Statistical Scaling of Geometric Characteristics in Millimeter Scale Natural Porous Media
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We analyze statistical scaling of structural attributes of two millimeter scale rock samples, Estaillades limestone and Bentheimer sandstone. The two samples have different connected porosities and pore structures. The pore-space geometry of each sample is reconstructed via X-ray micro-tomography at micrometer resolution. Directional distributions of porosity and specific surface area (SSA), which are key Minkowski functionals (geometric observables) employed to describe the pore-space structure, are calculated from the images, and scaling of associated order-\(q\) sample structure functions of absolute incremental values is analyzed. Increments of porosity and SSA tend to be statistically dependent and persistent (tendency for large and small values to alternate mildly) in space. Structure functions scale as powers \(\xi (q)\) of directional separation distance or lag, \(s\), over an intermediate range of \(s\), displaying breakdown in power law scaling at large and small lags. Powers \(\xi \!\!\left( q \right) \) of porosity and SSA inferred from moment and extended self-similarity (ESS) analyses of limestone and sandstone data tend to be quasi-linear and nonlinear (concave) in \(q\), respectively. We observe an anisotropic behavior for \(\xi (q)\), which appears to be mild for the porosity of the sandstone sample while it is marked for both porosity and SSA of the limestone rock sample. The documented nonlinear scaling behavior is amenable to analysis by viewing the variables as samples from sub-Gaussian random fields subordinated to truncated fractional Brownian motion or fractional Gaussian noise.
KeywordsMicrostructure Scaling Extended self-similarity Structure functions Pore scale characterization
Funding from MIUR (Italian Ministry of Education, Universities and Research-PRIN2010-11; project: “Innovative methods for water resources under hydro-climatic uncertainty scenarios”) is acknowledged. The authors are grateful to M. Siena for providing the algorithm and code for the extraction of porosity and SSA from the digitized binary images. The authors thank S.P. Neuman for stimulating discussions.
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