Hydraulic properties of porous sintered glass bead systems
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Abstract
In this paper, porous sintered glass bead packings are studied, using Xray Computed Tomography (XRCT) images at \(16\,\upmu \hbox {m}\) voxel resolution, to obtain not only the porosity field, but also other properties like particle sizes, pore throats and the permeability. The influence of the sintering procedure and the original particle size distributions on the microstructure, and thus on the hydraulic properties, is analyzed in detail. The XRCT data are visualized and studied by advanced image filtering and analysis algorithms on to the extracted subsystems (cubes of different sizes) to determine the correlations between the microstructure and the measured macroscopic hydraulic parameters. Since accurate permeability measurements are not simple, special focus lies on the experimental set up and procedure, for which a new innovative multipurpose cell based on a modular concept is presented. Furthermore, segmented voxelbased images (defining the microstructure) are used for 3D (threedimensional) lattice Boltzmann simulations to directly compute some of the properties in the creeping flow regime. A very good agreement between experimental and numerical porosity and permeability could be achieved, in most cases, validating the numerical model and results. Porosity and permeability gradients along the sample height could be related to gravity acting during sintering. Furthermore, porosity increases in the outer zones of the samples due to the different contact geometry between the beads and the confining cylinder wall during sintering (which is replaced by a membrane during permeability testing to close these pores at the surface of the sample). The influence of different filters on the gray scale distributions and the impact of the segmentation procedure on porosity and permeability is systematically studied. The complex relationships and dependencies between numerically determined permeabilities and hydraulic influence parameters are investigated carefully. In accordance to the wellknown Kozeny–Carman model, a similar trend for local permeability values in dependence on porosity and particle diameter is obtained. Other than statistical models, which estimate the pore throat distribution on the basis of the particle size distribution, in this study XRCT scans are used to determine the pore throats in sintered granular systems, which are finally linked to the intrinsic permeability through the lattice Boltzmann simulations. From the \(\mu \)XRCT analysis two distinct peaks in pore throat distributions could be identified, which can be clearly assigned to typical pore throat areas occurring in polydisperse granular systems. Moreover, a linear dependency between average pore throat diameter and porosity as well as between permeability and pore throat diameter is reported. Furthermore, almost identical mean values for porosity and permeability are found from subsystem and fullsystem REV analysis. For sintered granular systems, the empirical constant in the classical Kozeny–Carman model is determined to be 131, while a value of 180 is expected for perfect monodisperse sphere packings.
Keywords
Intrinsic permeability Darcy’s law Tortuosity Porosity Sintered glass beads \(\mu \)XRCT scan Lattice Boltzmann method XRCT data processing Image segmentation REV analysis Pore throat Kozeny–Carman1 Introduction
 1.
the \(\mu \)XRCTscanning and reconstruction of the porous material;
 2.
the filtering and segmentation of the image data;
 3.
the implementation of image data in numerical simulations;
 4.
validation of numerical calculations with experimental data;
 5.
finding new correlations between microstructure and macroscopic properties by combining the previous steps and insights.
In this work we investigate, both experimentally and numerically, the intrinsic permeability of artificial produced samples composed of sintered glass beads showing different particle diameters, porosities and degree of polydispersity. In contrast to common rock samples, like e.g. dolomite, sintered glass bead samples are characterized by their chemical stability and inertness, in addition to their relatively simple pore structure. Nevertheless, the pore structure, and thus the intrinsic hydraulic permeability can be influenced by the selection of certain glass beads and special sintering treatments. Another crucial advantage of sintered glass bead samples, in contrast to most rock samples, is the improved grayscale contrast between the pore space and the solid phase, which considerably simplifies the image segmentation process and thus ensures a better comparability of experimentally and numerically determined permeability values. The filtering and segmentation procedure of sintered glass bead packings is further simplified by single phase composition of the solid matrix. In this respect, sintered glass bead samples can serve as replacement material for soil and rock specimen to provide a benchmark in permeability calculations.
The present paper focuses on the hydraulic properties of porous sintered glass bead systems. The influence of the sintering process on the microstructure, and thus on the permeability of the sintered samples are analyzed in detail by using appropriate XRCT data analysis and visualization methods (AVIZO Fire 8.0.1 and 9.0). In Sect. 2 the wellknown Darcy’s law is introduced to define the intrinsic permeability of porous materials in general. Furthermore, the semiempirical Kozeny–Carman equation, which is often used to determine the intrinsic permeability of granular systems, is presented and discussed in terms of microstructural parameters. In Sect. 3 the Lattice Boltzmann (LB) method and the numerical setup, which is used to determine the local intrinsic permeabilities of the extracted data sets, is described briefly. For a better understanding of the underlying microstructure, the sintering procedure is described in Sect. 4.1. For validation of numerical data based on discretized \(\mu \)XRCT/voxel data sets of the produced sintered glass bead samples, the experimental setup and procedure of permeability measurements are described in Sect. 4.2. The developed multipurpose measuring cell is proposed in Sect. 4.3. Section 5 focuses on the elaborate processing of \(\mu \)XRCT scans, whereby possible ways are introduced, how to filter, segment and extract essential features, which highly influences the hydraulic properties of porous sintered granular systems. Section 6 starts with presentations and discussion of results obtained from numerical and experimental porosity and permeability measurements. In addition, the results from REV analysis for porosity and permeability and the peripheral porosity development of the samples are depicted and discussed in Sect. 6.1. Moreover, the numerically determined local permeability values are qualitatively and quantitatively compared with the theoretical predictions according to the Kozeny–Carman model. The results for pore throats and their dependency on the porosity and intrinsic permeability is presented and discussed in Sect. 6.2. The study of hydraulic properties of porous sintered granular packings is concluded inSect. 7.
2 Theory
Several approaches have been introduced in recent years to describe the tortuosity (fluid path) in Eq. (2) through porous media [11, 23, 24, 27].
3 The lattice Boltzmann method
4 Experiments
4.1 Sintering
Material parameters and characteristic particle diameters of the investigated glass beads in their state before sintering. The sample with the biggest particles had a larger sintering time and larger size (50mm) than the others (30mm), which possibly explains their deviation. The REV sizes from the larger bead samples, as used for LBsimulations, are maybe not representative since there are not enough particles (and thus poor statistics) for a proper comparison with experimental effective permeability results
Material description  Diameter\(^{1}\) (mm)  \(\hbox {D}_{10}{}^{2}\,(\upmu \hbox {m})\)  \(\hbox {D}_{50}{}^{2}\,(\upmu \hbox {m})\)  \(\hbox {D}_{90}{}^{2}\,(\upmu \hbox {m})\)  \(\hbox {D}[4,3]^{2}\,(\upmu \hbox {m})\)  \(\hbox {D}[3,2]^{2}\,(\upmu \hbox {m})\) 

Minibeads\(^\mathrm{a}\)  0.4–0.6  388  519  696  532  505 
0.6–0.8  507  683  916  702  666  
Minibeads\(^\mathrm{b}\)  0.8–1.0  646  883  1222  915  862 
1.0–1.2  766  1045  1415  1073  1016  
Glass beads Q1\(^\mathrm{c}\)  1.5–2.0  857  1190  1581  1209  1143 
2.0–2.5  –  –  –  –  –  
“Diamond” Pearls\(^\mathrm{d}\)  3.0 (± 0.2)  –  –  –  –  – 
Depending on the composition, the deformation temperature of the used glass particles varied between 575 and \(680~^\circ \hbox {C}\). The used glass beads showed different particle sizes and degree of polydispersity, cf. Table 1. The specific density of the sintered glass beads was at \(2.5~\hbox {g/cm}^3\).
The sintering of the glass beads were performed in a tubular furnace with heat power of 0.7 kW and a nominal target temperature of \(1000~^\circ \hbox {C}\) under atmospheric conditions. The induction furnace was equipped with three programmable temperature controllers (type West 5010) regulating the inner temperature at three different furnace zones. The temperature progression within the furnace was monitored continually at five different places during the sintering process using thermocouples (type K), see Fig. 1a. The measured temperature curves are depicted in Fig. 1b. The glass beads were filled in a quartz glass cylinder (with inner diameters of approximately 30 and 50 mm) and completely enclosed with graphite paper. The melting temperature of the quartz glass cylinder is around \(1713~^\circ \hbox {C}\) and thus certainly higher than the deformation temperature of the glass beads (\({\approx } 690~^\circ \hbox {C}\)). The graphite paper prevented an adhesion or sticking between the beads and the cylinder. The glass beads were manually shaken prior to sintering to reach the closest glass bead packing. The samples were subsequently loaded from the top with different masses ranging between 100 and 300 g corresponding to pressures of 1.39 and \(4.16~\hbox {kN/m}^2\) for samples with 30 mm diameter or 0.50 and \(1.50~\hbox {kN/m}^2\) for samples with 50 mm diameter, respectively. As can be seen in Fig. 1b the glass beads were heated up with a constant temperature rate of \(300~^\circ \hbox {C}\) per hour until the required sintering temperature of \(695~^\circ \hbox {C}\) was reached. Holding the beads at this temperature for approximately 2.5 h, the specimen was finally cooled down in an uncontrolled manner by switching off the furnace, cf. temperature curve at position 3 in Fig. 1b. Since the prepared samples were placed centrally in the furnace, only the temperature curve at position 3 is of major relevance. The preprogrammed target temperature of \(695~^\circ \hbox {C}\) in the outer zones (position 1 and 5) was not reached due to large heat losses at the top and bottom of the furnace whereas the target temperature at positions 2 and 4 was reached, but delayed by approximately 1–2 h.
Depending on the chemical composition, bead diameter, dead loads of the used masses and sintering duration, the initial heights of the untreated specimens shrank by 1–5 mm. After sintering the samples were cut by a diamond disc to the desired length of 50 mm.
4.2 Permeability measurement setup
To minimize the content of air bubbles and to guarantee for reproducible experimental results, use was made of filtered and deaired water. For this purpose, the water was mechanically filtered in various filter stages until reaching a degassing tank, where the filtered water was deaired.
The measuring cell including the hose connections were rinsed with carbon dioxide before the cell was flooded with filtered and deaired water. The carbon dioxide easily dissolves in water. In this way, the content of air bubbles in the cell was minimized and an optimal water saturation of the sample was achieved.
4.3 Measuring cell
The developed measuring cell has been designed according to a modular concept in order to use it in various applications. Figure 2b shows an illustration of the measuring cell in operating mode for stationary permeability measurements.
It consists of an inner and outer cylinder. The inner cylinder is made of an acrylic glass (PMMA) and produced in various sizes, whereas the outer cylinder is made of aluminium (AlCu4PbMgMn alloy) to stabilize the measuring cell. As can be seen from Fig. 2b, the sintered sample was positioned at the center of the cell and pneumatically fixed by a specially developed specimen holder coated with a 1mmthick latex membrane. During flow measurements a static air pressure on the latex membrane was applied which fixed the specimen in the current position and prevented a surrounding fluid flow. At the same time, the latex membrane ensured a hermetically sealing off the measuring cell to the outside. The pressure difference, mainly resulting from the viscous friction of the fluid passing through the porous sintered sample, could be taken via the upper and lower pressure port and measured by a highprecision differential pressure transducer (type FDW2JA, ALTHEN, Germany) capturing differential pressures up to 35 mbar (with an accuracy of 0.25%).
5 \(\mu \)XRCT data processing
\(\mu \)XRCT data processing is a crucial step towards understanding of fluid flow through complex morphologies like sintered glass beads. It is a important tool for visualization and quantification of parameters, such as porosity, particle sizes or pore throats, determining the hydraulic conductivity of a porous medium [43, 44, 45]. The correct procedure of the \(\mu \)XRCTdata including an adequate filtering and thresholding method is essential for a proper comparison between experimental and numerical determined permeabilities [17, 46].
Figure 4b shows the main image processing steps. Starting from raw data, the image file is filtered in several stages until the desired gray scale distribution is reached. For better visualization of the filter effect, Fig. 5 demonstrates exemplarily the filtering procedure on the basis of slices applied onto the original raw data with the corresponding grayscale value distribution. For the sake of clarity, during filtering the image is interpreted and processed as a three dimensional volume, and the grayvalues of each voxel are numbers of decimals obtained from 16bit binary representation of the gray level. In the initial state (state A), the grayscale distribution of the raw data show two peaks which can be clearly attributed to the pore space and the glass beads. Starting from the original raw data a simple logical operator with the socalled “arithmetic” module is applied to remove bright spots from the image file. These bright spots highlighted by a red circle in state A are caused by density fluctuations and chemical impurities of the beads and can be clearly assigned to greater grayscale values. In the first filtering step, the gray values belonging to these bright spots are lowered artificially by setting a defined maximum threshold for grayscale values. As highlighted in the example, a maximum threshold of 26,818 is used, cf. Fig. 5b.
Since the grayscale values belonging to the glass beads and pore space overlap due to their wide distributions, the segmentation procedure becomes difficult. Therefore, in the second step of the filtering procedure the socalled “delineate” filter is applied to enhance the edges of the glass beads and to adjust the contrast between the pore space and the glass beads. Local changes in intensity of grayscale values constitute a common issue during segmentation. The “delineate” filter, which is based on a phase contrast method, can detect sharp transitions between different phases to finally enhance and contrast the edge of an object to be segmented. As a result, the grayscale distributions between the glass beads and the pore space are clearly separated from each other (state C), which simplifies the further separation of the glass beads from the pore space.
In accordance with the process flow chart shown in Fig. 4b, the filtered grayscale image data is thresholded to generate a binary map. The threshold is selected manually for each sample in a way that the solid glass beads are assigned to values of unity and the pore space voxels are set to zero. For the determination of porosity and visualization of the pore space the binarized image data is inverted.
5.1 Particle number distribution
Characteristic parameters from particle number distributions gained from \(\mu \)XRCT (after sintering) for the sample with \(d_{p}=0.4{}0.6\,\hbox {mm}\)
Unit  256 voxel cube  512 voxel cube  1024 voxel cube  Initial cuboid  

Number of particles N  (–)  705  5144  39,290  78,375 
Arithmetic mean value \(\langle d_{p} \rangle \)  (\(\upmu \hbox {m}\))  383.33  499.59  499.59  499.59 
Standard deviation \(\sigma _{d_{p}}\)  (\(\upmu \hbox {m}\))  216.74  286.06  286.06  286.06 
5.2 Pore throat determination
A decisive factor, which determines the intrinsic permeability in granular porous systems is the pore throat area. Micro tomographic imaging techniques enable to localize, visualize and quantify such determining areas.
6 Results and discussion
In this section the results of the determining parameters of the intrinsic permeability, described in previous sections, obtained from \(\mu \)XRCT analysis and LB simulations are presented successively, discussed and compared with experimental results. Moreover, the dependency of the intrinsic permeability on different parameters and on the localizations and sizes of the investigated subsets are analyzed qualitatively and quantitatively. Please note that the results are only extensively showed for the sintered sample with \(d_p=0.4{}0.6\) mm due to the large amount of data. All other samples were produced in the same way as the sample with \(d_p=0.4{}0.6\) mm and thus show qualitatively the same results.
6.1 Porosity and permeability
Please note that the geometrical tortuosity determined from the centroids of twodimensional slices of 256 voxelsided cubes obtained from XRCT scans shows no spatial gradient along the sample height and no correlation is found with the porosity and permeability, cf. Ref. [53].
Figure 10a shows the permeabilities of the subsets as function of their porosities, for three different theoretical estimates, increasing according to Kozeny–Carman.^{1} Figure 10b shows the intrinsic permeabilities normalized by the square of the mean particle diameter, confirming the nonlinearly increasing trend of the intrinsic permeability with increasing porosity. The colored data points in Fig. 10a represent the predictions according to the Kozeny–Carman model, cf. Eq. (3), whereby the arithmetic (red), harmonic (green) and effective (black) diameter according to Eq. (4) from the particle size distribution are used as representative values to predict the permeabilities of the differently sized subsets. The permeabilities determined from the lattice Boltzmann simulations (open data points) and the predictions according to Kozeny–Carman using the arithmetic mean diameter show a similar increasing trend, whereby the numerically determined permeabilities are higher by a factor of approximately 1.3. The permeability predictions according to Kozeny–Carman using the harmonic or effective diameter according to Eq. (4) (green and black data points) show stronger fluctuations and clearly underestimate the numerically determined permeabilities. Fitting the predicted permeability values depicted in red into numerical results by using the empirical constant \(c_1\) as fit parameter, yields a value of 131, see Fig. 10b. This constant contains and reflects the effect of the microstructure (particle shape, tortuosity) as a result of the sintering process on the intrinsic permeability of the glass bead samples.
Figure 11c shows the resulting porosity distribution in dependence on the mean pipe crosssection radius of the investigated specimens showing different glass bead diameters and degree of polydispersity. Starting from the center of the investigated samples, the porosities remain relatively constant up to a mean crosssection radii of 11.5 mm and then increase to higher porosities for the largest investigated pipes in the external area of the produced samples. In these outer zones a clear increase of the porosity values for investigated samples can be seen due to the different contact between beads and cylinder wall during the sintering procedure and the gaps remaining between particles and walls. The clear porosity increase on the edges of the sintered samples is a consequence of the sintering procedure caused mainly by the presence of the walls.
The bar diagrams in Fig. 13 show a direct comparison between experimentally and numerically determined porosities and permeabilities of the investigated samples with different glass bead diameters and degrees of polydispersity. The numerical porosity and permeability results, represented by black bars, are obtained from 1024 voxelsided cubes taken from the center part of the entire scanned regions of the samples and used in LB simulations. The effective permeability results obtained from laboratory experiments, represented by white bars, are averaged values from 5 (or 8) independent permeability measurements, whereby 10 to 15 different measuring ranges for volume fluxes and pressure differences are run during each measurement, cf. Fig. 3. While the porosity values show a good agreement, the permeability results show much larger nonsystematic deviations. It should be noticed that small deviations in porosity and particle size can lead to significant deviations in permeabilities due their exponential influence, see Eq. (3). Larger deviations between experimental and numerical permeability results occur for 1.5–2.0 mm and larger particles, since the representativeness of the investigated subsets for numerical permeability calculation decreases with increasing particle diameter. Furthermore, it can be seen from Fig. 13b that the error bars in experimental permeability determinations are small compared to their averaged values, which indicates high reliability, repeatability and robustness of the used experimental permeability setup.
Moreover, the good agreement between voxelbased local and experimental determined global porosity values of the investigated different sintered samples in Fig. 13a proves the correctness of the filtering and segmentation procedure despite the underlying porosity gradient across the sample height. A proper filtering and segmentation procedure is surely essential for subsequent permeability simulations. Incorrect porosity determination during segmentation procedure can lead to substantial errors in permeability calculations, see Eq. (3). In this context sintered glass bead samples as simple replacement material for natural sandstones are suitable to set a benchmark for permeability calculations. Taking into account the porosity and permeability gradients with the sample height, the deviations between experimental and numerical results is insignificant. The maximum deviation factor of 2.4 that is obtained for glass bead diameters of 3.0 mm due to low representativity of the investigated subset is quite reasonable.
From the REV analysis, we have come to the important conclusion that the average value for porosity and permeability of smaller sided cubes is (almost) identical to results obtained from the initial cuboid, cf. Fig. 8. We have found out that the averaged values for permeability are almost equal independent from the size of the investigated subsets.
6.2 Pore throats
Figure 14a illustrates the local distribution of mean pore throat diameter from 256 voxelsided cubes within the cuboid for the sintered sample with bead diameters between 0.4 and 0.6 mm. The normalized pore throat distribution gained from the initial cuboid in Fig. 14b depicted in red shows two distinct peaks which can be clearly assigned to pore throat areas formed by either three or four particles. The larger peak at smaller diameters results from pore throat areas which are formed by three particles whereas the smaller peak at larger diameters is due to fourparticleconstellations. The probability of occurrence of pore throats formed by three particles is certainly higher compared to pore throat areas resulting from fourparticleconstellations. Downscaling to smaller volume elements (up to 256 voxelsided cubes) showed qualitatively the same pore throat distributions. In accordance with local distributions for porosity and permeability, the equivalent mean pore throat diameter also shows a spatial gradient along the sample height z, see Fig. 14a, decreasing from top to bottom about 14% due to compaction in deeper layers.
The correlation of the permeability, the porosity and the mean pore throat diameter for the sintered sample with bead diameters of 0.4–0.6 mm is plotted in Fig. 15, displaying the porosities \(\phi \) (a) and the normalized intrinsic permeabilities \(k_{z}^s/\langle d_{p}\rangle ^2\) (b) of differently sized subsets taken from the initial cuboid in dependence on their normalized mean pore throat diameters \(\langle d_{pt} \rangle /\langle d_{p} \rangle \). For both, the porosity and the normalized permeability a clear linear dependency on the normalized mean pore throat diameter \(\langle d_{pt} \rangle /\langle d_{p} \rangle \) can be seen, with correlation coefficients of the fits of 0.9378 and 0.9675, respectively. The higher mean pore throat diameters correlating with the higher porosity and the higher permeability values are located at the top of the investigated cuboid and decrease towards deeper layers.
7 Conclusion
In summary, we have demonstrated that the hydraulic properties of sintered glass beads are highly affected by the pore throats, in addition to the porosities and particle sizes. In this respect, the nondestructive Xray Computed Tomography is an efficient tool, which enables the visualization and quantification of internal (microstructural) parameters including the (voxelbased) morphometry of the porous sample. The way \(\mu \)XRCT data are processed is crucial. In this study, different ways have been introduced in order to identify the essential control parameters for the hydraulic properties of the sintered glass bead systems. The presented framework can in future be used to evaluate also the hydraulic characteristics of other kinds of porous media.

The intrinsic permeability in sintered granular packings depends not only on the porosity, but also correlates with the equivalent pore throat mean diameter.

The pore throat distributions in polydisperse packings show two distinct peaks arising from typical three or fourparticle constellations.

In the given narrow porosity and permeability range, a linear dependency of the permeability with the equivalent pore throat diameter is observed.

The quantitative comparison between experimental and numerical permeability values requires a proper filtering and segmentation procedure and an appropriate domain size, and then can lead to good agreement, even there are a few unexplained outliers, possibly due to channeling (higher permeability) or due to particularly dense samples (lower porosity).
Local REV analysis has revealed that the averaged values for porosity and intrinsic permeability of smaller sided cubes are (almost) identical to results of larger subsets obtained from the cuboid. A consequence of this result is that, we can either perform many cost effective and timesaving computations on small subsets, which are less representative, or compute one cost and time intensive large subset. For the calculations of effective quantities new strategies can be developed on the basis of results gained from the REV analysis.
From the REV data, the porosity development across the radial direction was analyzed. A slight increase of porosity at peripheral outer zones could be identified due to the contacts of the beads with the cylinder wall during sintering. Rotational sintering under pressurecontrolled gas atmosphere may eventually lead to more homogeneous samples avoiding spatial gradients of porosity and permeability. From the experimental point of view, the presence of air bubbles in the measuring cell poses a big challenge, which can highly affect the measurement of pressure differences, and thus of the intrinsic permeability. Taking into account these difficulties, a comparison between numerical and experimental determined intrinsic permeability values gives us the possibility to better understand the system. Eventually, we have achieved qualitatively and quantitatively a good agreement between experiment, microstructural analysis and numerical simulations.
We have demonstrated through 3D LB simulations that a proper filtering and segmentation procedure during \(\mu \)XRCT analysis is essential for porosity determinations and to obtain accurate permeability results. We have found that an almost linear correlation between both intrinsic permeability and porosity with the pore throat equivalent diameter can be observed if systematic errors during filtering and segmentation are avoided. Our numerical and experimental permeabilities are in general agreement with permeability predictions according Kozeny–Carman, although the validity of the Kozeny–Carman equation is limited to monodisperse and nonsintered granular packings. In contrast to the factor \(c_{1}=180\) for idealized sphere packings, a value of \(c_{1}=131\) for the empirical constant, which takes into account microstructural effects resulting from the sintering treatment of the polydisperse glass bead packing, could be determined by fitting the predicted permeability values to numerical results from LB simulations.
In this study XRCT structural and permeability analysis have been performed for several polydisperse sintered glass bead samples with different primary particle average diameters. The observed micromacro relations scale with the average diameters of the rather large particles. Further work is needed to extend the study towards systems with smaller particles, higher polydispersity or systems with longer sintering duration time and thus lower porosity. It has to be seen if the presented approach for the processing of the XRCT data and the resulting main findings, for instance the linear dependency of the intrinsic permeability with the pore throats, can be also found in those systems, and in natural porous materials like sandstone.
Footnotes
 1.
With second and third power of porosity.
Notes
Acknowledgements
We acknowledge support from the Stichting voor Fundamenteel Onderzoek der Materie (FOM), which was financially supported by the “Nederlandse Organisatie voor Wetenschappelijk Onderzoek” (NWO) (Project Number: 09iPOG141/2), and partial support by the NWO STWVICI (Project Number: 10828). We also thank Matthias Halisch from the Leibniz Institute for Applied Geophysics (LIAG) in Hannover, Germany for performing and providing \(\mu \)XRCT scans, as well as the Jülich Supercomputing Centre for providing the CPU time for the lattice Boltzmann simulations.
Compliance with ethical standards
Conflict of interest
There is no conflict of interest.
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