Retraction: Phase transitions, percolation, fracture of materials, and deep learning [Phys. Rev. E 102, 011001(R) (2020)].

Retraction of DOI: 10.1103/PhysRevE.102.011001.

Single-Station Coda Wave Interferometry: A Feasibility Study Using Machine Learning.

Coda wave interferometry usually is applied with pairs of stations analyzing the signal transmitted from one station to another. A feasibility study was performed to evaluate if one single station could be used. In this case, the reflected coda wave signal from a zone to be identified was analyzed. Finite-difference simulations of wave propagation were used to study whether ultrasonic measurements could be used to detect velocity changes in such a zone up to a depth of 1.6 m in a highly scattering medium. For this aim, 1D convolutional neural networks were used for prediction. The crack density, the crack length, and the intrinsic attenuation were varied in the considered background material. The influence of noise and the sensor width was elaborated as well. It was shown that, in general, the suggested single-station approach is a possible way to identify damage zones, and the method was robust against the studied variations. The suggested workflow also took advantage of machine-learning techniques, and can be transferred to the detection of defects in concrete structures.

Phase transitions, percolation, fracture of materials, and deep learning.

Percolation and fracture propagation in disordered solids represent two important problems in science and engineering that are characterized by phase transitions: loss of macroscopic connectivity at the percolation threshold p_{c} and formation of a macroscopic fracture network at the incipient fracture point (IFP). Percolation also represents the fracture problem in the limit of very strong disorder. An important unsolved problem is accurate prediction of physical properties of systems undergoing such transitions, given limited data far from the transition point. There is currently no theoretical method that can use limited data for a region far from a transition point p_{c} or the IFP and predict the physical properties all the way to that point, including their location. We present a deep neural network (DNN) for predicting such properties of two- and three-dimensional systems and in particular their percolation probability, the threshold p_{c}, the elastic moduli, and the universal Poisson ratio at p_{c}. All the predictions are in excellent agreement with the data. In particular, the DNN predicts correctly p_{c}, even though the training data were for the state of the systems far from p_{c}. This opens up the possibility of using the DNN for predicting physical properties of many types of disordered materials that undergo phase transformation, for which limited data are available for only far from the transition point.

Wettability control on deformation: Coupled multiphase fluid and granular systems.

Understanding of multiphase flow in porous media is important for a wide range of applications such as soil science, environmental remediation, energy resources, and CO_{2} sequestration. This phenomenon depends on the complex interplay between the fluid and solid forces such as gravitational, capillary, and viscous forces, as well as wettability of the solid phase. Such interactions along with the geometry of the medium give rise to a variety of complex flow regimes. Although much research has been done in the area of wettability, its mechanical effect is not well understood, and it continues to challenge our understanding of the phenomena on macroscopic and microscopic scales. In this paper, therefore, the effect of wettability on the deformation of porous media and fluid-fluid patterns is studied through a series of three-dimensional (3D) simulations. To this end, the discrete element method (DEM) and volume of fluid (VOF) are coupled to accurately model free-surface flow interaction in a cylindrical pack of spheres. The fluid-particle interactions are modeled by exchanging information between DEM and VOF, while the effect of wettability is considered to study how it controls fluid displacement. The results indicate that the drag force and deformation in the pack vary with the change in wettability and capillary number. To demonstrate the effect of both wettability and capillary number, a series of numerical experiments were conducted with two capillary numbers and three wettability conditions. Our results show that the drag force was greatest for near extreme wettability conditions, which resulted in a larger deformation.

Coupled fine-scale modeling of the wettability effects: Deformation and fracturing.

Multiphase flow in porous media has been thoroughly studied over the years and its importance is encountered in several areas related to geo-materials. One of the most important parameters that control multiphase flow in any complex geometry is wettability, which is an affinity of a given fluid toward a surface. In this paper, we have quantified the effects of wettability on deformation in porous media, along with other parameters that are involved in this phenomenon. To this end, we conducted numerical simulations on a porous medium by coupling the exchanged forces between the fluid and solid. To include the effect of wettability in the medium, we used the Fictitious Domain methodology and coupled it with volume of fluid through which one can model more than one fluid in the system. To observe the effect of wettability on dynamic processes in the designated porous medium, such as deformation, particle-particle contact stresses, particle velocity, and injection pressure, a series of systematic computations were conducted where wettability is varied through five different contact angles. We found that wettability not only controls the fluid propagation patterns but also affects drag forces exerted on the particles during injection such that larger deformations are induced for particles with lower wettability. Our results are also verified against experimental tests.

Quantifying accuracy of stochastic methods of reconstructing complex materials by deep learning.

Time and cost are two main hurdles to acquiring a large number of digital image I of the microstructure of materials. Thus, use of stochastic methods for producing plausible realizations of materials' morphology based on one or very few images has become an increasingly common practice in their modeling. The accuracy of the realizations is often evaluated using two-point microstructural descriptors or physics-based modeling of certain phenomena in the materials, such as transport processes or fluid flow. In many cases, however, two-point correlation functions do not provide accurate evaluation of the realizations, as they are usually unable to distinguish between high- and low-quality reconstructed models. Calculating flow and transport properties of the realization is an accurate way of checking the quality of the realizations, but it is computationally expensive. In this paper a method based on machine learning is proposed for evaluating stochastic approaches for reconstruction of materials, which is applicable to any of such methods. The method reduces the dimensionality of the realizations using an unsupervised deep-learning algorithm by compressing images and realizations of materials. Two criteria for evaluating the accuracy of a reconstruction algorithm are then introduced. One, referred to as the internal uncertainty space, is based on the recognition that for a reconstruction method to be effective, the differences between the realizations that it produces must be reasonably wide, so that they faithfully represent all the possible spatial variations in the materials' microstructure. The second criterion recognizes that the realizations must be close to the original I and, thus, it quantifies the similarity based on an external uncertainty space. Finally, the ratio of two uncertainty indices associated with the two criteria is considered as the final score of the accuracy of a stochastic algorithm, which provides a quantitative basis for comparing various realizations and the approaches that produce them. The proposed method is tested with images of three types of heterogeneous materials in order to evaluate four stochastic reconstruction algorithms.

Enhancing images of shale formations by a hybrid stochastic and deep learning algorithm.

Accounting for the morphology of shale formations, which represent highly heterogeneous porous media, is a difficult problem. Although two- or three-dimensional images of such formations may be obtained and analyzed, they either do not capture the nanoscale features of the porous media, or they are too small to be an accurate representative of the media, or both. Increasing the resolution of such images is also costly. While high-resolution images may be used to train a deep-learning network in order to increase the quality of low-resolution images, an important obstacle is the lack of a large number of images for the training, as the accuracy of the network's predictions depends on the extent of the training data. Generating a large number of high-resolution images by experimental means is, however, very time consuming and costly, hence limiting the application of deep-learning algorithms to such an important class of problems. To address the issue we propose a novel hybrid algorithm by which a stochastic reconstruction method is used to generate a large number of plausible images of a shale formation, using very few input images at very low cost, and then train a deep-learning convolutional network by the stochastic realizations. We refer to the method as hybrid stochastic deep-learning (HSDL) algorithm. The results indicate promising improvement in the quality of the images, the accuracy of which is confirmed by visual, as well as quantitative comparison between several of their statistical properties. The results are also compared with those obtained by the regular deep learning algorithm without using an enriched and large dataset for training, as well as with those generated by bicubic interpolation.

Image-based velocity estimation of rock using Convolutional Neural Networks.

Digital images of rock samples have been using extensively in Digital Rock Physics (DRP) to evaluate physical parameters of rock such as permeability, P- and S-wave velocities and formation factor. The parameters are numerically computed by simulation of the corresponding physical processes through segmented image of rock, which provide a direct and accurate evaluation of rock properties. However, recent advances in machine learning and Convolutional Neural Networks (CNN) allow using images as input. Such networks, however, require a considerable number of images as input. In this paper, CNNs are used to estimate the P- and S-wave velocities from images of rock medium. To deal with lack of input data, a hybrid pattern- and pixel-based simulation (HYPPS) is used as an efficient data augmentation method to increase the training data set. For each input image, 10 stochastic realizations are produced. Compare to the case wherein the stochastic models are not used, the new results from the enhanced network indicate a sharp improvement in the estimations such that R2 is increased to 0.94. Furthermore, the newly developed CNN network, unlike the one with the small data set (R2=0.75), manifests no over/underestimation. The estimated properties, in comparison with the computational results, indicate that CNNs perform outstandingly in predicting the physical parameters of rock without conducting any time-demanding forward modeling if enough input data are provided.

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.

Risk of fire occurrence in arid and semi-arid ecosystems of Iran: an investigation using Bayesian belief networks.

Identifying areas that have a high risk of burning is a main component of fire management planning. Although the available tools can predict the fire risks, these are poor in accommodating uncertainties in their predictions. In this study, we accommodated uncertainty in wildfire prediction using Bayesian belief networks (BBNs). An influence diagram was developed to identify the factors influencing wildfire in arid and semi-arid areas of Iran, and it was populated with probabilities to produce a BBNs model. The behavior of the model was tested using scenario and sensitivity analysis. Land cover/use, mean annual rainfall, mean annual temperature, elevation, and livestock density were recognized as the main variables determining wildfire occurrence. The produced model had good accuracy as its ROC area under the curve was 0.986. The model could be applied in both predictive and diagnostic analysis for answering "what if" and "how" questions. The probabilistic relationships within the model can be updated over time using observation and monitoring data. The wildfire BBN model may be updated as new knowledge emerges; hence, it can be used to support the process of adaptive management.

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.

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

Nonstationary disordered materials and media, those for which the probability distribution function of any property varies spatially when shifted in space, are abundant and encountered in astrophysics, oceanography, air pollution patterns, large-scale porous media, biological tissues and organs, and composite materials. Their reconstruction and modeling is a notoriously difficult and largely unsolved problem. We propose a method for reconstructing a broad class of such media based on partitioning them into locally stationary zones. Two methods are used for the partitioning. One is based on the Shannon entropy, while the second method utilizes a watershed transform. The locally stationary zones are then reconstructed based on a cross-correlation function and one-dimensional raster path that we recently introduced [P. Tahmasebi and M. Sahimi, Phys. Rev. Lett. 110, 078002 (2013)], with overlaps between the zones to ensure seamless transition from one zone to another. A large number of examples, including porous media, ecological systems, disordered materials, and biological tissues and organs, are reconstructed and analyzed to demonstrate the accuracy of the method.

Cross-correlation function for accurate reconstruction of heterogeneous media.

Porous media, heterogeneous materials, and biological tissues are examples of ubiquitous disordered systems, the understanding of which and any physical phenomenon in them entails having an accurate model. We show that a new reconstruction method based on a cross-correlation function and a one-dimensional raster path provides an accurate description of a wide variety of such materials and media. The reconstruction uses a single 2D slice of data to reconstruct an entire 3D medium. Seventeen examples are reconstructed accurately, as indicated by two connectivity functions that we compute for them. The reconstruction method may be used for both unconditioned and conditioned problems, and is highly efficient computationally.

Reconstruction of three-dimensional porous media using a single thin section.

The purpose of any reconstruction method is to generate realizations of two- or multiphase disordered media that honor limited data for them, with the hope that the realizations provide accurate predictions for those properties of the media for which there are no data available, or their measurement is difficult. An important example of such stochastic systems is porous media for which the reconstruction technique must accurately represent their morphology--the connectivity and geometry--as well as their flow and transport properties. Many of the current reconstruction methods are based on low-order statistical descriptors that fail to provide accurate information on the properties of heterogeneous porous media. On the other hand, due to the availability of high resolution two-dimensional (2D) images of thin sections of a porous medium, and at the same time, the high cost, computational difficulties, and even unavailability of complete 3D images, the problem of reconstructing porous media from 2D thin sections remains an outstanding unsolved problem. We present a method based on multiple-point statistics in which a single 2D thin section of a porous medium, represented by a digitized image, is used to reconstruct the 3D porous medium to which the thin section belongs. The method utilizes a 1D raster path for inspecting the digitized image, and combines it with a cross-correlation function, a grid splitting technique for deciding the resolution of the computational grid used in the reconstruction, and the Shannon entropy as a measure of the heterogeneity of the porous sample, in order to reconstruct the 3D medium. It also utilizes an adaptive technique for identifying the locations and optimal number of hard (quantitative) data points that one can use in the reconstruction process. The method is tested on high resolution images for Berea sandstone and a carbonate rock sample, and the results are compared with the data. To make the comparison quantitative, two sets of statistical tests consisting of the autocorrelation function, histogram matching of the local coordination numbers, the pore and throat size distributions, multiple-points connectivity, and single- and two-phase flow permeabilities are used. The comparison indicates that the proposed method reproduces the long-range connectivity of the porous media, with the computed properties being in good agreement with the data for both porous samples. The computational efficiency of the method is also demonstrated.

A hybrid neural networks-fuzzy logic-genetic algorithm for grade estimation.

The grade estimation is a quite important and money/time-consuming stage in a mine project, which is considered as a challenge for the geologists and mining engineers due to the structural complexities in mineral ore deposits. To overcome this problem, several artificial intelligence techniques such as Artificial Neural Networks (ANN) and Fuzzy Logic (FL) have recently been employed with various architectures and properties. However, due to the constraints of both methods, they yield the desired results only under the specific circumstances. As an example, one major problem in FL is the difficulty of constructing the membership functions (MFs).Other problems such as architecture and local minima could also be located in ANN designing. Therefore, a new methodology is presented in this paper for grade estimation. This method which is based on ANN and FL is called "Coactive Neuro-Fuzzy Inference System" (CANFIS) which combines two approaches, ANN and FL. The combination of these two artificial intelligence approaches is achieved via the verbal and numerical power of intelligent systems. To improve the performance of this system, a Genetic Algorithm (GA) - as a well-known technique to solve the complex optimization problems - is also employed to optimize the network parameters including learning rate, momentum of the network and the number of MFs for each input. A comparison of these techniques (ANN, Adaptive Neuro-Fuzzy Inference System or ANFIS) with this new method (CANFIS-GA) is also carried out through a case study in Sungun copper deposit, located in East-Azerbaijan, Iran. The results show that CANFIS-GA could be a faster and more accurate alternative to the existing time-consuming methodologies for ore grade estimation and that is, therefore, suggested to be applied for grade estimation in similar problems.