Prediction of the Anisotropic Effective Moduli of Shales based on the Mori–Tanaka Model and the Digital Core Technique

Abstract

Natural rocks belong to the class of polymineral composite materials with complex microstructures. Such strong heterogeneity of rocks makes it difficult to estimate the effective moduli by traditional models in theory. In the present study, a Mori–Tanaka (MT) model considering the shape and orientation of inclusion minerals obtained by micro-computed tomography (micro-CT) is established, and then it is applied to evaluate the anisotropic parameters of shales. In the MT model, the principal radii and Eulerian angles of the ellipsoidal inclusion are obtained by solving its inertia matrix through micro-CT. According to this inclusion information, we determined the statistics on the ratio of average principal radii and the distribution of Eulerian angles of inclusions with different minerals. In what follows, the effective elastic stiffness matrix of shale samples is predicted by the MT model, and the corresponding digital core is input for finite element method (FEM) analysis to verify the accuracy of the theoretical results. It is shown that the anisotropy of the elastic stiffness matrix predicted by the MT model and FEM is consistent under two sizes of representative volume elements. These findings highlight the potential of this approach for applications in rock mechanics, civil engineering, and oil exploitation, among others.

Appendices

Appendix A

The three most typical components of the Hill tensor P on the general ellipsoid in the local coordinate system O-x₁x₂x₃ can be expressed as

$$\left. \begin{gathered} P_{1111} { = }\frac{{3a_{1}^{2} }}{{16\pi G_{0} \left( {1 - \nu_{0} } \right)}}W_{11} + \frac{{1 - 4\nu_{0} }}{{16\pi G_{0} \left( {1 - \nu_{0} } \right)}}W_{1} \hfill \\ P_{1212} { = }\frac{1}{{16\pi G_{0} \left( {1 - \nu_{0} } \right)}}\left( {a_{1}^{2} W_{12} + W_{1} } \right) - \frac{{\nu_{0} }}{{16\pi G_{0} \left( {1 - \nu_{0} } \right)}}\left( {W_{1} + W_{2} } \right) \hfill \\ P_{1122} { = }\frac{{a_{1}^{2} }}{{16\pi G_{0} \left( {1 - \nu_{0} } \right)}}W_{12} - \frac{1}{{16\pi G_{0} \left( {1 - \nu_{0} } \right)}}W_{2} \hfill \\ \end{gathered} \right\}$$

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and the other components can be obtained by the corner mark rotation, such as

$$\left. \begin{gathered} P_{3232} { = }\frac{1}{{16\pi G_{0} \left( {1 - \nu_{0} } \right)}}\left( {a_{3}^{2} W_{32} + W_{3} } \right) - \frac{{\nu_{0} }}{{16\pi G_{0} \left( {1 - \nu_{0} } \right)}}\left( {W_{3} + W_{2} } \right) \hfill \\ P_{3322} = \frac{{a_{3}^{2} }}{{16\pi G_{0} \left( {1 - \nu_{0} } \right)}}W_{32} - \frac{1}{{16\pi G_{0} \left( {1 - \nu_{0} } \right)}}W_{2} \hfill \\ \end{gathered} \right\}$$

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In addition, the tensor of P has the symmetry of P_ijkl = P_jikl = P_ijlk = P_klij. In Equations (A1) and (A2), there are

$$\left. \begin{gathered} W_{12} { = }W_{21} { = }\frac{{W_{2} - W_{1} }}{{a_{1}^{2} - a_{2}^{2} }}, \, W_{13} { = }W_{31} { = }\frac{{W_{3} - W_{1} }}{{a_{1}^{2} - a_{3}^{2} }}, \, W_{23} { = }W_{32} { = }\frac{{W_{3} - W_{1} }}{{a_{2}^{2} - a_{3}^{2} }} \hfill \\ W_{11} = \frac{4\pi }{{3a_{1}^{2} }} - \frac{1}{3}\left( {W_{12} + W_{13} } \right), \, W_{22} = \frac{4\pi }{{3a_{2}^{2} }} - \frac{1}{3}\left( {W_{12} + W_{23} } \right) \hfill \\ W_{33} = \frac{4\pi }{{3a_{3}^{2} }} - \frac{1}{3}\left( {W_{13} + W_{23} } \right) \hfill \\ \end{gathered} \right\}$$

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where

$$\left. \begin{gathered} W_{1} { = }\frac{{4\pi a_{1} a_{2} a_{3} }}{{\left( {a_{1}^{2} - a_{2}^{2} } \right)\sqrt {a_{1}^{2} - a_{3}^{2} } }}\left[ {F\left( {\xi ,\zeta } \right) - E\left( {\xi ,\zeta } \right)} \right] \hfill \\ W_{2} { = }4\pi a_{1} a_{2} a_{3} \left[ {\frac{{\sqrt {a_{1}^{2} - a_{3}^{2} } }}{{\left( {a_{2}^{2} - a_{3}^{2} } \right)\left( {a_{1}^{2} - a_{2}^{2} } \right)}}E\left( {\xi ,\zeta } \right)} \right. \hfill \\ \, - \left. {\frac{{F\left( {\xi ,\zeta } \right)}}{{\left( {a_{1}^{2} - a_{2}^{2} } \right)\sqrt {a_{1}^{2} - a_{3}^{2} } }} - \frac{{a_{3} }}{{a_{1} a_{2} \left( {a_{2}^{2} - a_{3}^{2} } \right)}}} \right] \hfill \\ W_{3} = \frac{{4\pi a_{1} a_{2} a_{3} }}{{\left( {a_{2}^{2} - a_{3}^{2} } \right)\sqrt {a_{1}^{2} - a_{3}^{2} } }} - \left[ {\frac{{a_{2} \sqrt {a_{1}^{2} - a_{3}^{2} } }}{{a_{1} a_{3} }}} \right] - E\left( {\xi ,\zeta } \right) \hfill \\ \end{gathered} \right\}$$

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The symbols F ($\xi$,$\zeta$) and E ($\xi$,$\zeta$) are the elliptic integrals of type I and type II, respectively, which are

$$\left. \begin{gathered} F\left( {\xi ,\zeta } \right){ = }\int_{0}^{\xi } {\frac{{{\text{d}}y}}{{\sqrt {1 - \zeta^{2} \sin^{2} y} }}} \hfill \\ E\left( {\xi ,\zeta } \right) = \int_{0}^{\xi } {\sqrt {1 - \zeta^{2} \sin^{2} y} {\text{d}}y} \hfill \\ \end{gathered} \right\}$$

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where $\xi { = }\arcsin \sqrt {1 - \left( {a_{3} /a_{1} } \right)^{2} }$ and $\zeta = \sqrt {\left( {a_{1}^{2} - a_{2}^{2} } \right)/\left( {a_{1}^{2} - a_{3}^{2} } \right)}$.

Appendix B

The coordinates of the geometric center of the inclusion domain Ω in the coordinate system O-XYZ are

$$M_{X} = \frac{1}{{V_{\Omega } }}\int_{\Omega } {X{\text{d}}X{\text{d}}Y{\text{d}}Z} ,\,\,M_{Y} = \frac{1}{{V_{\Omega } }}\int_{\Omega } {Y{\text{d}}X{\text{d}}Y{\text{d}}Z} ,\,\,M_{Z} = \frac{1}{{V_{\Omega } }}\int_{\Omega } {Z{\text{d}}X{\text{d}}Y{\text{d}}Z}$$

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where M_X, M_Y, and M_Z are the coordinate values of the geometric center of inclusion domain Ω in O-XYZ, and V_Ω is the volume of the inclusion domain. Then, the components of the inertia matrix of the inclusion domain are

$$\begin{aligned} M_{2X} & = \frac{1}{{V_{\Omega } }}\int_{\Omega } {\left( {X - M_{X} } \right)^{2} {\text{d}}X{\text{d}}Y{\text{d}}Z} , \\ M_{2Y} & = \frac{1}{{V_{\Omega } }}\int_{\Omega } {\left( {Y - M_{Y} } \right)^{2} {\text{d}}X{\text{d}}Y{\text{d}}Z} , \\ M_{2Z} & = \frac{1}{{V_{\Omega } }}\int_{\Omega } {\left( {Z - M_{Z} } \right)^{2} {\text{d}}X{\text{d}}Y{\text{d}}Z} , \\ M_{2XY} & = \frac{1}{{V_{\Omega } }}\int_{\Omega } {\left( {X - M_{X} } \right)\left( {Y - M_{Y} } \right){\text{d}}X{\text{d}}Y{\text{d}}Z} , \\ M_{2XY} & = \frac{1}{{V_{\Omega } }}\int_{\Omega } {\left( {X - M_{X} } \right)\left( {Y - M_{Y} } \right){\text{d}}X{\text{d}}Y{\text{d}}Z} , \\ M_{2ZX} & = \frac{1}{{V_{\Omega } }}\int_{\Omega } {\left( {Z - M_{Z} } \right)\left( {X - M_{X} } \right){\text{d}}X{\text{d}}Y{\text{d}}Z} \\ \end{aligned}$$

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