## 4 Conclusions  
In this work, for the first time, we evaluate the Lattice-Boltzmann method for permeability prediction on substantial collections of sandstones and carbonates, both highly porous with open porosity and tight pore (6-10 um) sandstones of the Achimov suite. We also analyze the pore sizes and connectivity and relate them to the permeability. Our main conclusion is that LB with multiple relaxation times and half-bounce-back boundary conditions at the pore walls quantitatively predict permeabilities with a reasonable accuracy even in tight pore sandstones of low porosity. For samples of high  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{ \begin{tabular}{c} **Sample** \\ **Name** \\ \end{tabular}  } & \begin{tabular}{c} _k,_ **mD** \\ **(ICL)** \\ **(exp.)** \\ \end{tabular}  & \begin{tabular}{c} _k,_ **mD** \\ **(DHD)** \\ \end{tabular}  & \begin{tabular}{c} _k,_ **mD** \\ **(ICL PNM)** \\ \end{tabular}  & \begin{tabular}{c} _k,_ **mD** \\ **(ICL DNS)** \\ \end{tabular}  & \ \begin{tabular}{c} _k,_ **mD** \\ **(LBM)** \\ \end{tabular}  & \begin{tabular}{c} _k,_ **mD** \\ **Error, \%** \\ \end{tabular}  \\ \hline \multicolumn{8}{c}{Group 1 (sandstones from ICL database [37])} \\ \hline \(1\)\(\_\)BS & 1286 & – & – & 1111 & – & 1302 & \(1.24\%\)\({}^{\text{(a)}}\) \\ \(1\)\(\_\)LV60A & 35,300 & – & – & 27,200 & – & 30,246 & \(-14.32\%\)\({}^{\text{(a)}}\) \\ \(1\)\(\_\)S1 & 1678 & – & – & 1486 & – & 1525 & \(-9.09\%\)\({}^{\text{(a)}}\) \\ \(1\)\(\_\)S2 & 3898 & – & – & 3951 & – & 3741 & \(-4.02\%\)\({}^{\text{(a)}}\) \\ \(1\)\(\_\)S9 & 2224 & – & – & 3640 & – & 2097 & \(-5.71\%\)\({}^{\text{(a)}}\) \\ \(1\)\(\_\)C1 & 1102 & – & – & 556 & – & 1213 & \(8.67\%\)\({}^{\text{(a)}}\) \\ \(1\)\(\_\)A1 & 7220 & – & – & 8076 & – & 6024 & \(-16.57\%\)\({}^{\text{(a)}}\) \\ \hline \multicolumn{8}{c}{Group 2 (select sandstones and carbonates from ICL dataset [37])} \\ \hline \(2\)\(\_\)BS (a) & – & – & – & 3595 & 2805 & \(-22\%\)\({}^{\text{(b)}}\) \\ \(2\)\(\_\)DS (b) & – & – & – & 3812 & 3673 & \(-3.6\%\)\({}^{\text{(b)}}\) \\ \(2\)\(\_\)EC (c) & – & – & – & 221.4 & 313.3 & 41\%\({}^{\text{(b)}}\) \\ \hline \multicolumn{8}{c}{Group 3 (tight pore Achimov sandstones [43])} \\ \hline \(3\)\(\_\)A & – & 0.67 & 0.84 & – & – & 1.14 & \(70\%\)\({}^{\text{(c)}}\) \\ \(3\)\(\_\)B & – & 1 & 0.87 & – & – & 1.24 & \(24\%\)\({}^{\text{(c)}}\) \\ \(3\)\(\_\)C & – & 0.32 & 0.66 & – & – & 0.87 & \(172\%\)\({}^{\text{(c)}}\) \\ \(3\)\(\_\)D & – & 10.2 & 9.38 & – & – & 12.88 & \(26\%\)\({}^{\text{(c)}}\) \\ \(3\)\(\_\)E & – &porosity and wider pores (1_S1, 1_S2, 1_S9,1_C1) LBM results showed a somewhat better agreement with the experiments than the state-of-the-art direct Navier-Stokes solvers. For tight-pore samples, the agreement between LBM and other state-of-the-art calculations is satisfactory: the difference between LB permeabilities and those obtained by DiMP simulations is of the same order as the difference between two direct simulation methods, DiMP and DHD. Generally LBM tends to:  
* Somewhat overpredict the permeability of very tight pore least permeable samples due to underestimations of fluid-wall friction caused by the application of half-bounce-back boundary conditions;
* Somewhat underestimate the permeabilities of more permeable samples with open porosity due to a neglect of the compressibility effects. The permeability shows, as expected, a strong dependence on the effective porosity (and therefore with the average pore size that in our samples is strongly correlated with porosity, as well) and the pore volume distribution, in particular. However, the parameters of the network connectivity, in particular the pore throat coordination numbers, do not appear to influence the permeability in the sample considered in this work. It is not clear how general this observation is, because the pore structure is related to the process of the formation of geological samples and might be quite different for different types of rock formations.