Powder Technology 200 (2010) 69–77 Contents lists available at ScienceDirect 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 70 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 71 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]. 72 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 73 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. 74 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 75 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 76 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. References [1] D. Russo, W.A. Jury, A theoretical study of the estimation of the correlation scale in spatially varied fields, 2. Nonstationary fields. Water Resources Research 23 (1987) 1269–1279. [2] R. Webster, Is soil variation random? Geoderma 97 (2000) 149–163. [3] C.S. Chang, J. Gao, Second-gradient constitutive theory for granular material with random packing structure, International Journal of Solids and Structures 32 (1995) 2279–2293. [4] H.-B. Muhlhaus, F. Oka, Dispersion and wave propagation in discrete and continuous models for granular materials, International Journal of Solids and Structures 33 (1996) 271–283. [5] R.H.J. Peerlings, N.A. Fleck, Numerical analysis of strain gradient effects in periodic media, Journal De Physique IV (11) (2001) 153–160. [6] F. Feyel, J.-L. Chaboche, FE2 multiscale approach for modelling the elastoviscoplastic behaviour of long fiber SiC/Ti composite materials, Computer Methods in Applied Mechanics and Engineering 183 (2000) 309–330. [7] V. Kouznetsova, W.A.M. Brekelmans, F.P.T. Baaijens, An approach to micro–macro modelling of heterogeneous materials, Computational Mechanics 27 (2001) 37–48. [8] V. Kouznetsova, M.G.D. Geers, W.A.M. Brekelmans, Multi-scale constitutive modelling of heterogeneous materials with a gradient-enhanced computational scheme, International Journal for Numerical Methods in Engineering 54 (2002) 1235–1260. [9] J Bear, Y. Bachmat, Introduction to Modeling of Transport Phenomena in Porous Media, Kluwer, 1990. [10] V. Clausnitzer, J.W. Hopmans, Determination of phase-volume fractions from tomographic measurements in two-phase systems, Advances in Water Resources. 22 (6) (1999) 577–584. [11] G.O. Brown, H.T. Hsieh, D.A. Lucero, Evaluation of laboratory dolomite core sample size using representative elementary volume concepts, Water Resources Research 36 (5) (2000) 1199–1208. [12] P. Baveye, H. Rogasik, O. Wendroth, I. Onasch, J.W. Crawford, Effects of sampling volume on the measurement of soil physical properties: simulation with X-ray tomography data, Measurement Science and Technology 13 (2002) 775–784. [13] R. Al-Raoush, C.S. Willson, Extraction of physically-representative pore network from unconsolidated porous media systems using synchrotron microtomography, Journal of Hydrology 23 (3) (2005) 274–299. [14] M.R.B. Razavi, B. Muhunthan, O. Al Hattamleh, Representative elementary volume analysis using X-ray computed tomography, Geotechnical Testing Journal 30 (3) (2007), doi:10.1520/GTJ100164. [15] A.J. VandenBygaart, R. Protz, The representative elementary area (REA) in studies of quantitative soil micromorphology, Geoderma 89 (1999) 333–346. [16] Z.-Y. Ren, Q.-S. Zheng, A quantitative study of minimum sizes of representative volume elements of cubic poly crystals —numerical experiments, Journal of Mechanics and Physics of Solids 50 (2002) 881–893. [17] Z. Shan, A.M. Gokhale, Representative volume element for non-uniform microstructure, Computation Materials Science 24 (2002) 361–379. [18] T. Kanti, S. Forest, I. Gallient, V. Mounoury, D. Jeulin, Determination of the size of the representative volume element for random composites: statistical and numerical approach, International Journal of Solids and Structures 40 (2003) 3647–3679. [19] S. Graham, N. Yang, Representative volumes of materials based on micro structural statistics, Scripta Materialia. 48 (2003) 269–274. R. Al-Raoush, A. Papadopoulos / Powder Technology 200 (2010) 69–77 [20] M Ostoja-Starzewski, Scale effects in plasticity of random media: status and challenges, International Journal of Plasticity 21 (6) (2005) 1119–1160. [21] M. Nicodemi, A. Coniglio, H.J. Hermann, Frustration and slow dynamics of granular packings, Physical Review E 55 (1997) 3962–3969. [22] S.F. Edwards, D.V. Grinev, Compactivity and stress transmission of stress in granular materials, Chaos 9 (3) (1999) 551–558. [23] P. Richard, P. Philippe, F. Barbe, S. Bourles, X. Thibault, D. Bideau, Analysis by X-ray microtomography of a granular packing undergoing compaction, Physical Review E 68 (2) (2003) 020301–020304. [24] W.B Lindquist, S. Lee, Medial axis analysis of void structure in three-dimensional tomographic images of porous media, Journal of Geophysical Research 101 (B4) (1996) 8297–8310. [25] W. Oh, W.B. Lindquist, Image thresholding by indicator Kriging, IEEE. Transactions on Pattern Analysis and Machine Intelligence 21 (1999) 590–601. 77 [26] W.B. Lindquist, A. Venkatarangan, Investigating 3D geometry of porous media from high resolution images, Physics and Chemistry of the Earth (A) 25 (7) (1999) 593–599. [27] R. Al-Raoush, Microstructure characterization of granular materials, Physica A 377 (2007) 545–558. [28] R.C. Gonzalez, R.E. Woods, S.L. Eddins, Digital Image Processing Using Matlab, Prentice Hall, 2004. [29] R. Al-Raoush, K.A. Alshibli, Distribution of local void ratio in porous media systems from 3D X-ray microtomography images, Physica A 361 (2006) 441–456. [30] T. Aste, M. Saadatfar, T.J. Senden, Geometrical structure of disordered sphere packings, Physical Review E 71 (2005) 061302–061316. [31] R. Al-Raoush, M. Alsaleh, Simulation of random packings of polydisperse particles, Powder Technology 176 (1) (2007) 47–55.