## 1 Introduction
The main obstacle to the routine engineering applications of LBM in fluid flow simulations is a lack of understanding of the applicability in \"tight\" systems, that is, when the feature size (be it the pore width, fiber thickness, etc.) becomes comparable with the effective voxel size. It should be noted that NS solvers often also succumb to poor convergence in tight media [16]. Recent efforts invested in the development of direct methods [17; 18; 19], such as Direct Hydrodynamic (DHD) simulations [20] and Dimp-Hydro Solver (DiMP) [17], substantially improved flow predictions in tight pore samples at a certain cost in computational efficiency and versatility. As for LBM, most efforts were traditionally invested in DRP. A classical Lattice-Boltzmann method with the Bhatnagar-Gross-Krook collision operator (BGK-LBM) showed very good (up to 95%) agreement with experimental data on the permeabilities of wide pore samples [21; 22]. However, the sample size and mesh resolution, which were up to the standards of 2010, are outdated in 2023. The relaxation time dependencies for the wall boundary conditions may cause poor stability. The multiple-relaxation-time (MRT) [23; 24; 25; 26] approach, the most general and advanced relaxation model of today, has become common practice and improved the accuracy due to the generalization of the equation in terms of momenta. It also allows the bulk viscosity to be chosen independently of the shear viscosity, thus supporting the Galilean invariance. However, it is also more laborous to code, and it becomes computationally more expensive if not coded efficiently. In Ref. [16], LBM-MRT simulations overestimated the permeability of rock samples with moderate pore sizes by 30 to 100%, which is comparable to the errors of different experimental methods and sufficient for practical purposes. MRT and the off-lattice boundary [27; 28] improved the accuracy of permeability predictions to 7-8% for moderate pore sizes [29]. However, this approach requires an exact determination of the wall interfaces, which are unavailable in most digitized rock structures, excepting the usage of the marching cubes algorithm. Taking into account the pore size effects [30] and coupling LBM with finite differences calculations for the refinement of the pre-boundary layers [31], is also quite promising. However, success has been almost exclusively reported with structures of moderate to high porosity (\(\varphi>0.2\)), with developed networks of reasonably large pores (0.1-2 mm) [16; 22; 32; 33]. The applicability limits of LBM in tight pore media remain unexplored, and are of substantial interest, accompanied by recent progress in LBM simulations of complex fluids and the trend towards the development of tight pore hydrocarbon reservoirs with a very little open porosity, where structure-permeability relationships may be decisive for efficient filtration. The pore structure-permeability relationships were studied in the literature. Attention was paid to characteristic pore length, overall volume and tortuosity [34]; Euler characteristics [35]; connectivity and pore size distribution [36]. However, again, practically all of these studies deal with more porous systems. The structure of tight sandstone collectors and the relationship between topology and permeability remain unclear. In this paper, we evaluate the accuracy of single phase LBM simulations in sandstone and carbonate samples of different pore sizes, digital sample sizes, and resolution. We compare samples from different sources and validate the LBM simulations against experiments on permeability and direct NS solvers, which are considered here as state-of-the-art. The other goal of the paper is to relate the permeabilities to the structural features of the rock pore network, namely throat sizes and coordination numbers that characterize the pore connectivity along the pore size distributions and pore network tortuosity. This paper is structured as follows. First, we discuss the available samples, data sources, and digital representation (Section 2.1).