General Statistics-Based Methodology for the Determination of the Geometrical and Mechanical Representative Elementary Volumes of Fractured Media

Abstract

The upscaling of mechanical properties of fractured media requires the definition of an appropriate size for the Representative Elementary Volume (REV). Because of the stochastic nature of the fracture networks, the REV size is not deterministic and should be defined based on the variability of the equivalent properties. This work presents a new general methodology to define the size of the REV for the geometrical and elastic moduli of fractured media. Following previous works on heterogeneous materials, the decision criterion is based on the precision error that arises from the statistical theory of samples. The proposed methodology also relies on the use of the Central Limit Theorem (CLT) to assess the REV of fractured rocks. The CLT is shown to theoretically apply to both the geometrical and the elastic equivalent properties. From that observation, a general equation is drawn to predict the variance of an equivalent property for any REV candidate size, provided that the variance for one size only is known. These concepts are tested using numerous finite element simulations to obtain the distribution of the equivalent elastic moduli of two-dimensional samples containing two fracture networks previously studied for their elastic properties. These properties are confirmed to tend to a normal distribution, as stated by the CLT. Also, the standard deviations associated with the tested REV sizes were predicted with accuracy from the standard deviation obtained in the numerical simulations of only one proper reference volume. The mechanical REV was compared with the geometrical REV, which is based on the first invariant of the fracture tensor. In addition, to reduce computational costs, a procedure to reduce the number of simulations of the reference volume was proposed. A preliminary verification of the applicability of the methodology to non-elastic problems was made. Proper predictions were obtained for the standard deviation of the compression strength calculated in two studies that considered, altogether, both two-dimensional and three-dimensional samples, as well as plastic and damage models.