A Study of Uniaxial Compressive Strength of Shale Based on Homogenization Method

Abstract

Uniaxial compressive strength of shale is one of primary input parameters during shale gas development. In order to analyze uniaxial compression failure mechanism and obtain strength formula, the present study regards shale as heterogeneous material composed of pores, brittle minerals and clays. Pores are distributed as oblate ellipsoidal inclusion in clays, while brittle minerals are distributed as spheroidal inclusion in the equivalent clay matrix formed by pores and clays. With the help of the basic solution to Eshelby’s matrix–inclusion problem and the homogenization method, a two-step homogenization framework for shale was established. The influences of pore fluid on the rock mechanical behavior were also discussed by categories. Based on the composite mechanics, the failure strength of shale under uniaxial compression was deduced at last. The results show that the shale strength is affected by the mechanical properties of diagenetic minerals and enhanced with the increase of brittle mineral content. Since pore water promotes the initiation and propagation of cracks, shale strength decreases non-linearly with increasing water saturation.

Appendix

The object of this research is heterogeneous materials with ellipsoid inclusions, which exhibit transversely isotropic symmetry. Plane $z_{1} - z_{2}$ refers to the isotropic plane, and z₃ refers to the symmetry axis. In order to facilitate the inner product and inversion of tensor, the Walpole’s base is introduced to represent the transversely isotropic fourth-order tensor ${\mathbf{U}}$ (Walpole 1981), which can be expressed as:

$${\mathbf{U}} = \left[ {\begin{array}{*{20}c} c & d & e & f & g & h \\ \end{array} } \right]$$

(41)

The inverse of ${\mathbf{U}}$ can be written as:

$${\mathbf{U}}^{ - 1} = \left[ {\begin{array}{*{20}c} {\frac{d}{l}} & {\frac{c}{l}} & {\frac{1}{e}} & {\frac{1}{f}} & { - \frac{g}{l}} & { - \frac{h}{l}} \\ \end{array} } \right]$$

(42)

where $l = cd - 2gh$. Similarly, definition of ${\mathbf{U}}^{\prime}$ is $[\begin{array}{cccccc}c^{\prime}& d^{\prime} &e^{\prime} &f^{\prime} &g^{\prime} &h^{\prime}\end{array}]$, and the inner product can be noted in the following symbolic form:

$${\mathbf{UU}}^{\prime} = [\begin{array}{cccccc}cc^{\prime} + 2hg^{\prime} &d d^{\prime} + 2gh^{\prime} &ee^{\prime} &ff^{\prime} &gc^{\prime} + dg^{\prime} &hd^{\prime} + ch^{\prime} \end{array}]$$

(43)

Some standard fourth-order tensors are introduced here:

$${\mathbf{I}} = [\begin{array}{cccccc}1& 1 & 1 & 1 & 0 & 0 \end{array}]$$

(44)

$${\mathbf{J}} = \left[ {\begin{array}{*{20}c} {\frac{2}{3}} & {\frac{1}{3}} & 0 & 0 & {\frac{1}{3}} & {\frac{1}{3}} \\ \end{array} } \right]$$

(45)

$${\mathbf{K}} = {\mathbf{I}} - {\mathbf{J}} = \left[ {\begin{array}{*{20}c} {\frac{1}{3}} & {\frac{2}{3}} & 1 & 1 & { - \frac{1}{3}} & { - \frac{1}{3}} \\ \end{array} } \right]$$

(46)

Therefore, the stiffness tensor of material can be expressed as:

$${\mathbf{D}} = 3k{\mathbf{J}} + 2\mu {\mathbf{K}}$$

(47)

where $k$ and $\mu$ represent the bulk and shear moduli, respectively. The Eshelby tensor of ellipsoid inclusion can be written as Walpole’s base:

$${\mathbf{S}}^{ *} = \left[ {\begin{array}{*{20}c} {S_{1111} + S_{1122} } & {S_{3333} } & {2S_{1212} } & {2S_{2323} } & {S_{3311} } & {S_{1133} } \\ \end{array} } \right]$$

(48)

The corresponding components of Hill polarization tensor for Eshelby tensor ${\mathbf{S}}^{ *}$ can easily read:

$$S_{ijmn}^{*} = P_{ijpq} D_{pqmn}$$

(49)

where $D_{pqmn}$ represent the component of stiffness tensor.