Reconstruction of nonstationary disordered materials and media: Watershed transform and cross-correlation function

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.

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.

Data-driven discovery of the governing equations for transport in heterogeneous media by symbolic regression and stochastic optimization

With advances in instrumentation and the tremendous increase in computational power, vast amounts of data are becoming available for many complex phenomena in macroscopically heterogeneous media, particularly those that involve flow and transport processes, which are problems of fundamental interest that occur in a wide variety of physical systems. The absence of a length scale beyond which such systems can be considered as homogeneous implies that the traditional volume or ensemble averaging of the equations of continuum mechanics over the heterogeneity is no longer valid and, therefore, the issue of discovering the governing equations for flow and transport processes is an open question. We propose a data-driven approach that uses stochastic optimization and symbolic regression to discover the governing equations for flow and transport processes in heterogeneous media. The data could be experimental or obtained by microscopic simulation. As an example, we discover the governing equation for anomalous diffusion on the critical percolation cluster at the percolation threshold, which is in the form of a fractional partial differential equation, and agrees with what has been proposed previously.

A Two-Stage Reconstruction of Microstructures with Arbitrarily Shaped Inclusions

The main goal of our research is to develop an effective method with a wide range of applications for the statistical reconstruction of heterogeneous microstructures with compact inclusions of any shape, such as highly irregular grains. The devised approach uses multi-scale extended entropic descriptors (ED) that quantify the degree of spatial non-uniformity of configurations of finite-sized objects. This technique is an innovative development of previously elaborated entropy methods for statistical reconstruction. Here, we discuss the two-dimensional case, but this method can be generalized into three dimensions. At the first stage, the developed procedure creates a set of black synthetic clusters that serve as surrogate inclusions. The clusters have the same individual areas and interfaces as their target counterparts, but random shapes. Then, from a given number of easy-to-generate synthetic cluster configurations, we choose the one with the lowest value of the cost function defined by us using extended ED. At the second stage, we make a significant change in the standard technique of simulated annealing (SA). Instead of swapping pixels of different phases, we randomly move each of the selected synthetic clusters. To demonstrate the accuracy of the method, we reconstruct and analyze two-phase microstructures with irregular inclusions of silica in rubber matrix as well as stones in cement paste. The results show that the two-stage reconstruction (TSR) method provides convincing realizations for these complex microstructures. The advantages of TSR include the ease of obtaining synthetic microstructures, very low computational costs, and satisfactory mapping in the statistical context of inclusion shapes. Finally, its simplicity should greatly facilitate independent applications.

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

Stochastic reconstructions based on universal correlation functions allow obtaining spatial structures based on limited input data or to fuse multiscale images from different sources. Current application of such techniques is severely hampered by the computational cost of the annealing optimization procedure. In this study we propose a novel hierarchical annealing method based on rescaled correlation functions, which improves both accuracy and computational efficiency of reconstructions while not suffering from disadvantages of existing speeding-up techniques. A significant order of magnitude gains in computational efficiency now allows us to add more correlation functions into consideration and, thus, to further improve the accuracy of the method. In addition, the method provides a robust multiscale framework to solve the universal upscaling or downscaling problem. The novel algorithm is extensively tested on binary (two-phase) microstructures of different genesis. In spite of significant improvements already in place, the current implementation of the hierarchical annealing method leaves significant room for additional accuracy and computational performance tweaks. As described here, (multiscale) stochastic reconstructions will find numerous applications in material and Earth sciences. Moreover, the proposed hierarchical approach can be readily applied to a wide spectrum of constrained optimization problems.

Effect of the image resolution on the statistical descriptors of heterogeneous media

The characterization and reconstruction of heterogeneous materials, such as porous media and electrode materials, involve the application of image processing methods to data acquired by scanning electron microscopy or other microscopy techniques. Among them, binarization and decimation are critical in order to compute the correlation functions that characterize the microstructure of the above-mentioned materials. In this study, we present a theoretical analysis of the effects of the image-size reduction, due to the progressive and sequential decimation of the original image. Three different decimation procedures (random, bilinear, and bicubic) were implemented and their consequences on the discrete correlation functions (two-point, line-path, and pore-size distribution) and the coarseness (derived from the local volume fraction) are reported and analyzed. The chosen statistical descriptors (correlation functions and coarseness) are typically employed to characterize and reconstruct heterogeneous materials. A normalization for each of the correlation functions has been performed. When the loss of statistical information has not been significant for a decimated image, its normalized correlation function is forecast by the trend of the original image (reference function). In contrast, when the decimated image does not hold statistical evidence of the original one, the normalized correlation function diverts from the reference function. Moreover, the equally weighted sum of the average of the squared difference, between the discrete correlation functions of the decimated images and the reference functions, leads to a definition of an overall error. During the first stages of the gradual decimation, the error remains relatively small and independent of the decimation procedure. Above a threshold defined by the correlation length of the reference function, the error becomes a function of the number of decimation steps. At this stage, some statistical information is lost and the error becomes dependent on the decimation procedure. These results may help us to restrict the amount of information that one can afford to lose during a decimation process, in order to reduce the computational and memory cost, when one aims to diminish the time consumed by a characterization or reconstruction technique, yet maintaining the statistical quality of the digitized sample.

Accurate modeling and evaluation of microstructures in complex materials

Accurate characterization of heterogeneous materials is of great importance for different fields of science and engineering. Such a goal can be achieved through imaging. Acquiring three- or two-dimensional images under different conditions is not, however, always plausible. On the other hand, accurate characterization of complex and multiphase materials requires various digital images (I) under different conditions. An ensemble method is presented that can take one single (or a set of) I(s) and stochastically produce several similar models of the given disordered material. The method is based on a successive calculating of a conditional probability by which the initial stochastic models are produced. Then, a graph formulation is utilized for removing unrealistic structures. A distance transform function for the Is with highly connected microstructure and long-range features is considered which results in a new I that is more informative. Reproduction of the I is also considered through a histogram matching approach in an iterative framework. Such an iterative algorithm avoids reproduction of unrealistic structures. Furthermore, a multiscale approach, based on pyramid representation of the large Is, is presented that can produce materials with millions of pixels in a matter of seconds. Finally, the nonstationary systems-those for which the distribution of data varies spatially-are studied using two different methods. The method is tested on several complex and large examples of microstructures. The produced results are all in excellent agreement with the utilized Is and the similarities are quantified using various correlation functions.

Reconstruction of a digital core containing clay minerals based on a clustering algorithm

It is difficult to obtain a core sample and information for digital core reconstruction of mature sandstone reservoirs around the world, especially for an unconsolidated sandstone reservoir. Meanwhile, reconstruction and division of clay minerals play a vital role in the reconstruction of the digital cores, although the two-dimensional data-based reconstruction methods are specifically applicable as the microstructure reservoir simulation methods for the sandstone reservoir. However, reconstruction of clay minerals is still challenging from a research viewpoint for the better reconstruction of various clay minerals in the digital cores. In the present work, the content of clay minerals was considered on the basis of two-dimensional information about the reservoir. After application of the hybrid method, and compared with the model reconstructed by the process-based method, the digital core containing clay clusters without the labels of the clusters' number, size, and texture were the output. The statistics and geometry of the reconstruction model were similar to the reference model. In addition, the Hoshen-Kopelman algorithm was used to label various connected unclassified clay clusters in the initial model and then the number and size of clay clusters were recorded. At the same time, the K-means clustering algorithm was applied to divide the labeled, large connecting clusters into smaller clusters on the basis of difference in the clusters' characteristics. According to the clay minerals' characteristics, such as types, textures, and distributions, the digital core containing clay minerals was reconstructed by means of the clustering algorithm and the clay clusters' structure judgment. The distributions and textures of the clay minerals of the digital core were reasonable. The clustering algorithm improved the digital core reconstruction and provided an alternative method for the simulation of different clay minerals in the digital cores.

Reconstructing the microstructure of polyimide-silicalite mixed-matrix membranes and their particle connectivity using FIB-SEM tomography

Properties of a composite material made of a continuous matrix and particles often depend on microscopic details, such as contacts between particles. Focusing on processing raw focused-ion beam scanning electron microscope (FIB-SEM) tomography data, we reconstructed three mixed-matrix membrane samples made of 6FDA-ODA polyimide and silicalite-1 particles. In the first step of image processing, backscattered electron (BSE) and secondary electron (SE) signals were mixed in a ratio that was expected to obtain a segmented 3D image with a realistic volume fraction of silicalite-1. Second, after spatial alignment of the stacked FIB-SEM data, the 3D image was smoothed using adaptive median and anisotropic nonlinear diffusion filters. Third, the image was segmented using the power watershed method coupled with a seeding algorithm based on geodesic reconstruction from the markers. If the resulting volume fraction did not match the target value quantified by chemical analysis of the sample, the BSE and SE signals were mixed in another ratio and the procedure was repeated until the target volume fraction was achieved. Otherwise, the segmented 3D image (replica) was accepted and its microstructure was thoroughly characterized with special attention paid to connectivity of the silicalite phase. In terms of the phase connectivity, Monte Carlo simulations based on the pure-phase permeability values enabled us to calculate the effective permeability tensor, the main diagonal elements of which were compared with the experimental permeability. In line with the hypothesis proposed in our recent paper (Čapek, P. et al. (2014) Comput. Mater. Sci. 89, 142-156), the results confirmed that the existence of particle clusters was a key microstructural feature determining effective permeability.

Multiscale and multiresolution modeling of shales and their flow and morphological properties

The need for more accessible energy resources makes shale formations increasingly important. Characterization of such low-permeability formations is complicated, due to the presence of multiscale features, and defies conventional methods. High-quality 3D imaging may be an ultimate solution for revealing the complexities of such porous media, but acquiring them is costly and time consuming. High-quality 2D images, on the other hand, are widely available. A novel three-step, multiscale, multiresolution reconstruction method is presented that directly uses 2D images in order to develop 3D models of shales. It uses a high-resolution 2D image representing the small-scale features to reproduce the nanopores and their network, a large scale, low-resolution 2D image to create the larger-scale characteristics, and generates stochastic realizations of the porous formation. The method is used to develop a model for a shale system for which the full 3D image is available and its properties can be computed. The predictions of the reconstructed models are in excellent agreement with the data. The method is, however, quite general and can be used for reconstructing models of other important heterogeneous materials and media. Two biological examples and from materials science are also reconstructed to demonstrate the generality of the method.

Universal Stochastic Multiscale Image Fusion: An Example Application for Shale Rock

Spatial data captured with sensors of different resolution would provide a maximum degree of information if the data were to be merged into a single image representing all scales. We develop a general solution for merging multiscale categorical spatial data into a single dataset using stochastic reconstructions with rescaled correlation functions. The versatility of the method is demonstrated by merging three images of shale rock representing macro, micro and nanoscale spatial information on mineral, organic matter and porosity distribution. Merging multiscale images of shale rock is pivotal to quantify more reliably petrophysical properties needed for production optimization and environmental impacts minimization. Images obtained by X-ray microtomography and scanning electron microscopy were fused into a single image with predefined resolution. The methodology is sufficiently generic for implementation of other stochastic reconstruction techniques, any number of scales, any number of material phases, and any number of images for a given scale. The methodology can be further used to assess effective properties of fused porous media images or to compress voluminous spatial datasets for efficient data storage. Practical applications are not limited to petroleum engineering or more broadly geosciences, but will also find their way in material sciences, climatology, and remote sensing.