Pore Structure and Permeability of Tight-Pore Sandstones: Quantitative Test of the Lattice–Boltzmann Method

Abstract

This paper presents a characterization of the pore structure of tight-pore sandstones of the Achimov suite and examines the application of Lattice–Boltzmann method (LBM) simulations to estimate the permeabilities of rock formations with a single-scale porosity. Porosity is characterized by pore volume distribution, pore throat connectivity, and tortuosity, which are calculated from 3D computer tomography pore network maps. The tight sandstones are poorly permeable, with permeabilities from 0.7 to 13 mD. For comparison, sandstones and carbonates with higher porosity and permeability from the existing database are also considered. For the more permeable reference samples with wider pores (250 µm), LBM simulations show good agreement with the experiments and somewhat outperform the selected state-of-the-art direct simulations from the literature. For samples with the tightest pores and lowest porosity, LBM simulations tend to somewhat overestimate the permeability in comparison with the direct simulation methods, whereas for samples of higher porosity, a slight underestimation is obtained. We explain the inconsistencies by an interplay between the compressibility effects neglected by our LBM simulations in wider pores and the friction at the pore-wall interface, which is underestimated due to the use of the bounce-back conditions. However, the general agreement with experimental and direct simulation methods is very reasonable and suitable for practical use, which means that LBM is fast, highly parallel, and computationally sound even in tight pores.

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]. 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 permabilities 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 ($\phi >$ 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). The simulations are briefly described in Section 2.2. The structure of the samples is explored in Section 3.1, and the permeabilities are calculated in Section 3.2. The Appendix A and Appendix B describe the computational performance and the tortuosity calculations, and Section 4 concludes the report.

2. Materials and Methods

2.1. The Samples

Considered in this paper are mostly sandstones, chosen because of their relatively simple isotropic porosity, which makes the comparison of simulations (performed on small samples, 1–20 mm) to experimental studies (3–50 cm) the most straightforward and valid. The relative simplicity of the sandstone pore structure was evidenced by a 3D pore network reconstruction from 2D images using machine learning in a limited database [10]. Another reason to choose sandstones is the availability of reconstructed 3D maps of the pore space, for which experimental and simulation results have been reported in the literature.

We start with relatively wide pore sandstones from the Imperial College of London (ICL) collection [37]. The resolution of the 3D maps is 2.85 to 10.0 µm/voxel (lower than the other samples considered here), and the physical size ranges from 1.155 mm to 4.5 mm. The structures are mapped on relatively coarse grids, from ${300}^3$ to ${450}^3$ voxels. For these samples, both experimental data and Pore Network Model (PNM) simulations are available from the literature [38,39]. The experiments were carried out in the most classical manner: by injecting fluid (water) mass flow with the pressure measured on the inlet and outlet of each sample.

Group 2 also contains selected samples from the ICL collection [37]: two sandstones and two carbonates. For these samples, we did not find experimental data, but their permeability was extensively modeled in the literature [40,41,42]. On average, they are less porous and less permeable than the samples from group 1. They are selected for this study because the pore structure maps available for them are much larger (1000${}^3$ cubic voxels) than for the samples of group 1, making the maps suitable for pore network characterization. As a benchmark for permeability calculations, we used the finite volume CFD simulations from Raeini et al. [40].

Our target samples are classical tight sandstones from the Achimov formation [43]. They form two groups, here referred to as group 3 and group 4. For both groups, CT images were obtained with a spatial resolution 1.2 µm/voxel (that is, substantially finer compared to the maps for groups 1 and 2). For group 3, the 3D pore maps were reconstructed using an algorithm that utilizes the spatial covariance of the image in conjunction with indicator kriging to determine object edges [7] (Figure 1). The use of indicator kriging makes the thresholding local and guarantees smoothness in the threshold surface. The samples of group 4 are approximately twice as large and mapped with the same spatial resolution. They were reconstructed using a combination of “heavy” band-pass (Fourier transform) and bilateral filters for denoising and local thresholding. Random walk-based algorithms were applied in the binarization of the 3D maps of the samples in both groups 3 and 4. For both groups, we do not have experimental data, but direct simulations (DHD, DiMP) have been reported [7].

2.2. Lattice–Boltzmann Simulations

The LBM simulations of the permeability of samples are performed in Palabos [44], an open source computational fluid dynamics software, which is a well-established software package tested in multiple LBM applications [45,46,47]. Its modern architecture and storage organization [44] allow efficient parallelization in massive computations. A large library of models and boundary conditions allows for the facile adaptation to a particular phenomenon. The methodology is straightforward: The flow of fluid through the reconstructed digital pore matrix is imposed by a fixed pressure gradient between the inlet and outlet faces of the sample. All simulations were held in LB units, represented as a two-step conversion from physical units to dimensionless, which do not depend on real physical sizes of the system, then from dimensionless to discretized, in order to convert to space and time grid-related units [48]. The gradient is implemented using the common relationship between density and pressure in LBM: $p_{\mathrm{d}}=\rho c_{\mathrm{s}}^2$ [49], where $p_{\mathrm{d}}$—is dimensionless pressure, $c_{\mathrm{s}}$ is the speed of sound in LB units, and $\rho$—density in LB units.

Since the permeability is independent of the pressure, the density at the inlet boundary is constant and equal to ${\rho }_{inlet}=1.0$, and the density at the outlet is varied in the interval ${\rho }_{outlet}=[0.995;0.95]$ depending on the sample. The resulting gradient values were primarily aimed at the stability and accuracy of the LBM [49]. The methodology implies that the system obeys the Darcy law, which requires $\mathrm{Re}\le 1$.

To avoid the artificial anisotropic velocity slip at the solid boundaries caused by the choice of relaxation time(s), we apply the MRT collision operator [23], which provides a more precise ratio between the kinematic and bulk viscosities of the flow. At the solid wall, the half-bounce-back boundary condition (HBB-BC) [50] is applied. HBB-BC effectively places an impenetrable boundary exactly between the cells. This approach helps improve the momentum exchange algorithm and maintain the Galilean invariance. It also alleviates the effect of the “staircase” approximation of complex curved boundaries [49], which is one of the major limitations of LBM in complex geometries. Because all of the samples from the dataset were already voxelized and there was no information about isosurfaces of the wall interfaces, the off-lattice BCs that could potentially improve the accuracy are not applicable. The faces, which are not the inlet or outlet, are padded by symmetry boundary condition.

Simulations are assumed to converge to a steady state when the mean kinetic energy deviation over 1000 iterative steps decreases below $ϵ={10}^{-4}$. The relaxation time is $\tau =1$, which provides optimal stability and accuracy of the simulation according to Asinari et al. [51]. In LB simulations of sample 3_C, we experienced problems with convergence because of the substantial number of narrow pores less than 4 voxels in diameter. That is, the CT resolution is insufficient for a permeability evaluation of such tight pores. To avoid the issue, we artificially increased the map resolution by scaling the matrix twofold and interpolating the boundary regions using the nearest-neighbor method (0-order interpolation). Such a technique is quite common [7,52].

3. Results

3.1. Pore Structure Characterization

The porosity of the samples is estimated as the fraction of the void space in the sample map. More important is the effective porosity that only accounts for the open pores’ space (all isolated voids are excluded). Both quantities are shown in

Table 1. Samples used in the present study: discretization, resolutions and porosity.

Sample	Discretization,  Voxels	CT Resolution,  µm/Voxel	Total  Porosity, %	Effective  Porosity, %	Mean Pore  Radius $\langle \mathit{d}\rangle$, µm	Mean Pore Radius $\langle {\mathit{d}}^{\prime }\rangle$, Voxel
Group 1 (sandstones from ICL database [37])
1_BS ${}^{\left(\mathrm{a}\right)}$	400${}^3$	5.345	19.6	19.6	31.67	5.92
1_LV60A	450${}^3$	10.0	37.7	37.7	62.074	6.21
1_S1	300${}^3$	8.683	14.1	14.1	52.96	6.09
1_S2	300${}^3$	4.956	24.6	24.6	31.59	6.37
1_S9	300${}^3$	3.398	22.2	22.2	24.42	7.18
1_C1	400${}^3$	2.85	23.3	23.3	17.47	5.14
1_A1	300${}^3$	3.85	42.9	42.9	24.658	6.40
Group 2 (select sandstones and carbonates from ICL dataset [37])
2_BS ${}^{\left(\mathrm{b}\right)}$	1000${}^3$	3.00	21.6	21.6	21.13	7.04
2_DS ${}^{\left(\mathrm{c}\right)}$	1000${}^3$	2.69	19.5	19.5	19.18	7.13
2_EC ${}^{\left(\mathrm{d}\right)}$	1000${}^3$	3.31	10.9	10.9	19.90	6.01
2_KC ${}^{\left(\mathrm{e}\right)}$	1000${}^3$	3.00	12.3	12.3	18.02	6.00
Group 3 (tight pore Achimov sandstones [43])
3_A	600${}^3$	1.2	6.5	5.8	3.15	2.63
3_B	600${}^3$	1.2	5.3	4.4	5.05	4.2
3_C	600${}^3$	1.2	5.8	5.3	3.75	3.13
3_D	600${}^3$	1.2	9.9	9.7	5.3	4.42
3_E	600${}^3$	1.2	7	6.4	4.65	3.88
Group 4 (tight pore Achimov sandstones [43])
4_A	1400${}^3$	1.2	8.9	8.1	7.15	5.95
4_B	1400${}^3$	1.2	6.5	5.1	7.01	5.84
4_D	1400${}^3$	1.2	10.8	10.2	7.12	5.93

^a Berea sandstone; ^b Bentheimer sandstone; ^c Doddington sandstone; ^d Estaillades carbonate; ^e Ketton carbonate.

. The samples from group 1 are the most porous, and the Achimov samples from groups 3 and 4 are the least porous among the sandstones considered here: for example, sample 1_LV60A is 8.5 times more porous than sample 3_B. Carbonates from the ICL database (group 2) are also relatively low in porosity compared to other samples from groups 1 and 2, but still more porous than the Achimov sandstones (groups 3, 4). We also estimate the mean pore size from PSD, calculated using the machine learning framework [53]. The mean pore radius ranges from 5 to 9 voxels, which is sufficient for performing the simulations [52].

As mentioned, the samples from group 1 are the most porous, and their digital maps are the smallest. The pore morphologies of wide pore sandstones were extensively studied by Dong et al. [54]. For these reasons, we do not discuss the pore characteristics of this group here. For finer discretized maps of the samples from groups 2, 3 and 4, we calculate pore volume distributions and characterize the pore connectivity and pore network tortuosity. Using the Euclidean distance transform (EDT) [55], for each pore voxel, the largest sphere non-overlapping with the walls is found. Each pore space voxel is assigned the radius of the maximum inscribed sphere (MISA). The field contains as many maximum spheres as the binary image contains pore space voxels. Clearly, some void voxels would be part of multiple spheres. The set of maximum spheres has to be reduced to a subset of containing spheres by searching for each void voxel for the largest sphere in which this voxel was contained. This implies that the spheres contained within a larger sphere are eliminated. This method enables the dissection of a continuous pore space to an integer number of effective pores and thus calculates the pore volume distribution (PVD), as well as characterizes the connectivity and tortuosity.

Figure 2 shows the pore volume distribution obtained from the MISA analysis.

There is a visible difference between the pore structure of the reference sandstones and carbonates from the ICL database (group 2) and the Achimov sandstones (groups 3 and 4). All Achimov sandstones have unimodal pore volume distributions with the mode at 420–900 µm³ (

Table A1. Pore structure characterization parameters.

Sample	3_A	3_B	3_C	3_D	3_E	2_BS	2_DS	2_EC	2_KC
Average pore volume (voxel), $\langle V\rangle$	260	561	256	527	458	2824	11,376	1846	2642
Average pore volume (µm3), $\langle V^{\prime }\rangle$	450	970	442	911	793	76,533	222,162	67,028	71,350
Max pore volume (voxel), $V^{max}$	34,855	45,593	101,760	93,878	84,046	674,770	659,394	438,012	1,548,038
$C_t$	2.222 ± 0.437	2.148 ± 0.36	2.204 + 0.419	2.271 ± 0.473	2.256 ± 0.459	2.328 ± 0.492	2.284 + 0.462	2.303 ± 0.480	2.240 ± 0.440
$C_{\mathrm{p}}$	6.462 ± 7.161	4.736 ± 4.265	6.209 ± 7.229	7.817 ± 9.854	7.175 ± 8.529	12.316 + 16.101	10.300 ± 13.227	8.465 ± 13.487	7.568 + 10.033

), which approximately corresponds to an 8 µm pore radius. The mode of the pore volume distribution only slightly varies with the overall porosity. The reference samples (group 2) are more diverse. 2_BS and 2_KC samples (Bentheimer sandstone and Ketton carbonate) have a large share of very small pores that have a low influence on overall permeability. At the same time, they have a higher fraction of pores of much high volume compared to Achimov sandstones. The other two samples, 2_EC and 2_DS (also one sandstone and one carbonate), have unimodal pore volume distributions similar to the Achimov sandstones, but the modes are shifted to volumes 2–3 orders of magnitude larger, with the 3_DS sandstone having the largest average pore size. It is peculiar that within groups 2 (we do not of course pretend on any generalization) neither the prevailing mineral nor the overall porosity correlate with the type of pore volume distribution.

The pore coordination number $C_{\mathrm{p}}$ refers to the compartmentalized representation of the pore space obtained with the MISA algorithm and measures the number of pores with which a given pore shares throat voxels [6] (Table A1). $C_t$ is the pore throat coordination number, which is the number of effective pores connected by each junction connects. Most junctions, of course, only connect two neighboring pores, which basically means a narrowing in the channel through which the fluid flows. Since the ratio between junction diameter and channel diameter is unknown, it is difficult to say how narrows affect flow. The connectivity R is characterized by the concentration of junctions connecting more than two pores, and can be represented as:

$$R\left(C_t\right)=\frac{{\sum }_{i=C_t}^j{\delta }_{ji}}{V_{\mathrm{s}}}\langle V_{\mathrm{p}}\rangle ,$$

(1)

where $V_{\mathrm{s}}$ is the total volume of the system, $\langle V_{\mathrm{p}}\rangle$ is the average pore volume, and j is the index of junction. Such a normalization by a characteristic pore volume makes the concentration independent of spatial resolution.

Figure 3 demonstrates the connectivity of the pore networks. $C_t$ mean values are available in Table A1. One can observe that the junctions with high coordination numbers (>4) are so rare that they can hardly influence the permeabilities. The most important junctions, therefore, are those that connect three different pores. The connectivity varies substantially even within the same group. For example, sample 3_B has a very low concentration of three-pore and four-pore junctions compared to other group 3 samples, 3_D in particular. In general, Achimov sandstones are poorly connected to the reference samples from group 2. Sample 1D has the best connected network, according to the EDT analysis. The same sample has the widest pores among the Achimov sandstones.

3.2. 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 Darcy’s law equation:

$$k^{*}=\langle v\rangle \mu \Delta \rho ,$$

(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. Comparison of the calculated absolute permeability k by different methods.

Sample Name	k, mD (ICL) (exp.)	k, mD (DHD)	k, mD (DiMP)	k, mD (ICL PNM)	k, mD (ICL DNS)	k, mD (LBM)	Relative Error, %
Group 1 (sandstones from ICL database [37])
1_BS	1286	–	–	1111	–	1302	1.24% ${}^{\left(\mathrm{a}\right)}$
1_LV60A	35,300	–	–	27,200	–	30,246	−14.32% ${}^{\left(\mathrm{a}\right)}$
1_S1	1678	–	–	1486	–	1525	−9.09% ${}^{\left(\mathrm{a}\right)}$
1_S2	3898	–	–	3951	–	3741	−4.02% ${}^{\left(\mathrm{a}\right)}$
1_S9	2224	–	–	3640	–	2097	−5.71% ${}^{\left(\mathrm{a}\right)}$
1_C1	1102	–	–	556	–	1213	8.67% ${}^{\left(\mathrm{a}\right)}$
1_A1	7220	–	–	8076	–	6024	−16.57% ${}^{\left(\mathrm{a}\right)}$
Group 2 (select sandstones and carbonates from ICL dataset [37])
2_BS (a)	–	–	–	–	3595	2805	−22% ${}^{\left(\mathrm{b}\right)}$
2_DS (b)	–	–	–	–	3812	3673	−3.6% ${}^{\left(\mathrm{b}\right)}$
2_EC (c)	–	–	–	–	221.4	313.3	41% ${}^{\left(\mathrm{b}\right)}$
Group 3 (tight pore Achimov sandstones [43])
3_A	–	0.67	0.84	–	–	1.14	70% ${}^{\left(\mathrm{c}\right)}$
3_B	–	1	0.87	–	–	1.24	24%${}^{\left(\mathrm{c}\right)}$
3_C	–	0.32	0.66	–	–	0.87	172% ${}^{\left(\mathrm{c}\right)}$
3_D	–	10.2	9.38	–	–	12.88	26% ${}^{\left(\mathrm{c}\right)}$
3_E	–	0.68	1.28	–	–	1.69	148% ${}^{\left(\mathrm{c}\right)}$
Group 4 (tight pore Achimov sandstones [43])
4_A	–	–	3.05	–	–	5.54	81%
4_B	–	–	1.6	–	–	3.03	89%
4_D	–	–	12.53	–	–	16.3	30%

^a Error related to the experimental data; ^b error related to the results of ICL simulations (depending on method); ^c error related to the DHD simulations.

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 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.

Comparison of the LBM permeabilities with those obtained with PoreNetwork models [39,40] for group 1 shows very reasonable agreement and no clear trends. The largest difference is obtained for the 1_C1 sample, where the LBM estimate is twice as high. On the contrary, the permeability of the 1_S9 sandstone sample was 1.5 times lower than the PNM result. The discrepancies appear random, with no clear trend. We should say that our results show somewhat better agreement with the experiment, and at least the LBM calculations showed no large discrepancies. The PNM solver underestimates the permeability of the 1_C1 sample by the factor of 2 and overestimates the permeability of the 1_S9 sample by 70%, while the LBM estimates fall with 10% from the experimental result for both systems. However, in general, the discrepancies are by no means dramatic for both methods.

The samples from group 2 are somewhat tighter. Since the experiments are not available, we can only compare them with the same simulation benchmark of the OpenFOAM finite volume solver [40]. Again, the agreement between the LBM and DNS calculations is reasonable and does not show any clear trend. Yet again, LB showed substantially higher permeability (compared to DNS) for the least permeable sample, 2_EC carbonate. The same tendency was noted above for the group 1 sample. At the same time, the permeability of the Bentheimer sandstone sample (2_BS) has a wide PVD with a high share of very narrow pores but higher share of wide pores (which contribute the most to the permeability); LBM underestimated the permeability by 20% compared to the ICL direct simulations [40].

We relate this difference to two important factors in the LBM modeling. First, at high pressure gradients applied in sandstone permeability measurements, the compressibility of the fluid becomes important for the overall flow. In most LBM simulations, including those reported here, the compressibility is ignored and, therefore, the permeability is somewhat overestimated. At the same time, an artificial and anisotropic slip caused by the bounce-back boundary condition applied in our LBM simulations effectively decreases the viscosity–dependent friction between the pore fluid and the wall. The tighter the pores, the more pronounced this distortion is, leading to an overall overestimation of the permeability of tight-pore samples by the LBM.

Considering the tightest samples of this work, the Achimov sandstones of groups 3 and 4, we observe that the overprediction of the permeability by LBM becomes somewhat systematic. Unfortunately, the rigorous test against the experimental data is not available. However, LBM shows a higher permeability compared to both DHD and DiMP for all Achimov sandstone samples considered here, while DHD and DiMP results are reasonably close and show no clear trends when compared with each other (for example, DHD permeability is higher for the 3_B sample and lower for the 3_C sample compared to the DiMP permeability). Overall, the largest discrepancies between LBM and other methods are obtained for the tightest pore samples with the lowest permeabilities (3_C and 3_E). The exception is sample 3_B, which is poorly permeable, and yet, the results obtained by LBM and DHD are quite close. In fact, the larger discrepancy between DiMP and LBM permeabilities (sample 4_B) is no worse than the diescrepancy between the PNM results from [40] and the experiment for wide-pore sandstone 1_S2. In no case did we observe a dramatic disagreement that would certainly prohibit the practical application of LBM as the method for permeability prediction.

It might have also been expected that the artificial effect of friction reduction in LB were more pronounced for physically large samples. The longer the path the fluid needs to take “from start to finish”, the more important the friction forces become. Thus, for larger samples with larger geometry discretization error and more wall boundary voxels, higher deviations of LB calculations from the benchmark results were expected. However, the permeability calculations for group 2 do not show a larger disagreement between the LBM and DHD/DiMP results compared to those for group 1.

In a comparison between the geometric characteristic of the pore structure and the permeabilities calculated with LBM, we can assume that the overall porosity and the pore volume distribution strongly influence the permeability, while the role of connectivity is not obvious. The general correlation between porosity and permeability is apparent. In group 3, samples 3_A, 3_B, 3_C, and 3_E have a comparable porosity. Among them, sample 3_B has the highest fraction of wider pores and has the highest permeability, while sample 3_C has the lowest share of wider pores and the lowest permeability. At the same time, the lower pore throat coordination number in sample 3_B compared to the other Achimov sandstones from group 3 does not appear to decrease the permeability. Thus, overall, it is the volume occupied by relatively wide pores that makes a difference in terms of permeability rather than the pore throat coordination.

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 µm) 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 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. For some manufactured materials, the connectivity characteristics proved very important for fluid transport [57,58].

Speaking of requirements to the resolution of the digital porosity maps, this work confirms that the pore width should be larger than at least four voxels. If this is not the case, the LB solver experiences stability problems and shows poor convergence. In order to obtain a more stable simulation, the map can be scaled by the factor of two and the walls interpolated to achieve a smoother structure. What remains unclear is the role of the box size in comparison with the pore network feature size. Our geometrical analysis showed that in all considered samples, the box is not much larger than the topological features of the pore network: all maps showed a substantial anisotropy. For wide pore sandstones, this did not seem to create a problem, as the calculated permeabilities compared well with the experimental values. For tighter samples, where the scale of the pore network geometric features is expected to be larger in comparison with the available box size, the effect of the limited map size is expected to be more severe. However, we cannot draw a clear conclusion on the importance of the anisotropy effect without experimental data.