Anomalous diffusion in two‐dimensional Euclidean and prefractal geometrical models of heterogeneous porous media

Kim, Jung-Woo; Perfect, Edmund; Choi, Heechul

doi:10.1029/2006wr004951

Cited by 18 publications

(15 citation statements)

References 55 publications

(61 reference statements)

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“…Their relatively simple geometry (e. g. if compared with stochastically generated fractals) facilitates quantitative or qualitative connections between structural and transport properties. Random walks were already intensively studied in fractal lattices with the geometry of SCs, with finite or infinite ramification, and in randomized versions of the SCs [7][8][9][10][11][12][13][14][15][16][17][18][19].…”

Section: Introductionmentioning

confidence: 99%

Scaling relations in the diffusive infiltration in fractals

Reis

2016

Phys. Rev. E

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In a recent work on fluid infiltration in a Hele-Shaw cell with the pore-block geometry of Sierpinski carpets (SCs), the area filled by the invading fluid was shown to scale as F ∼ t n , with n < 1/2, thus providing a macroscopic realization of anomalous diffusion [Filipovitch et al, Water Resour. Res. 52 5167 (2016)]. The results agree with simulations of a diffusion equation with constant pressure at one of the borders of those fractals, but the exponent n is very different from the anomalous exponent ν = 1/D W of single particle diffusion in the same fractals (D W is the random walk dimension). Here we use a scaling approach to show that those exponents are related as n = ν (D F − D B ), where D F and D B are the fractal dimensions of the bulk and of the border from which diffusing particles come, respectively. This relation is supported by accurate numerical estimates in two SCs and in two generalized Menger sponges (MSs), in which we performed simulations of single particle random walks (RWs) with a rigid impermeable border and of a diffusive infiltration model in which that border is permanently filled with diffusing particles. This study includes one MS whose external border is also fractal. The exponent relation is also consistent with the recent simulational and experimental results on fluid infiltration in SCs, and explains the approximate quadratic dependence of n on D F in these fractals. We also show that the mean-square displacement of single particle RWs has log-periodic oscillations, whose periods are similar for fractals with the same scaling factor in the generator (even with different embedding dimensions), which is consistent with the discrete scale invariance scenario. The roughness of a diffusion front defined in the infiltration problem also shows this type of oscillation, which is enhanced in fractals with narrow channels between large lacunas.

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Section: Introductionmentioning

confidence: 99%

Scaling relations in the diffusive infiltration in fractals

Reis

2016

Phys. Rev. E

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“…1). Kim et al (2007) pointed out that the presence of large solid squares in 2D PSF porous media induced greater pore cluster clumping compared to 2D MF porous media. Based on the 100 random realizations, the average pore cluster size distributions of the 3D MF and PSF structures for different φ values are plotted in Fig.…”

Section: Geometric Characteristics Of Prefractal Porous Mediamentioning

confidence: 98%

“…The porosity, φ, of the 3D prefractal porous media can be computed using the fractal parameters, b, n, N ex , and N gen (Kim et al 2007):…”

Section: Geometric Characteristicsmentioning

confidence: 99%

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Geometric and Hydrodynamic Characteristics of Three-dimensional Saturated Prefractal Porous Media Determined with Lattice Boltzmann Modeling

Kim

Sukop

Perfect³

et al. 2011

Transp Porous Med

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Fractal and prefractal geometric models have substantial potential for contributing to the analysis of flow and transport in porous media such as soils and reservoir rocks. In this study, geometric and hydrodynamic parameters of saturated 3D mass and pore-solid prefractal porous media were characterized using the lattice Boltzmann model (LBM). The percolation thresholds of the 3D prefractal porous media were inversely correlated with the fraction of micro-pore clusters and estimated as 0.36 and 0.30 for mass and pore-solid prefractal porous media, respectively. The intrinsic permeability and the dispersivity of the 3D pore-solid prefractals were larger than those of the 3D mass prefractals, presumably because of the occurrence of larger solid and pore cluster sizes in the former. The intrinsic permeability and dispersivity of both types of structure increased with increasing porosity, indicating a positive relationship between permeability and dispersivity, which is at odds with laboratory data and current theory. This discrepancy may be related to limitations of the convection dispersion equation at the relatively high porosity values employed in the present study.

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“…To this end, the term anomalous diffusion (also referred to as restricted diffusion) is introduced (Sen 2003;Kim et al 2007;Fomin et al 2010Fomin et al , 2012. Anomalous diffusion is usually further divided into subdiffusive (slower than Fickian diffusion, σ < 1) and superdiffusive (faster than Fickian diffusion, σ > 1) regimes.…”

Section: Introductionmentioning

confidence: 99%

Dissolution Dynamics of Liquid/Liquid Binary Mixtures Within a Micromodel

Stevar

Vorobev

2013

Transp Porous Med

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We report optical observations of the dissolution behaviour of glycerol/water, soybean oil/hexane, and isobutyric acid (IBA)/water binary mixtures within a glass micromodel built as a 2D regular network of capillary tubes with diameter of 0.2 mm. The micromodel is initially filled with solute and then is horizontally immersed into a thermostatic solvent-filled bath. The micromodel is open at its corners for solute dissolution to occur with no pressure gradients being applied. Our study shows that the solvent penetration into the micromodel is diffusion-dominated in completely miscible binary mixtures (glycerol/water and soybean oil/hexane). This is, however, non-Fickian diffusion with the dissolution rate, dV /dt, being proportional to D 1/3 t −0.4 for almost the entire duration of the experiment (V is the volume occupied by the solvent, D is the diffusion coefficient, and t is time). For the partially miscible IBA/water mixture the experiments performed at undercritical temperatures revealed that the diffusive transport was negligible despite the mixture being out of its thermodynamic equilibrium. The water phase penetrated into some of the channels, but IBA was never completely displaced/dissolved from the micromodel and numerous interfaces remained visible after very long-time periods.

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Anomalous diffusion in two‐dimensional Euclidean and prefractal geometrical models of heterogeneous porous media

Cited by 18 publications

References 55 publications

Scaling relations in the diffusive infiltration in fractals

Scaling relations in the diffusive infiltration in fractals

Geometric and Hydrodynamic Characteristics of Three-dimensional Saturated Prefractal Porous Media Determined with Lattice Boltzmann Modeling

Dissolution Dynamics of Liquid/Liquid Binary Mixtures Within a Micromodel

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