## 1 Introduction  
Digital Rock Physics (DRP) complements laboratory studies in efforts to understand mechanical, thermal and transport properties as well as obtain characteristics of natural porous media important for water resources, oil recovery, composite manufactures, etc. The two fundamental components of the DRP are (1) a detailed characterization of the sample morphology and (2) property prediction from the structure. The typical workflow of DRP first involve digital reconstruction of the three-dimensional pore space matrix with computer tomography (CT), nowadays based on artificial intelligence [1]. Then, the methods of computational physics are applied to calculate the structure-property relationships. The analysis of the geometry of the pore space is traditionally based on the theoretical interpretation of gas adsorption data [2, 3], mercury or another liquid intrusion [4], positron annihilation, etc. Each of those methods is optimal for a particular type of sample and sensitive to pores of a certain diameter range. The primary result is the pore size distribution (PSD), although some methods are sensitive to finer geometric features, such as the pore throat distributions and ink-bottle effects [5]. Detailed 3D structures available from the segmentation of CT images allow a finer characterization of the pore geometry that is not limited to the overall porosity, surface area and PSD (at the resolution scale), but also include pore throat, pore volume distributions, and pore tortuosity characterization [6]. However, 3D maps are available for relatively small sample sizes. If the pore space resolution is in a micron-range, the sample mass is in the milligram range, while gas sorption data are available for approximately 10g samples. Thus, the representativeness of the structures remains an issue. Other problems are related to CT scanner resolution, as well as the segmentation artifacts of the images [7], although modern machine learning-based methods have improved reconstruction. The finer the pore structure, the more severe these issues become [8; 9]. After the 3D structure is processed, a variety of computational tools can be involved in predictions of the structure--property relationships. The most attention is paid to the permeability of single-phase and multi-phase flows through the pore network and the relationship of the flows to the pore network topology and mineral composition. Machine learning techniques can be applied to such relationships to restore the pore structure from 2D images [10; 11] and predict flows without any physics-based simulations. Such predictions for multi-component solid structures have greatly expanded the capacity for fluid flow simulations in geological solids, the manufacture of porous and fibrous materials, textiles, and pharmaceutical dosages, where fluid wicking is a topical problem [12]. This, however, is a difficult work that requires computationally efficient and standardized numerical simulation tools with precisely known applicability limits for the estimation of fluid flows. Single phase and multiphase fluids could be simulated with a variety of models, the choice of which is determined by the core of the applied problem. In the case of flows in the geometry of the pore space, the direct solution of the Navier-Stokes (NS) equations is the most common approach. It deals with systems of conservation equations for macroscopic properties and explicitly defined phase boundaries. On the contrary, the Lattice-Boltzmann method (LBM) has gained rapidly in popularity over the last few years [13]. In LBM, densities and velocities are computed for fictive particles on a lattice with streaming and collision processes using \"local rules\", which provide unmatched computational efficiency and versatility. Phase boundaries are calculated from the densities. LBM has been demonstrated to handle the dynamics of quite complex systems, processes, and interactions (e.g., disjoining forces), including foams and gas filtration [14; 15].