Hierarchical Optimization: Fast and Robust Multiscale Stochastic Reconstructions with Rescaled Correlation Functions

Measuring structural nonstationarity: The use of imaging information to quantify homogeneity and inhomogeneity

Heterogeneity is the concept we encounter in numerous research areas and everyday life. While "not mixing apples and oranges" is easy to grasp, a more quantitative approach to such segregation is not always readily available. Consider the problem from a different angle: To what extent does one have to make apples more orange and oranges more "apple-shaped" to put them into the same basket (according to their appearance alone)? This question highlights the central problem of the blurred interface between heterogeneous and homogeneous, which also depends on the metrics used for its identification. This work uncovers the physics of structural stationarity quantification, based on correlation functions (CFs) and clustering based on CFs different between image subregions. By applying the methodology to a wide variety of synthetic and real images of binary porous media, we confirmed computationally that only periodically unit-celled structures and images produced by stationary processes with resolutions close to infinity are strictly stationary. Natural structures without recurring unit cells are only weakly stationary. We established a physically meaningful definition for these stationarity types and their distinction from nonstationarity. In addition, the importance of information content of the chosen metrics is highlighted and discussed. We believe the methodology as proposed in this contribution will find its way into numerous research areas dealing with materials, structures, and measurements and modeling based on structural imaging information.

Computational design of anisotropic stealthy hyperuniform composites with engineered directional scattering properties

Disordered hyperuniform materials are an emerging class of exotic amorphous states of matter that endow them with singular physical properties, including large isotropic photonic band gaps, superior resistance to fracture, and nearly optimal electrical and thermal transport properties, to name but a few. Here we generalize the Fourier-space-based numerical construction procedure for designing and generating digital realizations of isotropic disordered hyperuniform two-phase heterogeneous materials (i.e., composites) developed by Chen and Torquato [Acta Mater. 142, 152 (2018)1359-645410.1016/j.actamat.2017.09.053] to anisotropic microstructures with targeted spectral densities. Our generalized construction procedure explicitly incorporates the vector-dependent spectral density function χ[over ̃]_{_{V}}(k) of arbitrary form that is realizable. We demonstrate the utility of the procedure by generating a wide spectrum of anisotropic stealthy hyperuniform microstructures with χ[over ̃]_{_{V}}(k)=0 for k∈Ω, i.e., complete suppression of scattering in an "exclusion" region Ω around the origin in Fourier space. We show how different exclusion-region shapes with various discrete symmetries, including circular-disk, elliptical-disk, square, rectangular, butterfly-shaped, and lemniscate-shaped regions of varying size, affect the resulting statistically anisotropic microstructures as a function of the phase volume fraction. The latter two cases of Ω lead to directionally hyperuniform composites, which are stealthy hyperuniform only along certain directions and are nonhyperuniform along others. We find that while the circular-disk exclusion regions give rise to isotropic hyperuniform composite microstructures, the directional hyperuniform behaviors imposed by the shape asymmetry (or anisotropy) of certain exclusion regions give rise to distinct anisotropic structures and degree of uniformity in the distribution of the phases on intermediate and large length scales along different directions. Moreover, while the anisotropic exclusion regions impose strong constraints on the global symmetry of the resulting media, they can still possess structures at a local level that are nearly isotropic. Both the isotropic and anisotropic hyperuniform microstructures associated with the elliptical-disk, square, and rectangular Ω possess phase-inversion symmetry over certain range of volume fractions and a percolation threshold ϕ_{c}≈0.5. On the other hand, the directionally hyperuniform microstructures associated with the butterfly-shaped and lemniscate-shaped Ω do not possess phase-inversion symmetry and percolate along certain directions at much lower volume fractions. We also apply our general procedure to construct stealthy nonhyperuniform systems. Our construction algorithm enables one to control the statistical anisotropy of composite microstructures via the shape, size, and symmetries of Ω, which is crucial to engineering directional optical, transport, and mechanical properties of two-phase composite media.

Robust surface-correlation-function evaluation from experimental discrete digital images

Correlation functions (CFs) are universal structural descriptors; surface-surface F_{ss} and surface-void F_{sv} CFs are a subset containing additional information about the interface between the phases. The description of the interface between pores and solids in porous media is of particular importance and recently Ma and Torquato [Phys. Rev. E 98, 013307 (2018)2470-004510.1103/PhysRevE.98.013307] proposed an elegant way to compute these functions for a wide variety of cases. However, their "continuous" approach is not always applicable to digital experimental 2D and 3D images of porous media as obtained using x-ray tomography or scanning electron microscopy due to nonsingularities in chemical composition or local solid material's density and partial volume effects. In this paper we propose to use edge-detecting filters to compute surface CFs in the "digital" fashion directly in the images. Computed this way, surface correlation functions are the same as analytically known for Poisson disks in case the resolution of the image is adequate. Based on the multiscale image analysis we developed a C_{0.5} criterion that can predict if the imaging resolution is enough to make an accurate evaluation of the surface CFs. We also showed that in cases when the input image contains all major features, but do not pass the C_{0.5} criterion, it is possible with the help of image magnification to sample CFs almost similar to those obtained for high-resolution image of the same structure with high C_{0.5}. The computational framework as developed here is open source and available within the CorrelationFunctions.jl package developed by our group. Our "digital" approach was applied to a wide variety of real porous media images of different quality. We discuss critical aspects of surface correlation functions computations as related to different applications. The developed methodology allows applying surface CFs to describe the structure of porous materials based on their experimental images and enhance stochastic reconstructions or super-resolution procedures, or serve as an efficient metrics in machine learning applications due to computationally effective GPU implementation.

A circle/sphere populating method to generate 2D/3D stochastic microstructures

A circle/sphere populating method is proposed to generate 2D/3D stochastic microstructures. The proposed method uses circles/spheres as the basic elements and generates microstructure features through the populating process of the circles/spheres. In the populating process, the cores are first generated randomly and circles/spheres start to populate around the cores or the previous generation's circles/spheres. The populating process is controlled by the input parameters including the volume fraction, core number, circle/sphere size distribution, circle/sphere populating distance distribution, circle/sphere populating number, and populating direction constraint angle. The proposed method was compared with the QSGS method and random circle/sphere method in 2-dimensional (2D) and 3-dimensional (3D) cases. The proposed method shows advantages in generating microstructures with clear feature geometries and boundaries. Furthermore, parametric studies are conducted in 2D and 3D to investigate the effect of input parameters on the generated microstructures. With the consideration of circle/sphere spatial distributions, the proposed method can achieve different degrees of feature clustering and agglomerating. A wide range of microstructure morphologies can be achieved by adjusting the input parameters. A more accurate description of the features in the microstructures can be achieved without the involvement of the annealing-based optimization process. As a case study, the proposed method was used to generate sandstone microstructures with different grain size distributions and spatial distributions, and the permeability of generated sandstone was analyzed. Furthermore, the proposed method was applied to generate the microstructure model with a target radial distribution function to demonstrate its computational efficiency by comparing it with the random sphere method and simulated annealing based method.

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.

Pore-network extraction using discrete Morse theory: Preserving the topology of the pore space

Pore-scale modeling based on the 3D structural information of porous materials has enormous potential in assessing physical properties beyond the capabilities of laboratory methods. Such capabilities are pricey in terms of computational expenses, and this limits the applicability of the direct simulations to a small volume and requires high-performance computational resources, especially for multiphase flow simulations. The only pore-scale technique capable of dealing with large representative volumes of porous samples is pore-network (PNM) based modeling. The problem of the PNM approach is that 3D pore geometry first needs to be simplified into a graph of pores and throats that conserve topological and geometrical properties of the original 3D image. While significant progress has been achieved in terms of geometry representation, no methodology provides full conservation of the topological features of the pore structure. In this paper we present a pore-network extraction algorithm for binary 3D images based on discrete Morse theory and persistent homology that by design targets topology preservation. In addition to methodological developments, we also clarify the relationship between topological characteristics of constructed Morse chain complex and pore-network elements. We show that the Euler numbers calculated for PNMs based on our methodology coincide with those obtained using the direct topological analysis. The characteristics of the extracted pore network are calculated for several 3D porous binary images and compared with the results of maximum inscribed balls-based and watershed-based approaches as well as a hybrid approach to support our methodology.

Ultraefficient reconstruction of effectively hyperuniform disordered biphase materials via non-Gaussian random fields

Disordered hyperuniform systems are statistically isotropic and possess no Bragg peaks like liquids and glasses, yet they suppress large-scale density fluctuations in a similar manner as in perfect crystals. The unique hyperuniform long-range order in these systems endow them with nearly optimal transport, electronic, and mechanical properties. The concept of hyperuniformity was originally introduced for many-particle systems and has subsequently been generalized to biphase heterogeneous materials such as porous media, composites, polymers, and biological tissues for unconventional property discovery. Existing methods for rendering realizations of disordered hyperuniform biphase materials reconstruction typically employ stochastic optimization such as the simulated annealing approach, which requires many iterations. Here, we propose an explicit ultraefficient method for reconstructing effectively hyperuniform biphase materials, based on the second-order non-Gaussian random fields where no additional tuning step or iteration is needed. Both the effectively hyperuniform microstructure and the latent material property field can be simultaneously generated in a single reconstruction. Moreover, our method can also incorporate hierarchical uncertainties in the heterogeneous materials, including both uncertainties in the disordered material microstructure and material property variation within each phase. The efficiency and feasibility of the proposed reconstruction method are demonstrated via a wide spectrum of examples spanning from isotropic to anisotropic, effectively hyperuniform to nonhyperuniform, and antihyperuniform systems. Our ultraefficient reconstruction method can be readily incorporated into material design, probabilistic analysis, optimization, and discovery of novel disordered hyperuniform heterogeneous materials.

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.

Quantifying microstructural evolution via time-dependent reduced-dimension metrics based on hierarchical n-point polytope functions

We devise reduced-dimension metrics for effectively measuring the distance between two points (i.e., microstructures) in the microstructure space and quantifying the pathway associated with microstructural evolution, based on a recently introduced set of hierarchical n-point polytope functions P_{n}. The P_{n} functions provide the probability of finding particular n-point configurations associated with regular n polytopes in the material system, and are a special subset of the standard n-point correlation functions S_{n} that effectively decompose the structural features in the system into regular polyhedral basis with different symmetries. The nth order metric Ω_{n} is defined as the L_{1} norm associated with the P_{n} functions of two distinct microstructures. By choosing a reference initial state (i.e., a microstructure associated with t_{0}=0), the Ω_{n}(t) metrics quantify the evolution of distinct polyhedral symmetries and can in principle capture emerging polyhedral symmetries that are not apparent in the initial state. To demonstrate their utility, we apply the Ω_{n} metrics to a two-dimensional binary system undergoing spinodal decomposition to extract the phase separation dynamics via the temporal scaling behavior of the corresponding Ω_{n}(t), which reveals mechanisms governing the evolution. Moreover, we employ Ω_{n}(t) to analyze pattern evolution during vapor deposition of phase-separating alloy films with different surface contact angles, which exhibit rich evolution dynamics including both unstable and oscillating patterns. The Ω_{n} metrics have potential applications in establishing quantitative processing-structure-property relationships, as well as real-time processing control and optimization of complex heterogeneous material systems.

Understanding degeneracy of two-point correlation functions via Debye random media

It is well known that the degeneracy of two-phase microstructures with the same volume fraction and two-point correlation function S_{2}(r) is generally infinite. To elucidate the degeneracy problem explicitly, we examine Debye random media, which are entirely defined by a purely exponentially decaying two-point correlation function S_{2}(r). In this work, we consider three different classes of Debye random media. First, we generate the "most probable" class using the Yeong-Torquato construction algorithm [Yeong and Torquato, Phys. Rev. E 57, 495 (1998)1063-651X10.1103/PhysRevE.57.495]. A second class of Debye random media is obtained by demonstrating that the corresponding two-point correlation functions are effectively realized in the first three space dimensions by certain models of overlapping, polydisperse spheres. A third class is obtained by using the Yeong-Torquato algorithm to construct Debye random media that are constrained to have an unusual prescribed pore-size probability density function. We structurally discriminate these three classes of Debye random media from one another by ascertaining their other statistical descriptors, including the pore-size, surface correlation, chord-length probability density, and lineal-path functions. We also compare and contrast the percolation thresholds as well as the diffusion and fluid transport properties of these degenerate Debye random media. We find that these three classes of Debye random media are generally distinguished by the aforementioned descriptors, and their microstructures are also visually distinct from one another. Our work further confirms the well-known fact that scattering information is insufficient to determine the effective physical properties of two-phase media. Additionally, our findings demonstrate the importance of the other two-point descriptors considered here in the design of materials with a spectrum of physical properties.

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.

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.

Descriptor-based method combined with partition to reconstruct three-dimensional complex microstructures

A descriptor-based method combined with a partition approach is proposed to reconstruct three-dimensional (3D) microstructures based on a set of two-dimensional (2D) scanning electron microscopy (SEM) images. The features in the SEM images are identified and partitioned into small features using the watershed algorithm. The watershed algorithm first finds the local gray-level maxima, and partitions the features through the gray-level local minima. The 3D size distribution and radial distribution of the small spherical elements are inferred, respectively, based on the 2D size distribution and radial distribution using stereological analysis. The 3D microstructures are reconstructed by matching the inferred size distribution and radial distribution through a simulated annealing-based procedure. Combining with the proposed partition approach, the descriptor-based method can be applied to complex microstructures and the computational efficiency of the reconstruction can be largely improved. A case study is presented using a set of 2D SEM images with nanoscale pore structure from the low-density CSH (calcium silicate hydrate) phase of a hardened cement paste. Cross sections were randomly selected from the reconstructed 3D microstructure and compared with the original SEM images using the pore descriptors and the two-point correlation function with satisfactory agreement. Using the 3D reconstructed model, the properties of the sample material can be investigated on such a small scale as demonstrated in this paper on quantifying the absolute permeability.

Application of percolation, critical-path, and effective-medium theories for calculation of two-phase relative permeability

There has been active development of numerical pore-network simulation of two-phase immiscible flows in porous media in recent years. These models allow for generation of capillary pressure and relative permeability curves. However, percolation models provide an efficient alternative, with reduced reliance on numerical techniques. Implementation of effective medium or critical path theory along with the percolation model allows for evaluation of the relative permeability curves. Both approximations failed to match the irreducible water saturation for water relative permeability. While the effective medium approximation poorly matches the pore network simulator, the critical path approximation is shown to match the result of the oil relative permeability. Despite the difference in end points, there is qualitative agreement between critical path approximation and the pore network simulator. Moreover, observed differences are not necessarily a drawback due to important boundary effects as discussed in the paper. Our results indicate that percolation-theory based predictions have the potential to become an efficient tool for upscaling by computing two-phase flow properties for numerous porosity subdomains.

Generation and structural characterization of Debye random media

In their seminal paper on scattering by an inhomogeneous solid, Debye and coworkers proposed a simple exponentially decaying function for the two-point correlation function of an idealized class of two-phase random media. Such Debye random media, which have been shown to be realizable, are singularly distinct from all other models of two-phase media in that they are entirely defined by their one- and two-point correlation functions. To our knowledge, there has been no determination of other microstructural descriptors of Debye random media. In this paper, we generate Debye random media in two dimensions using an accelerated Yeong-Torquato construction algorithm. We then ascertain microstructural descriptors of the constructed media, including their surface correlation functions, pore-size distributions, lineal-path function, and chord-length probability density function. Accurate semianalytic and empirical formulas for these descriptors are devised. We compare our results for Debye random media to those of other popular models (overlapping disks and equilibrium hard disks) and find that the former model possesses a wider spectrum of hole sizes, including a substantial fraction of large holes. Our algorithm can be applied to generate other models defined by their two-point correlation functions, and their other microstructural descriptors can be determined and analyzed by the procedures laid out here.

Probing information content of hierarchical n-point polytope functions for quantifying and reconstructing disordered systems

Disordered systems are ubiquitous in physical, biological, and material sciences. Examples include liquid and glassy states of condensed matter, colloids, granular materials, porous media, composites, alloys, packings of cells in avian retina, and tumor spheroids, to name but a few. A comprehensive understanding of such disordered systems requires, as the first step, systematic quantification, modeling, and representation of the underlying complex configurations and microstructure, which is generally very challenging to achieve. Recently, we introduced a set of hierarchical statistical microstructural descriptors, i.e., the "n-point polytope functions" P_{n}, which are derived from the standard n-point correlation functions S_{n}, and successively included higher-order n-point statistics of the morphological features of interest in a concise, explainable, and expressive manner. Here we investigate the information content of the P_{n} functions via optimization-based realization rendering. This is achieved by successively incorporating higher-order P_{n} functions up to n=8 and quantitatively assessing the accuracy of the reconstructed systems via unconstrained statistical morphological descriptors (e.g., the lineal-path function). We examine a wide spectrum of representative random systems with distinct geometrical and topological features. We find that, generally, successively incorporating higher-order P_{n} functions and, thus, the higher-order morphological information encoded in these descriptors leads to superior accuracy of the reconstructions. However, incorporating more P_{n} functions into the reconstruction also significantly increases the complexity and roughness of the associated energy landscape for the underlying stochastic optimization, making it difficult to convergence numerically.

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.

Multiscale modeling algorithm for core images

Computed tomography (CT) images of large core samples acquired by imaging equipment are insufficiently clear and ineffectively describe the tiny pore structure; conversely, images of small core samples are insufficiently globally representative. To alleviate these challenges, the idea of a super-resolution reconstruction algorithm is combined with that of a three-dimensional core reconstruction algorithm, and a multiscale core CT image fusion reconstruction algorithm is proposed. To obtain sufficient image quality with high resolution, a large-scale core image is used to provide global feature information as well as information regarding the basic morphological structure of a large-scale pore and particle. Then the texture pattern and the tiny pore distribution information of a small-scale core image is used to refine the coarse large-scale core image. A blind image quality assessment is utilized to estimate the degradation model of core images at different scales. A multilevel pattern mapping dictionary containing local binary patterns is designed to speed up the pattern matching procedure, and an adaptive weighted reconstruction algorithm is designed to reduce the blockiness. With our method, images of the same core at different scales were successfully fused. The proposed algorithm is extensively tested on microstructures of different rock samples; all cases of the reconstructed results and those of the actual sample were found to be in good agreement with each other. The final reconstructed image contains both large-scale and small-scale information that can provide a better understanding of the core samples and inform the accurate calculation of parameters.

Computed Tomography 3D Super-Resolution with Generative Adversarial Neural Networks: Implications on Unsaturated and Two-Phase Fluid Flow

Fluid flow characteristics are important to assess reservoir performance. Unfortunately, laboratory techniques are inadequate to know these characteristics, which is why numerical methods were developed. Such methods often use computed tomography (CT) scans as input but this technique is plagued by a resolution versus sample size trade-off. Therefore, a super-resolution method using generative adversarial neural networks (GANs) was used to artificially improve the resolution. Firstly, the influence of resolution on pore network properties and single-phase, unsaturated, and two-phase flow was analysed to verify that pores and pore throats become larger on average and surface area decreases with worsening resolution. These observations are reflected in increasingly overestimated single-phase permeability, less moisture uptake at lower capillary pressures, and high residual oil fraction after waterflooding. Therefore, the super-resolution GANs were developed which take low (12 µm) resolution input and increase the resolution to 4 µm, which is compared to the expected high-resolution output. These results better predicted pore network properties and fluid flow properties despite the overestimation of porosity. Relevant small pores and pore surfaces are better resolved thus providing better estimates of unsaturated and two-phase flow which can be heavily influenced by flow along pore boundaries and through smaller pores. This study presents the second case in which GANs were applied to a super-resolution problem on geological materials, but it is the first one to apply it directly on raw CT images and to determine the actual impact of a super-resolution method on fluid predictions.

Nested multiresolution hierarchical simulated annealing algorithm for porous media reconstruction

Microstructure strongly influences flow and transport properties of porous media. Flow and transport simulations within porous media, therefore, requires accurate three-dimensional (3D) models of the pore and solid phase structure. To date, no imaging method can resolve all relevant heterogeneities from the nano- to the centimeter scale within complex heterogeneous materials such as clay, reservoir rocks (e.g., travertine, chalk, ...), hardened cement paste, and concrete. To reconstruct these porous materials it is thus necessary to merge information from different 2D and potentially 3D imaging methods. One porous media reconstruction methodology that has been around for at least two decades is simulated annealing (SA). However, realizations with SA typically suffer an artificially reduced long-range connectivity, while multiphase reconstructions are not feasible in most cases because of a prohibitive computational burden. To solve these problems we propose a hierarchical multiresolution and multiphase simulated annealing algorithm. To decrease the computational cost of multiphase simulation, our algorithm sequentially simulates one phase after another, in a hierarchical way, which enables handling multimodal distributions and topological relations. Building upon recent work, our algorithm improves long-range connectivity and CPU efficiency by simulating larger particles using a coarser resolution that is subsequently refined compared to standard SA; our proposed extension not only offers the possibility to perform multiphase reconstruction but also allows us (i) to improve binary reconstruction quality, as quantified, e.g., by multiple-point histograms by up to one order of magnitude and (ii) to achieve an overall speed-up. The proposed algorithm is also shown to outperform the direct sampling multiple-point statistics method for the generation of cement paste microstructure with respect to both generation time and quality.

Calculation of tensorial flow properties on pore level: Exploring the influence of boundary conditions on the permeability of three-dimensional stochastic reconstructions

While it is well known that permeability is a tensorial property, it is usually reported as a scalar property or only diagonal values are reported. However, experimental evaluation of tensorial flow properties is problematic. Pore-scale modeling using three-dimensional (3D) images of porous media with subsequent upscaling to a continuum scale (homogenization) is a valuable alternative. In this study, we explore the influence of different types of boundary conditions on the external walls of the representative modeling domain along the applied pressure gradient on the magnitude and orientation of the computed permeability tensor. To implement periodic flow boundary conditions, we utilized stochastic reconstruction methodology to create statistically similar (to real porous media structures) geometrically periodic 3D structures. Stochastic reconstructions are similar to encapsulation of the porous media into statistically similar geometrically periodic one with the same permeability tensor. Seven main boundary conditions (BC) were implemented: closed walls, periodic flow, slip on the walls, linear pressure, translation, symmetry, and immersion. The different combinations of BCs amounted to a total number of 15 BC variations. All these BCs significantly influenced the resulting tensorial permeabilities, including both magnitude and orientation. Periodic boundary conditions produced the most physical flow patterns, while other classical BCs either suppressed crucial transversal flows or resulted in unphysical currents. Our results are crucial to performing flow properties upscaling and will be relevant to computing not only single-phase but also multiphase flow properties. Moreover, other calculation of physical properties such as some mechanical, transport, or heat conduction properties may benefit from the technique described in this study.

Reconstruction of porous media from extremely limited information using conditional generative adversarial networks

Porous media are ubiquitous in both nature and engineering applications. Therefore, their modeling and understanding is of vital importance. In contrast to direct acquisition of three-dimensional (3D) images of this type of medium, obtaining its subregion (s) such as 2D images or several small areas can be feasible. Therefore, reconstructing whole images from limited information is a primary technique in these types of cases. Given data in practice cannot generally be determined by users and may be incomplete or only partially informed, thus making existing reconstruction methods inaccurate or even ineffective. To overcome this shortcoming, in this study we propose a deep-learning-based framework for reconstructing full images from their much smaller subareas. In particular, conditional generative adversarial network is utilized to learn the mapping between the input (a partial image) and output (a full image). To ensure the reconstruction accuracy, two simple but effective objective functions are proposed and then coupled with the other two functions to jointly constrain the training procedure. Because of the inherent essence of this ill-posed problem, a Gaussian noise is introduced for producing reconstruction diversity, thus enabling the network to provide multiple candidate outputs. Our method is extensively tested on a variety of porous materials and validated by both visual inspection and quantitative comparison. It is shown to be accurate, stable, and even fast (∼0.08 s for a 128×128 image reconstruction). The proposed approach can be readily extended by, for example, incorporating user-defined conditional data and an arbitrary number of object functions into reconstruction, while being coupled with other reconstruction methods.

Dictionary optimization and constraint neighbor embedding-based dictionary mapping for superdimension reconstruction of porous media

The three-dimensional (3D) structure of a digital core can be reconstructed from a single two-dimensional (2D) image via mathematical modeling. In classical mathematical modeling algorithms, such as multipoint geostatistics algorithms, the optimization of pattern sets (dictionaries) and the mapping problems are important issues. However, they have rarely been discussed thus far. Pattern set (dictionary)-related problems include the pattern set (dictionary) size problem and the one-to-many mapping problem in a pattern set (dictionary). The former directly affects the completeness of the dictionary, while the latter is manifested such that a single to-be-matched 2D patch has multiple matching patterns in the library and it is hence necessary to select these modes to establish an optimal mapping relationship. Whether the two above-mentioned problems can be properly resolved is directly related to the accuracy of the reconstruction results. Super-dimension reconstruction is a new 3D reconstruction method proposed by introducing the concepts of training dictionary, prior model, and mapping into the reconstruction of the digital core from the field of super-resolution reconstruction. In addition, mapping relationship extraction and dictionary building are also key issues in super-dimension reconstruction. Therefore, this paper discusses these common dictionary-related problems from the perspective of super-dimension dictionaries. We propose dictionary optimization using augmentation dictionaries and clustering based on the boundary features of the dictionary elements to improve the completeness and expand the expression ability of the dictionary. Furthermore, we propose constraint neighbor embedding-based dictionary mapping to establish a more reasonable dictionary mapping relationship for super-dimension reconstruction, and we solve the one-to-many mapping problem in the dictionary. Our experimental results show that the performance of the super-dimension dictionary can be improved by the above-mentioned algorithm. Thus, through the optimized dictionary structure and mapping relationship determined by the above-mentioned methods, the 2D patch to be reconstructed can match a more accurate 3D block in the dictionary. Consequently, the reconstruction precision is improved.