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Size Effect Analysis for the Characterization of Marcellus Shale Quasi-brittle Fracture Properties

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Abstract

The fracture characterization of shale rocks requires understanding the scaling of the measured properties to enable the extrapolation from small-scale laboratory tests to field applications. In this study, the fracture properties of Marcellus shale were obtained through size effect tests. Fracture tests were conducted on three-point-bending specimens with increasing size. The test results show that the nominal strength decreases with increasing specimen size and it can be fitted well by Bažant’s size effect law. This demonstrates that shale fracture behavior deviates from classical linear elastic fracture mechanics (LEFM), and it has quasi-brittle characteristics. This implies, in turn, that the fracture toughness (or fracture energy) computed according to LEFM is size-dependent and, in general, cannot be considered a material property. Furthermore, the size effect analysis allows one to accurately identify the quasi-brittle fracture properties, namely the initial fracture energy and the effective fracture process zone length. A significant anisotropy was observed in the fracture properties determined with three principal notch orientations.

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Abbreviations

G :

Energy release rate

\(G_{\text {Ic}}\) :

Effective LEFM fracture energy

\(G_{\text {f}}\) :

Initial fracture energy

\(G_{\text {F}}\) :

Total fracture energy

\(K_{\text {IcA}}\) :

Apparent fracture toughness calculated using LEFM

\(K_{\text {I}}\) :

Mode I stress intensity factor

\(K_{\text {Ic}}\) :

Fracture toughness

\(K_{\text {IcA}}\) :

Apparent fracture toughness calculated using LEFM

\(\ell _{\text {FPZ}}\) :

Length of FPZ

\(c_{\text {f}}\) :

Effective FPZ length

\(\sigma (\delta )\) :

Cohesive stress as a function of crack opening \(\delta\)

\(f'_{\text {t}}\) :

Tensile strength of cohesive law

\(\beta\) :

Brittleness number

\(D_0\) :

Transitional size

\({\hat{D}}\) :

Normalized size

\(l_1\) :

Hillerborg’s characteristic length

L, D, t :

Specimen length, depth, and thickness

S :

Support span

a :

Crack length

\(a_0\) :

Notch length (equal to initial crack length)

\(\alpha\) :

Dimensionless crack length

\(\alpha _0\) :

Dimensionless notch length (equal to the initial value of \(\alpha\))

P :

Applied load

\(P_{\text {u}}\) :

Peak load

\(\sigma _{N}\) :

Nominal stress

\(\sigma _{Nu}\) :

Nominal strength (nominal stress at peak load)

\(E, E'\) :

In-plane and out-of-plane modulus of material

\(\nu , \nu '\) :

In-plane and out-of-plane Poisson’s ratio of material

\(G'\) :

Out-of-plane shear modulus of material

\(E_x, E_y\) :

Elastic constants (Young’s modulus) in the specimen coordinate system

\(\nu _{xy}, \nu _{yx}\) :

Elastic constants (Poisson’s ratio) in the specimen coordinate system

\(G_{xy}\) :

Elastic constants (shear modulus) in the specimen coordinate system

\(\lambda , \rho\) :

Dimensionless elastic constants

\(E^*\) :

Effective elastic modulus

\(g, g'\) :

Dimensionless energy release rate and its derivative

\(g_0, g'_0\) :

Dimensionless energy release rate at \(\alpha _0\) and its derivative value

k :

Dimensionless stress intensity factor

\(\xi\) :

Dimensionless function

\(\sigma _0\) :

A parameter in size effect law

MAPE\(_{a_0}\) :

Notch-machining error

\(R^2\) :

Coefficient of determination

RMSE:

Root-mean-squared error

SD:

Standard deviation

SE:

Standard error

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Acknowledgements

The authors would like to thank Professor Brad Sageman (Department of Earth and Planetary Sciences, Northwestern University) for providing the Marcellus shale samples used in this study and Professor Giuseppe Buscarnera (Department of Civil and Environmental Engineering, Northwestern University) for his assistance with the Mini-Tester. This work also made use of the Materials Characterization and Imaging Facility and the Center for Sustainable Engineering of Geological and Infrastructure Materials (SEGIM) at Northwestern University.

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  1. Theoretical and Applied Mechanics, Northwestern University, Evanston, IL, 60208, USA

    Weixin Li & Gianluca Cusatis

  2. Department of Civil and Environmental Engineering, Northwestern University, Evanston, IL, 60208, USA

    Weixin Li, Zhefei Jin & Gianluca Cusatis

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  1. Weixin Li
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Correspondence to Gianluca Cusatis.

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Li, W., Jin, Z. & Cusatis, G. Size Effect Analysis for the Characterization of Marcellus Shale Quasi-brittle Fracture Properties. Rock Mech Rock Eng 52, 1–18 (2019). https://doi.org/10.1007/s00603-018-1570-6

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