Rescaling Relations between Two- and Three-dimensional Local Porosity Distributions for Natural and Artificial Porous Media

TL;DR

This paper investigates how local porosity distributions in 2D and 3D porous media relate through rescaling, showing that with large enough measurement cells, their distributions can be matched by length scaling, especially in homogeneous, isotropic media.

Contribution

The study derives a length rescaling relation linking 2D and 3D local porosity distributions for porous media, applicable at finite scales and supported by theoretical and empirical analysis.

Findings

Good correspondence between 2D and 3D distributions when measurement cells exceed correlation length.

Matching variances of distributions allows for length rescaling between 2D and 3D.

Scaling behavior persists at smaller scales in examined systems.

Abstract

Local porosity distributions for a three-dimensional porous medium and local porosity distributions for a two-dimensional plane-section through the medium are generally different. However, for homogeneous and isotropic media having finite correlation-lengths, a good degree of correspondence between the two sets of local porosity distributions can be obtained by rescaling lengths, and the mapping associating corresponding distributions can be found from two-dimensional observations alone. The agreement between associated distributions is good as long as the linear extent of the measurement cells involved is somewhat larger than the correlation length, and it improves as the linear extent increases. A simple application of the central limit theorem shows that there must be a correspondence in the limit of very large measurement cells, because the distributions from both sets approach normal distributions. A normal distribution has two independent parameters: the mean and the variance. If the sample is large enough, LPDs from both sets will have the same mean. Therefore corresponding distributions are found by matching variances of two- and three-dimensional local porosity distributions. The variance can be independently determined from correlation functions. Equating variances leads to a scaling relation for lengths in this limit. Three particular systems are examined in order to show that this scaling behavior persists at smaller length-scales.