  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
