Direct relations between morphology and transport in Boolean models

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

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

Behind the scenes of cellular organization: Quantifying spatial phenotypes of puncta structures with statistical models including random fields

The cellular interior is a spatially complex environment shaped by nontrivial stochastic and biophysical processes. Within this complexity, spatial organizational principles-also called spatial phenotypes-often emerge with functional implications. However, identifying and quantifying these phenotypes in the stochastic intracellular environment is challenging. To overcome this challenge for puncta, we discuss the use of inference of point-process models that link the density of points to other imaged structures and a random field that captures hidden processes. We apply these methods to simulated data and multiplexed immunofluorescence images of Vero E6 cells. Our analysis suggests that peroxisomes are likely to be found near the perinuclear region, overlapping with the endoplasmic reticulum, and located within a distance of 1 µm to mitochondria. Moreover, the random field captures a hidden variation of the mean density in the order of 15 µm. This length scale could provide critical information for further developing mechanistic hypotheses and models. By using spatial statistical models including random fields, we add a valuable perspective to cell biology.

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.

Predicting permeability via statistical learning on higher-order microstructural information

Quantitative structure-property relationships are crucial for the understanding and prediction of the physical properties of complex materials. For fluid flow in porous materials, characterizing the geometry of the pore microstructure facilitates prediction of permeability, a key property that has been extensively studied in material science, geophysics and chemical engineering. In this work, we study the predictability of different structural descriptors via both linear regressions and neural networks. A large data set of 30,000 virtual, porous microstructures of different types, including both granular and continuous solid phases, is created for this end. We compute permeabilities of these structures using the lattice Boltzmann method, and characterize the pore space geometry using one-point correlation functions (porosity, specific surface), two-point surface-surface, surface-void, and void-void correlation functions, as well as the geodesic tortuosity as an implicit descriptor. Then, we study the prediction of the permeability using different combinations of these descriptors. We obtain significant improvements of performance when compared to a Kozeny-Carman regression with only lowest-order descriptors (porosity and specific surface). We find that combining all three two-point correlation functions and tortuosity provides the best prediction of permeability, with the void-void correlation function being the most informative individual descriptor. Moreover, the combination of porosity, specific surface, and geodesic tortuosity provides very good predictive performance. This shows that higher-order correlation functions are extremely useful for forming a general model for predicting physical properties of complex materials. Additionally, our results suggest that artificial neural networks are superior to the more conventional regression methods for establishing quantitative structure-property relationships. We make the data and code used publicly available to facilitate further development of permeability prediction methods.

Predicting Resistivity and Permeability of Porous Media Using Minkowski Functionals

Permeability and formation factor are important properties of a porous medium that only depend on pore space geometry, and it has been proposed that these transport properties may be predicted in terms of a set of geometric measures known as Minkowski functionals. The well-known Kozeny–Carman and Archie equations depend on porosity and surface area, which are closely related to two of these measures. The possibility of generalizations including the remaining Minkowski functionals is investigated in this paper. To this end, two-dimensional computer-generated pore spaces covering a wide range of Minkowski functional value combinations are generated. In general, due to Hadwiger’s theorem, any correlation based on any additive measurements cannot be expected to have more predictive power than those based on the Minkowski functionals. We conclude that the permeability and formation factor are not uniquely determined by the Minkowski functionals. Good correlations in terms of appropriately evaluated Minkowski functionals, where microporosity and surface roughness are ignored, can, however, be found. For a large class of random systems, these correlations predict permeability and formation factor with an accuracy of 40% and 20%, respectively.

Ubiquity of anomalous transport in porous media: Numerical evidence, continuous time random walk modelling, and hydrodynamic interpretation

Anomalous transport in porous media is commonly believed to be induced by the highly complex pore space geometry. However, this phenomenon is also observed in porous media with rather simple pore structure. In order to answer how ubiquitous can anomalous transport be in porous media, we in this work systematically investigate the solute transport process in a simple porous medium model with minimal structural randomness. The porosities we consider range widely from 0.30 up to 0.85, and we find by lattice Boltzmann simulations that the solute transport process can be anomalous in all cases at high Péclet numbers. We use the continuous time random walk theory to quantitatively explain the observed scaling relations of the process. A plausible hydrodynamic origin of anomalous transport in simple porous media is proposed as a complement to its widely accepted geometric origin in complex porous media. Our results, together with previous findings, provide evidence that anomalous transport is indeed ubiquitous in porous media. Consequently, attentions should be paid when modelling solute transport by the classical advection-diffusion equation, which could lead to systematic error.

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

To evaluate the variability of multiphase flow properties of porous media at the pore scale, it is necessary to acquire a number of representative samples of the void-solid structure. While modern x-ray computer tomography has made it possible to extract three-dimensional images of the pore space, assessment of the variability in the inherent material properties is often experimentally not feasible. We present a method to reconstruct the solid-void structure of porous media by applying a generative neural network that allows an implicit description of the probability distribution represented by three-dimensional image data sets. We show, by using an adversarial learning approach for neural networks, that this method of unsupervised learning is able to generate representative samples of porous media that honor their statistics. We successfully compare measures of pore morphology, such as the Euler characteristic, two-point statistics, and directional single-phase permeability of synthetic realizations with the calculated properties of a bead pack, Berea sandstone, and Ketton limestone. Results show that generative adversarial networks can be used to reconstruct high-resolution three-dimensional images of porous media at different scales that are representative of the morphology of the images used to train the neural network. The fully convolutional nature of the trained neural network allows the generation of large samples while maintaining computational efficiency. Compared to classical stochastic methods of image reconstruction, the implicit representation of the learned data distribution can be stored and reused to generate multiple realizations of the pore structure very rapidly.

Statistics of highly heterogeneous flow fields confined to three-dimensional random porous media

We present a strong relationship between the microstructural characteristics of, and the fluid velocity fields confined to, three-dimensional random porous materials. The relationship is revealed through simultaneously extracting correlation functions R_{uu}(r) of the spatial (Eulerian) velocity fields and microstructural two-point correlation functions S_{2}(r) of the random porous heterogeneous materials. This demonstrates that the effective physical transport properties depend on the characteristics of complex pore structure owing to the relationship between R_{uu}(r) and S_{2}(r) revealed in this study. Further, the mean excess plot was used to investigate the right tail of the streamwise velocity component that was found to obey light-tail distributions. Based on the mean excess plot, a generalized Pareto distribution can be used to approximate the positive streamwise velocity distribution.