM/FVM Simulations Rainer Kimmich, Andreas Klemm, Markus Weber, and Joseph D. Seymour1 Sektion Kernresonanzspektroskopie, Universität Ulm, 89069 Ulm, Germany ABSTRACT Based on computer-generated templates, percolation objects were fabricated. Random-site, semi-continuous swiss cheese, and semi-continuous inverse swiss-cheese percolation models above the percolation threshold were considered. The water-filled pore space was investigated by nuclear magnetic resonance (NMR) imaging and in the presence of a pressure gradient, by NMR velocity mapping. The percolation backbones were determined using velocity maps. The fractal dimension of the backbones turned out to be smaller by about 17 % than that of the complete cluster. As a further relation of interest, the volume-averaged velocity was calculated as a function of the probe volume radius. In a certain scaling window, the resulting dependence can be represented by a power law. The experimental results favorably compare to computer simulations with the finite-element method (FEM) or the finite-volume method (FVM). Thermal convection in percolation clusters of different porosities was studied using the NMR velocity mapping technique. The velocity distribution is related to the convection roll size distribution. The maximum velocity as a function of the porosity clearly visualizes a closed-loop percolation transition if the Rayleigh number conditions are appropriate. Percolation theory suggests a relationship between the anomalous diffusion exponent and the fractal dimension of the cluster, i.e. between a dynamic and a structural parameter. Interdiffusion between two compartments initially filled with H2O and D2O, respectively, was examined by proton imaging. The results confirm the theoretical expectation. Finally, advection driven dispersive transport was investigated in the large Péclet number limit. The superdiffusive transport anomaly was demonstrated and discussed in terms of the non-local advection-diffusion and the fractional diffusion theories. INTRODUCTION Transport of gases and liquids through porous media is largely determined by the geometry of the pore space apart from the influence of external fields and the characteristics of the fluid and its adsorption properties at pore walls. In this study, we examine the interrelation of transport and pore space microstructure of percolation clusters as the simplest class of pore space models. The term “transport” refers here to flow, diffusion, dispersion, and thermal convection. Model objects fabricated on the basis of numerically simulated templates have been studied in nuclear magnetic resonance (NMR) experiments. Three different, well-defined percolation models [1-3] are considered: a) Random site percolation: Sites on a square or cubic lattice with the lattice constant a are occupied with a probability (or porosity) p in the vicinity of the percolation threshold given by the critical value p=pc. Neighbour
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