In order to characterise a rock formation prior to subsurface operations, it is required to find a microscale rock volume for which the homogenised property does not fluctuate when the size of the sample is increased; the Representative Elementary Volume (REV). Its determination usually comes at the cost of a large number of simulations, making it overall a computationally expensive process. Therefore, many scientific studies have been dedicated to optimising the process of finding REV. Using statistical numerical methods, it is shown that the fluctuation of the effective property corresponds overall to a cone-like shape convergence. We suggest determining the generic evolution law of the cone of convergence, which can be used to predict the size of the REV and the effective physical property. This study is based on simulations of Stokes flow through idealised microstructures from which the permeability is upscaled. By tracing and plotting the convergence of permeability for multiple samples, the full cone of convergence appears. The cone shows exponential growth and decay, converging towards the effective permeability of the microstructure. By fitting a log-normal distribution on the collected data points, we show that the generic evolution law of the cone of convergence can always be described with two parameters, independently of the porosity. We show that the determined law of the cone also applies to real microstructures, despite the presence of natural heterogeneities. The new method allows us to reduce the computational costs of finding all characteristics related to REV by simulating several subsamples rather than the full-sized sample, unlocking thereby high-resolution samples which are often too computationally expensive. The use of a statistical model provides quantification of the precision level we can obtain on the REV determination.

The foundation of homogenisation methods rests on the postulate of Hill–Mandel, describing energy consistency throughout the transition of scales. The consideration of this principle is therefore crucial in the discipline of Digital Rock Physics which focuses on the upscaling of rock properties. For this reason, numerous studies have developed numerical schemes for porous media to enforce the Hill–Mandel condition to be respected. The most common method is to impose specific boundary conditions, such as periodic ones. However, these boundary conditions influence both the effective property and the size of the REV. The recent study of Thovert and Mourzenko (2020) has shown that most boundary conditions still result in the same intrinsic effective physical property if the averaging is applied outside the range of the boundary layer. From this discovery, it becomes logical to question the status of Hill–Mandel postulate in porous media when homogenising away from the boundary. In this contribution, we simulate Stokes flow through random packings of spheres and a range of rock microstructures. For each, we plot the evolution of the ratio micro- vs macro-scale of the energy of the fluid transport outside the boundary layer, for a growing subsample size of porous media. Here, we prove that we naturally find energy consistency across scales when reaching the size of the Representative Elementary Volume (REV), which is a known condition for rigorous upscaling. Furthermore, we show that this index for the energy consistency is a more accurate indicator of REV convergence since the mean value is already known to be unitary.