Gas Breakthrough Pressure (GBP) through Claystones: Correlation with FIB/SEM Imaging of the Pore Volume
Corrélations entre Pression de Percée de Gaz (PPG) et imagerie FIB/MEB de roches argileuses (argilites)
^{1}
École Centrale de Lille, Cité Scientifique, CS20048, 59651
Villeneuve d’Ascq Cedex – France
^{2}
Changzhou Institute of Technology, Chang Zhou Gong Xue Yuan, 213000
Changzhou Shi, Jiangsu Sheng – China
^{3}
Institut d’Électronique, de MicroÉlectronique et de Nanotechnologie (IEMN), UMR CNRS 8520, BP 60069, 59652
Villeneuve d’Ascq – France
email: catherine.davy@eclille.fr
^{*} Corresponding author
This contribution uses six claystone samples imaged by FIB/SEM (Focused Ion Beam/Scanning Electron Microscopy), within micrometric volumes located in the clay matrix; their 3D connected pore network is identified down to 1722 nm pore size. All samples are gently dried to minimize damage, and several are impregnated with Poly(Methyl MethAcrylate) (PMMA) resin to avoid further damage during FIB/SEM observations. Three pore volumes out of six are connected between two parallel end surfaces through cracklike pores; two are not connected between any two parallel end surfaces; only one sample has a connected pore network distinct from cracks.
By assuming varied pathways for gas to migrate by capillarity through the connected pore volumes (either by taking the shortest path, or through the largest path, or through the most frequent pore size, or by simulating the ingress of a non wetting fluid), we determine the Gas Breakthrough Pressure (GBP) through the initially fully liquid saturated claystone, from these micrometric volumes. The scale change (from the micrometric to the macroscopic scale) is assumed possible without changing the GBP value, and clay/water interactions are not accounted for. By comparison with GBP values measured in the laboratory on centimetricsized claystone samples, it is concluded that breakthrough occurs most probably by capillary digitation; microcracks are the most probable pathways for gas, so that gas does not progress in a homogeneous manner through the claystone, as standard macroscopic finite element models would represent it. For intact claystone, predictions based on the capillary ingress of a non wetting fluid provide a GBP value ranging between 714 MPa.
Résumé
Six échantillons de roches argileuses (argilites) sont observés au FIF/MEB (Faisceau d’Ions Focalisé couplé à un Microscope Électronique à Balayage) au sein de volumes micrométriques situés dans la matrice argileuse (seule partie poreuse du matériau). Les échantillons sont séchés de façon à minimiser leur endommagement, et imprégnés ou non de résine polyméthacrylate de méthyle (PMMA) pour les maintenir lors des observations. Le réseau poreux connecté en 3D de ces échantillons est mesuré jusqu’à des tailles de pores de 1722 nm de diamètre. Trois réseaux poreux sur six sont connectés par des pores de morphologie identique à des fissures ; deux réseaux poreux ne sont pas connectés entre deux faces paralléles de l’échantillon ; un seul réseau poreux (sur six) est connecté par des pores qui ne correspondent pas à des fissures.
En supposant différents modes de transport capillaire (via le chemin le plus court, via le chemin le plus large, via la taille de pore la plus fréquente, ou en simulant l’intrusion de fluide non mouillant) à l’échelle de ces réseaux poreux micrométriques, nous déterminons la Pression de Percée de Gaz (PPG) au travers de la roche initialement saturée de liquide. On suppose que le changement d’échelle est possible sans altération de la PPG, et que les interactions eau/argile sont négligeables. En comparant nos prédictions avec les valeurs expérimentales mesurées au laboratoire, nous montrons que la percée se produit avec la plus grande probabilité par digitation capillaire, via un volume limité de pores, qui ont une morphologie de fissures, et pas de façon homogène au travers de la roche argileuse, comme cela peut être représenté de façon simplifiée dans un modèle standard par Élements Finis. Pour le matériau intact, les prédictions basées sur l’intrusion capillaire de fluide non mouillant donnent une PPG variant entre 7 et 14 MPa.
© Y. Song et al., published by IFP Energies nouvelles, 2016
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Introduction
Industrial Context
In industrialized countries, a major societal issue is the longterm management of longlived and medium to high activity radioactive waste. In France, their deep underground storage is planned within a CallovoOxfordian (COx) claystone formation located at about 500 m depth in the East of the Paris Basin [1]. An Underground Research Laboratory (URL) exists there, with several kilometers tunnels already drilled and instrumented; Andra (Agence Nationale pour la gestion des Déchets RAdioactifs) is in charge of managing the site and the research regarding the safety assessment of the system. A second URL exists in the South of France (at Tournemire, Aveyron), where the host rock is a Toarcian claystone; it is managed by IRSN (Institut de la Radioprotection et de la Sûreté Nucléaire).
Ensuring a safe underground storage is managed through the socalled engineered barrier, which comprises waste vitrification, metallic or concrete casing around the storage tunnels, sealing with swelling clay plugs, and a whole specificallydesigned structure. The COx claystone host rock is also a determining component in retaining any radionucleide within the structure.
In particular, after the drilling of storage tunnels, macrofracturing is observed in a socalled Excavated Damaged Zone (EDZ) located close to the tunnel surface. Underground water seepage is shown to allow selfsealing of the EDZ [1], so that water permeability recovers values on the order of the undisturbed claystone (i.e. 10^{−20}10^{−21} m^{2}i.e. 110 nD). However, after the filling and closure of the storage tunnels, hydrogen gas may develop and progressively pressurize inside the repository, due to varied physicochemical phenomena [2–4]. In this context, one of the aims of the Engineered Barrier and of the host rock is to retain the gas.
As a consequence, the conditions for proper gas immobilization within the underground site, in the selfsealed zone and in the undisturbed matter, away from the repository, need to be assessed. This may be investigated (among other methods) by:

identifying experimentally the Gas Breakthrough Pressure (GBP) across fully watersaturated claystone, as in [2, 3, 5], or

predictions using an accurate characterization of the pore network combined to a wellchosen physical model for fluid migration.
It is the latter that we propose to do in this contribution.
StateoftheArt
Characterization of the Pore Network
The adequate characterization of the pore network of claystones is complex to gain. Indeed, the COx claystone is made of a range of minerals, comprising mainly tectosilicates, carbonates, pyrite, and, most importantly, clays. Whereas tectosilicates, carbonates, and pyrite are reputed non porous, they display porous interfaces with clays, and clays themselves display a wide range of pore sizes^{1}. On the whole, the pore size distribution of COx claystone ranges from below 1 nm (micropores) and up to several hundred nanometers (i.e. both meso and macropores) [4, 6–8].
While indirect methods span the whole pore size range, with only one or two techniques (Mercury Intrusion Porosimetry (MIP), nitrogen or CO_{2} adsorption), these generally require to model the pore network as the assembly of individual non connected circular cylindrical pores of varying diameter. This oversimplification is not necessary when using direct imaging techniques, such as XRay Computed MicroTomography (XRay CMT), Broad or Focused Ion Beam/Scanning Electron Microscopy (BIB/SEM or FIB/SEM), or Transmission Electron Microscopy (TEM) in order to image the pores in 2D or 3D [6–16]. However, any imaging technique is limited by the fact that the smaller the pixel (or voxel) size is, the smaller the sample size is. The representativeness of the imaged zone is not ensured with a single sample up to the macroscopic scale. A solution is to image as many zones as possible, and provide pore properties in a statistical sense [8, 15].
In the particular case of claystones, an additional difficulty occurs, which is the fineness of their pores. For instance, although using a voxel size of 700 nm, XRay CMT has been able to image only 0.5% of COx claystone porosity, whereas its total porosity given by indirect techniques is of 18±4% [17, 18]; this 0.5% relative pore volume is only made of scattered, non connected pores, so that no fluid transport may be predicted through them without further assumptions. Former research on COx claystone and Opalinus clay [6–8, 16] has shown that FIB/SEM combined to TEM provides a reliable assessment of the connected 2D and 3D pore networks of these claystones down to less than one nm, which is consistent with indirect measurements; these pores are located within the clay matrix (in contrast to the non porous carbonate and tectosilicate grains that claystones contain). For the COx, two peak pore sizes are determined by a combination of imaging techniques at 46 nm (TEM) and 50 nm (FIB/SEM) [8], i.e. within mesopores [19], in good agreement with Andra reference data [4].
Gas Migration Mechanisms
Extensive research exists on the experimental identification of gas breakthrough properties of low permeability media [3, 20–23]. In general, at the laboratory scale, the initial conditions are a fully watersaturated medium subjected to a gas pressure of given amplitude, until gas passage is detected by varying means. However, no consensus exists as the exact phenomena involved in the gas migration (or breakthrough) through watersaturated clayrocks [3, 20, 22, 24–29]. Rather, several mechanisms are considered.
The wellknown description by Marschall et al. [28] considers four different mechanisms leading to gas migration. These may occur simultaneously or separately, depending on the porous material or on the poromechanical loading considered.
The first migration mechanism consists in gas dissolution within pore water, followed by the transport of dissolved gas by diffusion and/or convection through the pore network. This phenomenon occurs systematically, yet to a limited extent for laboratory experiments, owing to the relatively short time needed to reach breakthrough (35 days for 13 cm long COx claystone samples, whereas first order calculations provide 13123 days for a passage by dissolution/diffusion) [3, 5].
Secondly, a viscocapillary twophase flux is considered. It is associated with negligible changes in the solid skeleton volume. In such instance, gas starts to penetrate the pore network; this occurs at the socalled gas entry pressure; then, gas progresses through the pore network due to capillary and viscous forces, and creates channels or pathways. Together with dissolution and diffusion through the pore fluid, this description by capillary action is privileged by [22, 24] for finely porous clayey materials; it does not mention the necessity of claystone damage for gas to migrate. In this contribution, we assume that the viscocapillary mechanism explains Gas Breakthrough (GB) through initially fully liquid saturated COx claystone. In particular, the Laplace’s (or Washburn’s) law is used to predict the location of the interface between gas and liquid as:(1)
where P_{gas} is the pressure in the gas phase, P_{water} is the pressure in the water phase, P_{c} is the socalled capillary pressure, γ is the surface tension at the interface between gas and liquid and r is the pore radius at the gas/water interface. In the following, in a first approach, γ is taken equal to 7.275×10^{−2} N/m, which is the usual value for the interface between air and water.
In the third mechanism, gas passage is allowed by pore network dilatancy. Gas migrates by progressive tensile failure and microcracking of the solid skeleton, and by dilating the width of gas pathways to increase its passage [25, 26, 30]. This mechanism requires gas pressures of sufficient magnitude to counterbalance the local compressive stress state (reproducing the in situ loading), and to reach the material tensile strength. It is favored in the case of claystones, which possess very low tensile strength [29].
The fourth mechanism is an amplification of the third one: it is hydraulic fracturing, whereby breakthrough occurs along macroscopic tensile fractures.
Aims and Scopes
Our former research [8, 31] has identified the pore volumes of both COx and Tournemire claystones, down to 1722 nm pore size, for six micrometric volumes. These are specifically located in the clay matrix, which is the porous part of these claystones.
Among the six micrometric volumes (Tab. 1), one is of Tournemire claystone (Tournemire1 sample), whereas the five others are taken in the COx, from two distinct cores (EST26095 and EST27405). Their clay content is 37% (EST26095), 50% (EST27405) or 66.6% (Tournemire1); their total porosity (at the macroscopic scale) is 12.914.8% (EST26095), 17.520.0% (EST27405) or 8%±2 (Tournemire).
To allow FIB/SEM imaging and limit sample damage (by microcracking), the Tournemire1 sample is dried at 100°C and impregnated in epoxy resin, whereas the COx EST26095 samples are more gently dried at 80°C and impregnated with PMMA [17]. The COx EST27405 samples are also gently dried at 80°C, but they are imaged without any preliminary resin impregnation. PMMA is chosen for its very low viscosity before polymerization (it is close to that of liquid water), whereas the absence of resin impregnation is expected to yield greater sample microcracking. The non impregnated claystone (EST27405) is assumed more representative of the EDZ than PMMAimpregnated matter (EST26095); PMMAimpregnated claystone is only subjected to desiccation and potential cracking during the gentle 80°C drying, whereas non impregnated matter sustains the continuous drying effect of the microscope vaccuum pumps. Moreover, former research by Robinet et al. [17] uses PMMAimpregnated COx claystone of the same core as here (EST26095), yet for XRay CMT imaging; we aim to provide comparable datasets to [17], although on a smaller scale (down to 1722 nm instead of 700 nm). In [8], we have also chosen to use PMMAimpregnated COx claystone to perform TEM imaging, because sample thinning is significantly eased by this strengthening means.
Following image segmentation [8] by the same procedure for all samples, we observe that four out of the six FIB/SEM samples have a connected pore volume, of 0.68 to 2.10% (Tab. 1, penultimate column to the right); the two other samples (of COx claystone) have no connected pore volume down to 17 nm pore size (8.49 nm voxel size, i.e. two voxels are used to define the smallest pore diameter). Out of these four connected pore volumes, three (EST274051, EST274052 and Tournemire1) have the typical elongated morphology of cracks over a significant part of the pore volume; without these microcracks, the pore volumes are not connected from one surface to the other. The EST27405 core being made of non impregnated COx claystone, and the Tournemire1 being epoxyimpregnated (impregnated mainly on its surface), it is concluded that sample preparation is bound to have yielded these microcracks, owing to drying shrinkage. On the whole, over five samples of the same claystone, imaged down to pores of 17 nm size, only one (the EST260952) has a connected pore volume, which is distinct of cracks (it may be assumed to be natural porosity, also present in the in situcored medium).
Although several of them are microcracked, the FIB/SEM pore volumes have been used to predict macroscopic fluid permeability K with KatzThompson equation (permeability to neutral fluids with respect to the clayey matter). The value of K for the intact EST260952 sample (K = 5.4 × 10^{−21} m^{2}) is significantly smaller than that of microcracked FIB/SEM samples (EST274051 and EST274052, K = 1.6 × 10^{−20} m^{2}). These values are all on the order of magnitude measured on centimetric plugs [8].
It is analyzed that FIB/SEM samples provide the biggest connected part of the pore volume (with pores bigger than 1722 nm), which are useful for describing fluid transport at the macroscopic scale on the first order. Due to the limited volume of this pore network (2.1% at most), and to its scarcity (one over five micrometric pore volumes is naturally connected), fluid transport of nonwetting fluids is bound to occur through a very limited proportion of discrete pathways (i.e. by a socalled fingering mechanism), rather than in a homogeneous manner through the whole centimetric plug volume [31].
Besides, in [31], we show that the porosity measured in clayey aggregates of the clay matrix with TEM imaging range between 10.2 and 25.2%. For a claystone with 3767% clay matrix and a total porosity on the order of 18%, this explains only partly the location of all the pores: by a simple averaging rule (assuming that no other parts of the medium are porous), the clay content combined to the porosity at the TEM scale yields a porosity ranging between 3.816.9%. Another potentially important source of pores, which has not been imaged herein, is that located at the interface between the non porous minerals (e.g. calcite, tectosilicates, pyrite). This will require further investigations.
In this context, where several micrometric pore volumes of the clay matrix of the COx (and Tournemire) claystones are available, this contribution investigates the prediction of GBP through COx and Tournemire claystones. Experimental validation is proposed with data obtained on centimetricsized COx claystone samples [3]. The main method and results are recalled hereafter.
For GBP prediction, a number of assumptions is made regarding the flowpaths, i.e. either through the shortest pore path, or through the largest pore path, or through the most frequent pore size obtained by different methods, or through microcracks defined by their average width or by simulating the ingress of nonwetting fluid. Predicted GBP are analyzed and discussed, so that the most probable breakthrough paths are circumscribed.
1 Methods
This contribution uses both FIB/SEM imaging and GBP experimental results. In this section, the FIB/SEM imaging technique and analysis are described, followed by the method for GBP assessment in the laboratory.
1.1 FIB/SEM Imaging and Analysis
Our FIB/SEM (FEI Strata DB235 instrument^{TM}) possesses a FIB, which first aim is to create a parallelepipoid out of the sample, by cutting a Ushaped hole from the polished surface (Fig. 1). The beam is along the vertical y axis. The parallelepipoid is then imaged on its face perpendicular to the sample surface (in a (x, y) plane) by an electron beam at an angle of 52° to the vertical y axis; the parallelepipoid is then cut by the ion beam, in order to remove a slice parallel to the imaged face, with a constant thickness, ranging from 10 to 50 nm (Tab. 1); the sample face is then imaged again (in a (x, y) plane), prior to being cut by the ion beam. This procedure is automated (with the Slice and View^{TM} software package) in order to provide 200300 2D grayscale images (an image stack) for each sample. While the imaged parallelepipoid sample is destroyed by the ion beam, the distance between two successive image is constant, which allows 3D image reconstruction.
Figure 1. SEM image of the sample polished top surface (in the (x, z) plane) after creation of the Ushaped hole by FIB. The isolated parallelepipoid is regularly sliced in a (x, y) plane (by the FIB) and imaged (by the electron beam) at a 1050 nm thickness until obtaining a 200300 grayscale image stack. 
1.1.1 Image Processing
Once the grayscale images are obtained, the aim is to isolate the pores (in black) from the rest (in white), i.e. to derive binary images, from which a 3D pore network can be reconstructed. The method described here is strictly identical to that used in [8], where further details are available.
To this purpose, the FIB/SEM images are:

tilted in order to account for the 52° angle of the electron beam with respect to the imaged (x, y) plane,

aligned and cropped to select areas strictly aligned along the z axis (without translation along the (x, y) plane).
A minimal filtering is then performed: the image background is removed (it is mainly the dark shadowing effects due to the bottom of the Ushaped hole), the vertical curtain artifacts are removed (with the FFT transform), and the Median Filter 3D is used with a 1.5 pixel size (to remove salt and pepper effects). Further, the histograms of all the images of the stack are normalized in order to span the whole grayscale range. Finally, segmentation (i.e. thresholding) is performed with each of the 17 algorithms available in the free software ImageJ. Discussion of the choice of the adequate segmentation algorithm is proposed in [8, 31].
All forthcoming analysis are performed on a single binary image stack (pores in black and solids in white) per sample, obtained with the Yen algorithm.
1.1.2 3D Pore Network, Connectivity and Ientification of Major Cracks
Segmented image stacks are combined in FEI Avizo Standard^{TM} software. A 3D reconstruction tool (SurfaceGen or Surface Generate) is available to image the pore surface. Whenever a part of the pore volume has one dimension significantly smaller than the others (of at least one tenth smaller than the other in the (x, y) plane), it is identified as a crack and its volume is selected by the Magic Wand tool in Avizo software. The Magic Wand tool selects all contiguous pixels to that selected; the ‘all slices’ option, which we use, expands the selection to as many contuguous voxels as there exists in the stack. Contiguous means that at least one pixel (or voxel) face is common with the other. Identified crack volumes are then considered as distinct objects from the rest of the pore network. This method is also used to determine the connected and non connected parts of the 3D pore volumes. Connectivity is determined between any two parallel sample end surfaces. When connectivity exists, it is observed consistently between the closest end surfaces. In the following, only the connected parts of the pore volumes are considered for GBP prediction.
1.1.3 Sample Representativeness
Each pore volume is assessed relatively to its representativeness towards the claystone microstructure, and most particularly towards its clay matrix. To this purpose, two complementary methods are used to determine the variability of porosity in any subset of the stack, depending on the subset size considered.
In both cases, the box counting method provides mean porosity values for any subset (or box) of size L. The box location is varied all over the 2D images, for the whole stack; L is varied from a two voxel size to the entire sample size. This method also provides standard deviation, and the relative error err(L) (expressed in %), defined as:(2)where std(L) is the standard deviation at a given box size L, and ϕ_{mean}(L) is the mean porosity (for the whole stack) determined at a given box size L.
The local porosity distribution ϕ is given for a box of size L, centered on a pixel of coordinates x_{i} [16, 32, 33], by:(3)where m is the number of positions of the box of given size L, and δ(t) is the Dirac delta function. At given box size L, should this distribution display a single narrow peak centered on sample porosity (rather than a wide distribution), the pore space, at this scale, is considered to present a certain spatial homogeneity [16].
A full discussion of sample representativeness is proposed in [8, 31], where it is shown that our micrometric claystone volumes may be considered representative of the sole clay matrix (not of the macroscopic scale, which also comprises non porous grains of tectosilicates and carbonates, pyrite, and their interfaces).
In this contribution, we show relative error err(L) results for all the five COx claystone image stacks (Fig. 2a), and an example of local porosity distribution for EST260953 (Fig. 2b, not previously published in [8]). For the bigger box sizes, the relative error err(L) may remain on the order of 7080% for Tournemire1 and EST260951 samples, whereas it can go down below 20% for EST274051 and EST274052; intermediary values on the order of 35% are obtained for EST260952 and EST260953. It is interesting to note that the bigger the sample volume, the bigger the relative error err(L). These results highlight the significant variability of porosity within the FIB/SEM samples, located within the sole clay matrix. It is a difficulty inherent to the imaging technique, which requires to image small sample volumes (of 27.8553 μm^{3}) in order to visualize small pore sizes (down to 1722 nm). This justifies a statistical approach, whereby each sample is a realization of the clay matrix porosity.
Figure 2. a) Relative error err(L) results for all the five COx claystone image stacks. (b) Local porosity distribution μ(ϕ, L) for EST260953 (COx claystone). 
As for the local porosity distribution of the EST260953 sample (Fig. 2b), no distinct pore peak exists for a small box size L of 0.19 μm: no sufficient homogeneity of the pore volume exists at this scale. For bigger box sizes L, on the order of 1.91 to 2.87 μm, a single peak is obtained. In such instance, a single average porosity exists, which is the signature of a certain homogeneity of the pore space. For a bigger box size L of 4.78 μm, two peaks are observed, which are attributed to the presence of two distinct pore features. Indeed, this sample is made of a natural pore volume, which does not resemble cracks (first pore feature), mixed with a distinct microcrack with one dimension significantly smaller than the others (second pore feature) [31]. The local porosity distribution provides a means to distinguish this microcrack (at big box sizes) or to provide a single average porosity gathering all the distinct pore objects present down to 1722 nm (i.e. a single peak at intermediary box sizes of 1.912.87 μm).
1.1.4 Pore Shape and Size Distribution (PSD)
2D and 3D Continuous Pore Size Distribution (3D CPSD) are proposed by [34, 35], and used herein. Indeed, CPSD are based on the assumption that the complex pore network is a continuum, which may be invaded by a fluid as in MIP, yet not from the outer sample surfaces (as with MIP) but by gradual invasion from the widest zone in the pore volume. This resembles more the invasion mechanisms of nitrogen or CO_{2} within a dry porous medium. CPSD calculations progress by measuring the pore volume (respectively area in 2D) invaded by a smaller and smaller sphere (in 3D) or disk (in 2D). Pore sizes are quantified by the Euclidian distance to the nearest boundary, so that CPSD corresponds to the relative pore volume (respectively area in 2D) at a given sphere (respectively disk in 2D) size interval. In practice, 2D and 3D CPSD are performed by using the Beat plugin available at ftp://ftp.empa.ch/pub/empa/outgoing/BeatsRamsch/lib. This toolbox is also able to simulate the intrusion of a fluid (mercury) from one end surface of the sample to its parallel opposite end surface, by progressing through the 3D connected pore volume; this MIP simulation is used hereafter for comparison purposes.
1.1.5 Skeletonization and Characterization of Pore Paths
Skeletonization consists of thinning the pore volume to a minimum, i.e. down to lines of one voxel thickness. A skeleton is made of an assembly of nodes and segments (or branches), each segment being charaterized by an average pore diameter. The distribution of the average pore diameter over the whole skeleton is a method to quantify the PSD (Fig. 6). Several algorithms are able to provide a skeleton. In Avizo^{TM}, two such algorithms are available: the AutoSkeleton and the Centerline Tree [36, 37]. Contrarily to the AutoSkeleton, the Centerline Tree skeletonization algorithm provides a tree topology, without any loop.
When the treelike skeleton is determined, each path between two end nodes is carefully identified by a recursive loop. Our method is derived from the connectedcomponent analysis for graphs [38, 39], combining a unionfind and a recursive algorithm. This approach follows research findings by Lindquist et al. [40–43] on the 3D geometry of the pore space in rocks. In practice, after the identification of a first path, its end node is artificially removed from the skeleton used for the analysis, to ensure that the path is not identified twice. Also, in order to avoid that a path should be identified twice (in one direction and then the other), the start node is imposed to have a strictly smaller label number than the end node. Moreover, this procedure alone does not ensure that all identified paths are located between two parallel end surfaces, to ensure connectivity. This requires to validate the location of the start and end nodes of the paths (one on a given end surface, and the other on the opposite parallel surface). Once each connected path is identified by a set of node labels, we determine its length as the sum of the lengths of each branch of the path (located between two neighbouring nodes), its main orientation (as that of the vector formed by the start and end nodes of the path) and also its tortuosity [8]. From the path length distribution, the shortest path length is deduced by minimization. The whole method for the extraction of particular paths is coded through an inhouse Fortran program.
Nota: Other approaches exist, which do not require any skeletonization prior to determining the connected pore paths [32, 4446]. Comparison of these approaches with that chosen here, for its simplicity, will be the subject of future research.
1.2 GBP Experiments
The laboratory experiments used as a validation of our GBP predictions are detailed in [3, 5]. They aim to reproduce the in situ loading and gas breakthrough conditions through initially watersaturated claystone. In brief, a centimetricsized claystone sample (37 mm diameter, 10 to 30 mm height) is placed in a hydrostatic cell [1], and subjected to a confining pressure P_{c} of 6 or 12 MPa (smaller than or on the order of in situ principal stress levels). It is injected with synthetic water (of in situ composition [47]) on its upstream side at 4 MPa pressure, i.e. at the lithostatic water pressure. Full saturation is achieved when water permeability K_{water} falls below 10^{−20}10^{−21} m^{2} (110 nD), which is the expected permeability of undisturbed claystone.
Following water saturation, upstream pipes of the hydrostatic cell are emptied from water, while the sample is kept at a constant confinement P_{c} = 6 or 12 MPa, in order to avoid its premature failure. The downstream chamber placed right after the sample is closed by a dedicated valve, and its pressure P_{downstream} is recorded with a pressure transducer accurate to ±100 Pa (Fig. 3). On the upstream side, argon gas pressure P_{upstream} is increased very slowly, at a rate of 0.51 MPa per day. The upstream gas pressure is given by a pressure transducer accurate to 1 kPa. Gas detection on the downstream side is performed using both the downstream pressure transducer and a dedicated argon gas detector accurate to ±10^{−7} L/s.
Figure 3. Configuration of the GBP experiment after sample water saturation, from [5]. Argon gas injection pressure is stabilized through an upstream buffer reservoir, and all measurements are performed on the sample downstream side, in a 2 cl chamber closed by a valve, which is opened regularly (every 2472 h) for GBP assessment. 
At constant imposed P_{upstream} value, gas detection is performed every 24 to 72 h, as follows: the downstream chamber valve is opened and gas is detected (or not) by placing the detector nozzle at about 5 mm from the valve opening. At first, gas appears discontinuously on the sample downstream side, and then, without changing the imposed upstream gas pressure, gas finally flows continuously, when it is given sufficient time. The pressure at which gas appears discontinuously, and then continuously, is the GBP.
After a first GBP test, the claystone sample is either subjected to a new water saturation phase for another GBP experiment (e.g. samples 1, 3 and 5), or it is dismounted.
This method alone does not ensure whether gas actually passes through the porous medium by dissolution and diffusion in pore water, or by capillarity. However, from first order calculations (see Aims and Scopes) and from separated measurements with a mix of gases (helium/argon), we show that dissolution and diffusion of gases do not explain discontinuous passage. Moreover, gas breakthrough occurs at a much lower pressure than that required to equilibrate the compressive stress state. The mechanism by pathway dilation is not a plausible explanation for breakthrough. Rather, in the following, capillarity is assumed to explain gas breakthrough.
2 Results and analysis
2.1 GBP Laboratory Results
Table 2 summarizes our main GBP results on COx claystone. At this stage, it is assumed that values on the same order would be obtained with Tournemire claystone. From Table 2, it is observed that several GBP experiments yield the same water permeability at saturation, whereas, for the same sample, GBP varies significantly from one experiment to the other, with values ranging from 1.45 to 5.3 MPa. For instance, for sample 5, although water permeability at saturation is very stable with values of 1.61.9×10^{−20} m^{2}, GBP varies from 2.2 to 3.6 MPa, i.e. it increases by 73%. Also, GBP is greater with greater sample height, which is a partial explanation to its huge variations, at least when comparing different samples.
For a given sample, the huge variations in GBP are attributed to the gas breakthrough mechanism, i.e. to capillary fingering, with different pathways possibly followed from one test to the other. Greater GBP with greater sample height means that capillary fingering progresses with more difficulty when the overall path length increases.
In brief, all these observations are in good agreement with a capillary fingering phenomenon. Let determine how micrometric and connected pore volumes within the clay matrix (as given by FIB/SEM) may help further validate this interpretation, through the prediction of GBP by Laplace’s law.
2.2 FIB/SEM Micrometric Pore Volumes
Figure 4 presents the 3D pore volumes of three COx claystone samples and of Tournemire1 sample. In green are represented the connected pore parts (at this scale), and the rest of the pore volume (which is disconnected at this scale) is either represented in red or in yellow. Table 1 shows that the total pore volumes identified at this scale (down to 1722 nm pore size) are of only 1.705.90%, for an expected 18%±4 at the macroscopic scale for COx claystone and 8%±2 for Tournemire. We have explained earlier (see Aims and Scopes) that such limited porosity down to 1722 nm may be attributed to the presence of a significant amount of pores below 17 nm (imaged by TEM) [8] and of pores located at the interface between the clay matrix and non porous aggregates. The connected pore volumes of our FIB/SEM samples are even smaller, with values ranging from 0.682.10%. However, their geometry is complex enough to justify different possible migration modes to predict gas breakthrough, as follows.
Figure 4. 3D pore volume of a) EST274051, b) EST274052, c) EST260952 COx claystone samples, and d). Tournemire1 sample. In red are the non connected and non cracklike pores – these are likely to be connected by micro and small mesopores below the resolution of the FIB/SEM device; in green, the connected porosity between the end surfaces perpendicular to the Zaxis; in yellow and bluegreen are the cracklike pores (Tournemire1 sample only). 
2.3 Predictions of GBP
The following assumptions are made (on a first order prediction basis):

the interaction of water with the clay aggregates (due to hydration or swelling) is not taken into account;

to account for GBP, the connectivity of the micrometric pore volumes given by FIB/SEM is assumed representative of the macroscopic behavior, although located solely within the clay matrix (i.e. within clayey solids mixed with submicrometric non porous grains);

the breakthrough mechanism is due to viscocapillary flow, i.e. it is describable by Laplace’s law for the interface between air/water (Eq. (1).
Several subsequent assumptions are made to describe simply the gas breakthrough through a 3D pore volume, without requiring 3D numerical simulations. Firstly, we only use the data provided by the skeleton of the connected pore volume, in terms of shortest path or largest path, or of most frequent pore size. This means that results in this contribution will be slightly different from [8], where we used the whole pore volume (connected and non connected). Secondly, we propose a prediction based on pore entry size, as determined by MIP simulations. Finally, we propose to deduce GBP rather from the characteristic size of cracks, which are identified and separated from the rest of the pore network, i.e. by using median crack widths.
2.3.1 From the Shortest Path  After Skeletonization
In this subsection, we assume that gas passes through the shortest pore path, so that the determining pore diameter is the smallest pore throat along the shortest pore path. The shortest pore path is determined for the skeleton of the four connected pore volumes, as given by the centerline tree algorithm. This algorithm is preferred because it provides a more realistic skeleton for fluid movements, without loops, for the considered pore volume. From the pore size distribution along the shortest path (Fig. 5a), we determine the pore throat as the minimum pore size value for each sample (Tab. 3). Figure 5a shows how hugely varying the pore size distribution is along the shortest path. We also observe that the smallest pore size is equivalent to the smallest pore size available with FIB/SEM, i.e. between 1726 nm, and that it is reached several times along the path, i.e. the gas passage is made difficult several times along its progression through this shortest path. These are flow constrictions representing ‘bottlenecks’ for gas flow.
Figure 5. For all four connected claystone samples, distribution of pore diameters along a) the shortest path or b) the largest path; c): along the largest path, but limited to microcracked EST27405 samples, presented when taking the cracks into account (or not). 
Figure 6. For all four connected claystone samples, Pore Size Distributions defined as the distribution of average pore diameters of all the segments, after skeletonization obtained a) by the Auto Skeleton method or b) by the Centerline Tree algorithm. 
Pore throat diameter and predicted GBP along the shortest path
From these smallest pore throat values along the shortest paths, GBP is predicted (Tab. 3), and clearly overestimated, with values ranging from 11.2 to 17.1 MPa, when compared to the experimental GBP values (Tab. 2). If all assumptions taken are correct, this means that gas does not pass through the shortest connected path, i.e. it passes through larger pores than 1726 nm.
2.3.2 From the Largest Path  After Skeletonization
An alternative is that gas does not pass through the shortest path, but through the largest, though longer, connected path between two parallel end surfaces at the micrometric scale (Fig. 5b). Except for the Tournemire1 and EST260952 samples, we observe that the smallest pore diameter along the largest path is, as expected, bigger than the smallest pore size available with FIB/SEM (i.e. bigger than 1722 nm) (Tab. 4). For the two samples EST274051 and EST274052, the largest pore path is hardly affected when taking into account the whole connected pore volume or only its cracklike part (Fig. 5c). It means that the largest pathway connection for these samples occurs through their microcracks.
Pore throat diameter and predicted GBP along the path with the largest pore throat
On the whole, this second assumption provides smaller GBP than when assuming a passage through the shortest pore path, as GBP values now range between 6.2215.06 MPa. However, this is still overestimating GBP when comparing with macroscopic experimental values (Tab. 2).
2.3.3 From PSD Given by Skeletonization
The peak pore size is the most frequent pore size encountered in the pore network. If gas is assumed to try and progress on several fronts at a time, i.e. through as many pathways as possible (and not only through the largest or the shortest), a realistic assumption is that gas passes mainly through the most frequent pore size. With this assumption, the two available skeletons have been determined, because they provide slightly different estimations. These are readily observed on the PSD defined as the distribution of the average pore diameters on each skeleton segment (Fig. 6). With the skeleton given by the autoskeleton algorithm, the GBP predictions range between 3.23 and 9.7 MPa (Tab. 5), which is only partly overestimating it when compared to experimental macroscopic values (Tab. 2). With the centerline tree skeletonization, smaller or equivalent values are predicted for GBP, when compared to the use of the autoskeleton (Tab. 5).
Predicted GBP using the PSD processed after two different skeletonizations of the connected pore network
With both skeletons, we conclude that we have found a plausible GBP prediction for both COx claystone cores, and for the Tournemire claystone too. For the EST27405 core, breakthrough rather occurs through the EST274052 sample (with a predicted GBP = 3.23 MPa) rather than through the EST274051 one (predicted GBP = 5.829.7 MPa). For the EST26095 core, the GBP value is of 4.164.85 MPa, and for Tournemire1 the value is as low as 4.16 MPa, i.e. both are within the measured range for COx claystone. This means that it is plausible that gas migrates by capillarity through the natural connected pore volume identified by FIB/SEM, although it is located within the clay matrix, and of a few microns size.
However, while assuming that gas passes through a multitude of pathways (and not only through the largest or the shortest) may be relatively adequate, this method presents several limitations. Firstly, as the largest pore path passes through a smallest pore diameter of about 19.346.8 nm, any other path requires the passage of gas through smaller pores (than 19.346.8 nm), so that the use of the most frequent pore diameter is not realistic. Indeed, there is no reason for it to be the smallest pore diameter of any of the paths followed. Secondly, using the skeleton to derive PSD requires to simplify the connected pore volume. Let propose GBP predictions without resorting to a skeleton.
2.3.4 Using Crack Widths as Determined from 2D and 3D CPSD
The simplest idea is to use the 2D and 3D CPSD (Fig. 7) and assume that gas migrates through the most frequent pore size (i.e. through the peak pore size of these distributions) (Tab. 6). In particular, in 2D, the CPSD reflects the crack widths, which are visible in the (x, y) planes of the 2D binary images. In such instance, the peak pore sizes are close to the average crack widths determined manually from several images, and as the geometrical tortuosity is 1 (cracks are straight paths), using a peak pore size may be considered a correct assumption.
Figure 7. PSD given by a) the 2D CPSD method or b) the 3D CPSD method. 
Predicted GBP using the PPS from the 2D/3D CPSD
In particular, for EST274051, the peak pore size (and average crack width) is of 90 nm, whereas it is of 80 nm for the EST274052 sample, and an identical 90 nm for Tournemire1 sample. For these connected volumes, the peak values are identical when using the 3D CPSD instead of the 2D. For the EST260952 sample however, the peak pore size given by CPSD is also of 90 nm, although its connected pore volume is not cracklike. In this case, the use of actual paths instead of peak pore sizes would be more adequate to describe gas breakthrough, and the sole 90 nm peak pore size may not be a reliable way to determine the main gas pathway (tortuosity is significantly greater than 1, contrarily to cracks).
With these peak pore sizes, GBP is predicted consistently as plausible, whatever the sample considered, with values of 3.233.64 MPa. However, as with PSD derived from the skeleton, this method does not ensure that single paths are followed, i.e. it is not physically realistic.
2.3.5 Using the Peak Pore Size Given by MIP Simulations
Let try a more physical method, which accounts for the progression of gas through the porous medium from an entry surface; in particular, this method considers the succession of smaller or larger pores as the use of a skeleton and its associated paths did.
Mercury is a non wetting fluid (with a contact angle generally of 130°), just as gas is, when trying to migrate through the waterwet solid. The Beat plugin available in ImageJ provides 3D simulation of mercury intrusion through the connected pore volumes (without any use of a skeleton), either from the bottom end surface (Fig. 8a) or along the top end surface (Fig. 8b). For a given sample, the variations in MIP simulations between the top and bottom end surfaces is related to the way the fluid progresses, from the chosen surface and through the complex 3D pore network, through to the other end; it is impacted by the succession of pores encountered along this progression.
Figure 8. PSD given by MIP simulations a) from the bottom surface perpendicular to the z axis for the EST26095 and EST27405 samples (respectively perpendicular to the x axis for Tournemire1) or b) from the top surface perpendicular to the z axis for the EST26095 and EST27405 samples (respectively perpendicular to the x axis for Tournemire1). 
With this method, predicted GBP range between an acceptable 3.23 MPa and an overestimated 14.55 MPa (Tab. 7). No prediction is within the acceptable range (given by macroscopic experiments) for the EST260952 sample (GBP = 7.2814.55 MPa), but gas breakthrough is plausible by capillarity through both EST274051 and EST274052 (GBP = 3.234.85 MPa are acceptable), and through the Tournemire1 sample (GBP = 3.234.16 MPa are acceptable).
Predicted GBP using the PPS from MIP simulations (entry from the top surface perpendicular to the x axis, or from the top surface perpendicular to the z axis) of the connected pore network
It is concluded that with this most plausible gas migration mode (similar to mercury intrusion), gas breakthrough is predicted for the EST27405 core (and for Tournemire claystone) by using micrometric pore volumes located in the clay matrix and given by FIB/SEM, which are microcracked. On the opposite, if non microcracked EST260952 sample is representative of the macroscopic scale, it does not allow gas breakthrough as measured in the laboratory. Rather, for EST26095, breakthrough with values similar to those presented in Table 2 is bound to occur through larger pores, and very possibly through microcracks. In situ, these are bound to be present in the EDZ, despite claystone selfsealing. On the opposite, if capillary digitation is the proper mechanism for gas breakthrough, intact COx claystone has a GBP ranging between 7.314.5 MPa, when using the naturally connected pore volume of EST260952.
Conclusion
This contribution predicts macroscopic GBP experimental results by using micrometric and connected pore volumes of the clay matrix of COx claystone and of Tournemire claystone. To this purpose, a set of assumptions is necessary:

the interaction of water with the clay aggregates is not taken into account;

to account for GBP, the micrometric pore volumes given by FIB/SEM are representative of the macroscopic behavior, although located solely within the clay matrix;

the breakthrough mechanism is due to viscocapillary flow, i.e. it is describable by Laplace’s law for the interface between air/water.
By using data from the skeletons derived from the connected pore volumes, we show that plausible GBP predictions by capillarity require that the gas does not pass solely through the shortest pore path, nor through the sole largest pore path, but through a multitude of pathways. Although providing realistic GBP predictions (3.234.85 MPa for both COx claystone cores) when resorting to the most frequent pore size, this does not reproduce the progression through a path within the material, so that this is not considered physically relevant.
Without simplifying the pore volumes with skeletons, 3D simulations of non wetting fluid intrusion is performed from any of the connected end surfaces of the samples. In such instance, which is considered the most physically plausible, GBP is adequately predicted by capillarity for the COx EST27405 core and for the Tournemire1 claystone. In both cases, the main pathways are cracklike. On the opposite, for the naturally connected pore volume of the EST260952 sample, the capillary digitation mechanism predicts a GBP ranging between 7.314.5 MPa. This means that none of our GBP experiments on 13 cm thick samples have been performed on intact matter. Indeed, sample preparation to such small thickness is bound to create microcracks in the medium.
This also means that for these claystones, if the capillary fingering mechanism is the proper breakthrough mechanism, the only way to account for the low GBP values measured at the macroscopic scale is to accept the presence of microcracks. An alternative to sufficiently big pores (that microcracks represent) would be the interfaces between the clay matrix and non porous grains. Identifying the latter would require to benefit from a 3D imaging device allowing to span greater volumes than our FIB/SEM, and yet smaller pore sizes than XRT, i.e. pore sizes ranging between 50700 nm. This is being rendered possible by the currently developed FIB/SEM instruments. This will hopefully be the subject of future research.
Another conclusion of this contribution is that gas passage through a liquidsaturated claystone (intact or microcracked) occurs through a limited amount of pathways, and not in a homogeneous manner through the whole volume, as standard macroscopic finite element models would represent it. This mechanism is usually termed capillary fingering.
Acknowledgments
We acknowledge funding by NEEDS MIPOR (Grand Défi InterDisciplinaire Nucléaire: Énergie, Environnement, Déchets, Société  MIlieux POReux, funded by CNRS, CEA, EDF, ANDRA, AREVA, IRSN, BRGM, France) in 2014/2015, under the DiphLBM2 project label (kindly coordinated by M. Dimitrowska, IRSN).
The authors are most grateful to L.M. Keller for making the Beat plugin (ImageJ) available to the scientific community.
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All Tables
Pore throat diameter and predicted GBP along the path with the largest pore throat
Predicted GBP using the PSD processed after two different skeletonizations of the connected pore network
Predicted GBP using the PPS from MIP simulations (entry from the top surface perpendicular to the x axis, or from the top surface perpendicular to the z axis) of the connected pore network
All Figures
Figure 1. SEM image of the sample polished top surface (in the (x, z) plane) after creation of the Ushaped hole by FIB. The isolated parallelepipoid is regularly sliced in a (x, y) plane (by the FIB) and imaged (by the electron beam) at a 1050 nm thickness until obtaining a 200300 grayscale image stack. 

In the text 
Figure 2. a) Relative error err(L) results for all the five COx claystone image stacks. (b) Local porosity distribution μ(ϕ, L) for EST260953 (COx claystone). 

In the text 
Figure 3. Configuration of the GBP experiment after sample water saturation, from [5]. Argon gas injection pressure is stabilized through an upstream buffer reservoir, and all measurements are performed on the sample downstream side, in a 2 cl chamber closed by a valve, which is opened regularly (every 2472 h) for GBP assessment. 

In the text 
Figure 4. 3D pore volume of a) EST274051, b) EST274052, c) EST260952 COx claystone samples, and d). Tournemire1 sample. In red are the non connected and non cracklike pores – these are likely to be connected by micro and small mesopores below the resolution of the FIB/SEM device; in green, the connected porosity between the end surfaces perpendicular to the Zaxis; in yellow and bluegreen are the cracklike pores (Tournemire1 sample only). 

In the text 
Figure 5. For all four connected claystone samples, distribution of pore diameters along a) the shortest path or b) the largest path; c): along the largest path, but limited to microcracked EST27405 samples, presented when taking the cracks into account (or not). 

In the text 
Figure 6. For all four connected claystone samples, Pore Size Distributions defined as the distribution of average pore diameters of all the segments, after skeletonization obtained a) by the Auto Skeleton method or b) by the Centerline Tree algorithm. 

In the text 
Figure 7. PSD given by a) the 2D CPSD method or b) the 3D CPSD method. 

In the text 
Figure 8. PSD given by MIP simulations a) from the bottom surface perpendicular to the z axis for the EST26095 and EST27405 samples (respectively perpendicular to the x axis for Tournemire1) or b) from the top surface perpendicular to the z axis for the EST26095 and EST27405 samples (respectively perpendicular to the x axis for Tournemire1). 

In the text 