Reconstruction of three-dimensional porous media using generative adversarial neural networks

Predicting compressive stress-strain behavior of elasto-plastic porous media via morphology-informed neural networks

Porous media, ranging from bones to concrete and from batteries to architected lattices, pose difficult challenges in fully harnessing for engineering applications due to their complex and variable structures. Accurate and rapid assessment of their mechanical behavior is both challenging and essential, and traditional methods such as destructive testing and finite element analysis can be costly, computationally demanding, and time consuming. Machine learning (ML) offers a promising alternative for predicting mechanical behavior by leveraging data-driven correlations. However, with such structural complexity and diverse morphology among porous media, the question becomes how to effectively characterize these materials to provide robust feature spaces for ML that are descriptive, succinct, and easily interpreted. Here, we developed an automated methodology to determine porous material strength. This method uses scalar morphological descriptors, known as Minkowski functionals, to describe the porous space. From there, we conduct uniaxial compression experiments for generating material stress-strain curves, and then train an ML model to predict the curves using said morphological descriptors. This framework seeks to expedite the analysis and prediction of stress-strain behavior in porous materials and lay the groundwork for future models that can predict mechanical behaviors beyond compression.

Structure-property relationships in fibrous meniscal tissue through image-based augmentation

This study introduces an adaptive three-dimensional (3D) image synthesis technique for creating variational realizations of fibrous meniscal tissue microstructures. The method allows controlled deviation from original geometries by modifying parameters such as porosity, pore size and specific surface area of image patches. The unbiased reconstructed samples matched the morphological and hydraulic properties of original tissues, with relative errors generally below 10%. Additional samples were generated with predefined deviations to increase dataset diversity. Analysis of 1500 synthesized geometries revealed relationships between microstructural features, hydraulic permeability and mechanical properties. Empirical correlations were derived to predict longitudinal and transverse hydraulic permeability as functions of porosity, with R2 values of 0.98 and 0.97, respectively. Finite-element simulations examined mechanical behaviour under compression, showing stress concentrations at fibre cross-links and permeability reductions that varied with porosity and flow direction. These results led to a porosity-dependent model for normalized Young's modulus ([Formula: see text]). The proposed correlations and data augmentation technique aid in investigating structure-property relationships in meniscal tissue, potentially benefiting biomimetic implant design. This approach may help bridge data gaps where obtaining numerous real samples is impractical or unethical.This article is part of the theme issue 'Uncertainty quantification for healthcare and biological systems (Part 1)'.

3D microstructure reconstruction and characterization of porous materials using a cross-sectional SEM image and deep learning

Accurate assessment of the three-dimensional (3D) pore characteristics within porous materials and devices holds significant importance. Compared to high-cost experimental approaches, this study introduces an alternative method: utilizing a generative adversarial network (GAN) to reconstruct a 3D pore microstructure. Unlike some existing GAN models that require 3D images as training data, the proposed model only requires a single cross-sectional image for 3D reconstruction. Using porous ceramic electrode materials as a case study, a comparison between the GAN-generated microstructures and those reconstructed through focused ion beam-scanning electron microscopy (FIB-SEM) reveals promising consistency. The GAN-based reconstruction technique demonstrates its effectiveness by successfully characterizing pore attributes in porous ceramics, with measurements of porosity, pore size, and tortuosity factor exhibiting notable agreement with the results obtained from mercury intrusion porosimetry.

Sandification degree classification of sandy dolomite base on convolutional neural networks

Sandification can degrade the strength and quality of dolomite, and to a certain extent, compromise the stability of a tunnel's surrounding rock as an unfavorable geological boundary. Sandification degree classification of sandy dolomite is one of the non-trivial challenges faced by geotechnical engineering projects such as tunneling in complex geographical environments. The traditional methods quantitatively measuring the physical parameters or analyzing some visual features are either time-consuming or inaccurate in practical use. To address these issues, we, for the first time, introduce the convolutional neural network (CNN)-based image classification methods into dolomite sandification degree classification task. In this study, we have made a significant contribution by establishing a large-scale dataset comprising 5729 images, classified into four distinct sandification degrees of sandy dolomite. These images were collected from the vicinity of a tunnel located in the Yuxi section of the CYWD Project in China. We conducted comprehensive classification experiments using this dataset. The results of these experiments demonstrate the groundbreaking achievement of CNN-based models, which achieved an impressive accuracy rate of up to 91.4%. This accomplishment underscores the pioneering role of our work in creating this dataset and its potential for applications in complex geographical analyses.

Spiking neural networks for nonlinear regression

Spiking neural networks (SNN), also often referred to as the third generation of neural networks, carry the potential for a massive reduction in memory and energy consumption over traditional, second-generation neural networks. Inspired by the undisputed efficiency of the human brain, they introduce temporal and neuronal sparsity, which can be exploited by next-generation neuromorphic hardware. Energy efficiency plays a crucial role in many engineering applications, for instance, in structural health monitoring. Machine learning in engineering contexts, especially in data-driven mechanics, focuses on regression. While regression with SNN has already been discussed in a variety of publications, in this contribution, we provide a novel formulation for its accuracy and energy efficiency. In particular, a network topology for decoding binary spike trains to real numbers is introduced, using the membrane potential of spiking neurons. Several different spiking neural architectures, ranging from simple spiking feed-forward to complex spiking long short-term memory neural networks, are derived. Since the proposed architectures do not contain any dense layers, they exploit the full potential of SNN in terms of energy efficiency. At the same time, the accuracy of the proposed SNN architectures is demonstrated by numerical examples, namely different material models. Linear and nonlinear, as well as history-dependent material models, are examined. While this contribution focuses on mechanical examples, the interested reader may regress any custom function by adapting the published source code.

Bayesian geochemical correlation and tomography

To accurately reconstruct palaeoenvironmental change through time it is important to determine which rock samples were deposited contemporaneously at different sites or transects, as erroneous correlation may lead to incorrectly inferred processes and rates. To correlate samples, current practice interpolates geological age between datable units along each transect, then temporal signatures observed in geochemical logs are matched between transects. Unfortunately spatiotemporally variable and unknown rates of sedimentary deposition create highly nonlinear space-time transforms, significantly altering apparent geochemical signatures. The resulting correlational hypotheses are also untestable against independent transects, because correlations have no spatially-predictive power. Here we use geological process information stored within neural networks to correlate spatially offset logs nonlinearly and geologically. The same method creates tomographic images of geological age and geochemical signature across intervening rock volumes. Posterior tomographic images closely resemble the true depositional age throughout the inter-transect volume, even for scenarios with long hiatuses in preserved geochemical signals. Bayesian probability distributions describe data-consistent variations in the results, showing that centred summary statistics such as mean and variance do not adequately describe correlational uncertainties. Tomographic images demonstrate spatially predictive power away from geochemical transects, creating novel hypotheses attributable to each geochemical correlation which are testable against independent data.

GAN-based generation of realistic 3D volumetric data: A systematic review and taxonomy

With the massive proliferation of data-driven algorithms, such as deep learning-based approaches, the availability of high-quality data is of great interest. Volumetric data is very important in medicine, as it ranges from disease diagnoses to therapy monitoring. When the dataset is sufficient, models can be trained to help doctors with these tasks. Unfortunately, there are scenarios where large amounts of data is unavailable. For example, rare diseases and privacy issues can lead to restricted data availability. In non-medical fields, the high cost of obtaining enough high-quality data can also be a concern. A solution to these problems can be the generation of realistic synthetic data using Generative Adversarial Networks (GANs). The existence of these mechanisms is a good asset, especially in healthcare, as the data must be of good quality, realistic, and without privacy issues. Therefore, most of the publications on volumetric GANs are within the medical domain. In this review, we provide a summary of works that generate realistic volumetric synthetic data using GANs. We therefore outline GAN-based methods in these areas with common architectures, loss functions and evaluation metrics, including their advantages and disadvantages. We present a novel taxonomy, evaluations, challenges, and research opportunities to provide a holistic overview of the current state of volumetric GANs.

Generating 3D images of material microstructures from a single 2D image: a denoising diffusion approach

Three-dimensional (3D) images provide a comprehensive view of material microstructures, enabling numerical simulations unachievable with two-dimensional (2D) imaging alone. However, obtaining these 3D images can be costly and constrained by resolution limitations. We introduce a novel method capable of generating large-scale 3D images of material microstructures, such as metal or rock, from a single 2D image. Our approach circumvents the need for 3D image data while offering a cost-effective, high-resolution alternative to existing imaging techniques. Our method combines a denoising diffusion probabilistic model with a generative adversarial network framework. To compensate for the lack of 3D training data, we implement chain sampling, a technique that utilizes the 3D intermediate outputs obtained by reversing the diffusion process. During the training phase, these intermediate outputs are guided by a 2D discriminator. This technique facilitates our method's ability to gradually generate 3D images that accurately capture the geometric properties and statistical characteristics of the original 2D input. This study features a comparative analysis of the 3D images generated by our method, SliceGAN (the current state-of-the-art method), and actual 3D micro-CT images, spanning a diverse set of rock and metal types. The results shown an improvement of up to three times in the Frechet inception distance score, a typical metric for evaluating the performance of image generative models, and enhanced accuracy in derived properties compared to SliceGAN. The potential of our method to produce high-resolution and statistically representative 3D images paves the way for new applications in material characterization and analysis domains.

Microstructure reconstruction of 2D/3D random materials via diffusion-based deep generative models

Microstructure reconstruction serves as a crucial foundation for establishing process-structure-property (PSP) relationship in material design. Confronting the limitations of variational autoencoder and generative adversarial network within generative models, this study adopted the denoising diffusion probabilistic model (DDPM) to learn the probability distribution of high-dimensional raw data and successfully reconstructed the microstructures of various composite materials, such as inclusion materials, spinodal decomposition materials, chessboard materials, fractal noise materials, and so on. The quality of generated microstructure was evaluated using quantitative measures like spatial correlation functions and Fourier descriptor. On this basis, this study also achieved the regulation of microstructure randomness and the generation of gradient materials through continuous interpolation in latent space using denoising diffusion implicit model (DDIM). Furthermore, the two-dimensional microstructure reconstruction was extended to three-dimensional framework and integrated permeability as a feature encoding embedding. This enables the conditional generation of three-dimensional microstructures for random porous materials within a defined permeability range. The permeabilities of these generated microstructures were further validated through the application of the lattice Boltzmann method. The above methods provide new ideas and references for material reverse design.

Review of modern models of porous media for numerical simulation of fluid flows

The current models of porous media for numerical simulation of multiphase fluid flows are reviewed and classified herein. Methods for building these models and problems of creating the same are examined, and examples where the models were used are given. This review has been made as part of hydrocarbon deposit hydrodynamic simulation problems arising in the oil-and-gas producing industry. The approaches and possibilities for creating a digital twin of actual core have been evaluated. It is noted that the porous medium models used cannot presently meet the industry demands to the full and the reasons behind that have been analyzed. A conclusion was made that an in-depth fundamental study on processes and phenomena affecting the multiphase flow structure at the micro-scale is required to take them into account properly in the description of the flow at a deposit scale.

Three-dimensional reconstruction of granular porous media based on deep generative models

Reconstruction of microstructure in granular porous media, which can be viewed as granular assemblies, is crucial for studying their characteristics and physical properties in various fields concerned with the behavior of such media, including petroleum geology and computational materials science. In spite of the fact that many existing studies have investigated grain reconstruction, most of them treat grains as simplified individuals for discrete reconstruction, which cannot replicate the complex geometrical shapes and natural interactions between grains. In this work, a hybrid generative model based on a deep-learning algorithm is proposed for high-quality three-dimensional (3D) microstructure reconstruction of granular porous media from a single two-dimensional (2D) slice image. The method extracts 2D prior information from the given image and generates the grain set as a whole. Both a self-attention module and effective pattern loss are introduced in a bid to enhance the reconstruction ability of the model. Samples with grains of varied geometrical shapes are utilized for the validation of our method, and experimental results demonstrate that our proposed approach can accurately reproduce the complex morphology and spatial distribution of grains without any artificiality. Furthermore, once the model training is complete, rapid end-to-end generation of diverse 3D realizations from a single 2D image can be achieved.

Coupled Lattice Boltzmann Modeling Framework for Pore-Scale Fluid Flow and Reactive Transport

In this paper, we propose a modeling framework for pore-scale fluid flow and reactive transport based on a coupled lattice Boltzmann model (LBM). We develop a modeling interface to integrate the LBM modeling code parallel lattice Boltzmann solver and the PHREEQC reaction solver using multiple flow and reaction cell mapping schemes. The major advantage of the proposed workflow is the high modeling flexibility obtained by coupling the geochemical model with the LBM fluid flow model. Consequently, the model is capable of executing one or more complex reactions within desired cells while preserving the high data communication efficiency between the two codes. Meanwhile, the developed mapping mechanism enables the flow, diffusion, and reactions in complex pore-scale geometries. We validate the coupled code in a series of benchmark numerical experiments, including 2D single-phase Poiseuille flow and diffusion, 2D reactive transport with calcite dissolution, as well as surface complexation reactions. The simulation results show good agreement with analytical solutions, experimental data, and multiple other simulation codes. In addition, we design an AI-based optimization workflow and implement it on the surface complexation model to enable increased capacity of the coupled modeling framework. Compared to the manual tuning results proposed in the literature, our workflow demonstrates fast and reliable model optimization results without incorporating pre-existing domain knowledge.

Prediction of the Soil Permeability Coefficient of Reservoirs Using a Deep Neural Network Based on a Dendrite Concept

Changes in the pore water pressure of soil are essential factors that affect the movement of structures during and after construction in terms of stability and safety. Soil permeability represents the quantity of water transferred using pore water pressure. However, these changes cannot be easily identified and require considerable time and money. This study predicted and evaluated the soil permeability coefficient using a multiple regression (MR) model, adaptive network-based fuzzy inference system (ANFIS), general deep neural network (DNN) model, and DNN using the dendrite concept (DNN−T, which was proposed in this study). The void ratio, unit weight, and particle size were obtained from 164 undisturbed samples collected from the embankments of reservoirs in South Korea as input variables for the aforementioned models. The data used in this study included seven input variables, and the ratios of the training data to the validation data were randomly extracted, such as 6:4, 7:3, and 8:2, and were used. The analysis results for each model showed a median correlation of r = 0.6 or less and a low model efficiency of Nash–Sutcliffe efficiency (NSE) = 0.35 or less as a result of predicting MR and ANFIS. The DNN and DNN−T both have good performance, with a strong correlation of r = 0.75 or higher. Evidently, the DNN−T performance in terms of r, NSE, and root mean square error (RMSE) improved more than that of the DNN. However, the difference between the mean absolute percent error (MAPE) of DNN−T and the DNN was that the error of the DNN was small (11%). Regarding the ratio of the training data to the verification data, 7:3 and 8:2 showed better results compared to 6:4 for indicators, such as r, NSE, RMSE, and MAPE. We assumed that this phenomenon was caused by the DNN−T thinking layer. This study shows that DNN−T, which changes the structure of the DNN, is an alternative for estimating the soil permeability coefficient in the safety inspection of construction sites and is an excellent methodology that can save time and budget.

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

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.

End-to-end three-dimensional designing of complex disordered materials from limited data using machine learning

Precise 3D representation of complex materials, here the lithium-ion batteries, is a critical step toward designing optimized energy storage systems. One requires obtaining several such samples for a more accurate evaluation of uncertainty and variability, which in turn can be costly and time demanding. Using 3D models is crucial when it comes to evaluating the transport and heat capacity of batteries. Further, such models represent the microstructures more precisely where connectivity and heterogeneity can be detected. However, 3D images are hard to access, and the available images are often collected in two dimensions (2D). Such 2D images, on the other hand, are more accessible and often have higher resolution. In this paper, a deep learning method has been applied to take advantage of 2D images and build 3D models of heterogeneous materials through which more accurate characterization and physical evaluations can be achieved. While being trained using only 2D images, the proposed framework can be utilized to generate 3D images. The proposed method is applied to a few realistic 3D images of lithium-ion battery electrodes. The results indicate that the implemented method can reproduce important structural properties while the flow and heat properties are within an acceptable range.

Improved recurrent generative model for reconstructing large-size porous media from two-dimensional images

Modeling the three-dimensional (3D) structure from a given 2D image is of great importance for analyzing and studying the physical properties of porous media. As an intractable inverse problem, many methods have been developed to address this fundamental problems over the past decades. Among many methods, the deep learning-(DL) based methods show great advantages in terms of accuracy, diversity, and efficiency. Usually, the 3D reconstruction from the 2D slice with a larger field-of-view is more conducive to simulate and analyze the physical properties of porous media accurately. However, due to the limitation of reconstruction ability, the reconstruction size of most widely used generative adversarial network-based model is constrained to 64^{3} or 128^{3}. Recently, a 3D porous media recurrent neural network based method (namely, 3D-PMRNN) (namely 3D-PMRNN) has been proposed to improve the reconstruction ability, and thus the reconstruction size is expanded to 256^{3}. Nevertheless, in order to train these models, the existed DL-based methods need to down-sample the original computed tomography (CT) image first so that the convolutional kernel can capture the morphological features of training images. Thus, the detailed information of the original CT image will be lost. Besides, the 3D reconstruction from a optical thin section is not available because of the large size of the cutting slice. In this paper, we proposed an improved recurrent generative model to further enhance the reconstruction ability (512^{3}). Benefiting from the RNN-based architecture, the proposed model requires only one 3D training sample at least and generates the 3D structures layer by layer. There are three more improvements: First, a hybrid receptive field for the kernel of convolutional neural network is adopted. Second, an attention-based module is merged into the proposed model. Finally, a useful section loss is proposed to enhance the continuity along the Z direction. Three experiments are carried out to verify the effectiveness of the proposed model. Experimental results indicate the good reconstruction ability of proposed model in terms of accuracy, diversity, and generalization. And the effectiveness of section loss is also proved from the perspective of visual inspection and statistical comparison.

On the generation of realistic synthetic petrographic datasets using a style-based GAN

Deep learning architectures have transformed data analytics in geosciences, complementing traditional approaches to geological problems. Although deep learning applications in geosciences show encouraging signs, their potential remains untapped due to limited data availability and the required in-depth knowledge to provide a high-quality labeled dataset. We approached these issues by developing a novel style-based deep generative adversarial network (GAN) model, PetroGAN, to create the first realistic synthetic petrographic datasets across different rock types. PetroGAN adopts the architecture of StyleGAN2 with adaptive discriminator augmentation (ADA) to allow robust replication of statistical and esthetical characteristics and improve the internal variance of petrographic data. In this study, the training dataset consists of > 10,000 thin section images both under plane- and cross-polarized lights. Here, using our proposed novel approach, the model reached a state-of-the-art Fréchet Inception Distance (FID) score of 12.49 for petrographic images. We further observed that the FID values vary with lithology type and image resolution. The generated images were validated through a survey where the participants have various backgrounds and level of expertise in geosciences. The survey established that even a subject matter expert observed the generated images were indistinguishable from real images. This study highlights that GANs are a powerful method for generating realistic synthetic data in geosciences. Moreover, they are a future tool for image self-labeling, reducing the effort in producing big, high-quality labeled geoscience datasets. Furthermore, our study shows that PetroGAN can be applied to other geoscience datasets, opening new research horizons in the application of deep learning to various fields in geosciences, particularly with the presence of limited datasets.

Performance Evaluation of Convolutional Auto Encoders for the Reconstruction of Li-Ion Battery Electrode Microstructure

Li-ion batteries play a critical role in the transition to a net-zero future. The discovery of new materials and the design of novel microstructures for battery electrodes is necessary for the acceleration of this transition. The battery electrode microstructure can potentially reveal the cells’ electrochemical characteristics in great detail. However, revealing this relation is very challenging due to the high dimensionality of the problem and the large number of microstructure features. In fact, it cannot be achieved via the traditional trial-and-error approaches, which are associated with significant cost, time, and resource waste. In search for a systematic microstructure analysis and design method, this paper aims at quantifying the Li-ion battery electrode structural characteristics via deep learning models. Deliberately, here, a methodology and framework are developed to reveal the hidden microstructure characteristics via 2D and 3D images through dimensionality reduction. The framework is based on an auto-encoder decoder for microstructure reconstruction and feature extraction. Unlike most of the existing studies that focus on a limited number of features extracted from images, this study concentrates directly on the images and has the potential to define the number of features to be extracted. The proposed methodology and model are computationally effective and have been tested on a real open-source dataset where the results show the efficiency of reconstruction and feature extraction based on the training and validation mean squared errors between 0.068 and 0.111 and from 0.071 to 0.110, respectively. This study is believed to guide Li-ion battery scientists and manufacturers in the design and production of next generation Li-ion cells in a systematic way by correlating the extracted features at the microstructure level and the cell’s electrochemical characteristics.

GEOENT: A Toolbox for Calculating Directional Geological Entropy

Geological entropy is based on Shannon information entropy and measures order in the structure of a spatial random variable. Metrics have been defined to quantify geological entropy in multidimensional (2D and 3D) heterogeneous systems, for instance, porous and fractured geological media. This study introduces GEOENT, a toolbox that can efficiently be used to calculate geological entropy metrics for any kind of input-gridded field. Additionally, the definition of geological entropy metrics is updated to consider anisotropy in the structure of the heterogeneous system. Directional entrograms provide more accurate descriptions of spatial order over different Cartesian directions. This study presents the development of the geological entropy metrics, a description of the toolbox, and examples of its applications in different datasets, including 2D and 3D gridded fields, representing a variety of heterogeneous environments at different scales, from pore-scale microtomography (μCT) images to aquifer analogues.

Generative adversarial networks for reconstruction of three-dimensional porous media from two-dimensional slices

In many branches of earth sciences, the problem of rock study on the microlevel arises. However, a significant number of representative samples is not always feasible. Thus the problem of the generation of samples with similar properties becomes actual. In this paper we propose a deep learning architecture for three-dimensional porous medium reconstruction from two-dimensional slices. We fit a distribution on all possible three-dimensional structures of a specific type based on the given data set of samples. Then, given partial information (central slices), we recover the three-dimensional structure around such slices as the most probable one according to that constructed distribution. Technically, we implement this in the form of a deep neural network with encoder, generator, and discriminator modules. Numerical experiments show that this method provides a good reconstruction in terms of Minkowski functionals.

Digital Rock Reconstruction with User-Defined Properties Using Conditional Generative Adversarial Networks

Uncertainty is ubiquitous with multiphase flow in subsurface rocks due to their inherent heterogeneity and lack of in-situ measurements. To complete uncertainty analysis in a multi-scale manner, it is a prerequisite to provide sufficient rock samples. Even though the advent of digital rock technology offers opportunities to reproduce rocks, it still cannot be utilized to provide massive samples due to its high cost, thus leading to the development of diversified mathematical methods. Among them, two-point statistics (TPS) and multi-point statistics (MPS) are commonly utilized, which feature incorporating low-order and high-order statistical information, respectively. Recently, generative adversarial networks (GANs) are becoming increasingly popular since they can reproduce training images with excellent visual and consequent geologic realism. However, standard GANs can only incorporate information from data, while leaving no interface for user-defined properties, and thus may limit the representativeness of reconstructed samples. In this study, we propose conditional GANs for digital rock reconstruction, aiming to reproduce samples not only similar to the real training data, but also satisfying user-specified properties. In fact, the proposed framework can realize the targets of MPS and TPS simultaneously by incorporating high-order information directly from rock images with the GANs scheme, while preserving low-order counterparts through conditioning. We conduct three reconstruction experiments, and the results demonstrate that rock type, rock porosity, and correlation length can be successfully conditioned to affect the reconstructed rock images. The randomly reconstructed samples with specified rock type, porosity and correlation length will contribute to the subsequent research on pore-scale multiphase flow and uncertainty quantification.

Deep-learning-based porous media microstructure quantitative characterization and reconstruction method

Microstructure characterization and reconstruction (MCR) is one of the most important components of discovering processing-structure-property relations of porous media behavior and inverse porous media design in computational materials science. Since the algorithms for describing and controlling the geometric configuration of microstructures need to solve a large number of variables and involve multiobjective conditions, the existing MCR methods have difficulty in gaining a perfect trade-off among the quantitative generation and characterization capability and the reconstruction quality. In this work, an improved 3D Porous Media Microstructure (3DPmmGAN) generative adversarial network based on deep-learning algorithm is demonstrated for high-quality microstructures generation with high controllability and high prediction accuracy. The proposed 3DPmmGAN allows the model to utilize unlabeled data for complex high-randomness microstructures end-to-end training within an acceptable time consumption. Further analysis shows that the trained network has good adaptivity for microstructures with different random geometric configurations, and can quantitatively control the generated structure according to semantic conditions, and can also quantitatively predict complex microstructure features. The key results suggest the proposed 3DPmmGAN is a powerful tool to accelerate the preparation and the initial characterization of 3D porous media, and potentially maximize the design efficiency for porous media.

Fast and scalable earth texture synthesis using spatially assembled generative adversarial neural networks

The earth texture with complex morphological geometry and compositions such as shale and carbonate rocks, is typically characterized with sparse field samples because of an expensive and time-consuming characterization process. Accordingly, generating arbitrary large size of the geological texture with similar topological structures at a low computation cost has become one of the key tasks for realistic geomaterial reconstruction and subsequent hydro-mechanical evaluation for science and engineering applications. Recently, generative adversarial neural networks (GANs) have demonstrated a potential of synthesizing input textural images and creating equiprobable geomaterial images for stochastic analysis of hydrogeological properties, for example, the feasibility of CO₂ storage sites and exploration of unconventional resources. However, the texture synthesis with the GANs framework is often limited by the computational cost and scalability of the output texture size. In this study, we proposed a spatially assembled GANs (SAGANs) that can generate output images of an arbitrary large size regardless of the size of training images with computational efficiency. The performance of the SAGANs was evaluated with two and three dimensional (2D and 3D) rock image samples widely used in geostatistical reconstruction of the earth texture and Lattice-Boltzmann (LB) simulations were performed to compare pore-scale flow patterns and upscaled permeabilities of training and generated geomaterial images. We demonstrate SAGANs can generate the arbitrary large size of statistical realizations with connectivity and structural properties and flow characteristics similar to training images, and also can generate a variety of realizations even on a single training image. In addition, the computational time was significantly improved compared to standard GANs frameworks.

Three-dimensional reconstruction of porous media using super-dimension-based adjacent block-matching algorithm

As porous media play an essential role in a variety of industrial applications, it is essential to understand their physical properties. Nowadays, the super-dimensional (SD) reconstruction algorithm is used to stochastically reconstruct a three-dimensional (3D) structure of porous media from a given two-dimensional image. This algorithm exhibits superiority in accuracy compared with classical algorithms because it learns information from the real 3D structure. However, owing to the short development time of the SD algorithm, it also has some limitations, such as inexact porosity characterization, long run time, blocking artifacts, and suboptimal accuracy that may be improved. To mitigate these limitations, this study presents the design of a special template that contains two parts of data (i.e., adjacent blocks and a central block); the proposed method matches adjacent blocks during reconstruction and assigns the matched central block to the area to be reconstructed. Furthermore, we design two important mechanisms during reconstruction: one for block matching and the other for porosity control. To verify the effectiveness of the proposed method compared with an existing SD method, both methods were tested on silica particle material and three homogeneous sandstones with different porosities; meanwhile, we compared the proposed method with a multipoint statistics method and a simulated annealing method. The reconstructed results were then compared with the target both visually and quantitatively. The experimental results indicate that the proposed method can overcome the aforementioned limitations and further improve the accuracy of existing methods. This method achieved 4-6 speedup factor compared with the traditional SD method.

2D-to-3D image translation of complex nanoporous volumes using generative networks

Image-based characterization offers a powerful approach to studying geological porous media at the nanoscale and images are critical to understanding reactive transport mechanisms in reservoirs relevant to energy and sustainability technologies such as carbon sequestration, subsurface hydrogen storage, and natural gas recovery. Nanoimaging presents a trade off, however, between higher-contrast sample-destructive and lower-contrast sample-preserving imaging modalities. Furthermore, high-contrast imaging modalities often acquire only 2D images, while 3D volumes are needed to characterize fully a source rock sample. In this work, we present deep learning image translation models to predict high-contrast focused ion beam-scanning electron microscopy (FIB-SEM) image volumes from transmission X-ray microscopy (TXM) images when only 2D paired training data is available. We introduce a regularization method for improving 3D volume generation from 2D-to-2D deep learning image models and apply this approach to translate 3D TXM volumes to FIB-SEM fidelity. We then segment a predicted FIB-SEM volume into a flow simulation domain and calculate the sample apparent permeability using a lattice Boltzmann method (LBM) technique. Results show that our image translation approach produces simulation domains suitable for flow visualization and allows for accurate characterization of petrophysical properties from non-destructive imaging data.

Adaptive phase-retrieval stochastic reconstruction with correlation functions: Three-dimensional images from two-dimensional cuts

Precise characterization of three-dimensional (3D) heterogeneous media is indispensable in finding the relationships between structure and macroscopic physical properties (permeability, conductivity, and others). The most widely used experimental methods (electronic and optical microscopy) provide high-resolution bidimensional images of the samples of interest. However, 3D material inner microstructure registration is needed to apply numerous modeling tools. Numerous research areas search for cheap and robust methods to obtain the full 3D information about the structure of the studied sample from its 2D cuts. In this work, we develop an adaptive phase-retrieval stochastic reconstruction algorithm that can create 3D replicas from 2D original images, APR. The APR is free of artifacts characteristic of previously proposed phase-retrieval techniques. While based on a two-point S_{2} correlation function, any correlation function or other morphological metrics can be accounted for during the reconstruction, thus, paving the way to the hybridization of different reconstruction techniques. In this work, we use two-point probability and surface-surface functions for optimization. To test APR, we performed reconstructions for three binary porous media samples of different genesis: sandstone, carbonate, and ceramic. Based on computed permeability and connectivity (C_{2} and L_{2} correlation functions), we have shown that the proposed technique in terms of accuracy is comparable to the classic simulated annealing-based reconstruction method but is computationally very effective. Our findings open the possibility of utilizing APR to produce fast or crude replicas further polished by other reconstruction techniques such as simulated annealing or process-based methods. Improving the quality of reconstructions based on phase retrieval by adding additional metrics into the reconstruction procedure is possible for future work.

Stochastic characterization and reconstruction of material microstructures for establishment of process-structure-property linkage using the deep generative model

In material design, microstructure characterization and reconstruction are indispensable for understanding the role of a structure in a process-structure-property relation. The significant contribution of this paper is to introduce a methodology for the characterization and generation of material microstructures using deep generative networks as the first step in the establishment of a process-structure-property linkage for forward or inverse material design. Our approach can be divided into two parts: (i) characterization of material microstructures by a vector quantized variational auto-encoder, and (ii) determination of the correlation between the extracted microstructure characterizations and the given conditions, such as processing parameters and/or material properties, by a pixel convolutional neural network. As an example, we tested our framework in the generation of low-carbon-steel microstructures from the given material processing. The results were in satisfactory agreement with the experimental observation qualitatively and quantitatively, demonstrating the potential of applying the proposed method to forward or inverse material design. One of the advantages of the proposed methodology lies in the capability to capture the stochastic nature behind the microstructure generation. As a result, this methodology enables us to build a process-structure-property linkage while quantifying uncertainties, which not only makes a prediction more robust but also shows a way toward enhancing our understanding of the stochastic competitive phenomena behind the generation of material microstructures.

Can Deep Learning Extract Useful Information about Energy Dissipation and Effective Hydraulic Conductivity from Gridded Conductivity Fields?

We confirm that energy dissipation weighting provides the most accurate approach to determining the effective hydraulic conductivity (Keff) of a binary K grid. A deep learning algorithm (UNET) can infer Keff with extremely high accuracy (R2 > 0.99). The UNET architecture could be trained to infer the energy dissipation weighting pattern from an image of the K distribution, although it was less accurate for cases with highly localized structures that controlled flow. Furthermore, the UNET architecture learned to infer the energy dissipation weighting even if it was not trained directly on this information. However, the weights were represented within the UNET in a way that was not immediately interpretable by a human user. This reiterates the idea that even if ML/DL algorithms are trained to make some hydrologic predictions accurately, they must be designed and trained to provide each user-required output if their results are to be used to improve our understanding of hydrologic systems.

Mesoscopic and multiscale modelling in materials

The concept of multiscale modelling has emerged over the last few decades to describe procedures that seek to simulate continuum-scale behaviour using information gleaned from computational models of finer scales in the system, rather than resorting to empirical constitutive models. A large number of such methods have been developed, taking a range of approaches to bridging across multiple length and time scales. Here we introduce some of the key concepts of multiscale modelling and present a sampling of methods from across several categories of models, including techniques developed in recent years that integrate new fields such as machine learning and material design.

Computationally Efficient Multiscale Neural Networks Applied to Fluid Flow in Complex 3D Porous Media

The permeability of complex porous materials is of interest to many engineering disciplines. This quantity can be obtained via direct flow simulation, which provides the most accurate results, but is very computationally expensive. In particular, the simulation convergence time scales poorly as the simulation domains become less porous or more heterogeneous. Semi-analytical models that rely on averaged structural properties (i.e., porosity and tortuosity) have been proposed, but these features only partly summarize the domain, resulting in limited applicability. On the other hand, data-driven machine learning approaches have shown great promise for building more general models by virtue of accounting for the spatial arrangement of the domains’ solid boundaries. However, prior approaches building on the convolutional neural network (ConvNet) literature concerning 2D image recognition problems do not scale well to the large 3D domains required to obtain a representative elementary volume (REV). As such, most prior work focused on homogeneous samples, where a small REV entails that the global nature of fluid flow could be mostly neglected, and accordingly, the memory bottleneck of addressing 3D domains with ConvNets was side-stepped. Therefore, important geometries such as fractures and vuggy domains could not be modeled properly. In this work, we address this limitation with a general multiscale deep learning model that is able to learn from porous media simulation data. By using a coupled set of neural networks that view the domain on different scales, we enable the evaluation of large ($$>512^3$$ > 512 3 ) images in approximately one second on a single graphics processing unit. This model architecture opens up the possibility of modeling domain sizes that would not be feasible using traditional direct simulation tools on a desktop computer. We validate our method with a laminar fluid flow case using vuggy samples and fractures. As a result of viewing the entire domain at once, our model is able to perform accurate prediction on domains exhibiting a large degree of heterogeneity. We expect the methodology to be applicable to many other transport problems where complex geometries play a central role.

A Three-Dimensional Microstructure Reconstruction Framework for Permeable Pavement Analysis Based on 3D-IWGAN with Enhanced Gradient Penalty

Owing to the increasing use of permeable pavement, there is a growing need for studies that can improve its design and durability. One of the most important factors that can reduce the functionality of permeable pavement is the clogging issue. Field experiments for investigating the clogging potential are relatively expensive owing to the high-cost testing equipment and materials. Moreover, a lot of time is required for conducting real physical experiments to obtain physical properties for permeable pavement. In this paper, to overcome these limitations, we propose a three-dimensional microstructure reconstruction framework based on 3D-IDWGAN with an enhanced gradient penalty, which is an image-based computational system for clogging analysis in permeable pavement. Our proposed system first takes a two-dimensional image as an input and extracts latent features from the 2D image. Then, it generates a 3D microstructure image through the generative adversarial network part of our model with the enhanced gradient penalty. For checking the effectiveness of our system, we utilize the reconstructed 3D image combined with the numerical method for pavement microstructure analysis. Our results show improvements in the three-dimensional image generation of the microstructure, compared with other generative adversarial network methods, and the values of physical properties extracted from our model are similar to those obtained via real pavement samples.

Machine learning-based microstructure prediction during laser sintering of alumina

Predicting material’s microstructure under new processing conditions is essential in advanced manufacturing and materials science. This is because the material’s microstructure hugely influences the material’s properties. We demonstrate an elegant machine learning algorithm that faithfully predicts the microstructure under new conditions, without the need of knowing the governing laws. We name this algorithm, RCWGAN-GP, which is regression-based conditional generative adversarial networks with Wasserstein loss function and gradient penalty. This algorithm was trained with experimental SEM micrographs from laser-sintered alumina under various laser powers. The RCWGAN-GP realistically regenerates the SEM micrographs under the trained laser powers. Impressively, it also faithfully predicts the alumina’s microstructure under unexplored laser powers. The predicted microstructure features, including the morphology of the sintered particles and the pores, match the experimental SEM micrographs very well. We further quantitatively examined the prediction accuracy of the RCWGAN-GP. We trained the algorithm with computer-created micrograph datasets of secondary-phase growth governed by the well-known Johnson–Mehl–Avrami (JMA) equation. The RCWGAN-GP accurately regenerates the micrographs at the trained time series, in terms of the grains’ shapes, sizes, and spatial distributions. More importantly, the predicted secondary phase fraction accurately follows the JMA curve.

Determination of the throat size distribution of a porous medium as an inverse optimization problem combining pore network modeling and genetic and hill climbing algorithms

The pore size distribution of a porous medium is often estimated from the retention curve or the invading fluid flow rate curve using simple relationships more or less explicitly based on the consideration that the porous medium is made of a bundle of cylindrical parallel tubes. This type of determination is tested using pore network simulations. Starting from two- or three-dimensional networks, the characteristics of which are known a priori, the estimation of the throat size distribution (TSD) is performed using the standard methods in the case of drainage. Results show a significant discrepancy with the input data. The disagreement is more pronounced when the fluid flow rate curve is employed together with the parallel tubes assumption. The physical origins of these shortcomings are identified. A method, based on pore network simulations combined with a genetic algorithm and the hill climbing algorithm, is then designed, which makes simultaneous use of the nonwetting fluid flow rate curve and the retention curve of the medium. Very significant improvement is achieved in the estimation of the TSD using this procedure.

Upscaling the porosity-permeability relationship of a microporous carbonate for Darcy-scale flow with machine learning

The permeability of a pore structure is typically described by stochastic representations of its geometrical attributes (e.g. pore-size distribution, porosity, coordination number). Database-driven numerical solvers for large model domains can only accurately predict large-scale flow behavior when they incorporate upscaled descriptions of that structure. The upscaling is particularly challenging for rocks with multimodal porosity structures such as carbonates, where several different type of structures (e.g. micro-porosity, cavities, fractures) are interacting. It is the connectivity both within and between these fundamentally different structures that ultimately controls the porosity–permeability relationship at the larger length scales. Recent advances in machine learning techniques combined with both numerical modelling and informed structural analysis have allowed us to probe the relationship between structure and permeability much more deeply. We have used this integrated approach to tackle the challenge of upscaling multimodal and multiscale porous media. We present a novel method for upscaling multimodal porosity–permeability relationships using machine learning based multivariate structural regression. A micro-CT image of Estaillades limestone was divided into small 60³ and 120³ sub-volumes and permeability was computed using the Darcy–Brinkman–Stokes (DBS) model. The microporosity–porosity–permeability relationship from Menke et al. (Earth Arxiv, https://doi.org/10.31223/osf.io/ubg6p, 2019) was used to assign permeability values to the cells containing microporosity. Structural attributes (porosity, phase connectivity, volume fraction, etc.) of each sub-volume were extracted using image analysis tools and then regressed against the solved DBS permeability using an Extra-Trees regression model to derive an upscaled porosity–permeability relationship. Ten test cases of 360³ voxels were then modeled using Darcy-scale flow with this machine learning predicted upscaled porosity–permeability relationship and benchmarked against full DBS simulations, a numerically upscaled Darcy flow model, and a Kozeny–Carman model. All numerical simulations were performed using GeoChemFoam, our in-house open source pore-scale simulator based on OpenFOAM. We found good agreement between the full DBS simulations and both the numerical and machine learning upscaled models, with the machine learning model being 80 times less computationally expensive. The Kozeny–Carman model was a poor predictor of upscaled permeability in all cases.

Connectivity-informed drainage network generation using deep convolution generative adversarial networks

Stochastic network modeling is often limited by high computational costs to generate a large number of networks enough for meaningful statistical evaluation. In this study, Deep Convolutional Generative Adversarial Networks (DCGANs) were applied to quickly reproduce drainage networks from the already generated network samples without repetitive long modeling of the stochastic network model, Gibb's model. In particular, we developed a novel connectivity-informed method that converts the drainage network images to the directional information of flow on each node of the drainage network, and then transforms it into multiple binary layers where the connectivity constraints between nodes in the drainage network are stored. DCGANs trained with three different types of training samples were compared; (1) original drainage network images, (2) their corresponding directional information only, and (3) the connectivity-informed directional information. A comparison of generated images demonstrated that the novel connectivity-informed method outperformed the other two methods by training DCGANs more effectively and better reproducing accurate drainage networks due to its compact representation of the network complexity and connectivity. This work highlights that DCGANs can be applicable for high contrast images common in earth and material sciences where the network, fractures, and other high contrast features are important.

RockFlow: Fast Generation of Synthetic Source Rock Images Using Generative Flow Models

Image-based evaluation methods are a valuable tool for source rock characterization. The time and resources needed to obtain images has spurred development of machine-learning generative models to create synthetic images of pore structure and rock fabric from limited image data. While generative models have shown success, existing methods for generating 3D volumes from 2D training images are restricted to binary images and grayscale volume generation requires 3D training data. Shale characterization relies on 2D imaging techniques such as scanning electron microscopy (SEM), and grayscale values carry important information about porosity, kerogen content, and mineral composition of the shale. Here, we introduce RockFlow, a method based on generative flow models that creates grayscale volumes from 2D training data. We apply RockFlow to baseline binary micro-CT image volumes and compare performance to a previously proposed model. We also show the extension of our model to 2D grayscale data by generating grayscale image volumes from 2D SEM and dual modality nanoscale shale images. The results show that our method underestimates the porosity and surface area on the binary baseline datasets but is able to generate realistic grayscale image volumes for shales. With improved binary data preprocessing, we believe that our model is capable of generating synthetic porous media volumes for a very broad class of rocks from shale to carbonates to sandstone.

Using Generative Adversarial Networks as a Fast Forward Operator for Hydrogeological Inverse Problems

Subsurface characterization using inverse techniques constitutes one of the fundamental elements of hydrogeological modeling applications. Available methods to solve inverse problems rely on a forward operator that predicts state variables for a given set of subsurface parameters. As the number of model parameters to be estimated increases, forward operations incur a significant computational demand. In this paper, we investigate the use of conditional generative adversarial networks (cGAN) as an emulator for the forward operator in the context of a hydrogeological inverse problem. We particularly investigate if the cGAN can be used to replace the forward operator used in the adaptive importance sampling method posterior population expansion (PoPEx) with reasonable accuracy and feasible computation requirement. The cGAN model trained on channelized geological structures has shown that the cGAN is able to reproduce the state variables corresponding to a certain parameter field. Hence, its integration in PoPEx yielded satisfactory results. In terms of the computational demand, the use of cGAN as a surrogate forward model reduces the required computational time up to 80% for the problem defined in the study. However, the training time required to create a model seems to be the major drawback of the method.

Efficient uncertainty quantification for permeability of three-dimensional porous media through image analysis and pore-scale simulations

In this paper, we propose an efficient coupled approach for uncertainty quantification of permeability for randomly reconstructed three-dimensional (3D) pore images, where the porosity and two-point correlations of a realistic sandstone sample are honored. The Joshi-Quiblier-Adler approach and Karhunen-Loève expansion are utilized for quick reconstruction of 3D pore images with reduced random dimensionality. The eigenvalue problem for the covariance matrix of 3D intermediate Gaussian random fields is solved equivalently by a kernel method. Then, the lattice Boltzmann method is adopted to simulate fluid flow in reconstructed pore space and evaluate permeability. Lastly, the sparse polynomial chaos expansion (sparse PCE) integrated with a feature selection method is employed to predict permeability distributions incurred by the randomness in microscopic pore structures. The feature selection process, which is intended to discard redundant basis functions, is carried out by the least absolute shrinkage and selection operator-modified least angle regression along with cross validation. The competence of our proposed approach is validated by the results from Monte Carlo simulation. It reveals that a small number of samples is sufficient for sparse PCE with feature selection to produce convincing results. Then, we utilize our method to quantify the uncertainty of permeability under different porosities and correlation parameters. It is found that the predicted permeability distributions for reconstructed 3D pore images are close to experimental measurements of Berea sandstones in the literature. In addition, the results show that porosity and correlation length are the critical influence factors for the uncertainty of permeability.

Three-dimensional multiscale fusion for porous media on microtomography images of different resolutions

Accurately acquiring the three-dimensional (3D) image of a porous medium is an imperative issue for the prediction of multiple physical properties. Considering the inherent nature of the multiscale pores contained in porous media such as tight sandstones, to completely characterize the pore structure, one needs to scan the microstructure at different resolutions. Specifically, low-resolution (LR) images cover a larger field of view (FOV) of the sample, but are lacking small-scale features, whereas high-resolution (HR) images contain ample information, but sometimes only cover a limited FOV. To address this issue, we propose a method for fusing the spatial information from a two-dimensional (2D) HR image into a 3D LR image, and finally reconstructing an integrated 3D structure with added fine-scale features. In the fusion process, the large-scale structure depicted by the 3D LR image is fixed as background and the 2D image is utilized as training image to reconstruct a small-scale structure based on the background. To assess the performance of our method, we test it on a sandstone scanned with low and high resolutions. Statistical properties between the reconstructed image and the target are quantitatively compared. The comparison indicates that the proposed method enables an accurate fusion of the LR and HR images because the small-scale information is precisely reproduced within the large one.

Microstructure synthesis using style-based generative adversarial networks

This work considers the usage of StyleGAN architecture for the task of microstructure synthesis. The task is the following: Given number of samples of structure we try to generate similar samples at the same time preserving its properties. Since the considered architecture is not able to produce samples of sizes larger than the training images, we propose to use image quilting to merge fixed-sized samples. One of the key features of the considered architecture is that it uses multiple image resolutions. We also investigate the necessity of such an approach.