Samples were extracted from three consecutive layers of different colours from a stratigraphic sequence (Fig. 1c and d). The lower layer (3) is ∼1.5 m thick and consists of sandstone that is light (pale) red–purple in colour with undulating bedding planes between the sublayers. The middle layer (2) is composed of grey–green shaly sandstone that is 20 cm thick with dark horizons at the bottom and top. The upper layer (1) comprises 1.5 m thick homogenous brown–yellow sandstone. Large sample blocks of ∼10–20 cm size were collected from these three layers, and the directions perpendicular to the bedding planes (defined as the z directions in our study) were noted. Subsequently, in the laboratory, smaller subsamples (described below) were prepared from these large samples for textural observations and various analytical measurements and computations. In total, seven subsamples from the top layer, eight subsamples from the middle layer and four subsamples from the bottom layer were investigated in the laboratory (Table 2).

The integrated analytical programme designed for this study includes the following laboratory measurements and computations conducted at different scales (from the microscale reflecting the scale of individual pores and grains to the core scale reflecting the scale of the laminas at the outcrop) (Table 1). Specimens ∼5–7 cm in size were investigated by petrographic and petrophysical lab methods. Subsamples ∼1 cm in size were retrieved from the aforementioned plugs for investigation by 3-D imaging, digital image analysis and simulation techniques (described in detail below).

Petrographic and petrophysical analysis (nos. 1–7 in Table 1) were conducted following the RP40 guidelines (Recommended Practices for Core Analysis, API, 1998), giving detailed information on theory, advantages and drawbacks of each method. An extended computational workflow (no. 8 in Table 1) combines several methods that may contain some variability in their application for the rock characterization. Those are described in more detail below.

Petrographic descriptions of rock compositions and textures at the microscale, notably those of the fine fraction, were performed using scanning electron microscopy (JCM-6000 Bench Top SEM device; e.g. Krinsley et al., 2005) using both backscatter and secondary electron modes.

Thin-section optical microscopy (Olympus BX53 device; e.g. MacKenzie et al., 2017) was used to estimate the mineral abundance and surface features of the grains and the mineralogical and textural features of matrix and cement. Grain size distributions were determined by a laser diffraction particle size analyser (LS 13 320; e.g. Wang et al., 2013). X-ray diffraction (Miniflex 600 device by Rigaku; e.g. Asakawa et al., 2020) was applied to powdered samples to determine their mineralogical composition.

Effective porosity and permeability were evaluated on dried cylindrical samples (2.5 cm in diameter and 5–7 cm in length). Effective porosity (ϕ) was measured using a steady-state nitrogen gas porosimeter produced by Vinci Technologies (HEP-E, Vinci Technologies; e.g. Viswanathan et al., 2018). Absolute permeability (κ) was measured by using a steady-state nitrogen gas permeameter (GPE, Vinci Technologies; e.g. Tidwell and Wilson, 1999).

Mercury intrusion porosimetry (Micromeritics AutoPore IV 9505, which considers pore throats larger than 0.006 µm; e.g. Giesche, 2006), was applied to dried cylindrical samples ∼1 cm3 in size to evaluate the following parameters (Table 1):

pore throat size distribution (PTSD) (Lenormand, 2003);

An extended computational workflow (similar to the procedure presented by Boek and Venturoli, 2010; Andrä et al., 2013a, b) (Fig. 2) serves as one of the main methodologies in our study to upscale to permeability and derive microscopic rock descriptors. It includes 3-D micro-CT imaging of porous samples, digital image processing and segmentation, analyses for the determination of representative elementary volumes and pore-scale flow modelling through the 3-D pore geometry of the rock. First, cylindrical subsamples 4–8 mm in diameter and 5–10 mm in length were retrieved from the larger samples studied in the laboratory (Fig. 2a) and were scanned non-destructively (Fig. 2b) by using a Nanotom 180 S micro-CT device (GE Sensing & Inspection Technologies, Phoenix X-ray product line; Brunke et al., 2008). The achieved voxel size of the data sets was 2.5 or 5 µm (isotropic), suitable for imaging pore throats that effectively contribute to the flow in the studied type of sandstone (e.g. Nelson, 2009). Afterwards, all data sets were filtered for denoising, X-ray artefact removal and edge enhancement (Fig. 2c). The post-processed images scanned with 2.5 µm resolution had an edge length of 1180 voxels or 2950 µm. Image artefacts were processed as described by Wildenschild and Sheppard (2013). Beam hardening artefacts were removed by applying the best-fit quadratic surface algorithm (Khan et al., 2016) to each reconstructed 2-D slice of the image. Ring artefact reduction and image smoothing (with preservation of sharp edge contrasts) were performed using a non-local means filter (Schlüter et al., 2014). Segmentation was performed to convert the grey-scale images obtained after image filtering into binary images to distinguish between voids and solid phases (Fig. 2c). The local segmentation approach, which considers the spatial dependence of the intensity for the determination of a voxel phase, was used in addition to a histogram-based approach (Iassonov et al., 2009; Schlüter et al., 2014). Segmentation was performed by the converging active contour algorithm (Sheppard et al., 2004), a combination of a watershed (Vincent and Soille, 1991) with an active contour algorithm (Kass et al., 1988).

Simulations involving the real geometry of an imaged rock are computationally power and time consuming. Therefore, the determination of a representative elementary volume (REV) is required (Fig. 2d), assuming that porous media are homogeneous at REV dimensions (Bear, 2013), to perform fluid flow simulations. Porosity, a basic macroscopic structural property of porous media, is used here for the estimation of an REV (Bear, 2013; Halisch, 2013a; Tatomir et al., 2016) based on its correlation with permeability (Kozeny, 1927; Carman, 1937) (refer to the Discussion section).

A “classic” REV approach was used that claims that the REV is attained when porosity fluctuations in the subvolumes that grow isotropically in three orthogonal directions become sufficiently small (Bear, 2013; Halisch, 2013a, b). Practically, a large number of randomly distributed cubes were analysed through the entire 3-D sample for their image porosity (IP). The chosen initial cube size (with an edge length of 10 pixels in our case) was increased by 10–100 voxels. The REV size was specified when agreements between the mean and median IP values as well as saturation in the IP fluctuations were attained.

Further, the representative binary 3-D image (REV) of the pore space was spatially discretized by tetrahedrals with Materialize software (Belgium) (Fig. 2e). This step is required for importing the REV into the finite element method (FEM)-based modelling software (Comsol Multiphysics simulation environment, v5.2a). Stokes flow (Re≪1) is simulated (Table 1) in the pore network (Fig. 2f) by the following equations (e.g. Narsilio et al., 2009; Bogdanov et al., 2011): 2Stokes equation:-∇p+μ∇2u‾=03continuity equation:∇⋅u‾=0, where ∇p is the local pressure gradient, u‾ is the local velocity vector in the pore space, and μ is the dynamic fluid viscosity. Fixed pressures (p=const) were specified at the inlet and outlet boundaries of the fluid domain with a constant pressure gradient of 2.424 Pamm-1 over the domain, prescribed in all the simulations for consistency. At the internal pore walls and at the lateral domain boundaries, no-slip boundary conditions (u‾=0) were imposed (e.g. Guibert et al., 2016). These boundary conditions are also used to simulate the flow setup in a steady-state experimental permeameter (e.g. Renard et al., 2001). The macroscopic fluid velocity 〈v‾〉 was evaluated by volumetrically averaging the local microscopic velocity field (e.g. Narsilio et al., 2009; Guibert et al., 2016). Then, from the average macroscopic velocity vectors vij in three orthogonal i directions corresponding to the pressure gradients ∇pj imposed in the j direction, the full 3-D second-rank upscaled permeability tensor κ can be found: vxxvxyvxzvyxvyyvyzvzxvzyvzz=-1μϕκxxκxyκxzκyxκyyκyzκzxκzyκzz4×∇px000∇py000∇pz, where z direction of the CT specimen used in this analysis is perpendicular to the natural layering of the sandstone identified in the outcrop and in the petrographic observations. x and y orthogonal directions lie in the horizontal plane, with an azimuth chosen randomly. The permeability tensor is symmetrized by 5κsym=12κ+κT. Additional permeability tensor simulations in the multiple REV subvolumes of one of the investigated samples have been performed by using the FlowDict module (Linden et al., 2015, 2018) of the GeoDict toolbox (Wiegmann, 2019). Pre-processing as well as boundary conditions are identical to those used in Comsol setup.

Tortuosity (τ; Bear, 2013; Boudreau, 1996) was calculated separately in the x, y and z directions in the meshed domain using the particle tracing tool of Comsol Multiphysics software (an additional method for deriving τ is presented later in this section).

3-D image analysis (Table 1) was conducted on a high-quality, fully segmented micro-CT image (edge length of 2950 µm scanned at a 2.5 µm voxel size). Non-connected void clusters in the binary specimen were labelled and then separated into objects (single pores and grains) by using a distance map followed by the application of a watershed algorithm (e.g. Brabant et al., 2011; Dullien, 2012). Image analysis operations were assisted by Fiji-ImageJ software (Schindelin et al., 2012) and by the MorphoLibJ plug-in (Legland et al., 2016). The following geometrical descriptors were derived from the segmented image limited by the image resolution of 2.5 µm (Table 1):

micro-CT image porosity (IP);

Additionally, we propose a new simple method to estimate the image porosity at a given resolution. Multiplication of the mercury effective saturation at the capillary pressure corresponding to the micro-CT resolution (e.g. 2.5 µm) by the porosity of the same sample measured by a gas porosimeter yields the micro-CT-predicted image porosity from MIP at the given resolution limit (Table 1).

Three types of sandstone rocks were characterized by techniques 1–8 listed in Table 1. The results are presented in Figs. 3–8 and summarized in Table 2.

Visualized segmented subvolumes of samples S1, S2 and S3, depicting quartz, pores, clay and heavy minerals, are presented in Fig. 9. The main pore size population in PSD of S1 is at ∼100–500 µm range with majority at ∼194 µm (Fig. 10). A smaller population of pores of ∼30–100 µm was identified as well, which may refer to (Mode 1) pore throat size derived from the MIP experiment (Table 2). Image resolution of 2.5 µm limited the analysis. The PSA calculated from micro-CT images is 0.068µm-1. The tortuosity, measured from the whole CT image, indicates similar values in the x and y directions of 1.37 and 1.38, respectively, whereas in the z direction, the tortuosity is 1.48 (Table 2). As many paths were considered, this difference indicates the textural features that appear in horizontal plane (Fig. 3a).

For S2 (Fig. 9), the main pore size population is in the ∼15–50 µm range (Fig. 10), with majority at ∼21 µm. This may refer to the pore throat size derived from MIP. However, smaller pore throat sizes which were derived from the MIP (mode peak is at ∼3.5 µm) could not be visualized due to the limited resolution of the image (2.5 µm) and because of high uncertainty associated with size of pores smaller than 10 µm (four voxels). Accordingly, they were excluded from the PSD evaluation (Fig. 10). A large pore population is also recognized at ∼100 µm (Fig. 10), which corresponds to the pore size scale recognized from the petrography (Fig. 4), MIP (Fig. 7) and CT image (Fig. 9). The PSA calculated from micro-CT images is 0.136µm-1 (Table 2), which is twice as large as the PSA of S1.

For S3 (Fig. 9), the main pore size population is in the ∼ 100–500 µm range (Fig. 10), with majority at ∼223 µm. The geometry-based tortuosity values measured from the whole CT image with multiple paths is 1.32, 1.34 and 1.39 in the x, y and z directions, respectively. The tortuosity is lower for S3 than for S1 in all directions, which is a direct result of the smaller amount of cement in the pore throats. The PSA of S3 is 0.069µm-1, which is similar to that of S1.

Fluid flow was modelled at the pore scale in two different micro-CT-scanned geometries: (1) a full cube of sample S1 and (2) sample S3 within its REV dimensions (Table 3), imaged with 2.5 µm voxel size. Modelling of the 3-D geometry of sample S2 was not performed due to its non-stationarity, which did not allow for finding the REV in the investigated domain.

The porosity of the meshed domain of sample S1 is 13.6 % (in contrast to 17.5 % in the segmented image; Table 3), and the mesh edge length is 14 µm along the pore walls. The observed porosity loss results from disconnecting narrow pore throats from the connected cluster imaged with a 2.5 µm voxel size due to the use of a 14 µm mesh size (the lowest possible for our computational needs). A maximum Reynolds number of Re=0.084 was used to guarantee the simulation in a creeping flow regime.

The symmetrized permeability tensor, κ (Eq. 5), was obtained as follows (Table 2): 6κsym=42066.31.9166.334412.81.9112.8163. The permeability tensor is anisotropic, with κzz being more than twice as small as κxx and κyy. This result is in agreement with the appearance of horizontal banding with higher cementation (Fig. 3a).

The porosity of the meshed domain of sample S3 is 25.9 % (in contrast to 28.3 % in the segmented image; Table 3), and the mesh edge length is 5 µm along the pore walls. A maximum Reynolds number of Re=0.22 was used to guarantee the simulation in a creeping flow regime. The symmetrized permeability tensor is close to isotropic (Table 2): 7κsym=451753854808547385474085. Additional permeability tensor simulations on equivalently REV-sized segmented subvolumes of S3 and on the full S3 (Fig. 15, cube volume of 3×3×3mm3) have been performed with GeoDict toolbox to ensure consistency of the estimated REV size. Subvolume locations are presented. Symmetrized permeability tensors simulated in these domains (Fig. 16) are close to the former one (Eq. 7), being also nearly isotropic.

The tortuosity of S3 computed from particle tracing in the x, y and z directions is 1.44, 1.39 and 1.47, respectively (Table 2). The largest value is observed in the z direction, which is in agreement with the lowest permeability in the z direction.

Each of the evaluated micro- and macroscale rock descriptors supplies a qualitative information about the sample permeability (Tables 2 and 3), which is used to validate the multi-methodological approach presented in this paper. Specifically, the increasing mercury effective saturation with increasing pressure shows a similar pore throat size distribution curve slope for sandstone samples S1 and S3 in the macropore throat range (Fig. 7), suggesting that these samples have similar structural connectivity. However, S1 has a smaller imaged volume fraction of pore space available for fluid flow that is controlled by macropore throats (i.e. 81 % in S1 vs. 93 % in S3; Fig. 7) due to its higher contents of silt, clay and FeOx cement. The intermediate layer (S2) with 19 % porosity comprises more fines, which form a clay matrix (Table 2) due to poor grain sorting and smaller mechanical resistance of clay to pressure under the burial conditions. Only ∼15 % of the pore volume fraction in S2 is controlled by bottle-neck macropore throats (Fig. 7). However, the characteristic length of S2, 12.3 µm (Table 2), indicates that macropore connectivity is still possible even when the pore space consists mainly of sub-macroscale porosity. This 0.15 volume fraction is in agreement with Harter (2005), who estimated a volume fraction threshold of 0.13 for correlated yet random 3-D fields required for full interconnectivity.

The value of the connectivity index of S3 (10) evaluated from CT data is approximately 3 times higher than that of S1 (3.49), while both rocks are defined as moderately sorted sandstones (Table 2). This difference is due to S1 having a smaller number of inequivalent loops within the imaged pore network than S3, leading to smaller Euler characteristics (see the Supplement for more details). Inequivalent loops are positively correlated with pore throats; their number is affected by the resolution of the CT image and by the partial volume effect at grain surfaces (Cnudde and Boone, 2013; Kerckhofs et al., 2008), where some voxels could be identified as grains and thus “clog” the small pore throats. Artefact porosity loss is apparent for S1, where the IP is 17.5 % (in contrast to the CT porosity of 23.5 % predicted from MIP; Table 2). The connectivity index of S2 (0.94; Table 2) is lower than those of both S1 and S3 because of the clay matrix, which clogs pores with sizes below the image resolution. The effect of the partial volume effect on the image connectivity and on the preservation of small features was reviewed by Schlüter et al. (2014).

A correlation was found between the grain size and the amount of FeOx cement in S1 evaluated at each slice along the z direction (from the image analysis; Fig. 17). Exceptionally large grains are detected (indicated by the red rectangle) near the cemented region at ∼750 µm. Large grains and a relatively high amount of cement can also be observed in the S1 thin section (Fig. 3b). Large grains cause large pores and generate relatively permeable horizons through which water flow and solute transport can become focused (McKay et al., 1995; Clavaud et al., 2008), supplying iron solutes. We suggest that a vadose zone was formed after flooding events, where the water flow mechanism could have changed from gravity dominated to capillary dominated. Water then flowed due to capillary forces along grain surfaces towards regions with larger surface areas, and iron solutes precipitated in a reaction with oxygen available in the partly saturated zone. We suggest that with time, this cementation mechanism caused a decrease in the pore throat size near the preferential path, while the preferential path with a low surface area remained open, eventually generating the observed anisotropic flow pattern.

In this respect, permeability anisotropy in sandstones at a small scale is usually attributed to the shape or preferential orientation of grains and pores (e.g. Sato et al., 2019) and to a heterogeneous distribution of cementing material at grain contacts (Louis et al., 2005). Clay-free and cement-free layers in S1 thus constitute the main avenues for flow in the parallel direction, shown by variation in porosity in the z direction (Fig. 11c) that is correlated with anisotropic permeability tensor (Eq. 6). At a larger scale, a higher degree of permeability anisotropy is usually associated with the presence of localized beds, foliation and compaction bands that constitute barriers to flow in the perpendicular direction (see Halisch et al., 2009; Clavaud et al., 2008, and references therein).

Flow modelling in the specified REV of S1 shows anisotropy (Table 2) and an average permeability value of 310 mD that is close to that derived from MIP (330 mD). However, the average permeability is lower than the average experimental gas permeability (∼543 mD); this difference should be related to the loss of porosity due to limitations on the CT resolution, image processing and meshing (Table 3; see Sect. 5.2 for more details).

In contrast, flow modelling and upscaling to the macroscale indicate an isotropic S3 sample (Eq. 7). However, the modelled permeability (∼4500 mD) is 10 times higher than the MIP-derived permeability (∼466 mD; Table 2). Gas permeability measurements indicate anisotropy, yielding permeabilities of ∼4600 mD in the x–y plane and ∼220 mD in the z direction (with an anisotropy ratio of ∼20, defined here as κ∥/κ⊥, e.g. Tiab and Donaldson, 2004). For comparison, the values of this ratio obtained from experimental permeability measurements were ∼1.2 for Bentheim sandstone (Louis et al., 2005), ∼1.7–2.5 for a sandstone within the Cretaceous Virgelle Member, Alberta, Canada (Meyer and Krause, 2001), and ∼8.5 for Berea sandstone (Sato et al., 2019). However, in our laboratory measurements conducted parallel to the layering (in the x–y plane), poorly cemented grains in S3 could dislocate from the weakly consolidated sample due to the application of a pressure gradient. This could have resulted in a higher measured gas flux and thus a higher permeability parallel to the layering, yielding a high anisotropy.

Alternatively, the disagreement between the laboratory-determined permeability perpendicular to the layering, κ⊥, and the isotropic permeability obtained from the flow modelling (Table 2, Eq. 7) may also stem from the small dimension of the modelled REV domain (cube edge length of ∼0.875 mm), which may not have included the additional textural features (e.g. Fig. 5d) that constrain fluid flow on a larger scale of the lab sample of S3 (2.5 cm in diameter and 5–7 cm in length).

However, the consistency of the REV size in S3 by the additional permeability simulations on equivalently REV-sized segmented subvolumes and on the entire sample (Figs. 15 and 16) is confirmed by yielding nearly isotropic permeability tensors that are also in good agreement with previously simulated permeability tensor in the REV (Eq. 7). The average permeability derived from all REV-sized subvolumes is ∼4381 mD, compared to the average permeability of ∼ 4500 mD simulated over the entire S3 geometry. This, again, is in good accordance with the gas permeability of S3 measured parallel to the layering (∼4600 mD, Table 2). These additional simulation results (Fig. 16) strengthen our conclusion that those may not have included the textural features that constrain fluid flow on a larger scale of the sample S3 tested by the laboratory experiments. Similarly, the differences with the permeability estimated by MIP seem to originate from the same reason.

For sample S2, REV and slice-by-slice porosity analysis indicated an REV size larger than the investigated sample size (Figs. 13c and 14). For this reason, the analytical programme formulated in our study cannot entirely be applied to S2 due to the impossibility of determining a reliable REV and hence conducting pore-scale flow modelling. As a result, although sample S2 represents a common sandstone, it is very heterogeneous in nature, and a sample larger than at least 9 mm (which is a maximal length in the z direction of the tested domain; Fig. 13c) is required to capture its REV. The MIP-derived permeability is 4 mD; this low permeability is due to a clay-rich matrix that encloses substantial void space (Hurst and Nadeau, 1995; Neuzil, 2019). The gas permeability of the quartz wacke layer (S2, ∼4.6 mD on average) is approximately 2 orders of magnitude lower than that of the quartz arenite layers (S1 and S3; Table 2). The permeability anisotropy ratio of S2 is ∼2.8. The high inverse correlation between the porosity and clay matrix content enhanced in the z direction (Fig. 13c and f) suggests that the clay matrix pattern appears as horizontal layering, thus generating the observed anisotropy.

Finally, non-marine sandstones of Lower Cretaceous age (as well as sandstones in general) feature big complexity and variability in their characteristics, as immediately seen even from a comparison of our samples S1, S2 and S3 from the same outcrop (Table 2). For instance, low porosity of Wealden quartz arenite sandstones from the Weald Basin within the Ashdown and Wadhurst Clay formations in southeast England ranges between 6.3 % and 13.2 %, while permeability ranges between 0.4 and 11.9 mD (Akinlotan, 2016), suggested to be controlled mainly by grain sizes, grain shapes and sorting that are directly linked to their depositional environment. Average porosities of 3.06 % and 0.19 % were evaluated in medium- and fine-grained tight gas sandstones, respectively, from the Lower Cretaceous Denglouku Fm. in the Songliao Basin, China (Zhang et al., 2019). Alternatively, a secondary porosity of 4 % to 22 % was generated by acidic fluids acting in the compactional regime, destructing a high primary porosity in sandstone of Lower Cretaceous Shurijeh Fm. in the eastern Kopet Dagh Basin in NE Iran (Moussavi-Harami and Brenner, 1993). Significant average porosity and permeability of 20 % and 3700 mD, respectively, were quantified in the Masila Block, Upper Qishn Fm. of the Lower Cretaceous Age, Republic of Yemen (Harding et al., 2002). The multi-methodological approach suggested in this study is applicable to all those sandstones with broad ranges of their textural, topological and mineralogical characteristics and should lead to their accurate petrophysical characterization.

The extended computational workflow (Fig. 2) serves as the main tool in this study for upscaling permeability from the pore-scale velocity field. The accuracy of each step in the workflow affects the ultimate result.

Following the steps of the workflow, a micro-CT image resolution of 2.5 µm limits the reliability of the representation of the porous medium and defines the lower pore identification limit using this method. As an example of this limitation, the bulk SSA calculated by MIP is larger than the PSA calculated by micro-CT image analysis in all the samples (Table 2), although the pore volume is always smaller than the bulk volume. The PSA from micro-CT is limited by the image resolution and therefore does not consider relatively small pores with large surfaces. The PSAs of S1 and S3 are similar, but the SSA (from MIP) of S1 is 20 times larger than that of S3 because S1 has a larger surface area at small pores created mainly by FeOx cement (compare Fig. 3c–f for S1 to Fig. 5c for S3). In contrast to SSA, PSA in S2 is only twice as large as that of S1 due to the presence of clay and clay matrix with large surface areas.

Image processing and segmentation were applied in this study to recover the image geometry, which was blurred by noise or affected by the partial volume effect (see Sect. 3). Then, the loss of pore space due to the resolution limits was estimated in this study from the amount of mercury filling the pores with diameters equal to the resolution limit (Fig. 7a). After segmentation, sample S1 had a segmented image porosity of 17.5 % and a CT-predicted porosity of 23.5 % from MIP (Tables 2 and 3). Therefore, the difference in porosities generated by the partial volume effect in the image processing scheme (e.g. Cnudde and Boone, 2013) is a significant component of error, especially for small structures, such as pores with a large surface area-to-volume ratio. In contrast, the image porosity of S3 after segmentation was 28.3 %, which is close to the porosity of 30.4 % estimated from MIP (Tables 2 and 3). This is a result of the very small degree of cementation and the absence of FeOx flakes in the majority of the sample pores, leading to the small contribution of the partial volume effect. In comparison, a fine-grained and well-sorted Lower Cretaceous Fm. sandstone from the Heletz field (e.g. Fig. 1a) (Tatomir et al., 2016) comprising clay and calcite had MIP and micro-CT porosities of 26.7 % and 20.9 %, respectively.

An additional source of inaccuracy is the use of a porosity-based REV for permeability approximations. Mostaghimi et al. (2012) showed that for CT images of sandpacks (homogenous samples), the porosity-based REV had an edge length of 0.5 mm, whereas the permeability-based REV was twice as large. Moreover, the porosity- and permeability-based REVs in images of crushed bead packs derived by Zhang et al. (2000) had edge lengths of 1.71 and 2.57 mm, respectively. According to Mostaghimi et al. (2012), larger REV values for permeability rely on contributions from the tortuosity and connectivity of pore spaces, whereas the larger REV values of Zhang et al. (2000) might be related to the heterogeneity of the sample.

This discrepancy indicates a larger REV for a rock property evaluated using physics-based simulations than for those estimated using morphology-based methods (Saxena et al., 2018, and references therein). Furthermore, implementing the classic REV determination methodology (e.g. Callow et al., 2020) using very small search subvolumes is not in agreement with capturing a sufficient structural complexity (Saxena et al., 2018).

Flow simulations performed in subvolumes and full sample of S3 (Figs. 15 and 16) support this conclusion. Small dimensions of the evaluated REV (∼0.875 mm) of homogeneous S3 ensure efficient calculations. Both porosity and permeability demonstrate good agreement (Table 2, Fig. 16), thus confirming a representativeness of the estimated REV and a continuity of these characteristics over the chosen sample. However, the differences in porosity between the subsamples and the full sample are smaller than the corresponding differences in permeability (Fig. 16), as anticipated from the porosity-based REV derivation discussed above.

An additional verification of the REV size for flow simulations in S3 follows the approach given by Saxena et al. (2018). They demonstrated that for homogeneous sandstones, the smallest pore throats can be accurately resolved at NI>10, where NI=DD/Δx is a ratio of the pore throat size corresponding to mercury entry pressure, DD, to the voxel size, Δx. NI controls the lower bound on permeability that can be reliably calculated using a digital rock image, to capture sufficient structural complexity of rock microstructure which affects flow, attributed to DD visualized using Δx. For our subvolumes of sample S3 imaged with Δx=2.5 µm resolution and DD=35µm (Table 2), NI=14>10. In addition, there is a requirement for the minimal REV size for representativeness for permeability calculation, NREV=L/Deff>5 (Saxena et al., 2018), where L is a digital rock (i.e. domain) length and Deff is the effective grain diameter (e.g. Říha et al., 2018). For S3 subvolumes with L=875µm REV size and Deff=58.6µm (computed from laboratory grain size data which includes both sand and fines, Fig. 6), this requirement is achieved as well: NREV=15>5, which also proves the reliability of the subvolume permeability modelling with the presented approach. The calculations in subvolume performed with Comsol (Eq. 7, Table 2) demonstrate the smallest deviation in mean permeability compared to that in the full sample (0.85 %). In comparison, other subvolumes modelled with GeoDict (Fig. 16) have larger mean permeability deviations from the full sample (ranging between 5 % and 17 %), still demonstrating a very good agreement with those conducted on the full-scale S3 domain.

Further, according to Saxena et al. (2018), REV size supported by NREV for flow simulations should also be insensitive to the choice of boundary conditions, whose effect on tensorial flow properties diminishes with an increasing sample size (e.g. Guibert et al., 2016; Gerke et al., 2019). No-slip boundary conditions applied in our study at four lateral faces of the modelling domains correspond to those in the experimental permeability measurements and are also the most commonly used boundary conditions for the pore-scale flow simulations (Guibert et al., 2016 and references therein). However, they were recently shown to suppress the transversal flow through the simulation domain to some extent, resulting in deviation in alignment of the permeability tensor and in underestimation of its magnitude (Gerke et al., 2019) even at REV dimensions. Thus, the difference in the mean permeability derived from all REV-sized subvolumes (∼4381 mD) and that simulated over the entire S3 geometry (∼4500 mD) (Fig. 16) can also be attributed to this effect. For future studies, we suggest that determining REV size for the flow simulation from porosity is justified by acknowledging the typical ratio of 2 between those for permeability and porosity.

To upscale to permeability reliably, the REV domain should be sufficiently large such that it is bounded from below by the scale of the textural bedding but should not be larger than necessary to optimize the computational efficiency (while remaining within the same scale of heterogeneity, i.e. at the macroscale). As a result, a REV with an edge length of ∼2950 µm was chosen in the current study in sample S1, based on slice-by-slice porosity profiles that reveal millimetre-scale layering in the z direction (Fig. 11c) rather than on the classic isotropic REV approach. For comparison, in other studies, the edge lengths of REVs in sandstones were 0.68 mm (Ovaysi and Piri, 2010), 0.8 mm (Mostaghimi et al., 2012) and 1.2 mm (Okabe and Oseto, 2006; Tatomir et al., 2016) derived by the classic approach. In contrast to the classic REV estimation, where porosity analysis does not consider directionality, 1-D profiles from slice-by-slice porosity provide additional information on anisotropy and inhomogeneity of the sample which have implications on the ultimate determination of the REV. In the future studies, the 1-D profiles can be used to calculate spatial correlation length (e.g. using variogram analyses) of geological structures that include layering, as in S1 in the current study.

Another source of inaccuracy is the geometry used for the flow model. The geometry considered in this study included only the pore network connecting six faces of the REV cube. Other pore spaces in the REV disconnected from the main network were deleted (because all paths smaller than the resolution were prescribed as grain pixels due to the partial volume effect), thus resulting in the smaller effective size of the simulation domain. The image porosity of sample S1 was 17.5 %, whereas its connected porosity was estimated as 15.6 % (Table 3), while those of sample S3 were 28.3 % and 27.9 %, respectively.

Furthermore, the mesh was generated by taking a trade-off between the size of the mesh elements (four elements in the smallest pore throat) and computational limits into account, while coarsening the mesh elements towards the pore centre. The connectivity between pores with very fine pore throats that could not be replaced by mesh elements could be lost, resulting in the loss of those pores in the calculations. In sample S1, the porosity used in the simulation was approximately 50 % smaller than the porosity estimated by gas porosimetry (Tables 2 and 3). In contrast, the porosity used in the simulations in S3 was mostly preserved, comprising ∼84 % of that estimated in the laboratory.

For comparison, in the fine-grained sample of the Lower Cretaceous sandstone from Heletz field in Israel (Fig. 1a), which has grain size characteristics similar to those of S1 but with higher clay and additional calcite contents (Tatomir et al., 2016), the permeability upscaled from micro-CT flow modelling (conducted by the same simulation method as that in the current study) exceeded the gas permeability by a factor of ∼7. This could be related to the reduction in the specific surface area by image processing and meshing (Mostaghimi et al., 2012) conducted for the flow modelling.

Finally, the upscaling process from the flow modelling successfully predicted the permeability anisotropy ratio of ∼2.3 in S1, as discussed above. For comparison, the permeability anisotropy ratio evaluated using micro-CT flow monitoring in clay-free sandstones (Clavaud et al., 2008) had a mean value of ∼2.5 (ranging from ∼1.7 to ∼5.2), related to the presence of less permeable silty layers. This is consistent with the ratio estimated at the pore scale in Rothbach sandstone (∼5) (Louis et al., 2005), attributed to lamination due to differences both in the characteristics of the solid phase (grain size and packing) and in the content of the FeOx.