Abstract
The spatial resolution of 3D imaging techniques is often balanced by the achievable field of view. Since pore size in shales spans more than two orders of magnitude, a compromise between representativeness and accuracy of the 3D reconstructed shale microstructure is needed. In this study, we characterise the effect of imaging resolution on the microstructural and mass transport characteristics of shales using micro and nanocomputed tomography. 3D mass transport simulation using continuum and numerical physics respectively is also compared to highlight the significance of the Knudsen effect on the reconstructed solid surface. The result shows that porosity measured by microCT is 25% lower than nanoCT, resulting in an overestimated pore size distribution and underestimated pore connectivity. This leads to a higher simulated intrinsic permeability. An overestimated diffusive flux and underestimated permeability are obtained from the continuum mass transport simulation compared to the numerical ones when the molecularwall collision is accounted, evidenced by the large deviation of the measured Knudsen tortuosity factor and permeability correction factor. This study is believed to provide new knowledge in understanding the importance of imaging resolution and gas flow physics on mass transport in porous media.
Introduction
In recent years shale gas has attracted much attention due to the accessible energy reserves stored in lowpermeability organicrich shales and mudstones. These reservoirs contain a significant amount of hydrocarbons, and the successful exploitation of such resources plays a crucial role in meeting the world’s surging demand for natural gas. This has the potential to play a significant role in the transition to a cleaner energy future due to its high energy content, resulting in lower emissions of carbon and volatile organic compounds (VOCs) at combustion, relative to coal and oil^{1}. The gas is released with the help of hydraulic fracturing techniques also known as “fracking”^{2} and gas injection displacement^{3}, and is transported through pores of multiple length scales, eventually converging in the main wellbore^{4}. While the fracture network greatly determines the productivity of shale reservoirs^{5,6,7}, the transport of shale gas within the matrix also plays an important role^{8,9,10}. Valid pore structure analysis and imagebased computational fluid dynamics (CFD) simulation of the shale gas flow in the porous media rely heavily on a faithful 3D representation of the porous microstructure.
Nondestructive threedimensional Xray computed tomography (Xray CT) has been widely applied to the multiscale microstructure study of the shale gas^{7,11}. This technique provides more reliable and representative 3D microstructure compared to those reconstructed by discrete 2D SEM images^{12,13}, and helps to mitigate the artefacts of the pore phase potentially introduced from the 2D serial sectioning^{14,15}. However, like other imaging techniques, there is a tradeoff between the image resolution and the field of view (FOV), and therefore a compromise has to be made between the representativeness and the accuracy of the imaged microstructure, which could inevitably exclude small pores due to the hierarchical pore size distribution in the shale (i.e. ranging from tens of nanometre to micrometre)^{16}. A previous study^{17} characterized the gas flow in micro and nanopores using ideal cylindrical pore model, which however cannot account for the effect of complex surface roughness of the wall, the constriction and the arbitrary morphology.
Transport of gas molecules in porous media is mainly governed by two mechanisms: (1) continuum flow, in which the gas molecules interaction is dominant and is often modelled as a viscous effect in continuum physics and (2) the collisions between gas molecules and the wall, also known as molecular flow^{18}. The predominant mechanism(s) in the transport regime will depend on the gas species, temperature, pressure and microstructure^{19,20,21}. The Knudsen number K_{n}, calculated as the ratio between the mean free path of the gas molecules and the pore size, is widely used to assess the flow regime in porous media: If K_{n} < 0.01 (continuum regime), the flow is mainly governed by molecular diffusion and the Knudsen flow can be neglected; if K_{n} > 10 (Knudsen regime), the gas is highly rarefied and effect of molecular flow outweighs the viscous flow in the continuum regime because of the frequent collisions between the molecules and the porous medium. As for 0.01 < K_{n} < 10 (transitional regime), shale gas flow is governed by both mechanisms.
The wide distribution of the pore size causes two problems in the mass transport study: (1) it is not reliable to estimate the Knudsenbased diffusivity based on the averaged pore size, which could potentially overestimate the gas flow due to the constriction effect^{21,22}; (2) Viscous flow fails in smaller pore spaces as the diffusion flow mechanisms associated with porewall interactions become dominant^{23}, which leads to underestimating the permeability. This means conventional continuum physics can no longer describe the flow field in shales^{24}.
To account for the gas moleculeswall interaction (i.e. wall slippage effect), different theoretical models have been adopted to predict the apparent permeability of nanopores, which is a key property for shale gas production. Klinkenberg^{25} analytically addressed the gaswall collisions by introducing the slippage effect associated with the pressure. Beskok and Karniadakis^{26} mathematically integrated the Knudsen effect into the permeability measurement by comparing the apparent and intrinsic permeability. Tang et al.^{27} proved that the apparent permeability is nonlinearly related to the intrisinc permeability.
Direct Simulation Monte Carlo (DSMC) is a numerical method widely used to solve the thermodynamic states of the rarefied gas based on Boltzmann equation, which effectively overcomes the challenges in gaswall interaction by continuum modelling with conservation equations. Compared to other numerical methods such as molecular dynamics (MD)^{3}, DSMC is less computational expensive with high confidence^{28}.This method was validated either by experimental permeability^{29} or analytical solution^{30}. DSMC has been applied to study the gas flow in a variety of materials of distinct pore morphologies, such as in solid oxide fuel cells^{31}, cylindrical channels^{32,33}, random/aligned fiber orientations^{30,34,35} and ablative materials^{29}.
In this study, we aim to elucidate the effect of imaging resolution on the characterization of porous microstructure and mass transport properties in shales using multilength scale Xray CT, followed by the imagebased CFD simulation using both continuum and numerical method for the first time to highlight the effect of moleculeswall interaction on the extracted effective mass transport parameters (i.e. Knudsen tortuosity factor, apparent permeability) which could partially be neglected either by meanfield fluid dynamics (continuum flow) or resolution limitation. The particlebased CFD simulation methodology using reconstructed 3D microstructure proposed in this study is highly applicable not only to the shales, but also to be of wide interest across an increasingly broad range of mass transport studies in geological materials.
Methods
Xray Computed Tomography
A cylindrical sample pillar, already employed for previous investigations^{7}, was prepared from the shale sample using an A Series/Compact Laser Micromachining System (Oxford Laser, Oxford, UK) following the procedure explained by Bailey et al.^{36}. The threedimensional microstructure of shale sample was investigated using two Xray computed tomography microscopes (Carl Zeiss Xray Microscopy Inc., Pleasanton, CA): micronscale Zeiss Xradia 520 Versa (microCT) and nanoscale Zeiss Xradia 810 Ultra (nanoCT).
For microCT, a total of 1401 radiographs were acquired over a 360° sample rotation range with an exposure time of 35 seconds per radiograph. The shale sample was placed between the Xray source and a 2k × 2k detector providing a voxel resolution of 224 nm using the 20x objective magnification and a Field of View (FOV) of 448 μm. The instrument was operated at 80 kV. NanoCT employs posttransmission Fresnel zone plates to achieve resolution in the sub 100 nm range^{37}. A total of 1601 projections were collected per 180° sample rotation with an exposure time of 36 seconds. This allowed achieving a set of raw image data with an isotropic voxel resolution of 63 nm and a FOV of 65 μm.
The raw transmission images from both microand nanoscale CT imaging experiments were reconstructed using a commercial image reconstruction software package (Zeiss XMReconstructor, Carl Zeiss Xray Microscopy Inc., Pleasanton, CA), which employs a filtered backprojection algorithm. Tomographic scan details are shown in Table 1. The 3D reconstructed volume of the shale was segmented and analysed using Software Avizo Fire 9.2 (Thermo Fisher Scientific, USA). Due to the low Xray absorption coefficient difference, it is not possible to distinguish the organic matter (kerogen) from pores based on the reconstructed grayscale data, thus the combined phases are rendered together. This phenomenon is normal in processing Xray CT data and the same measure was taken in published research^{11}. The pore size distribution (PSD) was measured using the plugin ‘Beat’^{38} in opensource software FiJi^{39}.
Effective mass transport parameters by continuum fluid dynamics
The surface mesh (ASCII *.stl) file was generated after the segmentation of the porous phase and imported into the commercial computational fluid dynamics (CFD) software StarCCM+ (CDAdapco Inc., London). A mesh refinement procedure was undertaken to improve the mesh quality from asimported raw data (Fig. 1a) to the refined triangular surface mesh (Fig. 1b), to the final polyhedral volume mesh (Fig. 1c). The use of a polyhedral mesh has proven to be more accurate for fluidflow problems than a hexahedral or tetrahedral mesh of a similar size. The optimized mesh is closed and manifold, with no holes and free edge and the volume change of the porous phase is ensured not to exceed 1% to maintain the microstructural originality.
The tortuosity factor is an effective mass transport parameter representing the effect of complex porous gas pathways on the gas flow^{22,40}. In this study, it was measured by CH_{4} ordinary diffusive flow: the CH_{4} molar concentration was set as c = 1 mol m^{−3} at the inlet and c = 0 at the outlet. It is noted that the tortuosity factor measured by continuum physics is a material parameter and independent of the concentration gradient of the gas. The onedimensional gas flow Q_{e} can be described by Fick’s law as
where D is the intrinsic diffusivity, A is the crosssectional area of the fluid domain, Δc is the concentration change and x is the diffusion length. In the porous medium, Eq. (1) is modified as
where τ_{c} is the tortuosity factor, ε is the porosity and can be measured by CT data analysis. By dividing Q_{e} by Q_{p}, the effective transport parameter ε/τ can be obtained,
It is noted that in the continuum fluid model, the effective transport parameter is independent of the intrinsic diffusivity, indicating that it is a material parameter. The Reynold’s number^{41} of the shale gas flow is far less than unity, which suggests that viscous forces dominate over inertial forces and the permeability can be obtained according to Darcy’s law^{42},
where k is the permeability of the porous medium, v is the gas velocity, μ is the dynamic viscosity of the gas, P is the pressure and x is the distance in the flow direction. The intrinsic permeability was obtained by setting a pressure drop (50 Pa) from the inlet to the outlet according to Eq. (4). It is noted that for continuum fluid dynamics, the intrinsic permeability is independent of the pressure gradient.
Effective mass transport parameters by noncontinuum fluid dynamics
To highlight the significance of moleculeswall interactions (Knudsen effect) in the hierarchical porous shale, numerical simulation method Direct Simulation Monte Carlo (DSMC) was used on the 3D reconstructed shale from Xray CT scan, which is believed to provide a faithful representation of the wall roughness and pore morphology. A subvolume consisting of 168 × 200 × 200 voxel (10.6 × 12.6 × 12.6 μm) was used in this study.
The Stochastic PArallel Rarefiedgas Timeaccurate Analyzer (SPARTA)^{43} DSMC code developed at Sandia National Laboratory (USA) was used in this work. The generated surface mesh (i.e.stl file) of the shale was imported into the SPARTA software such that it was embedded in the fluid domain which is composed of an array of 3D Cartesian grids (1.5 million in total). Intermolecule and moleculewall collisions were performed following a notimecounter (NTC) procedure^{44}. Shale gas (CH_{4}) was simulated from slip flow regime (0.01 < K_{n} ≤ 0.1) to transitional regime (0.1 < K_{n} ≤ 10) with incremental pressure to obtain Knudsen tortuosity factor τ_{k} based on Eq.(4) and apparent permeability K_{a} based on Eq. Table as a combination of the ideal gas law, conservation of mass and the differential form of Darcy’s law.,
where J denotes the mass flux by DSMC; M, R, T, μ are molecular weight of the gas species, gas constant, the temperature and viscosity respectively. Buffer zones of at least 10% total flow domain were added. A total of 20 million simulation molecules were generated so that the average molecule number in each cell is above 20 to avoid statistical scattering^{28}. Each of these simulating molecules is regarded as the representative of a large number of real molecules, the ratio of which is known as scaling factor^{45}, to reduce the demand of computational resources. In this study, a scaling factor of 15 was used and small enough to provide accurate DSMC results. The validation of this technique was performed experimentally^{31}. The interaction between gas molecules and the porous media can be seen in the video (see Supplementary Video S1).
Results and Discussion
The effect of imaging resolution on the reconstructed volume is highlighted in Fig. 2. The top row (a–c) shows the grayscale virtual slices scanned and reconstructed using microCT, from which it is clear to see the blurred microstructure due to the resolution limits and it is impossible to extract the pore network with high confidence, particularly for the smallest pores; in the bottom row (d–f), the same region obtained from nanoCT was registered and shown as the superimposition of the microCT images. By comparing the obtained microstructures between the top and bottom row, it is found that nanoCT scan provides significantly sharper images so that more microstructural details such as edges and narrow pores which are missing in microCT scans can be captured in nanoCT data. In the next section, two case studies will be presented to highlight (1) the effect of imaging resolution on describing the microstructural characteristics and mass transport properties in the direction parallel to the horizontal natural bedding of the shale gas sample; (2) the disparity of obtained mass transport parameters vertical to the natural bedding between continuum and numerical CFD simulation attributed to the captured submicron 3D pore network.
Case study 1: effect of imaging resolution on pore structure and mass transport metrics
This study aims to compare the microstructural metrics and mass transport parameters (i.e. tortuosity factor and permeability) as a consequence of the extra porosity which can be imaged in the nanoCT. The same subvolume was extracted from micro and nanoCT and compared in Fig. 3. Figure 3a,d compare the morphology of the same pore under two resolutions. The details of the pore edges and curvatures which can be seen in nanoCT (Fig. 3a) are volumeaveraged in microCT (Fig. 3d). This could lead to an undersegmentation of the pore network from the reconstructed volume (Fig. 3b,e). This local homogenisation effect by microCT can result in two disadvantages which undermine further analysis: (1) the pore size distribution and porosity will be over and underestimated respectively; (2) the percolation will be underestimated as the extracted pore network does not include all of the subresolution pores.
After segmentation, a Continuous Pore Size Distribution (CPSD) analysis was carried out and the PSD is shown as a heatmap with the colourcoded according to its size (Fig. 3c,f). A highly complex pore structure is resolved using nanoCT in contrast to the smoothed singlepore feature using microCT. Figure 3g highlights the disparity of the extracted pore structure by overlaying the 3D rendered pore structure under the two resolutions.
The CPSD measurement is summarised in Fig. 4. It is observed that with finer features resolved in nanoCT data, the pore size can be quantified with a smaller step size compared to that in microCT. NanoCT scan allows to capture and quantify tinier pores (<1 μm). It is noted that large pores (>1 μm) that are dominant in microCT data are not observed in nanoCT measurement. On the other hand, a large amount of pore volume shifts to the low radius end of the histogram. This disparity is speculated as another disadvantage of coarse resolution scan of the shale gas: the complex curvature of the pore edge is homogenised in 3D so that the pore throat resolved in nanoCT (yellow arrow in Fig. 3a) is averaged with the slices above and underneath the plane, resulting in the pore with blurred edge and less dark grey value (Fig. 3d), as a consequence of which, the measured pore size distribution deviates from the practical value.
The segmented pores were then meshed and imported to the CFD software Simcenter STARCCM+ (Siemens, Plano, TX, USA) for the continuum simulation in order to assess the variation of the measured mass transport properties originating from different imaging resolutions. Figure 5a,b compare the concentration distribution of methane gas at steady state and it is found that for the first half of the flow field the nanoCT pore volume exhibits a much lower concentration gradient (Fig. 5b) compared to microCT pore (Fig. 5a), but for the second half of the pore volume the concentration distribution is identical. The reason for this phenomenon is that the nanoCT managed to resolve a larger lateral region of the first half of the pore, providing a higher crosssection for the flow thereby the reduced concentration drop, whilst for the second half more parallelconnected pores are captured in the nanoCT, which in essence would not alter the concentration distribution, instead yielding a higher flow rate. In order words, the morphological difference between two resolution scans mainly consists of laterallyresolved pores which may connect in parallel or in series with the existent pore. However, the velocity field of the viscous flow driven by a constant pressure difference (5 × 10^{−4} bar) is significantly different between the two samples: the microCT sample exhibits much higher velocity than the nanoCT one, which is considered as a consequence of the larger surface area and narrower pores in nanoCT, leading to a more remarkable viscous effect. The measured mass transport metrics between the microCT and nanoCT samples are summarised in Table 2. The continuum tortuosity factor τ_{c} between two samples are similar, as is consistent with the concentration field in Fig. 5a,b. The resultant permeability of microCT sample is almost one order of magnitude higher than the nanoCT value.
Case study 2: gas flow simulation perpendicular to the bedding direction
This case study aims to investigate the difference of the measured tortuosity factor and permeability when the gas molecules  wall collision is considered in the CFD simulation. This is important as in most of the cases the shale gas flow is governed by transitional and Knudsen regime and thus Darcy flow fails in smaller pore spaces in which the wallslippage effect becomes dominant. This means the conventional continuum CFD method with the nonslippage condition at the porewall interface can no longer faithfully describe the gas flow in the shale. However, few studies have compared the disparity of the extracted tortuosity factor and permeability obtained between continuum and numerical method, and thus the uncertainty is ambiguous. Different from Case Study 1, in which the boundary condition was applied so that the gas flew parallel to the natural bedding direction, Case Study 2 examines the gas transport property vertical to the natural bedding, in which direction the resistance is significantly higher and the gas molecules – wall interaction is more dominant.
Figure 6a shows the concentration distribution of CH_{4} simulated using continuum CFD method. It is observed that the gradient here is less smooth and uniform compared to Fig. 5a,b, evidenced by a sharp decrease of the concentration at the pore throat vertically connecting the top and bottom half of the horizontally aligned pores. This is the main reason for the strong anisotropy of gas transport in the shale. The narrow pore throat can be visualised in Fig. 6b in terms of the consequent local high velocity of the gas flow. Figure 6c shows some possible streamlines of the gas transport from the top to the bottom, via one of the pore throats. It is noted that the trajectory is highly convoluted and tortuous. The intrinsic permeability measured by the continuum flow is K_{i} = 7.1 × 10^{−19} m^{2}. Figure 6d is a snapshot showing the CH_{4} distribution (red spheres) by particlebased numerical simulation. To the author’s knowledge, this is the first time that DSMC method has been used on the study of shale gas based on the reconstructed 3D volume of the pore network which provides the poresolid boundary resolved with high confidence at 0.1 μm resolution. This is favourable to examine the collision between the gas molecules and the surface of the wall, as is shown in Fig. 6e. It is observed that the collision does not occur uniformly in the pore volume, instead, it is highly localised at the places where the pore size is smaller than the surrounding areas. This can be supported by comparing with the 3D distribution of the pore diameter Fig. 6f, in which the red arrows point out the corresponding areas of high collision frequency.
Figure 6g compares the Knudsen tortuosity factor τ_{k} as a function of the gas pressure with that obtained from continuum modelling, which is a constant value independent of the gas pressure. It is found that when the CH_{4} is highly rarefied, τ_{k} is measured to be as large as 50, then drops drastically with the pressure. At 1 bar, τ_{k} ≈ 27. The curve asymptotically converges to the continuum tortuosity factor τ_{c} = 10 when the pressure is above 8 bar, from which and onward continuum flow is dominant. Figure 6g proves that in reallife situations, as the gas pressure is much higher than 8 bar, continuum modelling can be safely applied to the investigated gas shale with the minimal pore size larger than 0.1 μm, which is the imaging resolution of this study. Figure 6h plots the ratio (η) of moleculeswall collision and intermolecules collision as a function of the K_{n} the two types of collision: moleculeswall and intermolecules collisions. It is found that the simulated points are linearly proportional to the Knudsen number, aligning with η = K_{n} very well. Finally, a correction factor f to the intrinsic permeability K_{i} as a function of Knudsen number is obtained based on the apparent permeability K_{a} using DSMC method (K_{a} = f∙K_{i}) (Fig. 6i). When K_{n} ≤ 0.01, the intrinsic and apparent permeability are in a good agreement, implying the little influence of viscous flow from the wall slippage; with the increase of K_{n} thus decreasing pressure, the apparent permeability diverges with the intrinsic one, for instance, when K_{n} = 1, the gas molecules – wall collision is so predominant that the apparent permeability is 9 times larger than the intrinsic one. This result is compared with the empirical solution derived from Klinkerberg’s model^{25} and Beskok/Karniadakis model^{26}. Generally, all three curves exhibit the exponential relationship between the correction factor and the Knudsen number, and the measured one by DSMC method is slightly larger than the other two empirical models. This could arise from a variety of factors related to the geometry of the pore structure, such as constriction, shape etc.
Conclusions
This study firstly compared the difference of reconstructed 3D volume of the shale scanned by Xray Computed Tomography (CT) using different resolutions (voxel size 224 nm for microCT and 63 nm for nanoCT), based on which the continuum CFD simulation was conducted to highlight the effect of imaging resolution on the obtained tortuosity factor and permeability of the shale. The second part of the study discussed the importance of gas moleculeswall collision and wall slippage effect by numerical Direct Monte Carlo Simulation (DSMC) which has been applied to the microstructureresolved shale model for the first time and then compared the disparity of the mass transport parameters obtained by the conventional continuum CFD modelling. It is found that lowresolution scan has two main disadvantages: (1) the pore size distribution and porosity are over and underestimated respectively; (2) the percolation is underestimated as the extracted pore network does not include all of the subresolution pores. These lead to a much larger intrinsic permeability of the microCT scan than the nanoCT one. The morphological difference of the pore structure between two resolution scans mainly consists of laterallyresolved pores which may connect in parallel or in series with the existent pore. The former would not change the concentration distribution but provide a higher mass flow whereas the latter would render a lower local concentration gradient due to the increased crosssectional flow area. When the surface collision (Knudsen effect) and slippage are considered, the tortuosity factor can be as large as 50 for the most rarefied gas and then significantly drop with the pressure until asymptotically reaching the value (10) obtained by continuum method, implying an overestimated diffusive flux when the Knudsen effect is not included. In addition, the apparent permeability showed an exponential relationship with the intrinsic one as a function of the Knudsen number, indicating that as the pressure decreases, the deviation of the apparent permeability is larger from the intrinsic one. It is also shown that the ratio of the frequency of the molecularwall and intermolecular collision can be estimated by the Knudsen number. As both numerical and continuum simulation methods are widely used in the shale gas study, this study is believed to provides new insights in emphasizing the validity and uncertainty level of the shale gas flow under a variety of conditions. The conclusion drawn can also be used as a reference for the gas flow in other porous media.
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Acknowledgements
This work was supported by the EPSRC (EP/N032888/1, EP/M014045/1, EP/K005030/1, EP/M008428/1), PRS acknowledges funding from the Royal Academy of Engineering (CIET1718/59), XL acknowledges the support of the NPL Measurement Fellowship.
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F.I. and X.L. conceived the project; P.R.S. and D.J.L.B. funded and managed the project as directors of the Electrochemical Innovation Lab (EIL); F.I. conducted the Xray CT experiments and the computer segmentation; X.L. conducted the CFD simulation using Direct Simulation Monte Carlo (DSMC); T.M.M. contributed to the discussion of the results; F.I. and X.L. analysed all results; F.I. and X.L. wrote the manuscript; F.I. and X.L. contributed equally to this work; all authors reviewed the manuscript.
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Iacoviello, F., Lu, X., Mitchell, T.M. et al. The Imaging Resolution and Knudsen Effect on the Mass Transport of Shale Gas Assisted by Multilength Scale XRay Computed Tomography. Sci Rep 9, 19465 (2019). https://doi.org/10.1038/s41598019559997
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DOI: https://doi.org/10.1038/s41598019559997
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