4.1 General
Figure 5 presents examples of typical CT slices from the samples. As observed in Fig. 5(a), skeletons of the loess sample were mainly composed of uniformly distributed coarse particles. As shown in Fig. 5(f), large aggregates comprising both coarse particles and fine clays were scarce in the loess sample. Most of the pores in the loess sample were interparticle pores (Fig. 5(d)) and overhead pores, which are large pores formed by lots of particles and are prone to collapse under loading or water (Fig. 5(e)). Compared with the loess sample, the paleosol sample had a considerably denser microstructure (Figs. 5(b) and 5(c)), which agrees with the larger bulk density and higher clay fraction of the paleosol sample (Table 1). Fewer overhead pores were present in this sample, and the interparticle pores were smaller (Fig. 5(g)). However, densely packed aggregates, which flocked together, were observed in this sample, along with large interaggregate pores (Figs. 5(c) and 5(h)).
The REV for the quantitative analysis of aggregates (30–220 µm) and interaggregate pores (15–200 µm) observed in the paleosol sample would beyond the micrometer scale. Thus these aggregates were not considered in this study. This issue can be addressed by selecting larger samples and lower resolutions. In addition, the studied samples generally exhibited a bimodal distribution, with boundary values of approximately 1–2 µm for coarse to fine components, as indicated by the particle size distributions in Fig. 1. The resolution of 1 µm was insufficient for analyzing the fine components. Hence the coarse components, which were the major constituent of samples, were the focus of this study.
Size, shape, and arrangement are the fundamental elements of soil microstructure (Assallay, 1998). The parameters of equivalent diameter, shape factor, and orientation angle, as explained in Tables 2 and 3, were used to represent the three aforementioned elements. Pore connectivity is a critical property of the pore structure, especially in the analysis of hydraulic behavior, and was represented by the pore throat equivalent diameter, coordinate number, and tortuosity (Tables 2 and 3). Thus, in this study, the quantitative characteristics of particles (equivalent diameter, shape factor, and orientation angle) and pores (porosity, equivalent diameter, orientation angle, pore throat equivalent diameter, pore coordinate number and tortuosity) were statistically investigated. The distributions of these parameters were also quantified using common probability distribution functions. The goodness of fit was estimated using the adjusted Rsquare value (Eq. (1)), which can be less than or equal to 1; a value closer to 1 indicates a better fit.
where yi represents the statistical data, ͞y the mean value, andthe fitting values. n is the number of statistical data and m the number of fitted coefficients in the function.
4.2 Representative elementary volume
Selection of the REV is the first step in quantitative analysis of microstructures. In this study, the REV was obtained based on porosity. The porosity was calculated from the CT images using the traditional cube method, as shown in Fig. 6(a). First, an initial point, at a random location in the samples, was taken as the starting point of a cube with a side of 100 µm. Then, the cube was enlarged equally in the three dimensions with fixed initial point. For each initial point, a group of cubes with sides of 100–1,000 µm in increments of 100 µm was extracted. The porosities for the groups were calculated and are presented in Figs. 6(b) and 6(c). The porosities of the smaller cubes exhibited large fluctuations, and the fluctuation decreased as the cube size increased. The porosity remained stable as the side length of the cubes increased up to 700–800 µm. Thus, the preferred REV was 800 µm × 800 µm × 800 µm.
The stationary porosity of the loess sample in Fig. 6(b) is 39–42%, while that of the paleosol samples in Fig. 6(c) is 32–35%. For the loess sample, the stationary porosity agreed with the calculated bulk porosity of 43% (Table 1). This was attributed to the homogeneous structure of the loess sample, as shown in Fig. 5(a). However, the paleosol sample exhibited a significantly lower stationary porosity than its calculated bulk porosity of 42% (Table 1). The large interaggregate pores of the paleosol sample were eliminated in the quantitative analysis of the images, as discussed in Sect. 4.1. The bulk porosity was calculated based on the bulk density, accordingly, a sample with size of a diameter of 61.8 mm and height of 20 mm was examined, wherein the presence of interaggregate pores were unavoidable. Consequently, it is reasonable that the stationary porosity of the paleosol sample was smaller than the calculated bulk porosity. Furthermore, the comparison between the stationary porosity of 32–35% and the calculated bulk porosity of 43% indicated that most of the pore volume in the paleosol sample was contributed by the interparticle and overhead pores.
To determine whether the REV obtained from the porosity was suitable for the other parameters listed in Table 2, each parameter was evaluated on several cubes. The cubes all had the same dimensions of 800 µm × 800 µm × 800 µm, and were randomly located in the samples. The discreteness of the parameters caused by different cubes was estimated in terms of the standard deviation and was expressed as an error bar. As it was difficult to define the volume of pore connectivity, frequency percentages for the pore connectivity parameters were counted by number (Figs. 7–11), while those for the other parameters were counted by volume (Fig. 12). The statistical results of eight parameters with the error bar—the particle equivalent diameter (Fig. 7), particle shape factor (Figs. 8(a) and 8(b)), particle orientation angle (Fig. 9(b)), pore equivalent diameter (Fig. 10), pore orientation angle(Fig. 11(a)), pore throat equivalent diameter(Fig. 12(a)), pore coordinate number (Fig. 12(b)) and tortuosity (Fig. 12(c))—showed that all parameters fluctuated within a narrow range. This phenomenon was consistent with the result of porosity at the stable stage (Figs. 6(b) and 6(c)). The fluctuations of these parameters may have been caused by the natural heterogeneity of the soil, which was unavoidable. However, these fluctuations had no impact on the overall distribution of the parameters for the loess and paleosol samples, as shown in Figs. 7–12. Thus, the REV obtained from the porosity was also appropriate for the analysis of other microstructural parameters listed in Table 2.
4.3 Quantitative characterization of 3D microstructure
The quantitative particle characteristics in terms of size, shape, and arrangement and the pore structure characteristics in terms of pore size, pore arrangement, and pore connectivity were discussed in the following subsections.
4.3.1 Particle characteristics
According to Fig. 7, the 3D particle size (EqD) was generally smaller than 70 µm. The mean values for the loess particles and paleosol particles were 27.2 µm and 21.7 µm, respectively. The side size of the REV was approximately more than 10 times the maximum particle size and 25–30 times the mean value. The fitting results in Fig. 7 show that particle sizes of all samples obeyed the Weibull distribution. Eq. (2) describes the probability density function. This distribution function was also demonstrated to be suitable for loess samples from seven sections across the Chinese Loess Plateau by Sun et al. (2004) and for 160 loess samples by Zhao et al. (2013).
where α and β are the function coefficients of the Weibull distribution.
Characterizing the particle morphology is a complex task. Many descriptors of particle morphology have been proposed in previous literatures (Howarth 2010; Blott and Pye 2008; Rogers and Smalley 1993), however, it is difficult to find one that describes particle morphology comprehensively. Generally, particle morphology involves two aspects, shape and surface texture (Blott and Pye 2008). This work mainly focuses on the basic shape of particles. A combination of three dimensions, L1, L2 and L3 (Table 2), was utilized to describe the 3D particle shape. The data of these dimensions were reduced to 2D data using the ratios of L2/L1 and L3/L2; the corresponding volume percentage distributions are shown in Figs. 8(a) and 8(b). The data (L2/L1, L3/L2, volume percentage) were fitted efficiently using the probability density function of a bivariate normal distribution (Eq. (3)). The volume percentage peaks in Figs. 8(a) and 8(b) for the loess and paleosol samples represent shape ratios of approximately 1.53:1.28:1 (L1:L2:L3) and 1.64:1.28:1(L1:L2:L3), respectively. These statistical values were different from the result (8:5:2) obtained by Rogers and smalley (1993), who used the random number theory. Four shape categories were defined according to the modified Zingg classification (Howarth 2010; Rogers and Smalley 1993)—disk (L1 = L2 > L3), sphere (L1 = L2 = L3), blade (L1 > L2 > L3), and rod (L1 = L2 < L3). A statistical result with an accuracy of 10% for calculation is presented in Fig. 8(c). The result shows that 56–57% of the particles were shaped like blades, 16–19% like rods, 5–6% like disks, and 17–21% like spheres.
where µ1, µ2, σ1, σ2, and ρ are the function coefficients of the bivariate normal distribution.
Particle arrangement was expressed as the distribution of the maximum Feret diameter orientations, which were displayed as the polar angle (0° ≤ φ ≤ 90°) and azimuthal angle (180° ≤ θ ≤ 180°) in a spherical coordinate system where the Z axis was perpendicular to the bedding (Fig. 9(a)). The orientation angles (φ, θ) in the upper hemisphere of the spherical coordinate system were divided evenly into 31 groups. These groups were expressed by 31 markers, as shown in Fig. 9 (a) and 9(c). Figure 9(c) showed the topdown view of Fig. 9(a) along the Z axis. The volume percentages for the 31 groups shown in Fig. 9 (b) are also expressed as the size of the markers in Fig. 9(c), i.e., a group with a bigger marker had a larger volume percentage. The preferred orientation can be clearly observed in Fig. 9(c). Groups with large markers all had φ values between 60° and 90°, but their θ values were random. The observations revealed that particles in the samples were oriented in the polar directions but not azimuthally, implying a transversely isotropic structure. This result is consistent with the fact that loess, a type of aeolian soil, has a sedimentary structure (Xu et al. 2019; Heller and Liu 1982). The anisotropy index (Ia) expressed in Eq. (4) is generally used to reflect the overall anisotropy of soil arrangement (Li et al. 2020; Tovey and Krinsley 1992). An increase in Ia from 0 to 100% represents a change in microstructure from random or isotropic to preferred aligned or completely anisotropic. The Ia of the analyzed particles was 74–76%, which indicated that they were generally preferred aligned. Figure 9(d) shows that Ia decreased as the particle size increased, which means that smaller particles are prone to being preferred oriented; specifically, particles smaller than 5 µm had an Ia of up to 96–98%.
where dmax and dmin are the largest and smallest volume percentages among those of all 31 groups of 3D orientation angles.
4.3.2 Pore structure characteristics
Figure 10 presents the 3D pore size distributions of the samples. The mean EqDs of pores for the loess and paleosol samples were 20.9 µm and 13.9 µm, respectively. The fitting results show that the pore sizes of all samples obeyed the gamma distribution (Eq. (5)):
where a and b are the function coefficients of the gamma distribution.
The analysis method for 3D pore arrangement was the same as that for particle arrangement. The volume percentages for the 31 groups of orientation angles shown in Fig. 11(a) are expressed using 31 markers with different sizes in a spherical coordinate system from a topdown view (Fig. 11(b)). The statistical results in Fig. 11(b) show that pores were generally randomly arranged. The Ia value of 47–48% also indicated that the pore direction was slightly isotropic. This implied that the anisotropy of interparticle pores was not dependent on that of particles.
The pore throat sizes of the samples also obeyed the gamma distribution, as shown in Fig. 12(a). The mean values of pore throat size for the loess and paleosol samples were 4.8 and 3.3 µm, respectively. The pores and pore throats of the loess sample were all larger than those of the paleosol sample. Figures 12(b) and 12(c) show that the distributions of pore coordinate number (CN) and tortuosity (τ) were irregular and therefore cannot be described statistically. The average CN was 5.5–5.8, indicating that most of the pores were connected to approximately five other pores or pore throats. The average τ was 1.213–1.216. The paleosol sample had a smaller CN but a larger τ than the loess sample did. In previous works on Chinese loess that used Xray CT, the macropores of samples from Jingyang, Shaanxi province had an average CN of 2.56 and an average τ of 1.12 at a resolution of 73.9 µm (Li et al. 2019), and those from Yuci, Shanxi province had an average CN of 3.15 at a resolution of 60 µm (Li et al. 2018). The aforementioned average CN and average τ values were smaller than those of the pores in the samples used in this study, the majority of which were interparticle and overhead pores at a resolution of 1 µm, with large interaggregate pores eliminated. This suggested these interparticle and overhead pores had a denser, more complex pore structure than the macropores.
4.4 Differences between 2D and 3D microstructures
Various sections of the 2D microstructure from different directions were quantitatively analyzed, as shown in Fig. 13(a). These sections were extracted after segmenting the 3D microstructure, to negate the differences induced by the image processing method. S3 was the section along the horizontal bedding direction. S1, S2, S3, and S4 had different dip angles with respect to the horizontal bedding and an angular spacing of 45° each. S1, S5, S6, and S7 were along the vertical direction and had an angular spacing of 45° each. Nonoverlapping sections, all parallel to S3 and with a size of 3,000 µm × 3,000 µm, were extracted randomly from different locations in the samples. The porosity values calculated from these sections was shown in Figs. 6(b) and 6(c). According to the results, the porosity values exhibited negligible discreteness, and thus, the size was within the REA for the 2D calculation. Since an elongated particle in 2D space can be rodlike or flaked in 3D space, the results with the 2D shape obtained from the sections cannot represent the overall 3D shape of particle. Therefore, only the 2D parameters from Table 3 were measured in the sections with a size of 3,000 µm × 3,000 µm. The statistical results were summarized in Figs .13–15.
4.4.1 Size
Figs. 13 (b)–(d) show the distributions of 2D size for the particles, pores and pore throats. The particle size distribution from all the sections could be well described using the Weibull function for the loess and paleosol samples with mean EqD2D values of 22.7–23.2 μm and 17.5–17.7 μm, respectively. The differences between the sections were small. Both the pore sizes and porethroat sizes from all sections obeyed the gamma distribution. The results of the statistical analysis are presented in Table 4. The mean sizes of the pore and pore throat of the loess sample were 17.8–18.3 μm and 5.0–5.2 μm, and those of the paleosol samples were 11.2–11.4 μm and 3.4–3.6 μm. The values of these parameters from different sections also exhibited minor differences. By comparison, it was verified that the 2D size distributions of the particles, pores and pore throats agreed with the corresponding 3D results, except that the mean values of the 2D size were smaller than those of the 3D size.
Table 4
Results of statistical analysis for size distributions of particles, pores and pore throats.
Sample type


Particle size

Pore size

Pore throat size


Weibull distribution

Gamma distribution

Gamma distribution


Mean

Rsquare

α

β

Mean

Rsquare

a

b

Mean

Rsquare

a

b

Loess sample

3D

27.19

0.9970

2.29

33.29

20.88

0.9956

5.98

3.82

4.76

0.9809

3.45

1.50

S1

23.09

0.9948

2.01

28.43

18.03

0.9927

4.64

4.32

4.99

0.9935

1.76

2.98

S2

23.04

0.9952

2.03

28.33

18.02

0.9924

4.73

4.23

5.22

0.9943

1.66

3.32

S3

23.19

0.9940

2.01

28.33

18.17

0.9934

4.70

4.26

4.98

0.9953

1.85

2.83

S4

22.84

0.9955

2.05

27.93

18.11

0.9931

4.64

4.31

5.22

0.9934

1.68

3.27

S5

23.12

0.9945

2.00

28.26

17.91

0.9926

4.66

4.25

5.16

0.9943

1.71

3.16

S6

23.02

0.9954

2.06

28.22

18.30

0.9928

4.62

4.36

5.00

0.9949

1.79

2.90

S7

22.73

0.9934

2.02

27.54

17.84

0.9925

4.71

4.18

5.24

0.9946

1.70

3.24

Paleosol sample

3D

21.66

0.9958

2.35

27.02

13.94

0.9862

5.10

2.83

3.27

0.9752

4.72

0.84

S1

17.49

0.9959

2.10

21.83

11.27

0.9808

5.36

2.17

3.42

0.9980

2.12

1.75

S2

17.64

0.9942

2.06

21.89

11.31

0.9817

5.27

2.22

3.53

0.9985

2.12

1.79

S3

17.65

0.9942

2.08

21.96

11.15

0.9803

5.34

2.18

3.44

0.9983

2.16

1.72

S4

17.62

0.9947

2.07

21.97

11.21

0.9834

5.32

2.20

3.57

0.9989

2.11

1.81

S5

17.61

0.9937

2.08

21.88

11.16

0.9840

5.35

2.17

3.53

0.9986

2.12

1.80

S6

17.56

0.9945

2.06

21.96

11.20

0.9784

5.30

2.20

3.42

0.9979

2.16

1.71

S7

17.51

0.9945

2.07

21.99

11.37

0.9816

5.23

2.25

3.63

0.9983

2.09

1.86

4.4.2 Arrangement
Fig. 14 summarizes the statistical results of 2D arrangement obtained from the seven sections. The orientations of the maximum Feret diameters for the 2D particles and pores were presented as a rose diagram in the polar coordinate system, where the orientation angle θ2D (0–180°) was divided into 10 groups, and the volume percentage for each group was represented by the polar radius (Figs. 14(a) and (b)). According to the statistical results, the particles tended to be preferred oriented, and different sections had different orientation distributions (Fig. 14(a)). The calculated Ia for particles from different sections varied from 53.1 to 79.4%, with the horizontal section (S3) having the smallest value (Fig. 14(c)). In addition, Fig. 14(b) shows that the pore orientation was generally random and that pores in all sections had consistent orientation distributions. The Ia of the pores differed only slightly among the sections, albeit much more than the corresponding 3D values did (Fig. 14(d)). These results demonstrated that sectional analysis cannot represent the 3D preferred orientation or anisotropy.
4.4.3 Pore connectivity
Fig. 15 presents the results of the average pore coordinate number and average tortuosity in both 2D and 3D space. The results showed that the paleosol sample had a smaller pore coordinate number and larger tortuosity than the loess sample did. The values were all slightly different in different sections. For all sections, the average CN2D and average τ2D were 2.4–2.8 and 1.231–1.271, respectively. The average CN2D was much smaller than the corresponding 3D value (5.5–5.9), whereas the average τ2D was larger than the 3D value (1.213–1.216). These differences between the 2D and 3D values can be explained by the pore structure characteristics. In the 3D space, the pores were connected to several other pores via long and narrow channels, as shown in Fig. 3. However, the 2D parameters were calculated considering the connected pores only in the 2D section. Consequently, the 3D tortuous and complex channels outside the section may have been omitted from the calculation. Therefore, CN2D was much smaller while τ2D much larger than the respective 3D values.
4.5 Implications for macroscopic behaviors
The instability of loess engineering and occurrence of geological hazards (such as landslides) can be easily caused by the failure of the contact zone between the loess and paleosol strata owing to their differences in geotechnical properties, particularly their hydraulic behavior (Dijkstra et al. 2015; Lei 2014). The microstructure of loess is different from that of paleosol in terms of particles and pores, as clearly depicted in Fig. 5 and revealed by the results in Sect. 4. This is the primary cause for the differences in their geotechnical properties. Hydraulic behavior is mainly determined by the pore structure of soil. The results indicated that the paleosol sample possesses a considerably denser microstructure, smaller pores, and smaller pore throats than those of the loess sample. This explains why the permeability coefficient of paleosol is smaller than that of loess, despite that their porosity values are close. In addition, the porosity of paleosol is mainly due to the interparticle and overhead pores, whereas the interaggregate pores are significantly large as discussed in Sect. 4.2. Thus the contributions of both types of pores to permeability cannot be neglected. However, the two types of pores should be studied at two different scales considering their size difference, and the results can then be integrated in the quantitative analysis of permeability and be used in further stability predictions.
Besides, the Xray CT images revealed that the paleosol sample developed aggregates (30–220 µm). The properties of these aggregates when subjected to loading and watering, as well as the contributions of the aggregates to the strength of loess and stability of loess engineering, may be different from those of the particles. However, these issues are rarely considered in the existing literatures and should be addressed in future work.