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.
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Acknowledgements
The authors greatly appreciate the financial support of the Strategic Priority Research Program of the Chinese Academy of Sciences (XDA14010303), the National Natural Science Foundation of China (11972375), and the Science and Technology Project in the West Coast New Area of Qingdao (2020-81).
Funding
This study was funded by the Strategic Priority Research Program of the Chinese Academy of Sciences (XDA14010303), the National Natural Science Foundation of China (11972375), and the Science and Technology Project in the West Coast New Area of Qingdao (2020-81).
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For author contributions, we declare that ZW, as the first author of this manuscript, is responsible for the theoretical and numerical calculations for all the models in the paper. GC has helped with the model and results analyses. JL has helped to modify the paper and all the result analyses. LF has discussed the details of the paper with the other authors.
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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
and the other components can be obtained by the corner mark rotation, such as
In addition, the tensor of P has the symmetry of Pijkl = Pjikl = Pijlk = Pklij. In Equations (A1) and (A2), there are
where
The symbols F (\(\xi\),\(\zeta\)) and E (\(\xi\),\(\zeta\)) are the elliptic integrals of type I and type II, respectively, which are
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
where MX, MY, and MZ are the coordinate values of the geometric center of inclusion domain Ω in O-XYZ, and VΩ is the volume of the inclusion domain. Then, the components of the inertia matrix of the inclusion domain are
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Wang, Z., Cao, G., Liu, J. et al. Prediction of the Anisotropic Effective Moduli of Shales based on the Mori–Tanaka Model and the Digital Core Technique. Pure Appl. Geophys. 180, 2981–2998 (2023). https://doi.org/10.1007/s00024-023-03298-8
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DOI: https://doi.org/10.1007/s00024-023-03298-8