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.

Appendices

Appendix

A Calculation of the equivalent elastic properties

The stress-strain relationship for linear elastic anisotropic media can be expressed as:

$$\begin{aligned} {\varepsilon }_{ij} = S_{ijkl} {\sigma }_{kl} \end{aligned}$$

(16)

We consider here the equivalent compliance tensor of a fractured rock mass where the intact rock has Young modulus $E_\mathrm{r}$ and Poisson ratio $\nu _\mathrm{r}$. In the two-dimensional space, the constitutive tensor $S_{ijkl}$ can be expressed in terms of the equivalent elastic moduli as:

$$\begin{aligned} S_{ijkl}= \,& {} \left[ \begin{array}{cccc} S_{11} &{} S_{12} &{} S_{13} &{} S_{14}\\ S_{21} &{} S_{22} &{} S_{23} &{} S_{24}\\ S_{31} &{} S_{32} &{} S_{33} &{} S_{34}\\ S_{41} &{} S_{42} &{} S_{43} &{} S_{44} \end{array} \right] \nonumber \\=\,& {} \left[ \begin{array}{cccc} \frac{1}{E_x} &{} -\frac{\nu _{yx}}{E_y} &{} -\frac{\nu _{zx}}{E_z} &{} \frac{\eta _{x,xy}}{G_{xy}}\\ -\frac{\nu _{xy}}{E_x} &{} \frac{1}{E_y} &{} -\frac{\nu _{zy}}{E_z} &{} \frac{\eta _{y, xy}}{G_{xy}}\\ -\frac{\nu _{xz}}{E_x} &{} -\frac{\nu _{yz}}{E_y} &{} \frac{1}{E_z} &{} \frac{\eta _{z,xy}}{G_{xy}}\\ \frac{\eta _{xy,x}}{E_{x}} &{} \frac{\eta _{xy,y}}{E_y} &{} \frac{\eta _{xy,z}}{E_z} &{} \frac{1}{G_{xy}} \end{array} \right] \end{aligned}$$

(17)

where $E_i$ are the elastic moduli, $\nu _{ij}$ are Poisson ratios, $\eta _{i,jk}$ are coefficients of mutual inflience of the first kind and $\eta _{ij,k}$ are coefficients of mutual influence of the second kind. Considering that the fractures have strikes in the direction z, they do not affect the deformations in this direction; thus, $E_z$ = $E_r$, $\nu _{xz} = \nu _{yz} = \nu _{r}$, and the components $S_{31}$, $S_{32}$ and $S_{33}$ are then equal to those of the compliance tensor of the intact rock. Also, since the shear stress $\sigma _{xy}$ does not affect deformations in z, $S_{34}$ is equal to zero. Considering the symmetry conditions, $S_{13} = S_{31}$, $S_{23} = S_{32}$ and $S_{34} = S_{43}$. Hence, there are 7 components of the tensor which are known a priori because of the assumption of bidimensionality.

For plane-strain conditions, the relationship in (16) reduces to:

$$\begin{aligned} \left[ \begin{array}{cc} \varepsilon _{x} \\ \varepsilon _{y} \\ 0 \\ \gamma _{xy} \end{array} \right] = \left[ \begin{array}{cccc} S_{11} &{} S_{12} &{} {S_{13}}^r &{} S_{14}\\ S_{21} &{} S_{22} &{} {S_{23}}^r &{} S_{24}\\ {S_{31}}^r &{} {S_{32}}^r &{} {S_{33}}^r &{} 0\\ S_{41} &{} S_{42} &{} 0 &{} S_{44} \end{array} \right] \ \left[ \begin{array}{cc} \sigma _{x} \\ \sigma _{y} \\ \sigma _{z} \\ \tau _{xy} \end{array} \right] \end{aligned}$$

(18)

Three linearly-independent boundary conditions are necessary to obtain the unknowns of the elastic compliance tensor. In this paper, we used the applied stresses illustrated in Fig. 7. The resulting displacements $u_i$ ($i= x, y$) at the boundaries were used to calculate the homogenized strains as:

$$\begin{aligned} \varepsilon _{ij} = \frac{u_{i,j} + u_{j,i}}{2} \end{aligned}$$

(19)

The stress $\sigma _z$ can be calculated from the applied stresses and the properties of the intact rock as:

$$\begin{aligned} \sigma _z = -\frac{{S_{31}}^r \sigma _x + {S_{32}}^r \sigma _y}{{S_{33}}^r} \end{aligned}$$

(20)

And the tensor components are calculated using (20) and the system formed by lines 1, 2 and 4 in (18)