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Published May 2023 | public
Journal Article

Computation of effective elastic moduli of rocks using hierarchical homogenization

Abstract

This work focuses on computing the homogenized elastic properties of rocks from 3D micro-computed-tomography (micro-CT) scanned images. The accurate computation of homogenized properties of rocks, archetypal random media, requires both resolution of intricate underlying microstructure and large field of view, resulting in huge micro-CT images. Homogenization entails solving the local elasticity problem computationally which can be prohibitively expensive for a huge image. To mitigate this problem, we use a renormalization method inspired scheme, the hierarchical homogenization method, where a large image is partitioned into smaller subimages. The individual subimages are separately homogenized using periodic boundary conditions, and then assembled into a much smaller intermediate image. The intermediate image is again homogenized, subject to the periodic boundary condition, to find the final homogenized elastic constant of the original image. An FFT-based elasticity solver is used to solve the associated periodic elasticity problem. The error in the homogenized elastic constant is empirically shown to follow a power law scaling with exponent -1 with respect to the subimage size across all five microstructures of rocks. We further show that the inclusion of surrounding materials during the homogenization of the small subimages reduces error in the final homogenized elastic moduli while still respecting the power law with the exponent of -1. This power law scaling is then exploited to determine a better approximation of the large heterogeneous microstructures based on Richardson extrapolation.

Additional Information

© 2023 Elsevier. We acknowledge Shell for financial support and for providing the digital rock images. We thank Dr. Nishank Saxena at Shell for his constant support of the project. The authors also would like to thank Math2Market for providing the GeoDict software at a discount, and providing technical support. CRediT authorship contribution statement. Rasool Ahmad: Conception and design of study, Methodology, Analysis and interpretation, Writing of the original manuscript, Revision of the manuscript. Mingliang Liu: Analysis and interpretation, Writing of the original manuscript. Michael Ortiz: Analysis and interpretation, Writing of the original manuscript. Tapan Mukerji: Analysis and interpretation, Writing of the original manuscript. Wei Cai: Conception and design of study, Methodology, Analysis and interpretation, Writing of the original manuscript, Revision of the manuscript. The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Additional details

Created:
August 22, 2023
Modified:
October 18, 2023