## 3 Results
### Pore Network Permeabilities  
Each LB simulation produces stationary velocity and and pressure fields (Figure 4). Typically, high-velocity zones are observed in the throats leading to larger pores and in the pores of low tortuosity. It is possible to visually distinguish several connected pores through which the main flow passes, as well as the dead-end pores. Generally, the flow in samples with wider pores is more uniform. The contrast between fast-moving and slow-moving sectors for wide pore samples is not as pronounced as in the tighter pore sandstones of low porosity. Examples are shown in Figure 4.  
Absolute permeability \(k^{*}\) is calculated with the [PERSON]'s law equation:  
\[k^{*}=\langle v\rangle\mu\Delta\rho, \tag{2}\]  
where \(\langle v\rangle\) is the mean velocity of flow, obtained from LBM simulation; \(\mu=1\) is the dynamic viscosity in LB units; \(\Delta\rho\) is the density difference between inlet and outlet in LB units. For conversion to physical units, the LB units should be multiplied by a characteristic scale. For permeability, it leads to \(k=k^{*}s^{2}\), where \(s\) is the sample resolution. Table 2 shows the results of the permeability calculations in comparison with the experimental data [38; 39] and the results obtained by other methods that are considered the most accurate today and applied as modern benchmarks [7; 38; 39]. First, we can observe that the permeability predictions for the wide pore sandstones made with the LBM methods agree with the experiments quite well. The largest discrepancy obtained is approximately 16%, and most of the results fall within 10% of the experimental value, which is comparable to the experimental error and certainly sufficient for practical purposes. We cannot clearly say that LBM systematically underestimates or overestimates the permeability, although our values tend to be lower compared to the experiments. In general, the comparison with experiments is affected by two artifacts that are related to the size of the digital map. First of all, for samples of low porosity, the percolation effects are quite important. The percolation of pore networks in a smaller body is typically achieved at lower porosity than in a larger body with a similar structure, unless the characteristic feature scale of the pore network is much smaller than the system size [56]. In our digitized rock structures, the size of the system and the scale of the characteristic geometric features of the pore network are somewhat comparable, which is evidenced by the tortuosity maps presented in Appendix B. In an infinite porous system, the ratio of the length of the shortest path through the porous network to the Euclidian distance between the same two points should converge to a constant as the distance increases. In the Achimov sandstone samples considered here, the tortuosity maps show non-uniform and anisotropic structures with no visible asymptotic relationship. This means that the characteristic scale of the pore network topology is comparable to the size of the digitized sample. This artifact should facilitate percolation and, therefore, increase the calculated permeability. On the other hand, the limited size of the 3D map effectively cuts off the channels that cross the side boundaries of the digitized sample. Therefore, the channels  
Figure 4: Magnitude of the velocity field for (**A**) tight-pore Achimov sandstone sample 3_E with 6.4%porosity (**B**) wide-pore sandstone sample 1_LV60A with higher porosity (37.7%) (see Table 1). that cross the side boundaries become dead-end pores, the effective porosity decreases, and the permeability decreases as well. The smaller the digitized sample size is, the higher the fraction of channels that are effectively cut off. To what extent each of the factors matters is hard to say, and a detailed study of the system size would be required for that. These considerations (which, surprisingly, have received little attention in the literature) apply to all CFD methods, not only to LBM.