In search for representative elementary volume (REV) within heterogeneous materials: A survey of scalar and vector metrics using porous media as an example

The Representative Elementary Volume (REV) concept, a cornerstone in porous system heterogeneity assessment, was initially conceived to determine the minimal domain volume suitable for homogenization and upscaling. However, the definition of REV and usability in continuum-scale models is vague. In this study, we conduct comprehensive REV analyses on multiple samples, encompassing a range of scalar and vector metrics. Our investigation probes the representativity of crucial medium characteristics, including porosity, permeability, and Euler density, alongside descriptors rooted in pore-network statistics, correlation functions, and persistence diagrams. We explore both deterministic and statistical REV sizes (dREV and sREV), facilitating a robust comparative assessment. Crucially, we introduce an novel methodology tailored for harnessing vector metrics, known for their ability to reveal intricate structural insights. Our results underscore the superiority of the sREV approach, particularly for low-content metrics, addressing inherent limitations of dREV in characterizing homogeneities in such cases. Furthermore, the sREV approach incorporates stationarity analysis into REV evaluation, ensuring result consistency between sREV and dREV under stationarity conditions. Encouragingly, our findings suggest that high-information-content metrics, notably correlation functions combined with persistence diagrams, have the potential to establish a universal REV for steady-state physical properties. This proposition warrants further verification through a comprehensive assessment and comparison of REV values across major physical properties. REV analysis plays a pivotal role not only in assessing medium properties but also in scrutinizing different descriptors of 3D images – we note that REV analysis and image/field stationarity analysis are ultimately the same techniques under the hood. The discussion based on obtained results and recent finding by other researchers advances the understanding of REV within porous media, introduces a versatile methodology with broader applications, and is expected to be useful in numerous fields including materials science, cosmology, machine learning, and more. We redefine the classical definition of REV by adding stationarity condition and upper/lower bounds on its volume. While for simplicity, in this work we shall mainly focus on porous media as immediately applicable to digital rock, petrophysics, hydrology and soil physics problems, the developed mythology can be applied to other material types - composites, biological tissues, granular matter, food engineering and numerous other types of matter.