Dual focused ion beam-scanning electron microscopy (FIB-SEM) is frequently being used to characterize nano-scale pore structures observed in carbonate and shale gas rocks. However, applications are limited to qualitative analysis of nanopore structures. Herein, the concept of statistical representative elementary volumes (SREV) is applied to FIB-SEM data of a Cretaceous chalk sample. Lattice-Boltzmann (LB) simulations with multiple relaxation time and topological analysis show that the size of the SREV for this chalk sample can be established at ~ 10 microns based on anisotropic permeability, tortuosity, and specific surface area. This work confirms that the FIB-SEM technique can be used for the quantitative analysis of nanopore structures and highlights nano-scale basis for strong anisotropy in the presence of fractures. In addition, nanopores and pore throats are not resolved at voxel dimensions less than ~ 80 nm, resulting in significant underestimation of surface area and permeability.
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 Geological storage of CO2 and exploration of unconventional resources pose a significant relevance on formation properties relevant to fluid flow and distribution at the submicron scale (i.e., nanopores). Complex nanopore topologies and wetting properties observed in carbonate and shale gas rocks must be characterized for enhanced analysis of petrophysical behavior [Hassenkam et al., 2009; Loucks et al., 2009]. Common techniques such as X-ray microtomographic imaging (MCT) and optical microscopic imaging lack necessary resolution to infer structures and properties of nano- to microscale pores, suggesting the need to combine micron-scale imaging (e.g., MCT) with other techniques such as focused ion beam-scanning electron microscopy (FIB-SEM) for multiscale imaging capabilities [Wildenschild and Sheppard, 2013]. Recently, characterization of pore structures using FIB-SEM has been performed with the aim of determining wetting properties of chalk [Hassenkam et al., 2009], characterizing multiscale pore structures of carbonate rocks [Sok et al., 2009], identifying pore networks in mudstones (i.e., shale formations) [Heath et al., 2012], and inferring pore network and transport properties of nanopore networks [Dewers et al., 2012].
 Macroscopic transport properties and constitutive behavior are associated with a representative elementary volume (REV), wherein macroscopic variables are defined as volume averages over the REV [Zhang et al., 2000]. In practice, FIB-SEM has been used at a scale smaller than that of an REV. As pore structures and fluid properties at the submicron scale imaged by FIB-SEM are likely to be highly variable between and within samples, it is necessary to evaluate the size of a so-called statistical REV (SREV) [Zhang et al., 2000]. SREV represents a scale where the mean of properties is constant and variation is small, but yet still smaller than the size of REVs. The concept of SREV has been used for quantifying microstructures of various materials including single phase flow in sandstones at the microscale [Zhang et al., 2000], mechanical properties of fiber-reinforced composites [Trias et al., 2006], and transport properties of fuel cell materials [Wargo et al., 2012]. However, the scale of SREV has not been evaluated for quantitative analysis of FIB-SEM data with nanopore structures observed in geo-materials (e.g., carbonate rocks and shale mudstones).
 The objectives of this work are to evaluate the size of SREV for transport properties of nano-porous chalk from 3-D reconstruction of FIB-SEM data and to analyze nanopore systems in order to account for the influence of nanopore structures on transport properties as a function of averaging scales. In this work, a sample of Cretaceous Selma Chalk is examined using FIB-SEM, pore-scale modeling, and topological analysis. Selma Group chalks have been proposed as a secondary seal for geological storage of CO2 and are also a current exploration target for unconventional gas. The segmented 3-D data set using image analysis techniques is used to examine scale dependency of porosity (φ), specific surface area (Ss), tortuosity (τ), permeability (k), and anisotropy (λ). Lattice-Boltzmann (LB) simulations are performed to obtain k and mean τ at several different scales, and 3-D topological analysis of medial axes (i.e., the skeleton of the connected pore networks) is also performed to estimate geometrical tortuosity. Uniquely, our analysis reveals a nano-scale basis for anisotropy in this chalk.
2.1 FIB-SEM Imaging
 A one inch diameter plug of Selma Group (Navarro Formation) chalk was obtained from core (1891.1 m below ground surface), in association with NETL's Southeast Regional Carbon Sequestration Partnership Phase II, near Escatawpa, Mississippi, USA. φ and k of this core plug are 12.5–16.7% and 0.012–0.108 mD, respectively [Petrusak et al., 2009]. A small piece with known orientation was mounted on a metal holder and coated with Au-Pd. A 10 micron-size portion was further coated with platinum to minimize curtaining effects during ion-beam slicing. Serial sectioning and imaging were done with an FEI Helios 600 NanolabDualBeamTM instrument and SliceandViewSoftwareTM. Ion milling of slices ~15 nm thick was done with a Ga beam. SEM imaging was taken in backscattered mode with through-the-lense detection. Nine hundred and sixty two slices were used in the data set analyzed here, equal to ~15 µm in the slicing (z-) dir., and ~16 µm by 10 µm per image.
2.2 Image Processing and Analysis
 The FIB-SEM image stack was segmented into either solid or pore phases. Steps in the image analysis include background subtraction, noise filtration (using median and fast Fourier transform (FFT) band-pass filters), segmentation, dilation and erosion, and interpolation in the z-direction. To enhance image quality, we applied several series of filters for four different groups of images based on image quality, abundance of submicron fracture, and degree of illumination gradient and image contrast. A review of image analysis for quantifying pore structure is provided by Wildenschild and Sheppard . Details of image analysis with example images are provided in Text S1 and Figures S1–S3 in the supporting information. In this work, small and large fractures are segmented separately with different filtering procedure and threshold values, and combined together to recover the complex pore structure and to counter electron charging especially visible in the microfracture portion of images (i.e., the white portions of the raw SEM image in Figure 1a). The 2-D segmented images (e.g., Figure 1b) were assembled into a 3-D stack (Figure S1c), which was interpolated to obtain cubic voxels of 15.6 nm dimension. Overall, the resulting 3-D volume of pore structure was 14.8(x) × 8.3(y) × 15.3(z) microns with15.6 nm resolution (930 × 520 × 962 voxels).
φ was calculated based on segmented 3-D binary data and Ss was determined using the two-point correlation function approach on the same segmented volumes [Berryman and Blair, 1987]. 3DMA [Lindquist et al., 2005] was also used to calculate pore throat sizes and pore size distributions, and to determine geometrical tortuosity (τ).
2.3 Lattice Boltzmann Simulations
 The lattice Boltzmann (LB) method [Zhang et al., 2000] is used to solve for single phase flow in complex geometries. LB simulations were performed on segmented data using Palabos software (Palabos, Parallel Lattice Boltzmann Solver, 2012, http://www.lbmethod.org/palabos/ (October 2012))—an open-source parallel LB solver. We employ a 3-D, 19 velocity (D3Q19) model with multiple relaxation time, which shows a superior performance compared to single-relaxation time in complex geometries. For LB simulations, the segmented image stack was used to establish internal boundaries (i.e., walls of porous medium) where simple bounce-back boundary conditions were used. For each directional k, a no-flow boundary condition was imposed to the boundaries parallel to the main flow direction. A fixed pressure boundary condition was used at the inlet and outlet boundaries. k was computed by relating the total flux to the pressure gradient over the length of sample and the mean τ was computed based on flux weighting average (see section 2 in TextS1). The tortuosity distribution in each direction was also computed using particle tracking based on LB-simulated flow fields [Chen et al., 2008].
 For LB simulations, k and τ are analyzed at seven different scales (1503, 3003, 4003, 5003, 600 × 520 × 600, 765 × 520 × 765, and 930 × 520 × 962 voxels), corresponding to geometric averaging scales of 2.34, 4.68, 6.24, 7.8, 8.92, 10.49, and 12.09 µm, respectively. The total number of subvolumes over the entire data at the averaging scales ranges from 10 to 42. For each subvolume, φ and Ss are also computed as described previously.
3 Results and Discussion
 To characterize statistical properties, five different properties (φ, Ss, τ, k, and λ) are presented in Figure 2 as a function of averaging length scale. The anisotropy (λ) is defined as the ratio of the horizontal k (kh = the square root of kx × ky) to the vertical k (kz). In addition, the coefficient of variation (CV), which is the standard deviation divided by the mean value, is used to analyze the variation independently of the mean (Figure 2d). All five properties fluctuate significantly at the 2.34 µm scale (150 voxels) and approach a constant value at > ~10 µm (~700 voxels). For all properties, the CV value at ~10 µm is less than 11%, demonstrating the size of an SREV at a length scale of tens of microns for this Selma sample. This suggests that it is possible to characterize key features of highly complex geomaterials (e.g., carbonate rocks and shale mudstones) from a limited set of selected small volumes (e.g., 3–10 samples obtained from a thin section) with FIB-SEM analysis. This has been recently done for fuel cell materials [Wargo et al., 2012].
 Importantly, results for τ and k also show a strong anisotropy at the SREV scale. The segmented images (Figures S1c and S2) show a key pore structure in the Selma, with a micron-scale microfracture with adjacent connected and unconnected nanopores. The presence of a microfracture imparts a large influence on the estimated mean and variation of permeability anisotropy (Figures 2c and 2d). Even at the smaller scale in subvolumes not involving the microfracture, there is an anisotropy evident in the disparity in τ and k in the x- and z-dir. compared to the y-dir. and this reflects the longer aspect ratio and better connectivity of pores in the x- and z-dir. compared to y. This shows that there is an inherent anisotropy at this micron scale in the chalk matrix, as well as another anisotropy imparted by the microfracture. Figure 2 also shows that the variation of k is greater than that for τ, Ss, and φ. This is because k is a more complex property of pore structure than properties represented by one or two-point correlation functions (Ss and φ). With a CV value of 15% as a criterion, the SREV scale for τ, Ss, and φ is at ~ 5 µm, but the SREV for k and λ is at ~ 10 µm. Hence, it is more desired to use k and/or λ as a parameter to determine an appropriate averaging volume. This is consistent with previous work where k was shown to require a longer length scale for SREVs [Zhang et al., 2000]. In FIB-SEM images with a typical sample volume at ~103–503 µm3, it will be important to consider the anisotropic features in order to apply results of FIB-SEM characterization of such small volumes to upscaled flow and transport modeling.
 To examine the impact of image resolution on the four textural properties and the associated effect on predicted flow and transport properties, a series of averaging steps are applied to the 3-D image stack. The 15.6 nm data are averaged at 31.2 to 624 nm scales using a simple rule such that the pore phase is assigned when the number of pixels for the pore space exceeds the half of pixels at an averaging scale. Property values at different resolutions normalized by those at 15.6 nm are shown in Figure 3a. φ first decreases by 5–10% at 31–78 nm and then linearly decreases with scale, while Ss decreases almost exponentially with increasing resolution. τx and τz are nearly constant up to ~450 nm while τy increases linearly with scale. kx and kz decrease linearly at a scale greater than ~ 80 nm, and ky experiences a rapid decline followed by very little change from 200 nm on to bigger scales (Figure 3b). These changes are a manifestation of complex changes in pore throat size and connectivity with the changes in voxel resolution.
 Pore throat and volume distributions from 3DMA results are also shown (Figure 3c). The cumulative (CDF) volume distribution at a pore size of 80 nm is ~8% and the cumulative pore throat diameter distribution is ~60%. This corresponds to the point of inflection in the kx and kz decline in Figure 3b, showing that the decline of k is due to the loss of connected pore throats with decline in resolution. Overall, the decrease of φ matches the loss of pore volume fraction well, and the decrease of Ss in Figure 3b also follows the probability density function (PDF) of pore throat size. Estimates of Ss and φ are strongly affected by the image resolution and are well correlated to the loss of small pores and pore throats seen in the distributions of Figure 3c, highlighting the importance of including nanopore structures obtained from FIB-SEM analysis in estimating flow and reactive properties from imaging methods.
 Comparison of flow-based (from LB simulation with particle tracking) and geometrical (from 3DMA) tortuosities is shown in Figure 4. τ distributions from both methods show a strong anisotropy, but the difference in the x- and z-dir. compared to y is greater with LB-based results. In particular, a higher anisotropy with particle tracking results suggests flow paths are more complex than those given by geometrical results. In x- and z-dir., fluid flows dominantly through the microfracture, resulting in lower τ than the geometrical analysis, but in the y-dir. the comparison would indicate that most fluid volume flows via slightly longer paths than given by the geometrical analysis (also seen by Duda et al. ). Primary flow paths obtained from LB-based flow fields can be used to distinguish advective- and diffusive-dominant domains, by showing preferred pathways for advective transport. Identifying Ss and pore size distribution for each domain type can be useful in developing multiscale reactive transport models [e.g., Lichtner and Kang, 2007].
 The increasing demand of unconventional oil and gas resources requires a better understanding of quantitatively characterizing nanopore structures. This work confirms that the FIB-SEM technique can be used for the quantitative analysis of nanopore structural features impacting pore-scale flow and transport properties and the size of the SREV for this chalk sample can be established at ~10 µm based on anisotropic k, τ, and Ss. LB simulations and topological analysis identify properties of complex pore structures at the submicron scale and also show nano-scale basis for strong anisotropy in the presence of fractures. Analysis of resolution highlights the importance of FIB-SEM data at submicron scale where petrophysical and multiphase flow properties in carbonate rocks and shales depend upon complex 3-D pore structure and connectivity. As a possible route for upscaling, a set of segmented 3-D FIB-SEM data at the SREV scale (e.g., 3–10 samples) can be mapped to lower-resolution 2-D thin section or 3-D MCT data. Textural properties determined at the FIB-SEM scale can then be used to reconstruct multiscale pore structure using stochastic methods (e.g., multiple-point simulation [Remy et al., 2009]) or dual-scale pore network models [Blunt et al., 2013], and then applied to upscaled single or multiphase flow or reactive transport, for core- and larger-scale simulations.
 This study was supported as part of the Center for Frontiers of Subsurface Energy Security, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Basic Energy Sciences under Award Number DE-SC0001114. We thank G. Koperna and R. Petrusak of ARI, and R. Esposito of the Southern Company, for use of the Selma Chalk core, M. Rye for obtaining the FIB image data set, and J. Heath for help with core preparation. We thank the editor M. B. Cardenas and two anonymous reviewers for their constructive reviews. Sandia National Laboratories is a multiprogram laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-AC04-94AL85000.
 The editor thanks two anonymous reviewers for their constructive reviews.