Deep learning for diffusion in porous media.

We adopt convolutional neural networks (CNN) to predict the basic properties of the porous media. Two different media types are considered: one mimics the sand packings, and the other mimics the systems derived from the extracellular space of biological tissues. The Lattice Boltzmann Method is used to obtain the labeled data necessary for performing supervised learning. We distinguish two tasks. In the first, networks based on the analysis of the system's geometry predict porosity and effective diffusion coefficient. In the second, networks reconstruct the concentration map. In the first task, we propose two types of CNN models: the C-Net and the encoder part of the U-Net. Both networks are modified by adding a self-normalization module [Graczyk et al. in Sci Rep 12, 10583 (2022)]. The models predict with reasonable accuracy but only within the data type, they are trained on. For instance, the model trained on sand packings-like samples overshoots or undershoots for biological-like samples. In the second task, we propose the usage of the U-Net architecture. It accurately reconstructs the concentration fields. In contrast to the first task, the network trained on one data type works well for the other. For instance, the model trained on sand packings-like samples works perfectly on biological-like samples. Eventually, for both types of the data, we fit exponents in the Archie's law to find tortuosity that is used to describe the dependence of the effective diffusion on porosity.

Pushing Droplet Through a Porous Medium.

I use a mechanical model of a soft body to study the dynamics of an individual fluid droplet in a random, non-wettable porous medium. The model of droplet relies on the spring-mass system with pressure. I run hundreds of independent simulations. I average droplets trajectories and calculate the averaged tortuosity of the porous domain. Results show that porous media tortuosity increases with decreasing porosity, similar to single-phase fluid study, but the form of this relationship is different. Supplementary Information: The online version contains supplementary material available at 10.1007/s11242-021-01705-z.

Predicting porosity, permeability, and tortuosity of porous media from images by deep learning.

Convolutional neural networks (CNN) are utilized to encode the relation between initial configurations of obstacles and three fundamental quantities in porous media: porosity ([Formula: see text]), permeability (k), and tortuosity (T). The two-dimensional systems with obstacles are considered. The fluid flow through a porous medium is simulated with the lattice Boltzmann method. The analysis has been performed for the systems with [Formula: see text] which covers five orders of magnitude a span for permeability [Formula: see text] and tortuosity [Formula: see text]. It is shown that the CNNs can be used to predict the porosity, permeability, and tortuosity with good accuracy. With the usage of the CNN models, the relation between T and [Formula: see text] has been obtained and compared with the empirical estimate.

Finite-size anisotropy in statistically uniform porous media.

Anisotropy of the permeability tensor in statistically uniform porous media of sizes used in typical computer simulations is studied. Although such systems are assumed to be isotropic by default, we show that de facto their anisotropic permeability can give rise to significant changes in transport parameters such as permeability and tortuosity. The main parameter controlling the anisotropy is a/L , being the ratio of the obstacle to system size. Distribution of the angle alpha between the external force and the volumetric fluid stream is found to be approximately normal, and the standard deviation of alpha is found to decay with the system size as (a/L);{d/2} , where d is the space dimensionality. These properties can be used to estimate both anisotropy-related statistical errors in large-scale simulations and the size of the representative elementary volume. For porous media types studied here, the anisotropy effect becomes negligible only if a/L < or = 0.01 . This constraint was apparently violated in many previous computer simulations that need now to be recalculated.

Tortuosity-porosity relation in porous media flow.

We study numerically the tortuosity-porosity relation in a microscopic model of a porous medium arranged as a collection of freely overlapping squares. It is demonstrated that the finite-size, slow relaxation and discretization errors, which were ignored in previous studies, may cause significant underestimation of tortuosity. The simple tortuosity calculation method proposed here eliminates the need for using complicated, weighted averages. The numerical results presented here are in good agreement with an empirical relation between tortuosity (T) and porosity (varphi) given by T-1 proportional, variantlnvarphi , that was found by others experimentally in granule packings and sediments. This relation can be also written as T-1 proportional, variantRSvarphi with R and S denoting the hydraulic radius of granules and the specific surface area, respectively. Applicability of these relations appears to be restricted to porous systems of randomly distributed obstacles of equal shape and size.

Power-exponential velocity distributions in disordered porous media.

Velocity distribution functions link the micro- and macro-level theories of fluid flow through porous media. Here we study them for the fluid absolute velocity and its longitudinal and lateral components relative to the macroscopic flow direction in a model of a random porous medium. We claim that all distributions follow the power-exponential law controlled by an exponent γ and a shift parameter u_{0} and examine how these parameters depend on the porosity. We find that γ has a universal value 1/2 at the percolation threshold and grows with the porosity, but never exceeds 2.

Anisotropy of flow in stochastically generated porous media.

Models of porous media are often applied to relatively small systems, which leads not only to system-size-dependent results, but also to phenomena that would be absent in larger systems. Here we investigate one such finite-size effect: anisotropy of the permeability tensor. We show that a nonzero angle between the external body force and macroscopic flux vector exists in three-dimensional periodic models of sizes commonly used in computer simulations and propose a criterion, based on the ratio of the system size to the grain size, for this phenomenon to be relevant or negligible. The finite-size anisotropy of the porous matrix induces a pressure gradient perpendicular to the axis of a porous duct and we analyze how this effect scales with the system and grain sizes. We also analyze how the size of the representative elementary volume (REV) for anisotropy compares with the REV for permeability.

Hydraulic tortuosity in arbitrary porous media flow.

Tortuosity (T) is a parameter describing an average elongation of fluid streamlines in a porous medium as compared to free flow. In this paper several methods of calculating this quantity from lengths of individual streamlines are compared and their weak and strong features are discussed. An alternative method is proposed, which enables one to calculate T directly from the fluid velocity field, without the need of determining streamlines, which greatly simplifies determination of tortuosity in complex geometries, including those found in experiments or three-dimensional computer models. Based on numerical results obtained with this method, (a) a relation between the hydraulic tortuosity of an isotropic fibrous medium and the porosity is proposed, (b) a relation between the divergence rate of T with the system size at percolation porosity and the scaling of the most probable traveling length at bond percolation is found, and (c) a range of porosities for which the shape factor is constant is identified.