Two-point Probability Function Research Articles

Controllable image expansion of rock castings based on deep learning

Abstract Digital rock physics (DRP) offers an effective method of deriving elastic parameters from digital rock images, but its practical application is always limited to limited datasets. Recently, deep learning techniques have presented a promising avenue for generating more extensive and cost-effective samples. However, generating controllable samples according to user definition remains very difficult due to high dependence on sufficient datasets. To resolve this problem, a new network was proposed based on the UNet framework through image translation (UNet-IT) to expand rock castings by given porosity in relatively fewer datasets. Practical tests on carbonate rock images demonstrate that the proposed method can generate samples tailored to specific porosity requirements, which achieved a minimum porosity relative error of less than 1%. Compared with the unextended samples, the generated ones have completely different pore structures in terms of two-point probability, two-point cluster, and lineal path functions. Furthermore, the elastic parameters of the generated images obtained through the finite element method (FEM) and practical logging data matched well, with an average relative error of ∼9%. This indicates that the generated samples can be used as effective data to estimate fine rock physics templates and then improve inversion accuracy.

A 3D grain‐based reconstruction method from a 2D surface image for the Distinct Lattice Spring Model

AbstractIn this work, a surface image‐based 3D grain‐based model (GBM) reconstruction method based on the digital image processing (DIP) technique, periodic random packing and the simulated annealing algorithm is proposed. Taking a surface digital image of Beishan granite as an example, a K‐means clustering algorithm is employed to extract the mineral compositions, the distribution of which is quantified by the two‐point probability function (TPPF). Given the 3D volume fraction of each mineral composition, their corresponding 3D two‐point probability functions can be evaluated from their 2D forms by the linear interpolation method. The 3D GBM reconstruction is generated by a simulated annealing algorithm, with the Monte Carlo algorithm extending the calculation of the two‐point probability function as the target function to the random particle model. To improve the computational efficiency, the periodic boundary condition is applied to both the random particle generator and the evaluation of the two‐point probability function, allowing large‐scale GBM to be generated for the Distinct Lattice Spring Model (DLSM). The elastic response and advantages of the random particle model for the DLSM are investigated. Finally, the DLSM is further enriched with a tension‐cutoff Mohr‐Coulomb failure criterion, and the GBM reconstruction method is successfully tested for reproducing the mechanical behaviours of granite and interpreting the failure mechanism at the mesoscale under different loading conditions.

The novel continuous reconstruction approach for reconstructing anisotropic porous rocks

In this paper, a novel continuous reconstruction method is proposed to generate a numerical porous rock model that is statistically equivalent to natural rock. First, the spatial and mechanical properties of pore microstructures in rocks are obtained by computed tomography (CT) and triaxial compression tests. Then, the reconstructed rock models are generated based on the novel continuous reconstruction method. The generated rock models preserve the interest characteristics of the microstructures in rock. Compared with the exchange of discrete and isolated elements in our previous contributions, the microstructures are directly continuously reconstructed to form the reconstruction model. With the continuous approach, the natural rock sample can be efficiently represented by the reconstruction model on a large scale. We also consider reconstructing the anisotropic microstructures by different spatial correlation functions, such as the two-point probability function (TPPF), the lineal-path function (LPF), the fractal function (FF), and the two-point cluster function (TPCF). In addition, we introduce the percolating cluster volume (PCV) to improve the characterization capabilities of the proposed reconstruction method. The comparisons among the reference model, reconstructed model and rock samples show that the proposed continuous reconstruction method is able to reconstruct porous rocks while preserving mechanical and topological properties.

A Step-by-Step Equivalent Microprediction Method for the Mechanical Properties of Composite Solid Propellants considering Dewetting Damage

Reliable prediction of the macromechanical properties of composite solid propellants in the microscale can accelerate the development of propellants with high mechanical properties. According to the characteristics of the composition ratio of a four-component hydroxyl-terminated polybutadiene (HTPB) propellant with the component ammonium perchlorate (AP), hydroxyl-terminated polybutadiene, aluminum powder (AL), and cyclotrimethylenetrinitramine (or RDX for short), an improved random delivery algorithm was developed to separately model filler particles with the different sizes. A step-by-step equivalent representative volume element (RVE) model was generated to reflect the microstructures of the propellant. The isotropy and uniformity of the RVE model were also tested using a two-point probability function. The Park-Paulino-Roesler (PPR) cohesive model was introduced to simulate the particle debonding (or dewetting) in solid propellant. The stress-strain curves of the propellant were obtained by the macroscopic test with the extension rate 200 mm/min at different temperatures. Based on these experimental data, the 8 characteristic parameters suitable for the microinterface of the propellant were obtained by using an inversion optimization method. A microscale finite element prediction model of the propellant considering dewetting damage was constructed to study the evolution process of the microdamage of the propellant. The predicted stress-strain curves of the propellant under different loading conditions were in good agreement with the test results.

Gradient-based phase segmentation method for characterization of hydrating cement paste microstructures obtained from X-ray micro-CT

X-ray micro-computed tomography (micro-CT) has been widely used in the non-destructive investigation of 3D microstructures of cementitious materials. Microstructures from micro-CT are reconstructed from the attenuated transmitted X-ray intensity through a sample or linear attenuation coefficient (LAC). Although uncertainties can be introduced during processes, such as data collection and microstructure reconstruction, the LAC values are determined based on the X-ray energy level and material composition so that they can be considered as almost absolute measures. In this study, the microstructural evolution of cement paste during hydration was investigated using microstructures with a 4.5 μm voxel size obtained from X-ray micro-CT. A new phase segmentation method based on the mean LAC values for the pore and unhydrated phases inside the binder was developed. The pore phase features were characterized by porosity, equivalent pore diameter, and two-point probability functions. The findings indicate that the proposed segmentation method is well adapted for micro-CT imaging to identify the characteristics of pores, hydrates, and clinkers.

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.

Representative elementary area of shale at the mesoscopic scale

The Representative Elementary Area (REA) of two shales (Callovo-Oxfordian claystone and Tournemire argillite) that are actively studied in the framework of the deep disposal of radioactive waste have been estimated from two mineral maps by classical methods (i.e., the box-counting and statistical approaches) and by different microstructural descriptors (i.e., the two-point probability function, lineal path function, percolation length, and variogram). The classical box-counting method provides estimates of the REA size of the clay fraction in the range from 129 μm to 441 μm, consistent with estimates obtained from the literature on other shales. However, these estimates show an extreme sensitivity to the chosen ε threshold or error and a wide scatter, thereby bringing the statistical homogeneity of both maps into question. Although the two-point probability function and lineal path function infer lower bounds of the REA size, these microstructural descriptors are relevant to demonstrate the microstructural anisotropy of both shales due to the alignment of nonclay grains parallel to bedding at the study scale. The results from the two-point probability function and variogram undoubtedly confirm that the Tournemire mineral map is not statistically homogeneous with regard to its mineral composition. This aspect it makes difficult to interpret the results and even questions the REA size determination of this particular map. Finally, our set of results allow us to recommend the use of the two-point probability function and variogram to preliminarily validate the statistical homogeneity of maps under study before calculating the REA size using conventional methods, e.g. the box-counting and statistical approaches.

Inverse modeling of inelastic properties of a two-phase microstructure

The prediction of material properties with the inclusion of morphology has been an area of increasing interest for material scientists in the past decades. A myriad of statistical continuum mechanics formulations have been developed to investigate the properties of a two-phase microstructure given its morphology. In this study, the structure-property model is inverted to create an inverse microstructure model for a two-phase Ti64 material to predict the microstructure required to achieve a desired property. For this purpose, an inverse formulation is developed using the two-point correlation function representation of the microstructure within the statistical continuum framework. Using this formulation the initial microstructure is then predicted by knowing a desired strength. This approach calculates the optimum values of the two-point probability functions which are associated with the minimum error in the predicted strength with respect to the desired strength. Finally, 2D microstructures are reconstructed using the predicted values of the two-point probability functions to represent the morphology of the initial microstructure at four different temperatures of Ti64 (850, 900, 950, 1000 °C).

An improved simulated annealing algorithm for reconstructing 3D large-scale porous media

Based on the morphological information obtained from 2D slice images of real porous media, an improved simulated annealing algorithm (SAA) was proposed to reconstruct 3D large-scale porous media, which are intractable to handle for conventional SAA. Three different statistical functions were introduced to characterize the morphological information of real sandstone, including the one-point probability function, the two-point probability function and the lineal-path function. By changing the update method of the two-point probability function and the lineal-path function, i.e., using incremental calculation instead of conventional global calculation, the efficiency of reconstructing 3D large-scale porous media was greatly improved. Besides, in the later stage of reconstruction that the basic structure of porous media had been formed, the pixel selection algorithm was performed to speed up the reconstruction process. To evaluate the accuracy of the improved SAA, the similarity between the 3D reconstructed volume and the reference image of prototype sandstone was examined. The results showed good agreement between the reconstructed model and the references. The efficiency of the improved SAA was verified by comparison with the conventional SAA, the results of which indicated that the improved SAA can significantly shorten the reconstruction time of 3D large-scale porous media.

Wang tiling for particle heterogeneous materials: Algorithms for generation of tiles/cubes via molecular dynamics

This paper aims at a reduction of periodicity artefacts during a generation of random heterogeneous material models. The traditional concept of the Periodic Unit Cell is compared with a novel approach of the stochastic Wang tiling. Since modelled structures consist of hard circular/spherical particles in a matrix, the algorithm for placement of inclusions is based on the modified molecular dynamics. We introduce two types of Wang tile boundary conditions to decrease periodicity artefacts. Tested samples for 2D applications form sets of both monodisperse and polydisperse microstructures. The overall volume fractions of these samples are approximately 0.2, 0.4, and 0.6, respectively. The generated sets are analysed both visually and statistically via a two-point probability function. An extension of the stochastic Wang tiling enables to create 3D structures, as well. Therefore, artificial periodicity is also investigated on a 3D sample consisting of spherical particles of identical radii distributed in a continuous phase.

On a pluri-Gaussian model for three-phase microstructures, with applications to 3D image data of gas-diffusion electrodes

A pluri-Gaussian model for three-phase microstructures is presented and relationships between model parameters and microstructure characteristics are discussed. In particular, analytical formulas for two-point coverage probability functions in terms of covariance functions of the underlying Gaussian random fields are considered, which allow for an efficient estimation of model parameters. The model is fitted to tomographic image data obtained by FIB-tomography, which represent porous gas-diffusion electrodes consisting of silver and polytetrafluorethylene. The considered type of electrode is used as oxygen depolarized cathode for the production of chlorine. In order to fit the microstructure model, the covariance functions of the Gaussian random fields are parameterized, which leads to a stochastic microstructure model with five parameters. It is shown that most microstructure characteristics of tomographic image data are well reproduced by the model despite the low number of model parameters. Finally, limitations of the model with respect to the fit of continuous phase size distributions are discussed. Combining stochastic microstructure modeling with numerical simulation of effective macroscopic properties will allow in future work for a model-based investigation of microstructure-property relationships for the considered gas-diffusion electrodes.

Three-dimensional numerical reconstruction method for irregular structures of granular geomaterials

Granular geomaterials are comprised of many geometrically irregular and randomly distributed particles on multi-scales that strongly affect their physical and or mechanical properties. However, providing an accurate representation or characterization of the complex structures of granular geomaterials remains a challenge. In this study, we propose a novel computerized method to numerically represent a three-dimensional (3D) irregular granular structure based on a natural conglomerate sample. To represent the 3D granular structure, the reconstruction algorithm incorporates a two-point probability function along both the principal and diagonal directions, as well as a linear-path function along the principal directions of the model, and a multithread parallel computing algorithm is used to accelerate the 3D reconstruction process of the irregular structures. To validate the accuracy of the proposed algorithm, the characteristics and mechanical properties of the reconstructed 3D granular structure are analyzed and compared with those of the prototype conglomerate sample. This comparison shows that the proposed algorithm improves the reconstruction efficiency of previous algorithms, and the reconstruction model exhibits greater similarity to the natural conglomerate structure than previous reconstruction models. This proposed computational method can also be used to reconstruct other granular structures.

A New Method of Porous Space Reconstruction Using Multipoint Histogram Technology

Pore-scale modeling is becoming a hot topic in overall reservoir characterization process. It is an important approach for revealing the flow behaviors in porous media and exploring unknown flow patterns at pore scale. Over the past few decades, many reconstruction methods have been proposed, and among them the simulated annealing method (SAM) is extensively tested and easier to program. However, SAM is usually based on the two-point probability function or linear-path function, which fails to capture much more information on the multipoint connectivity of various shapes. For this reason, a new reconstruction method is proposed to reproduce the characteristics of a two-dimensional (2D) thin section based on the multipoint histogram. First, the two-point correlation coefficient matrix will be introduced to determine an optimal unit configuration of a multipoint histogram. Second, five different types of seven-point unit configurations will be used to test the unit configuration selection algorithm. Third, the multipoint histogram technology is used for generating the porous space reconstruction based on the prior unit configuration with a different calculation of the objective function. Finally, the spatial connectivity, patterns reproduction, the local percolation theory (LPT), and hydraulic connectivity are used to compare with those of the reference models. The results show that the multipoint histogram technology can produce better multipoint connectivity information than SAM. The reconstructed system matches the training image very well, which reveals that the reconstruction captures the geometry and topology information of the training image, for instance, the shape and distribution of pore space. The seven-point unit configuration is enough to get the spatial characters of the training image. The quality of pattern reproduction of the reconstruction is assessed by computing the multipoint histogram, and the similarity is around 97.3%. Based on the LPT analysis, the multipoint histogram can describe the anticipated patterns of geological heterogeneities and reproduce the connectivity of pore media with a high degree of accuracy. The two-point correlation coefficient matrix and a new construction theory are proposed. The new construction theory provides a stable theory and technology guidance for the study of pore space development and multiphase fluid flow rule in the digital rock.

Identifying design parameters controlling damage behaviors of continuous fiber-reinforced thermoplastic composites using micromechanics as a virtual testing tool

In this paper, we propose a micromechanical approach to predict damage mechanisms and their interactions in glass fibers/polypropylene thermoplastic composites. First, a representative volume element (RVE) of such materials was rigorously determined using a geometrical two-point probability function and the eigenvalue stabilization of homogenized elastic tensor obtained by Hill–Mandel kinematic homogenization. Next, the 3D finite element models of the RVE were developed accordingly. The fibers were modeled with an isotropic linear elastic material. The matrix was modeled with an isotropic linear elastic, rate-independent hyperbolic Drucker–Prager plasticity coupled with a ductile damage model that is able to show pressure dependency of the yield and damage behavior often found in a thermoplastic material. In addition, cohesive elements were inserted into the fiber-matrix interfaces to simulate debonding. The RVE faces are imposed with periodical boundary conditions to minimize the edge effect. The RVE was then subjected to transverse tensile loading in accordance with experimental tensile tests on [90]8 laminates. The model prediction was found to be in very good agreement with the experimental results in terms of the global stress-strain curves, including the linear and nonlinear portion of the response and also the failure point, making it a useful virtual testing tool for composite material design. Furthermore, the effect of tailoring the main parameters of thermoplastic composites is investigated to provide guidelines for future improvements of these materials.

Multi-thread parallel algorithm for reconstructing 3D large-scale porous structures

Geomaterials inherently contain many discontinuous, multi-scale, geometrically irregular pores, forming a complex porous structure that governs their mechanical and transport properties. The development of an efficient reconstruction method for representing porous structures can significantly contribute toward providing a better understanding of the governing effects of porous structures on the properties of porous materials. In order to improve the efficiency of reconstructing large-scale porous structures, a multi-thread parallel scheme was incorporated into the simulated annealing reconstruction method. In the method, four correlation functions, which include the two-point probability function, the linear-path functions for the pore phase and the solid phase, and the fractal system function for the solid phase, were employed for better reproduction of the complex well-connected porous structures. In addition, a random sphere packing method and a self-developed pre-conditioning method were incorporated to cast the initial reconstructed model and select independent interchanging pairs for parallel multi-thread calculation, respectively. The accuracy of the proposed algorithm was evaluated by examining the similarity between the reconstructed structure and a prototype in terms of their geometrical, topological, and mechanical properties. Comparisons of the reconstruction efficiency of porous models with various scales indicated that the parallel multi-thread scheme significantly shortened the execution time for reconstruction of a large-scale well-connected porous model compared to a sequential single-thread procedure.

Packing of nonoverlapping cubic particles: Computational algorithms and microstructural characteristics.

Packing of cubic particles arises in a variety of problems, ranging from biological materials to colloids and the fabrication of new types of porous materials with controlled morphology. The properties of such packings may also be relevant to problems involving suspensions of cubic zeolites, precipitation of salt crystals during CO_{2} sequestration in rock, and intrusion of fresh water in aquifers by saline water. Not much is known, however, about the structure and statistical descriptors of such packings. We present a detailed simulation and microstructural characterization of packings of nonoverlapping monodisperse cubic particles, following up on our preliminary results [H. Malmir et al., Sci. Rep. 6, 35024 (2016)2045-232210.1038/srep35024]. A modification of the random sequential addition (RSA) algorithm has been developed to generate such packings, and a variety of microstructural descriptors, including the radial distribution function, the face-normal correlation function, two-point probability and cluster functions, the lineal-path function, the pore-size distribution function, and surface-surface and surface-void correlation functions, have been computed, along with the specific surface and mean chord length of the packings. The results indicate the existence of both spatial and orientational long-range order as the the packing density increases. The maximum packing fraction achievable with the RSA method is about 0.57, which represents the limit for a structure similar to liquid crystals.

Microstructural characterization of random packings of cubic particles.

Understanding the properties of random packings of solid objects is of critical importance to a wide variety of fundamental scientific and practical problems. The great majority of the previous works focused, however, on packings of spherical and sphere-like particles. We report the first detailed simulation and characterization of packings of non-overlapping cubic particles. Such packings arise in a variety of problems, ranging from biological materials, to colloids and fabrication of porous scaffolds using salt powders. In addition, packing of cubic salt crystals arise in various problems involving preservation of pavements, paintings, and historical monuments, mineral-fluid interactions, CO2 sequestration in rock, and intrusion of groundwater aquifers by saline water. Not much is known, however, about the structure and statistical descriptors of such packings. We have developed a version of the random sequential addition algorithm to generate such packings, and have computed a variety of microstructural descriptors, including the radial distribution function, two-point probability function, orientational correlation function, specific surface, and mean chord length, and have studied the effect of finite system size and porosity on such characteristics. The results indicate the existence of both spatial and orientational long-range order in the packing, which is more distinctive for higher packing densities. The maximum packing fraction is about 0.57.

Compression and reconstruction of random microstructures using accelerated lineal path function

Microstructure reconstruction and compression techniques are designed to identify microstructures with desired properties. While a microstructure reconstruction involves searching for a microstructure with prescribed statistical properties, a microstructure compression focuses on efficient representation of material morphology for the purpose of multiscale modelling. Successful application of these techniques, nevertheless, requires proper understanding of the underlying statistical descriptors quantifying morphology of a material. In this paper, we focus on a lineal path function designed to capture short-range effects and phase connectedness, which can hardly be handled by the commonly used two-point probability function. Usage of the lineal path function is, however, significantly limited because of huge computational requirements. So as to examine the properties of the lineal path function during computationally exhaustive compression and reconstruction processes, we start with an acceleration of the lineal path evaluation, namely by porting part of its code to a graphics processing unit using the CUDA (Compute Unified Device Architecture) programming environment. This allows us to present a unique comparison of the entire lineal path function with the commonly used rough approximation based on the Monte Carlo and/or sampling template. Moreover, this accelerated version of the lineal path function is then compared to the two-point probability function during the compression and reconstruction of two-phase morphologies. Their significant features are discussed and illustrated using a set of artificial periodic as well as real-world random microstructures.

Study on Multi-Objective Reconstruction of Random Media

This paper deals with a reconstruction of random media via multi-objective optimization. Two statistical descriptors, namely a two-point probability function and a two-point lineal path function, are repetitively evaluated for the original medium and the reconstructed image to appreciate the improvement in the optimization progress. Because of doubts of the weights setting in the weighted-sum method, purely multi-objective optimization routine Non-dominated Sorting Genetic Algorithm~II is utilized. Three operators are compared for creating new offspring populations that satisfy a prescribed volume fraction constraint. The main contribution is in the testing of the proposed methodology on several benchmark images.

Comparison of DEM-Based Wang Tilings and PUC

This contribution presents a statistical evaluation of microstructures assembled by means of2D Wang tiles, morphology of which is designed employing an algorithm based on Discrete ElementMethod. Comparison between realization of Wang tiling and Periodic Unit Cell representation is providedin terms of spatial descriptors. Namely, radial distribution function g2 and two-point probabilityfunction S2 are used to assess the artefacts related to periodicity in synthesized microstructures. Wedemonstrate significant reduction in periodicity achieved with Wang tiling (compared to the periodicsynthesis) in both the microstructures and results of numerical simulations.