Quantifying complex microstructures of earth materials: Reconstructing
higher-order spatial correlations using deep generative adversarial
networks
Abstract
Key to most subsurface processes is to determine how structural and
topological features at small length scales, i.e., the microstructure,
control the effective and macroscopic properties of earth materials.
Recent progress in imaging technology has enabled us to visualise and
characterise microstructures at different length scales and dimensions.
An approach to characterisation is the sampling of n-point correlation
functions - known as statistical microstructural descriptors (SMDs) -
from images. SMDs can then be used to generate statistically equivalent
structures having larger sizes and additional dimensions – this process
is known as $reconstruction$. We show that a deep-convolutional
generative adversarial network trained with Wasserstein-loss and
gradient penalty (WGAN-GP) results in a stable training and high-quality
reconstructions of two-dimensional electron microscopy images of complex
rock samples. To evaluate reconstruction performance, n-point polytope
functions are calculated in both reconstructed and original
microstructures and mean square error between them is used as a quality
metric. These n-point polytope functions provide statistical information
about symmetric, user-oriented higher-order geometrical patterns in
microstructures. Our results show that GANs can naturally capture these
higher-order statistics at short and long ranges. Furthermore, we
compare our model with a benchmark stochastic reconstruction method
based solely on two-point correlation. Our findings indicate that
although yielding the same two-point statistics, two microstructures can
be morphologically and structurally different, emphasising the need for
coupling higher-order correlation functions with reconstruction methods.
This is a critical step for future schemes that aim to reconstruct
complex heterogeneous systems and couple microstructures to macroscopic
phenomena.