The development of microfocused X-ray computed tomography (CT) devices enables digital imaging analysis at the pore scale. The applications of these devices are diverse in soil mechanics, geotechnical and geoenvironmental engineering, petroleum engineering, and agricultural engineering. In particular, the imaging of the pore space in porous media has contributed to numerical simulations for single-phase and multiphase flows or contaminant transport through the pore structure as three-dimensional image data. These obtained results are affected by the pore diameter; therefore, it is necessary to verify the image preprocessing for the image analysis and to validate the pore diameters obtained from the CT image data. Moreover, it is meaningful to produce the physical parameters in a representative element volume (REV) and significant to define the dimension of the REV. This paper describes the underlying method of image processing and analysis and discusses the physical properties of Toyoura sand for the verification of the image analysis based on the definition of the REV. On the basis of the obtained verification results, a pore-diameter analysis can be conducted and validated by a comparison with the experimental work and image analysis. The pore diameter is deduced from Young–Laplace's law and a water retention test for the drainage process. The results from previous study and perforated-pore diameter originally proposed in this study, called the voxel-percolation method (VPM), are compared in this paper. In addition, the limitations of the REV, the definition of the pore diameter, and the effectiveness of the VPM for an assessment of the pore diameter are discussed.
The estimation of pore dimensions and pore networks in soil is one of the most important studies to evaluate the mechanical and hydrodynamic properties for soil science (Topp and Miller, 1966), soil mechanics (Carman, 1939; Mualem, 1976; Dullien, 1992; Chanpus, 2004), geotechnical and geoenvironmental engineering, and petroleum engineering (Chatzis et al., 1983; Culligan et al., 2006; Gharbi and Blunt, 2012). Carman (1939) considered that water did not seep into straight channels but around irregularly shaped solid particles. Topp and Miller (1966) reported on the water-pressure changes, which shifted the main-branch wetting and drying curves, as well as the families of “rewet” and “redry” scanning loops in the water retention curve (WRC) of glass-bead media. Mualem (1976) proposed a simple model that predicted the unsaturated hydraulic conductivity curves by using WRCs. Chatzis et al. (1983) evaluated the effects of the particle size, the particle-size distribution, the macroscopic particle size, the macroscopic and microscopic heterogeneities, the microscopic dimensions such as the ratio of the pore body to the pore-throat size, and the pore-to-pore coordination number on the residual oil under water-wet conditions. In fact, it is difficult to define the pore dimensions in grains because the pores are surrounded by grains and are thus not isolated. The shape of a soil particle is not spherical but rather a complicated shape; thus, the pore dimensions are only defined on the basis of some assumptions. In the past decade, microfocused X-ray computed tomography (CT) scanners have been used to study the pore structure and residual fluid in porous media. Culligan et al. (2006) used an X-ray CT scanner to visualize an oil–water–glass-bead system. Gharbi and Blunt (2012) reported that the efficiency of water flooding as an oil recovery process in carbonates should depend on the wettability and connectivity of the pore space. Figure 1 shows an X-ray CT image of a grain sample in two dimensions. The X-ray CT shows the spatial distribution of density, which enables the soil particles and pores to be distinguished. Locally, the longest and shortest length of the pore, as shown in Fig. 1a, can be measured by using software for image analysis. However, it is partial property of the pore, and the required information is at least a property in representative volume. It should be required not to measure individual pore size by using software, but to estimate them by using a systematic method. Moreover, complicated pores have an aspect ratio, as shown in Fig. 1; thus, a discussion of connectivity of pores will be required for the study of the hydrodynamic issue in soils. The aim of this paper is to propose an evaluation method for the pore size.
Binary image of pores between grains (black indicates the pore space and white indicates the particles).
We now review the current techniques for pore analysis. The most popular
methods for measuring pores in soil are the indirect methods of a mercury
intrusion porosimetry (MIP) or the air intrusion method (AIM). These methods
are based on the concept that the pore structure is assumed to be a straight
tube. Through the recent developments of the scanning electron microscope
(SEM) and nondestructive testing methods such as CT and magnetic resonance
imaging (MRI), the pore structure in soil can be directly measured
(Tomotsune et al., 2015). The application of CT to soil mechanics with
respect to the localization of the soil under triaxial compression was
reported by Desrues et al. (1996). Otani et al. (2000) evaluated the crack
width of undisturbed clay using an industrial X-ray CT scanner. Likewise,
the applications of CT to geotechnical engineering have been diverse over
the past two decades. In particular, advanced CT has been developed to scan
at the microscale. For example, Altman et al. (2005) evaluated
synchrotron-source X-ray CT for the visualization of flow phenomena in rock
samples. Wildenschild et al. (2002; 2005a, b)
focused on the flow process in the pores between
sand particles using a microfocused X-ray CT scanner and successfully
visualized in granular materials. Higo et al. (2011) studied the deformation
of unsaturated sand using X-ray micro-CT scanner and discussed the pore size
and the meniscus in a pore. As for a discussion of a capillary in soil,
Andrew et al. (2014, 2015) developed a system that
conducts a CO
This paper discusses an evaluation method for pore structure of sand from microfocused CT scan data. In this paper, the authors distinguish a pore from the pore structure. In the first part of this paper, the authors propose the application of a mathematical morphology method for estimating the pores between sand particles. By showing the analysis results of simple subjects, the usefulness of proposed method will be validated. Next, the importance of the selection of a representative element volume (REV) for estimating the grain-size distribution and the averaged pore index, such as the porosity and specific surface, is discussed. The authors show there is an optimum REV in this paper. On the basis of the above fundamental examination of treating CT data, the authors propose a voxel-percolation method (VPM) to evaluate the pore structure of sand. The estimated results are compared with a WRC test, and the effectiveness of proposed method is described. It is concluded that the resolution required for evaluating the pore structure is almost equivalent to that for a pore. The final objective of this research is to develop a general method for soils. As the primary aim of this research, the evaluation of sand is treated in this study because it is natural material.
Specifications of the microfocused X-ray CT scanner.
Table 1 lists the specifications of the microfocused X-ray CT scanner
(TOSHIBA TOSCANNER 32300 FPD) installed at the X-Earth Center at Kumamoto
University in 2010. The generated X-ray beam is a polychromatic beam with a
wide range of frequencies; hence, corrections are made for the
beam-hardening effect. Radioscopic images alone are not sufficient to
accurately represent the internal components that have a complex structure.
Tomographic reconstruction allows the detection of fine flaws, foreign
matter, separation, and other phenomena. State-of-the-art technology allows
for high-precision and high-speed inspection, permitting new applications in
various fields. The detector is a flat-panel detector (FPD), which enables
three-dimensional scanning with a cone-shaped X-ray beam. During scanning,
the scan table was rotated to obtain a 360
Table 2 lists the scan conditions selected for this study. An X-ray tube
voltage of 60 kV and a current of 200
Scan conditions used in this study.
There may be some methods for processing an image of a pore in sand from X-ray CT data. In this paper, the applicability of the mathematical morphology is discussed for the definition of the pore structure. The mathematical morphology describes the complicated shape by using a spherical element whose diameter is known (Soille, 2003). The basic operations are available in many image analysis software packages (Luis et al., 2005).
The basic process for image analysis in this study consists of image segmentation to create a binary image from an original image (Sect. 3.1.1) and the determination of the pore diameter using a granulometric method based on the mathematical morphology (Sect. 3.1.2).
The most popular method to evaluate the porosity of porous materials from CT data is based on a statistical assumption. The parameters of a distribution function based on the CT data can be determined by an optimization technique (e.g., Kato et al., 2013; Mukunoki et al., 2014). The accuracy of these methods depends on the selection of distribution functions based on the CT data. The required number and type of functions are still under research, and there may be a number of solutions to these issues. As the first step of image analysis in this study, the pore and grain are identified from CT data and Fig. S1 in the Supplement shows a histogram of a CT image of Toyoura sand. The image segmentation method developed by Otsu (1979) was used because the Toyoura sand tested exhibited two distinct peaks, as shown in Fig. S1. The two peaks in the CT values indicate the two phases of the particles and pore space. During this process, the image is analyzed so that the grains are interpreted as black areas and the pores are white areas. By determining the number of voxels for each region, the averaged index associated with the pore, such as the void ratio, can be evaluated. However, by this process, the distribution of the pores according to size cannot be evaluated.
In the second step of the image analysis, for each pore identified from the
binary data, its shape is evaluated by using the mathematical morphology. On
the basis of this concept, a unit element
Figure S2 illustrates the spherical element. In this study, 13
spherical element sizes are used. The first three spherical elements for the
granulometric method are shown in Fig. S2. The maximum radius
Explanation of features of GIA.
The granulometric method can recreate the complicated pore space by overlapping spherical elements having several different diameters. Initially, the smallest spherical element, i.e., one voxel, should be applied to the analysis area, which is a pore space, and it should occupy the entire pore space. Then, the next larger spherical element with a diameter of three voxels, as shown in Fig. S2b, is applied to the same area of pore space, but the next larger spherical element cannot occupy the entire pore space, as shown in Fig. S2c. Likewise, the entire pore space is scanned by each spherical element. For a more complicated pore shape, more spherical elements having different diameters will be required. In the granulometric method, a spherical element may be partially overlapped; hence, the distribution of the pore diameter indicates that the nonoverlapping part of a spherical element is evaluated. In addition, the sum of the ratios of overlap of spherical elements at each step can be used to evaluate the pore volume and degree of saturation by the VPM, as explained in the following section.
It is noted that a spherical element is referred to as a circular element in
this section because the target subject is drawn in two dimensions. At the
end of the image analysis using the GIA, the number of voxels in each
circular element can be counted. The accuracy of the area measurement by the
GIA is verified by comparing the number of counted voxels with the
calculated areas of a circle and square. Figure 2a and b show a black
circle and a black square rotated 45
If the diameter of a spherical element is equivalent to the pore diameter,
its pore geometry must be a set of parallel lines or a rectangular shape.
The area enclosed in dotted lines, as shown in Fig. 2d and e, was
analyzed by the GIA, and both areas were found to be 19 801 voxels. The two
artificial images in Fig. 2d and e can provide an interesting
discussion. Certainly, the GIA estimates the width in the area of the
diamond; despite the fact that the definition of width is vague, GIA
produced Fig. 2d. Even if the target image is rotated by 45
X-ray CT image of Toyoura sand with different regions of image analysis in 3-D.
Verification results of GIA.
The REV of CT image should be discussed to estimate the geometrical
properties of sand by an image analysis. In this study, the porosity and
specific surface area of Toyoura sand were evaluated, and the REV was
assessed. The detailed steps can be found in Fujiki et al. (2014); hence,
the concept is only introduced. A subsampling region is defined; the porosity
(
Relative standard deviation by changing the dimension of image analysis.
The dimension of the CT image is 1024
Results of REV analysis.
Figure 5 shows the distributions of the grain size obtained from a sieving
test and an image analysis. The grain-size distribution can be obtained from
Image J for the image analysis, which is provided using the function
“object counter”. With the exception that the image analysis area is a
cube consisting of 100 voxels, the grain-size distribution obtained from the
CT image fits the results of the sieving test well. The results are
summarized in Table 4. The effect of the size of the reference sample is not
significant. This observation supports the fact that the sample with a voxel
size of 300 (or 1.5 mm) is larger than the REV when the size of one voxel is
5
Grain size distribution curve obtained from image analysis.
In this study, a water retention test with a reducing elevation head method
(WRT-REHM) was selected to conduct water drainage tests because it was
available to measure the moisture content of identical specimens at
different elevation heads during the water drainage process. The specimen
used for the WRT-REHM was identical to the scanned sample. Figure 6a and b show photographs of the setup used for the water drainage test system
with a suction method, and Fig. 6c shows a cross-sectional view of a
mold that was tested. The mold is made of acrylic, through which an X-ray
beam could be transmitted without strong beam-hardening. The dimensions of
the mold were a height of 120 mm, an inner diameter of 10 mm, and a
thickness of 1 mm. In order to measure the amount of drained water, a glass
syringe was used with a scale of 0.01 mL. A membrane filter with a mean pore
diameter of 0.2
Water retention test apparatus with the elevation head method.
The entire test system was set up in a room containing the installed
microfocused X-ray CT scanner, as shown in Fig. 6d, and the temperature
was controlled at 20
VPM analysis in grains (pore).
X-ray CT image of pore in Toyoura sand in 3-D.
In the process described in Sect. 3.2, the GIA produces a pore by overlapping many spherical elements. In this section, a pore-structure analysis based on the GIA is described. It is important to consider the three-dimensional continuity of the pore when analyzing the pore structure. In this study, a pore-structure analysis method that performs a vertical air-entry simulation with the imaged pore from the X-ray CT data is proposed. This method is called the VPM. For instance, there are 13 types of spherical elements, as shown in Fig. 7. For convenience, the images in Fig. 7 are drawn in two dimensions; therefore, sphere elements should be called circle elements in the explanation of Fig. 7. In order to start the percolation flow simulation, the water-drainage process is modeled as follows in this study. First, the original image is binarized into pore space (white) and soil particles (black; see Fig. 7a). Second, the pore space is analyzed by the GIA; thus, the distribution of the labeled spherical elements can be determined (Fig. 7b). Third, the VPM finds the labeled voxel of the circular element with the largest radius from the corner of the defined side. As in Fig. 7c, only the area of the spherical element with a radius of 13 is shown as white. Fourth,VPM finds the voxels labeled 13; then, it will keep painting those voxels until it recognizes no continuous circular element (Fig. 7c–o). Likewise, the labeled number of the largest circular element is scanned to the image obtained in the second step; thus, only the voxels corresponding to the largest circular element are counted. The third step should be repeated until the smallest circular element is used. Fifth, the results in the fourth step produce the degree of saturation by the sum of the counted voxels divided by the number of voxels in the entire pore. Lastly, the capillary pressure can be evaluated by using Young–Laplace's equation with the diameter of the circular element. In an actual analysis, the described process in the above can be processed in three dimensions.
Distribution of perforated pore size in 3-D.
The water retention property of a soil is a typical parameter influenced by
pore structure. In this section, the WRC can be reproduced by combining the
GIA and VPM. On the basis of Sect. 4.2, the water retention curve (WRC),
The diameter of the spherical element at each step contributes to the
calculation of the capillary pressure head (
In this study, a WRC test for the drainage process could be performed; thus,
it was simulated by the following treatments. In the first step, each
labeled spherical element is categorized, and the spherical element number
with the maximum diameter is recognized. Second, the number of voxels with
the spherical element number corresponding to the maximum diameter from the
direction of the air entry side is counted, and it should continue until the
discontinuous condition is reached. Third, the first step yields the
capillary pressure head using the Young–Laplace equation with the
substitution of the latest perforated diameter. The second step yields the
degree of saturation by dividing the number of voxels not counted by the
total number of pore spaces. This can be plotted on one WRC. Fourth, once
the counting process for the second step is finished, the next label of
spherical element should be checked on the basis of the same process
detailed in the first three steps. Lastly, the WRC can be created. In order
to verify the perforated pore diameter, WRCs were obtained from an
experiment at 20
CT images of percolated pore space as the drainage process.
Figure 8 shows a binary 3-D image of the pore space of Toyoura sand based on the method in Sect. 3.1. In this figure, the soil particles are invisible. Figure 9a–e show binarized X-ray CT images obtained from Fig. 8 in two dimensions after the GIA using 13 different spherical elements, and Fig. 9f shows the final analysis results by overlapping each result. White represents the pore space, and black represents the soil particles. Eventually, this image processing was conducted in three dimensions to obtain a 3-D map, as shown in Fig. 10. Visually, each color element is uniquely distributed, as shown in Fig. 10. Two neighboring elements from the 3-D map are found to have neighboring colors in the color bar. That is, the pore size has a continuous distribution. From the viewpoint of hydraulic behavior, the local velocities of the pore water are always different at each pore.
Figures 11a–f show X-ray CT images simulated from the VPM analysis in
dimensions of 300 voxel at each capillary pressure analyzed by Eq. (5).
Figure 11g shows the occupation ratio of the cumulative volume of the
spherical elements obtained from Fig. 11a–f. In fact, the
occupation ratio of the cumulative volume of the spherical element
countervails the volume ratio of air in the pore structure; therefore, the
degree of saturation can be evaluated by subtracting the cumulative volume
of the spherical elements from the entire pore volume. This is expressed as
Despite the small change in the capillary pressure between 30 and 40 cm,
Comparison of saturation degree measured and analyzed at each capillary pressure.
Water retention curves obtained from image analysis and experiment.
Mostaghimi et al. (2013) and Muljadi et al. (2015) evaluated the permeability using the image data of the pores. They also discussed the REV of their sample and analyzed the pore structure using image data. The studies that consider a pore structure, in geoenvironmental engineering and petroleum engineering have to evaluate the degree of saturation and the capillary pressure; a CT image can provide the degree of saturation without the use of any model. The capillary pressure is the driving force for transferring fluids in the unsaturated condition; however, it is difficult to define the capillary pressure in porous media on the basis of both experimental work and an image analysis if the relationship between the pore size and the capillary pressure is modeled. Blunt et al. (2002) discussed the capillary pressure in porous media using Poiseuille's law to evaluate the relative permeabilities of oil and water. Blunt et al. (2013) and Iglauer et al. (2013) analyzed the basic physical properties of pores from a CT image; in particular, Blunt et al. (2013) first estimated the relative permeability and then obtained the WRC from the relative permeability distribution. The issues of how to model the migration of oil in porous media such as rocks/soils and how to inject air for remediating contaminated soil by fuels require that the water–oil flow in the soil quantitatively understands the pore structure. Mayer and Miller (1993) visualized blobs of nonaqueous phase liquids (NAPLs) in a model test and deduced the blob diameter using the Young–Laplace equation, the number of capillaries, and the number of bonds; in short, the model was used to evaluate the pore size. The MIT, SEM, and AIM (Sato et al., 1992; Kamiya et al., 1996; Uno et al., 1998) have been used to measure the pore diameter. In general, the MIT is used to evaluate the pore size in clay. Sato et al. (1992) and Kamiya et al. (1996) attempted to develop a method to evaluate the pore size in sandy soil. Uno et al. (1998) included the moisture characteristic in the results obtained from an AIM (Kamiya et al., 1996) and proposed the moisture characteristic curve method (MCCM). The measurement principle of the AIM is similar to that of the MIT, and the obtained pore size is evaluated as the diameter of a pipe; however, the water contents were not measured. Uno et al. (1998) deduced the capillary pressure head using Darcy's equation for air permeation, and then they evaluated the pore size on the basis of a pipe model (Kamiya et al., 1996). A WRC is constructed using the degree of saturation and the capillary pressure head measured by the head method with suction or given by Young–Laplace's law with the diameter of a pipe as the representative pore diameter. The AIM gives the statistical distribution of a pore diameter model of sandy soil as a glass tube. In short, the pore diameter is defined as the diameter of the tube, as per the implicit agreement in a number of papers.
Perforated-pore size distribution for each image size.
Comparison of pore size obtained from AIM and VPM.
Water retention plots obtained from GIA and VPM.
Figure 14 shows the distribution of a perforated pore diameter for Toyoura sand. A spherical element is representative of the pore size in this study. The GIA counts the number of voxels for each size of the spherical elements; therefore, we can determine the total number of voxels, which can be multiplied by the volume of one voxel to obtain the total pore volume. Then, it is possible to calculate the percentage of the perforated spherical element for each size of the spherical element on the basis of the volume of the spherical element as the percent finer by volume in Fig. 14. Figure 15 presents a comparison between the pore diameters deduced by Uno et al. (1998) and those of the authors. In particular, the measured results between 0.065 and 0.85 mm have a better fit than those between 0.03 and 0.055 mm. This indicates that the AIM overestimated between 0.03 and 0.055 mm. The AIM by Uno et al. (1998) supposes that the pore space in sand exists in a tubular form. Since air wants to intrude a large pore space, it is thought that the evaluation of the pore size by the AIM has a high precision; however, the precision decreases for the evaluation of pore space with a small size, where air does not smoothly intrude. On the other hand, the VPM can easily evaluate the size of a complicated pore space regardless of the physical interaction and does not assume that the complicated pore is a straight tube. Hence, these results imply that the pore size obtained from AIM is not a Poiseuille distribution.
The VPM also evaluates the connectivity of the pore space. The GIA provides the number of voxels of the spherical element and the spatial distribution with the VPM. The AIM can also provide the pore diameter as the inner diameter of a pipe but not the spatial distribution of the pore diameter. This issue indicates that the VPM has the advantage of being able to estimate the WRC. In fact, the distribution in Fig. 14 can provide a pore-size distribution function (PDF) with respect to the perforated pore diameter. The PDF can also provide the degree of saturation by the sum of the spherical element. Figure 16 presents the WRC analyzed by the VPM and PDF in this study. The WRC obtained from the PDF was far from the results of the VPM in terms of the measured points. The definition of the pore size could be requested based on Fig. 16. As described in Sect. 4.1, the VPM considers percolation using cluster labeling based on the connectivity of the pore spaces. On the other hand, the PDF does not utilize percolation concepts. Figure 16 shows that a reasonable WRC can be obtained from the degree of saturation and the distribution of pore diameter using the properties of percolation. Therefore, it is significantly useful that the GIA and VPM can estimate the water retention property on the basis of the geometry of the pore structure without performing a WRC test.
In this study, a specimen of Toyoura sand was scanned using a microfocused
X-ray CT scanner, and the 3-D spatial distribution of a spherical element as
the pore size ( The size of a voxel affected the results of the image analysis. When the
size of one voxel was 5 The results of the GIA show that the perforated pore diameter was less than
the pore diameter from the AIM and less than 0.068 mm; moreover, mostly
similar pore diameters were evaluated near 0.085 mm. Hence, the AIM provided
partially different pore diameters from the results of the GIA. This issue
revealed that the pore diameter obtained from the AIM was not a Poiseuille
distribution. The AIM can estimate the pore diameter as the diameter of a pipe and the
occupation ratio; however, the spatial information is not included.
Therefore, it was difficult to assess the WRC based on the pore diameter and
its occupation ratio. In contrast, the newly proposed VPM in this study can
distinguish each spherical element by labeled number corresponding to the
radius of the spherical element. As a result, the connectivity of complex
pore spaces can be evaluated; therefore, the VPM provides a WRC close to the
measured result. It was concluded that the VPM can easily evaluate the size of a complicated
pore space regardless of the physical interaction and does not assume that
the complicated pore is a straight tube. Then, the capillary pressure head
can be deduced on the basis of the Young–Laplace equation as long as
precise image data of the pores in sand were obtained by a microfocused
X-ray CT scanner.
The second, third, and fourth conclusions are based on the first conclusion.
The appropriate dimension for the image analysis should be defined on the
basis of the particle diameter and voxel size. When using a microfocused
X-ray CT scanner, a smaller sample should be scanned if a greater resolution
is required. In future work, it will be necessary to verify the appropriate
dimensions (i.e., the REV) for several types of grains with round and
angular shapes and wide range of grain sizes. These features will further
clarify the issue of pore connectivity with respect to the aspect ratio,
which affects the results of the WRC.
This research was supported by a Grant-in-Aid for Scientific Research (C) no. 26420483. The authors would like to say thank you very much for the reviewers' careful check and precious comments to our paper. We also appreciate Laurent Oxarango, who is an associate professor at the University of Joseph Fourie, for his helpful comments. Then, we thank Hitomi Ohta, Chiaki Nagai, and Yusaku Fujiki, former bachelor's and graduate school students at Kumamoto University, and Toru Yoshinaga and Takahiro Yoshinaga, the technical staff at the X-Earth Center, for their sincere contribution to this research. Edited by: M. Halisch