Powder Technology 200 (2010) 69–77
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Powder Technology
j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / p ow t e c
Representative elementary volume analysis of porous media using X-ray
computed tomography
Riyadh Al-Raoush a,⁎, Apostolos Papadopoulos b
a
b
Department of Civil and Environmental Engineering, Southern University and A&M College, Baton Rouge, LA 70813, USA
Department of Environmental Science, Lancaster University, Lancaster, LA1 4YQ, UK
a r t i c l e
i n f o
Article history:
Received 26 August 2009
Received in revised form 3 February 2010
Accepted 15 February 2010
Available online 19 February 2010
Keywords:
Computed tomography
Representative elementary volume (REV)
Microscale
Particle size distribution
Local void ratio
Coordination number
a b s t r a c t
The concept of representative elementary volume (REV) is critical to understand and predict the behaviour
of effective parameters of complex heterogeneous media (e.g., soils) in a multiscale manner. Porosity is
commonly used to define the REV of a given sample. In this paper we investigated whether the use of a REV
for porosity can be used as a REV for other parameters such as particle size distribution, local void ratio and
coordination number. X-ray computed tomography was used to obtain 3D images (i.e., volumes) of natural
sand systems with different particle size distributions. 3D volumes of four different systems were obtained
and a REV analysis was performed for these parameters utilizing robust 3D algorithms.
Findings revealed that the REVmin for porosity may not be adequate to be considered as a REV for parameters
such as particle size distribution, local void ratio and coordination number. The REVmin for these parameters
was observed to be larger than the REVmin for porosity. Heterogeneity of the systems was found to be an
important factor to determine the REV for the parameters analyzed in this paper. The REV analysis revealed
that as the uniformity coefficient increased, a larger volume was required to obtain the REVmin for the
distribution of particle sizes and coordination number whereas a smaller volume was required to obtain the
REVmin for local void ratio. Therefore, determination of the REV for parameters described in this paper or any
microscale parameter of concern should not be derived based on REV for porosity and should be determined
based on their distributions over different volumes.
© 2010 Elsevier B.V. All rights reserved.
1. Introduction
The scale of observation is an important aspect in modelling the
behaviour of material (e.g., soil) or deriving its effective macroscale
parameters from the constitutive relations that are governed by the
spatial distribution of its components. A given soil sample can be
considered homogenous when the scale of observation is large
enough where parameters of concern are constant. Upon further
increase of the spatial scale, soil parameters may become nonstationary [1,2].
Macro and multiscale modelling are two approaches commonly
used to study the behaviour of materials and to predict effective
macroscale parameters. In the macroscale modelling approach, the
domain of concern is considered homogenous where assumptions
regarding the microscale behaviour of materials are usually imposed
to reach closed-form expressions [3–5]. In the multiscale approach, a
coupled micro and macroscale analysis is performed where a
microscale analysis is performed over volumes of finite sizes and
results are then incorporated into macroscale formulations to
⁎ Corresponding author. Tel.: + 1 225 771 5870; fax: + 1 225 771 4320.
E-mail address: riyadh@engr.subr.edu (R. Al-Raoush).
0032-5910/$ – see front matter © 2010 Elsevier B.V. All rights reserved.
doi:10.1016/j.powtec.2010.02.011
evaluate effective macroscale parameters [6–8]. A key issue in both
modelling approaches is the determination of a representative
elementary volume (REV). The REV can be defined as the minimum
volume of a soil sample from which a given parameter becomes
independent of the size of the sample [9]. The size of an REV ranges
from a minimum bound, which is the transition from the microscale
to the macroscale level, to a maximum bound, which is the transition
from a homogenous to a heterogeneous state.
There are two main approaches commonly used to determine the
REV of a given sample for prediction and analysis of effective
macroscale parameters. The first approach determines a REV based
on porosity regardless of the macroscale parameter of concern (i.e., a
sample is considered a REV if its porosity is constant over different
sizes of the sample). This approach is common in soil science and
hydrology literature. Clausnitzer and Hopmans [10] used porosity as a
base to determine REV of ideal systems (i.e., glass beads packs). Other
studies determined REVs of natural soil systems based on porosity
[11–14]. A similar approach was used to determine representative
elementary area (REA) for natural soil systems [15]. The second
approach determines a REV based on macroscale parameters without
prior determination of microscale parameters of the sample (e.g.,
porosity). This approach is commonly used in engineering mechanics
literature. The sample is considered a REV when the macroscale under
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R. Al-Raoush, A. Papadopoulos / Powder Technology 200 (2010) 69–77
consideration is constant over different volumes. Macroscale parameters such as elastic modules and peak stress are common in
determining the size of REV [16–20]. The determination of the REV
size is by no means a trivial task as it is a function of the nature of the
material being considered, the objective of the macroscale model and
microscale parameters that impact the effective macroscale parameters. Moreover, the difficulty of microscale characterization of real
porous media systems makes the determination of REV based on
microscale parameters even more challenging.
It has been recognized that geometry and topology of soil particles
control its thermal, fluid-like flow, stress transfer mechanisms and
other mechanical parameters. For instance, mechanical conditions of a
granular sand system such as shearing, compaction, transmission of
stress and statistical parameters of contact force distribution are
greatly influenced by its microscale characteristics (e.g., the distribution of the number of contacts of particles) [21–23]. Considering two
adjacent particles within the assembly, one needs to account for the
exact interaction characteristics so that the interparticle forces and
relative motions can be estimated. The interparticle interaction is
essentially controlled by sizes of particles and the nature of their
contacts. Furthermore, the macroscale behaviour of granular assemblies such as fluid-like flow, stress–strain response and multi-phase
flows (e.g., relative permeability) can be described by microscale
models. Such models require determination of microscale parameters
such as distributions of local void ratio, number of contacts and
particle size.
The overall objective of this paper is to test whether parameters
such as particle size distribution, local void ratio and coordination
number are constant over a REV for porosity (i.e., can a REV for
porosity be considered as REV for these microscale parameters?). This
objective was achieved through the following steps: (1) use of X-ray
microtomography to obtain non-destructive 3D images of different
sand systems; (2) development of robust and efficient algorithms to
compute pertinent parameters from 3D images at the microscale
level; and (3) REV analysis of each microscale parameter to observe its
correlation to the REV for porosity.
used to reconstruct the 3D volume. The maximum height and width of
each scan is 49 mm and 28 mm, respectively. The size of each voxel of
the reconstructed volume (i.e., the resolution) is (40 × 40 × 40 µm).
2.3. Image reconstruction and pre-processing
Natural sand systems were prepared and used for imaging and REV
analysis. To incorporate heterogeneity in the analysis, four different
samples of the following particle sizes were used: 1.4 – 1.7 mm, 1.0 –
1.2 mm, 0.4 – 0.6 mm, and 0.4 – 1.7 mm. These samples were labeled
in this paper as S1, S2, S3 and S4, respectively. The sand was carefully
packed under dry conditions in Perspex tubes (24 mm internal
diameter and 10 cm height). Sand particles were poured into the tube
through a funnel while continuously tapping the tube wall to ensure
homogenous distribution of particles. To ensure repeatability of the
packings, each system was packed and imaged three different times.
Porosity values obtained from images of replicas of each system were
identical. Porosity values obtained from images matched values
obtained from laboratory measurements of porosity. Therefore, only
one representative sample of each system was presented and
analyzed in this paper.
An image (3D volume) in this paper refers to a stack of
reconstructed cross-sections (i.e., 2D slices) of the sample. Each
cross-section was reconstructed by the filtered back-projection
algorithm using the radiograph projections. In the reconstructed
images, each voxel represents the mean linear attenuation coefficient
of the corresponding resolved volume of the sample. Ring artifacts
commonly appear close to the rotation axis due to noise in each
projection. These were eliminated using a linear translation mechanism embedded in the 5-axis manipulator table of the X-ray CT
equipment. Ring artifacts caused by pixels nonuniformity in the
imaging device were eliminated using an image processing algorithm
provided by CT scanner manufacturer. Beam hardening artifacts were
insignificant in the images due to the use of a copper filter and the
relative small size of the sample. Each projection was the average of a
number of frames dependent on the X-ray flux (the higher the X-ray
flux the less the number of frames required, 32 frames typically
acquired for each radiograph in this study) which reduced errors
caused by noise therefore increasing the quality of the reconstructed
3D volume.
Segmentation was used to convert 3D images to binary images by
identifying two populations (i.e., phases) in the image based on their
intensity values. In this paper we used the algorithm and software
(i.e., 3DMA) described in Refs. [24–26] for image segmentation, the
reader should refer to these references for more detailed description
of the technique. Segmentation was performed using local thresholding values based on an indicator Kriging approach utilizing two
threshold limits. Intensity values below the lower threshold were
identified as one phase whereas intensity values larger than the
higher threshold were identified as another phase. Values between
the two thresholds were assigned to either phase using the maximum
likelihood estimate of each phase based on the two-point correlation
function. The accuracy of the segmentation algorithm was verified by
comparing porosity values obtained from the images of the systems to
the values obtained from laboratory measurements. This segmentation approach has also proven to be a very accurate and effective
method of segmentation for different types of porous media systems
(e.g., [13]).
Factors such as the existence of spots of extreme intensity values in
the solid phase, segmentation process, and resolution of the image can
generate small clusters of isolated voxels and small gaps in the solid
phase in the binary image. Although such artifacts can be minimized
by performing intensity scaling to the raw image and utilizing
optimized thresholding values during the segmentation process, a few
can still exist. In this paper, specific filters were implemented to
remove small clusters of isolated voxels and fill artificial small gaps in
the solid phase before any computations of microscale parameters.
Top and bottom slices in the raw images of each sand system were
removed since they contain noise due to the scanning process, extra
rows and columns at boundaries of the raw images were also removed
to improve the computational efficiency.
2.2. X-ray computed tomography
2.4. Microscale parameters
Sand tubes were scanned using a Benchtop 160Xi (X-tek Ltd)
cone-beam X-ray computed tomographer at 130 kV and 240 µA. A
copper filter (0.5 mm thickness) was used to obstruct X-rays below
50 kV which typically contribute to noise in the acquired images. Each
2D projection collected by the X-ray intensifier (detector) consisted of
an averaging of 64 radiographs obtained at every 0.294° projection
angle, producing a total of 1225 projections for a 360° rotation and
A brief description of the algorithms used to compute microscale
parameters from 3D images is provided in the following sub-sections.
A more detailed description can be found in Al-Raoush [27].
2. Materials and methods
2.1. Sand samples
2.4.1. Recognition of individual particles
Identification of individual particles in a given sand image is the
first step required to determine its microscale parameters. Particle
R. Al-Raoush, A. Papadopoulos / Powder Technology 200 (2010) 69–77
identification algorithm was performed based on watershed transform
according to the following steps: (1) computing the distance map of
foreground voxels (i.e., particles) in the image by calculating the
Euclidean distance, Xab
euclidean, between two voxels aia, ja, ka and bib,jb,kb as:
ab
2
2
2 1 =2 ;
Xeuclidean = ðia −ib Þ + ðja −jb Þ + ðka −kb Þ
ð1Þ
where ia, ja and ka are the row, column, and depth indices of a given
voxel (e.g., a) in the 3D image; (2) identifying local maxima in the
distance map which are markers that distinguish individual particles
(catchment basins); (3) performing a flooding simulation (relabeling) starting from local maxima and simultaneously, at a
constant rate, labeling voxels that have the same values in the
distance map; (4) identifying dividing lines between distinct particles
(watershed lines) based on the direction of labeling in the flooding
simulations. While particle identification was implemented in 3D in
this paper, a 2D illustration is shown in Fig. 1 to simplify the
illustration. Fig. 1a shows a binary image of two touched particles.
Fig. 1b shows the distance map of the particles calculated by distance
transform to obtain local maxima. Fig. 1c shows the flooding
simulation that starts from local maxima shown in Fig. 1b. Fig. 1e
illustrates the directions of labeling that define voxels in dividing lines
between particles. Fig. 2 shows particle labeling (identification) of the
sand systems used in this paper.
2.4.2. Porosity and local void ratio
Porosity (referred as interconnected porosity in this paper) was
computed as the ratio of the total void space of the image to its total
volume. Distribution of local void ratio was obtained by computing a
local void ratio for each individual particle. Local void ratio of a particle
can be defined as the ratio of the volume of the void space associated
with the particle divided by the volume of the particle. For an assembly
of particles, this in turn requires determination of a network of polygons
that are equidistant from at least two points on the boundaries of the
particles. Different algorithms can be used to achieve this step (e.g.,
[28,29]). The algorithm developed by Al-Raoush [27] was used to
compute distributions of local void ratio in this paper. The local void
ratio of each particle was computed through the following steps: (1)
identification of individual particles; (2) creation of a distance map of
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background voxels in the image that contained all particles starting from
boundaries of the particles towards their centroids; (3) creation of a
distance map of background voxels of the image that only contained the
particle under consideration; (4) the image obtained in step 3 was
subtracted from the image obtained in step 2 where voxels that have “0”
value in the resulting image make up the local void region of the particle
of interest; (5) steps 3 and 4 were repeated for all particles in the system.
Fig. 3 shows 2D and 3D images of local void boundaries of a small portion
of the S1 sand system.
2.4.3. Particle size distribution
Particle size distribution (PSD) for an assembly of particles is
expressed as a cumulative probability density function (i.e., the
number of particles larger than a certain size). The PSD can be readily
determined from 3D images upon identification of individual particles
using the methodology described in previous sections. In this paper,
the PSD for a sand system was determined by computing the
diameters of individual particles. The diameter of a particle was
computed as:
P
D = 2⋅∏
B
B 2
Ci −Vi
B 2
B 2
+ Cj −Vj
+ Ck −Vk
1
=2 ;
ð2Þ
where VBi , VBj and VBk are row, column and depth indices of the
boundary voxel VB, Ci, Cj and Ck are the coordinates of the center of the
particle and computed as:
∑i
Ci =
i
volume of particle
;
ð3Þ
;
ð4Þ
;
ð5Þ
∑j
Cj =
j
volume of particle
∑k
Ck =
k
volume of particle
f(i,j,k) is the corresponding voxel value of the particle, f(i,j,k) = 1 in a
binary image. The volume of the particle is the total number of voxels
Fig. 1. The procedure of individual particle identification; (a) binary image of two touched particles, (b) distance transform of the two particles, (c) flooding simulation process starts
from local maxima to define watershed line that separates the particles, (d) a voxel is considered part of the watershed line if the labeling in the flooding simulation enters a voxel
from the indicated directions, (e) labeled (identified) particles (separated particles) [27].
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R. Al-Raoush, A. Papadopoulos / Powder Technology 200 (2010) 69–77
Fig. 2. 3D images of sand systems used in the analysis; from left to right: S1, S2, S3, and S4 sand systems. Images are output of the particle identification algorithm where each particle
is identified (labeled with a distinct color in the images for further REV analysis). Diameters and heights of images are (23.2 mm, 31.2 mm), (24.0 mm, 31.2 mm), (23.2 mm,
31.2 mm) and (24.0 mm, 31.2 mm) for the S1, S2, S3 and S4 sand samples, respectively.
that have the same label of the particle as obtained from the particle
identification algorithm. Fig. 4 is a simplified 2D illustration of the
variables in Eq. (2).
2.4.4. Particle coordination number (number of contacts)
The average number of particles in contact with a given particle is a
critical parameter in the analysis of a granular packing; it gives
information about several parameters in the system and used to
simulate its behaviour [22,30]. Determining the contact number
geometrically is difficult due to the uncertainty of computing the
accurate position of the center of a particle. Computing the exact
number of contacts geometrically becomes impossible in the case of
irregular particles such as granular materials. In this paper, an
accurate estimate of the number of particles in contact with a given
particle was computed through the following steps: (1) each particle
was identified and labeled with a distinct value using the particle
identification algorithm described previously; (2) values of the
twenty six neighboring voxels of boundary voxels of each particle
were identified. The number of contacts of a given particle was
determined as the number of distinct voxel values found in step 2
excluding the “zero” value, where “zero” values depict the void phase;
(3) steps 2 and 3 were repeated for all particles in the system.
2.5. Representative elementary volume (REV) analysis
REV analysis for each sample was performed by computing porosity
of different volume increments starting with a cylindrical volume
centered at the center of the image. The volume was then expanded
gradually by radial increments. Porosity was computed and plotted for
each volume increment. To test whether parameters such as particle
size distribution, local void ratio and coordination number are constant
over REV for porosity, frequency distributions of these parameters were
computed for five different volumes over which porosity was constant.
3. Results and discussion
Fig. 2 shows 3D images of the different sand systems. Sizes of the
images are 580 × 580 × 780, 600 × 600 × 780, 580 × 580 × 780 and
600 × 600 × 780 voxels for the S1, S2, S3 and S4 sand systems
respectively (ordered from left to right in Fig. 2). Where diameters
and heights of images are (23.2 mm, 31.2 mm), (24.0 mm,
31.2 mm), (23.2 mm, 31.2 mm) and (24.0 mm, 31.2 mm) for the
S1, S2, S3 and S4 sand samples, respectively. Note that the resolution
is constant in all images (40 μm/voxel) and the slight difference in
the sizes of the total images (i.e., 20 μm in rows and columns) is due
to the difference in the sizes of the tubes used for imaging. Each
volume, however, is large enough to obtain REV for porosity as
shown in Fig. 5 and discussed below. Therefore, the slight difference
in the sizes of the total images has no impact on the REV analysis.
The color scheme in the images reflects particle identification. Each
particle was identified and labeled by a distinct number (shown as a
color in the images) using the particle identification algorithm
described previously. The total number of particles identified in
each system is 5,177, 11,333, 129,200 and 39,320 in the S1, S2, S3
Fig. 3. 2D and 3D illustration of boundaries of local voids of a small portion of the S1 sand system. The green color depicts sand particles and the red color depicts the network of
boundaries of local voids. Diameter of the 2D and 3D samples is 23.2 mm and depth of the 3D images is 4.0 mm.
R. Al-Raoush, A. Papadopoulos / Powder Technology 200 (2010) 69–77
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Fig. 4. Illustration of particle diameter computation using Eq. (2) [27].
and S4, respectively. Note that specific characterization of these
systems in terms of porosity, local void ratio, coordination number
is not the scope of this paper; the objective is to observe the
distribution of these parameters over volumes that have constant
porosity to determine whether a REV for porosity is sufficient to
define REV for these parameters.
Fig. 5 shows REV analysis for porosity of the sand systems. The x-axis
represents a dimensionless ratio of a sub-volume of the image to the
volume of a sphere with a diameter equal to the D50 of the system (i.e.,
Vimage/VD50). Where D50 refers to the particle size corresponding to the
percent passing of 50%, i.e., 50% of particles by weight are smaller than
D50. In all systems, porosity fluctuates initially until it reaches a constant
Fig. 5. Determination of REV for porosity of the sand systems: In all systems, porosity is constant over volumes larger than volume A. Therefore, the REV for porosity is any volume
between volume A (i.e., REVmin) and volume E (i.e., the total sample size). Distributions of particle size distribution, local void ratio and coordination number were computed over
volumes A through E to test whether these parameters are constant over a REV for porosity as shown in Figs. 6–8.
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R. Al-Raoush, A. Papadopoulos / Powder Technology 200 (2010) 69–77
value over volumes larger than volume A. This indicates that the REV for
porosity can be any sample size between volume A (i.e., REVmin) and
volume E (i.e., the total sample size). Note that REVmin is different in
these systems due to the difference in their particle size distributions.
REVmin is typically smaller in well-graded systems. As shown in Fig. 6, S1
and S2 systems are well-graded systems. Therefore, smaller volume is
required to attain the REVmin for such systems compared to S3 and S4
systems. In all systems, however, a REV is attained when Vimage/
VD50= 2 × 104. This trend is similar to experimental observations
reported by Razavi et al. [14]. Mean values of porosity for all systems
are consistent with values for similar systems obtained experimentally
from computed tomography images [13] and numerically for simulated
random packings of spherical particles [31].
Distributions of particle size, local void ratio and coordination
number were obtained over volumes A through E to test whether these
parameters are constant over a REV for porosity as shown in Figs. 6–8.
Fig. 6 shows particle size distributions computed for volumes A through
E. Such distributions allowed computing the uniformity coefficient,
Cu=D60/D10, for each system. D60 and D10 refer to the particle size
corresponding to the percent passing of 60% and 10%, respectively.
Uniformity coefficient can be used, to a large extent, as a measure of
heterogeneity of the sample, i.e., the larger the value of the uniformity
coefficient, the better graded (more poorly sorted) is the sample. The
uniformity coefficient is used to explain the trends of distributions of the
parameters in the context of the REV analysis in terms of the
heterogeneity of a sand system. Cu values are 2.6, 2.1, 1.55, and 1.65
for the S1, S2, S3 and S4, respectively. These values indicate that the S1
system is the most well-graded system whereas the S3 system is the
most poorly-graded system. This characterization can be observed in the
distributions of particle sizes obtained for different volumes. As shown
in Fig. 6, the distributions of particle size in the S1 and S2 systems are
approximately constant over a range of REVs for porosity (volumes A
through E). This behaviour is mainly attributed to the homogeneity of
the systems as indicated by the uniformity coefficient, indicating that a
poorly-graded system tend to have the same particle size distribution
over any REV for porosity. On the other hand, particle size distributions
of the S3 and S4 systems are not constant over volumes that range from
the REVmin for porosity to the total size of the sample (volumes A
through E). This behaviour is mainly attributed to the heterogeneity of
the systems as indicated by the uniformity coefficient, indicating that a
well-graded system requires larger volume than the REVmin for porosity
to attain constant particle size distributions. In the S3 and S4 systems,
volume D is considered the REVmin for particle size distribution
compared to volume A in the S1 and S2 systems.
Fig. 7 shows distributions of local void ratio computed for volumes A
through E for all systems. The arrangement of particles influences the
void space associated with each particle and hence the distribution of
local void ratio. In the S1 system, local void ratio approaches constant
distributions (i.e., REV for local void ratio) at volume C which is larger
than the REVmin for porosity. Although the same behaviour can be seen
Fig. 6. Distributions of particle size over five different volumes selected from set volumes for which porosity is constant (volumes are shown in Fig. 5).
R. Al-Raoush, A. Papadopoulos / Powder Technology 200 (2010) 69–77
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Fig. 7. Distributions of local void ratio over five different volumes selected from set volumes for which porosity is constant (volumes are shown in Fig. 5).
in the S2 system, less variation in the distributions at volumes A and B is
observed due to less variability of particle sizes in the S2 system as
indicated by the uniformity coefficient. For both systems, although
volumes A and B are considered REV for porosity, a sample of this size
does not represent a REV for local void ratio since the distributions
fluctuate when the sample size increases, (e.g., to volume C in the case
of S1 and S2 systems). This behaviour is mainly attributed to the
heterogeneity of the systems as indicated by their uniformity
coefficients, indicating that a poorly-graded system may have different
distributions of local void ratio over a REV for porosity. On the other
hand, as the system becomes more well-graded as in the S3 and S4
systems, distributions of local void ratio approach constant distributions
over volumes that are REV for porosity. A well-graded system requires a
smaller volume than the volume required for a poorly-graded system to
obtain REV for local void ratio. In both cases, the REVmin for porosity may
not be adequate to be considered a REV for local void ratio. The REVmin
for local void ratio depends on heterogeneity of the system and should
be determined for each system.
Fig. 8 shows distributions of coordination number computed for
volumes A through E for all systems. Particle size distribution and
arrangement of particles influence the number of contacts of a given
particle and hence the distribution of coordination number of a given
system. In the S1 system, which has the highest uniformity coefficient
(poorly-graded systems), distribution of coordination numbers tend
to attain constant distributions over volumes close to the REVmin for
porosity. A REV for coordination number is obtained at volume C, no
change in the distribution is observed upon further increase of the
sample volume (e.g., volume D and E). As can be seen in the S2, S3, and
S4 systems, a well-graded system requires a larger volume than the
volume required for a poorly-graded system to obtain REV for
coordination number. In both cases, the REVmin for porosity may not
be adequate to be considered a REV for coordination number. The
REVmin for coordination number depends on heterogeneity of the
system and should be determined for each system.
4. Conclusions
The combined use of advanced non-destructive imaging and
robust 3D algorithms enabled obtaining accurate quantification of
local void ratio, particle size distribution and coordination number at
the microscale level from 3D X-ray microtomography images. Such
quantifications enabled performing REV analysis of these parameters
to test whether a REV for porosity can be considered REV for these
parameters.
Findings revealed that the REVmin for porosity may not be adequate
to be considered as a REV for parameters such as particle size
distribution, local void ratio and coordination number. The REVmin for
these parameters was observed to be larger than the REVmin for porosity.
The heterogeneity of the systems was found to be an important factor to
determine the REV for the parameters analyzed in this paper. The REV
analysis revealed that as the uniformity coefficient increased, a larger
volume was required to obtain the REVmin for the distribution of particle
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R. Al-Raoush, A. Papadopoulos / Powder Technology 200 (2010) 69–77
Fig. 8. Distributions of coordination number over five different volumes selected from set volumes for which porosity is constant (volumes are shown in Fig. 5).
sizes and coordination number whereas a smaller volume was required
to obtain the REVmin for local void ratio. Therefore, determination of the
REV for parameters described in this paper or any microscale parameter
of concern should not be derived based on REV for porosity and should
be determined based on their distributions over different volumes.
Further research is needed to investigate the impact of REV misidentification on microscale-based modelling of key macroscopic properties
and the impact of image resolution on the computed parameters.
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