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.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20

References

  1. Barton N (2002) Some new Q-value correlations to assist in site characterisation and tunnel design. Int J Rock Mech Min Sci 39:185–216

    Article  Google Scholar 

  2. Caspari E, Milani M, Rubino J, Müller T, Quintal B, Holliger K (2016) Numerical upscaling of frequency-dependent p- and s-wave moduli in fractured porous media. Geophys Prospect 64:1166–1179

    Article  Google Scholar 

  3. Duncan JM, Goodman RE (1968) Finite element analyses of slopes in jointed rock: a report of an investigation. Tech. Rep. S-68-3, U.S. Army Corps of Engineers

  4. Esmaieli K, Hadjigeorgiou J, Grenon M (2010) Estimating geometrical and mechanical REV based on synthetic rock mass models at Brunswick Mine. Int J Rock Mech Min Sci 47:915–926

    Article  Google Scholar 

  5. Farahmand K, Vazaios I, Diederichs MS, Vlachopoulos N (2018) Investigating the scale-dependency of the geometrical and mechanical properties of a moderately jointed rock using a synthetic rock mass (SRM) approach. Comput Geotech 95:162–179

    Article  Google Scholar 

  6. Goodman R, Taylor R, Brekke T (1968) A model for the mechanics of jointed rock. J Soil Mech Found Div 94:105–115

    Google Scholar 

  7. Harthong B, Scholtès L, Donzé FV (2012) Strength characterization of rock masses using a coupled DEM-DFN model. Geophys J Int 191(2):467–480

    Article  Google Scholar 

  8. Hoek E, Brown ET (1997) Practical estimates of rock mass strength. Int J Rock Mech Min Sci 34(8):1165–1186

    Article  Google Scholar 

  9. JianPing Y, WeiZhong C, DianSen Y, JingQiang Y (2015) Numerical determination of strength and deformability of fractured rock mass by FEM modeling. Comput Geotech 64:20–31

    Article  Google Scholar 

  10. Jones TA (1969) Skewness and kurtosis as criteria of normality in observed frequency distributions. J Sediment Res 39(4):1622–1627

    Article  Google Scholar 

  11. Kachanov M (1980) Continuum model of medium with cracks. J Eng Mech Div ASCE 106:1039–1051

    Google Scholar 

  12. Kanit T, Forest S, Galliet I, Mounoury V, Jeulin D (2003) Determination of the size of the representative volume element for random composites: statistical and numerical approach. Int J Solids Struct 40:3647–3679

    Article  Google Scholar 

  13. Kulatilake PHSW, Wang S, Stephansson O (1993) Effect of finite size joints on the deformability of jointed rock in three dimensions. Int J Rock Mech Min Sci Geomech 30:479–501

    Article  Google Scholar 

  14. Kulatilake PHSW, Malama B, Wang J (2001) Physical and particle flow modeling of jointed rock block behavior under uniaxial loading. Int J Rock Mech Min Sci 38:641–657

    Article  Google Scholar 

  15. Liu Y, Wang Q, Chen J, Zhan Song S J, Han X (2018) Determination of geometrical revs based on volumetric fracture intensity and statistical tests. Appl Sci 8(800):1–18

    Google Scholar 

  16. Min K, Jing L (2003) Numerical determination of the equivalent elastic compliance tensor for fractured rock masses using the distinct element method. Int J Rock Mech Min Sci 40:795–816

    Article  Google Scholar 

  17. Nguyen VP (2014) An open source program to generate zero-thickness cohesive interface elements. Adv Eng Softw 74:27–39

    Article  Google Scholar 

  18. Ni P, Wang S, Wang C, Zhang S (2017) Estimationf of the REV size for fractured rock mass based on damage coefficient. Rock Mech Rock Eng 50:555–570

    Article  Google Scholar 

  19. Oda M (1982) Fabric tensor for discontinuous geological materials. Soils Found 22:96–108

    Article  Google Scholar 

  20. Oda M (1988) A new method for evaluating the representative elementary volume based on joint survey of rock masses. Can Geotech J 25:440–447

    Article  Google Scholar 

  21. Oda M, Suzuki K, Maeshibu T (1984) Elastic compliance for rock-like materials with random cracks. Soils Found 24:27–40

    Article  Google Scholar 

  22. Pouya A, Ghoreychi M (2001) Determination of rock mass strength properties by homogenization. Int J Numer Anal Methods Geomech 25(13):1285–1303

    Article  Google Scholar 

  23. Rasmussen LL, de Farias MM, de Assis AP (2018) Extended Rigid Body Spring Network method for the simulation of brittle rocks. Comput Geotech 99:31–41

    Article  Google Scholar 

  24. Schultz R (1996) Relative scale and the strength and deformability of rock masses. J Struct Geol 18(9):1139–1149

    Article  Google Scholar 

  25. Shewchuk JR (1996) Triangle: engineering a 2D quality mesh generator and Delaunay triangulator. In: Applied computational geometry: towards geometric engineering, lecture notes in computer science, vol 1148. Springer, pp 203–222, from the First ACM workshop on applied computational geometry

  26. Wang X, Cai M (2020) A DFN-DEM multi-scale modeling approach for simulating tunnel excavation response in jointed rock masses. Rock Mech Rock Eng 53(3):1053–1077. https://doi.org/10.1007/s00603-019-01957-8

    Article  Google Scholar 

  27. Wang S, Huang R, Gamage Ni R P, Zhang M (2013) Fracture behavior of intact rock using acoustic emission: experimental observation and realistic modeling. Geotech Test J 36(6):903–914

    Google Scholar 

  28. Wu Q, Kulatilake PHSW (2012) REV and its properties on fracture system and mechanical properties, and an orthotropic constitutive model for a jointed rock mass in a dam site in china. Comput Geotech 43:124–142

    Article  Google Scholar 

  29. Yang JP, Chen WZ, Dai YH, Yu HD (2014) Numerical determination of elastic compliance tensor of fractured rock masses by finite element modeling. Int J Rock Mech Min Sci 70:474–482

    Article  Google Scholar 

  30. Zhang L, Einstein H (2000) Estimating the intensity of rock discontinuities. Int J Rock Mech Min Sci 37:819–837

    Article  Google Scholar 

  31. Zhang W, Chen JP, Liu C, Huang R, Li M, Zhang Y (2011) Determination of geometrical and structural representative volume elements at the Baihetan dam site. Rock Mech Rock Eng 45:409–419

    Article  Google Scholar 

Download references

Acknowledgements

This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior – Brasil (CAPES) – Finance Code 001.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Ana Carolina Loyola.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Loyola, A.C., Pereira, JM. & Cordão Neto, M.P. General Statistics-Based Methodology for the Determination of the Geometrical and Mechanical Representative Elementary Volumes of Fractured Media. Rock Mech Rock Eng (2021). https://doi.org/10.1007/s00603-021-02374-6

Download citation

Keywords

  • Representative volume
  • Discrete fracture networks
  • Upscaling
  • Finite element method
  • Statistical analysis