Quantifying microstructures of earth materials using higher-order spatial correlations and deep generative adversarial networks

Abstract

The 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. However, one limitation of these technologies is the trade-off between resolution and sample size (or representativeness). A promising approach to this problem is image reconstruction which aims to generate statistically equivalent microstructures but at a larger scale and/or additional dimension. In this work, a stochastic method and three generative adversarial networks (GANs), namely deep convolutional GAN (DCGAN), Wasserstein GAN with gradient penalty (WGAN-GP), and StyleGAN2 with adaptive discriminator augmentation (ADA), are used to reconstruct two-dimensional images of two hydrothermally rocks with varying degrees of complexity. For the first time, we evaluate and compare the performance of these methods using multi-point spatial correlation functions—known as statistical microstructural descriptors (SMDs)—ultimately used as external tools to the loss functions. Our findings suggest that a well-trained GAN can reconstruct higher-order, spatially-correlated patterns of complex earth materials, capturing underlying structural and morphological properties. Comparing our results with a stochastic reconstruction method based on a two-point correlation function, we show the importance of coupling training/assessment of GANs with higher-order SMDs, especially in the case of complex microstructures. More importantly, by quantifying original and reconstructed microstructures via different GANs, we highlight the interpretability of these SMDs and show how they can provide valuable insights into the spatial patterns in the synthetic images, allowing us to detect common artefacts and failure cases in training GANs.

True 2D-to-3D Reconstruction of Heterogeneous Porous Media via Deep Generative Adversarial Networks (GANs)

“…Essential to the reconstruction of heterogeneous and complex microstructures is the determination of a representative image size (RES) that captures the structural elements of the system under consideration. Larger training images pose higher computational demands, while overly small images fail to fully capture material behavior and heterogeneity, resulting in the generation of pore artifacts and unrealistic shapes (Amiri et al, 2023). Hence, a RES analysis should be conducted for heterogeneous and complex samples to best determine an appropriate image size for training the model (Volkhonskiy et al, 2019;Costanza-Robinson et al, 2011).…”

“…To determine the RES of our sample, we adopted the approach employed by Amiri et al (2023) which relies on the widely popular two-point correlation function, S 2 (r) (Torquato & Haslach Jr, 2002;Jiao et al, 2007Jiao et al, , 2008, defined as the probability, P , that two randomly selected points with a distance, r, fall within the same phase of interest (i), V i , in a d-dimensional space, R d (Yeong & Torquato, 1998):…”

“…The 2D-to-3D image reconstruction is an inverse problem in which limited microstructural data (e.g., 2D images) are used to generate statistically equivalent microstructures with larger sizes and/or additional dimensions. This provides representative 3D microstructural information at resolutions sufficient for detecting multi-scale features within the samples (Jiao et al, 2007;Yeong & Torquato, 1998;Amiri et al, 2023;Sahimi & Tahmasebi, 2021).…”

“…However, the statistical information captured by S 2 alone is not sufficient in the case of heterogeneous microstructures with complex structure and morphology (Jiao et al, 2010;Gommes et al, 2012;Amiri et al, 2023). To overcome the limitations, other studies (Hajizadeh et al, 2011;Tahmasebi & Sahimi, 2012) have turned to high-order n-point correlation functions (n ≥ 3) to more precisely quantify higher-order spatial patterns in complex microstructures, thus enhancing reconstruction accuracy.…”