Change in the Microstructure and Fractal Characteristics of Intact and Compacted Loess Due to Its Collapsibility

Abstract

This work aimed to examine the fractal dimension and the difference in the law between intact and compacted loess before and after collapse. Uniaxial compression tests were performed to obtain specimens under various vertical stresses, and mercury intrusion porosimetry (MIP) tests were conducted to determine the pore size distribution (PSD). Three models were selected to determine the fractal dimensions based on PSD. As a result, the pores were classified into ultra-micropores (d < 0.1 μm), micropores (0.1 μm < d < 2 μm), small pores (2 μm < d < 10 μm), and large pores (d > 10 μm). When the fractal dimensions were determined using the capillary pressure model, there were three fractal intervals (D_s1, D_s2, and D_s3), with only D_s1 and D_s2 meeting the definition of fractal dimension. D_s1 increased considerably after the collapse, but D_s2 declined. The thermodynamic law-based model presented the best linear fit, and there was only one fractal interval. The fractal dimension D_n increased dramatically after the specimen underwent wet collapse. In conjunction with fractal theory, it revealed that collapse changed the uniformity of the pore system, making the microscopic pores coarser and more intricate after collapse.

1. Introduction

Aeolian deposits give rise to yellow silt-sized loess, which is extensively dispersed in dry and semi-arid areas, represented by the Loess Plateau in northern China, spanning the Gansu, Shaanxi, and Shanxi provinces [1,2,3]. High pressures on the Asian continent brought chilly winds from Siberia towards southeastern China throughout the early Pleistocene and even late Pliocene. As it flows over the desert in the northwest, the enhanced northern monsoon gathers the gravels up and then erodes them into silt and clay particles of varying sizes. Subsequently, these particles constantly travel from northwest to southeast, reaching different locations, with the distance covered depending on the size and quality. As illustrated in Figure 1, this process can be classified into progressive sandy loess, silty loess, and clay loess. Besides the intact loess used for foundations or slopes in urbanization, compacted loess is employed in infrastructure projects like 100-m-high filled embankments. As a result, its exceptional collapsibility and structure have piqued interest and resulted in investigation [4,5,6,7,8].

The collapsibility of unsaturated loess has been extensively studied, and the micro and macro mechanics of compacted or intact loess have been partially elucidated. Mechanical tests typically include one-dimensional (1D) consolidation tests [9] and triaxial humidification tests [10]. Correspondingly, researchers have investigated and characterized the collapsing behavior of loess under various mechanical conditions using scanning electron microscope (SEM) [11], and mercury intrusion porosimetry (MIP) [12], and other techniques. They found that both exhibit a dominant macroscopic pore size, which becomes smaller after wet collapse, and inter-aggregated pores redistribution is caused by macroscopically mechanical changes [13,14]. This effect can also be observed in remolded loess [15]. As a result, the collapse behavior of compacted loess and its differing mechanism from intact loess, has attracted increasing attention. For example, Ge et al. (2021) investigated the effects of the compaction states and water content of compacted loess and revealed that collapse resulted from construction conditions at the dry of optimum state or at the wet of optimum state [8]. The findings revealed that dry density and compacted water content substantially affect the composition of compacted specimens [16]. As a result, the collapse behavior of compacted loess is likewise a comprehensive reaction after matrix suction decrease, intergranular cement breakdown, carbonate loss, and mechanical load rise [17]. Jiang et al. (2014) [18] compared the triaxial shear properties and microstructure changes of intact loess and remolded loess to clarify the significance of structure in loess deformation. They reported that the macroscopically mechanical properties are affected by the microstructure.

Currently, few studies have quantified the effects of the microstructure on pore size distribution (PSD) before and after collapse of loess. Li et al. (2020) [19] proposed a PSD predicting methodology for intact and remolded loess after the consolidation and shearing deformation of loess. This is of great significance to further quantitatively characterize the influence of macro-mechanics on the microscopic pore reorganization of the loess. Therefore, fractals are applied to analyze the self-similarity of pores during deformation to establish a more quantitative method for the microscopic characterization of clay based on the characterization of two-dimensional (2D) images and three-dimensional (3D) pore diameter. Experiments have revealed that the fractal dimension increases when the suction and dry density of compacted bentonite rise [20]. After triaxial and uniaxial humidification tests, Luo et al. (2018) [21] developed an end-element model based on SEM images of loess and divided the intact loess pores into super-large pores (>200 μm), large pores (50~200 μm), medium pores (5~50 μm), and small pores (<5 μm). They observed that proportion of pores larger than 5 μm affects the collapse of loess using the 2D images. As a result, this research aims to elucidate the fractal characteristics of loess microstructures using the 3D pore size measurements before and after wet collapse.

In Xi’an City, Shaanxi Province, China, the intact specimens were gathered from the bottom of a foundation pit (Figure 1). Both the intact and compacted specimens were subjected to a uniaxial compression collapse test, and their PSDs were measured with MIP. This research involves analyzing and comparing the correlation between changes in soil characteristics and microstructure deformation. In addition, three models were employed to figure out the fractal dimensions of loess specimens based on the PSD data. Afterward, the mechanism for the development of fractal evolution of loess was investigated, both before and after soaking.

2. Test Design and Methods

2.1. Test Material

The loess specimens were excavated at a depth of 5~6 m in the foundation pit of a construction site in the eastern suburbs of Xi’an (Figure 1). To conserve the in situ water content, they were removed by cutting the inner wall of the foundation pit to fragments (25 cm × 25 cm × 25 cm) and sealing them with black plastic bags. The basic physical properties of the loess were examined in a laboratory, and the results are presented in

Table 1. Some physical properties of the loess specimen in this research.

Property	Value
Specific gravity	2.71
Initial moist bulk density (g/cm3)	1.37
Maximum dry density, ρdmax (g/cm3)	1.65
Initial water content (%)	14.3
optimum water content (%)	20.8
Void ratio	0.9721
Atterberg limits
wI (%)	30.2
wp (%)	20.5
Ip	9.7
Particle size distribution
Gravel content (≥2 mm)	0
Sand 0.075~2 mm (%)	0.1
Silt 0.002~0.075 mm (%)	72.3
Clay < 0.002 mm	27.6
Unified Soil Classification System	CL
Main minerals
Quartz (%)	35.8
Feldspar (%)	12.5
albite (%)	21.4
Calcite (%)	19.6
Dolomite (%)	2.3
Chlorite (%)	1.2
Illite (%)	7.2

. As determined by the MS-2000 laser particle size analyzer, the loess specimens primarily comprised 72.3% silt and 27.3% clay. The soil was determined as clay with low plasticity, and the tests revealed that its liquid and plastic limits were 30.2% and 20.5%, respectively [22]. The compaction curve of the specimens is depicted in Figure 2. The loess was compacted at the standard compaction energy, i.e., 600 kN·m/m³. The optimum water content and maximum dry density after compaction were 17.0% and 1.65 g/cm³, respectively. The XRD examination revealed that the loess was mostly composed of minerals such as quartz (35.8%), albites (21.4%), and calcites (19.6%), with minor amounts of illite and chlorite (Table 1).

2.2. Preparation of Specimens

The specimens were designed with two structural states (intact and compacted) to examine the fractal alterations of loess before and after collapse. The oedometer specimens were extracted from whole block samples using a greased steel ring (61.8 mm in diameter and 20 mm in height). Once the steel ring was fully embedded in the soil block, the remaining soil could be cut and then flattened by the upper and bottom surfaces of the ring, allowing for the preparation of an undisturbed consolidated specimen.

After cutting the undisturbed specimen, the disturbed part, which was moisture-free and went through a conventional sieve (2 mm in diameter). To achieve the optimal moisture content (20.8%, which was determined using the Proctor compaction test by ASTM D1557 [22]), the dried soil after sieving was then mixed with airless water, as illustrated in Figure 2. The soil was then compressed twice into the steel ring using the sampler to match the bulk density with the same as the in situ density (1.37 g/cm³) in accordance with the size of the oedometer specimens. To prevent the soil stratification, the surface was roughened after compaction of the first layer. Then, it was dried to match the natural water content of the loess by 10.3% in a cool ventilated area. The dried specimens were stored in a humid chamber for 72 h to ensure homogeneous moisture.

2.3. Oedometer Tests

Both the intact and compacted loess specimens were tested with the oedometer tests. Ten specimens were subjected to various maximum vertical stresses in oedometer cells (i.e., 200, 400, 600, 800, and 1600 kPa) to evaluate the pore structure and fractal dimension before and after collapse. Specifically, five specimens were loaded under constant water (initial water content) to target stresses, with each loading maintained for 24 h. Simultaneously, the remaining five specimens underwent saturation using the same loading procedure. Figure 3 presents the compressed states of the specimens, and

Table 2. Initial and final states for intact and compacted specimens.

Sample ID	Type of Specimens	Vertical Stress	Initial State		Final State		Time Point
		kPa	Void Ratio	Degree of Saturation	Void Ratio	Degree of Saturation
I0	Intact	0	0.972	0.398	0.972	0.398
I200		200	0.985	0.393	0.937	0.414	Before soaking
IS200			0.972	0.398	0.867	0.981	After soaking
I400		400	0.977	0.396	0.888	0.436	Before soaking
IS400			0.964	0.402	0.756	0.985	After soaking
IS600		600	0.975	0.397	0.821	0.473	Before soaking
IS600			0.974	0.397	0.732	0.983	After soaking
I800		800	0.972	0.398	0.787	0.497	Before soaking
IS800			0.979	0.395	0.637	0.981	After soaking
I1000		1600	0.968	0.400	0.644	0.601	Before soaking
IS1000			0.977	0.396	0.509	0.988	After soaking
C0	Compacted	0	0.972	0.398	0.972	0.398
C200		200	0.972	0.398	0.919	0.421	Before soaking
CS200			0.972	0.398	0.671	0.983	After soaking
C400		400	0.972	0.398	0.900	0.430	Before soaking
CS400			0.972	0.398	0.597	0.982	After soaking
C600		600	0.972	0.398	0.860	0.451	Before soaking
CS600			0.972	0.398	0.540	0.985	After soaking
C800		800	0.972	0.398	0.849	0.456	Before soaking
CS800			0.972	0.398	0.492	0.981	After soaking
C1000		1600	0.972	0.398	0.729	0.531	Before soaking
CS1000			0.972	0.398	0.429	0.982	After soaking

provides the details on the vertical stress and the void ratio during loading and soaking.

2.4. MIP Tests

The oedometer specimens in the compressed state shown in Figure 3 were trimmed into 8 mm × 8 mm × 8 mm cubes for MIP tests. To ensure that the material for microscopic examination was entirely dry and unaffected by drying temperature on the pore structure of specimens, all cubes were freeze-dried [23]. The dried cubes were taken and immersed in mercury. The low-pressure unit was set as 0.5~30 psi (approximately 207 kPa), and the high-pressure unit was 60,000 psi (approximately 414 MPa). Mercury gradually penetrated the pores (360 nm > d > 3.0 nm) of loess after the pressure gradually increased. In the tests, the internal pores were assumed to be cylindrical. According to the results of the MIP tests, an equation between the invasion pressure P_Hg of mercury and the entrance or throat pore diameter d can be obtained based on the law of Laplace’s capillarity [24], as follows:

$$P_{Hg}=-\frac{4{\sigma }_{Hg}\cos \theta }{d}$$

(1)

where P_Hg is the maximum pressure required to invade the pore, σ_Hg refers to the surface tension of mercury, and θ represents the contact angle between the pore walls and the mercury. In this research, σ_Hg = 0.485 N/m and θ = 130° [14,19,25,26].

3. Models for Determining the Fractal Dimension of Loess

Fractal geometry, developed in the 1970s, provides a means to represent the roughness and regularity of porous media [27]. The fractal dimension, a crucial mathematical parameter independent of the observed size, can reflect the complexity of the soil microstructure [28,29,30]. The PSD obtained by MIP tests serves as a basis to characterize the pore throat distribution pattern. Furthermore, the microscopic pore fractal characteristics of soil can also be analyzed with the fractal theory. Various typical fractal models were suggested based on the PSD, as illustrated below.

3.1. Capillary Pressure Model

Zhang et al., proposed that the logarithm of the capillary pressure possess a linear relationship with the logarithm of the corresponding aqueous phase saturation, based on which the fractal dimension of the pore structure of loess can be calculated [31]:

$$\lg S=\left(3-D_S\right)\lg P_{\min }+\left(D_S-3\right)\lg P$$

(2)

where, S is the cumulative pore volume fraction with a pore size less than r, which is the saturation of the aqueous phase at capillary pressure P in the MIP tests (%); P represents the capillary pressure (MPa) corresponding to the pore size r; P_min denotes the capillary pressure corresponding to the maximum pore size in loess, that is, the inlet capillary pressure (MPa) in the mercury pressure test; and Ds refers to the fractal dimension.

3.2. Menger Sponge Model

Friesen and Mikula proposed a new method to calculate the fractal dimension based on the equation for the Menger sponge model [32]. There were micropore throat structures of different sizes inside the loess, and their random characteristics can be simulated according to the idea of the Menger sponge. Under the Menger sponge mode, the fractal dimension can be calculated with the equation below:

$$\log \frac{d{V}_m}{dP}\propto \left(D_m-4\right)\log P$$

(3)

where, V_m is the cumulative volume intruded by mercury with a pore size larger than r; P refers to the capillary pressure (MPa) corresponding to the pore size r; and D_m represents the fractal dimension. The above equation revealed a linear relationship between log(dV_m/dP) and logP. The fractal dimension D_m = a + 4, where a is the slope of the straight line. The fitted linear relationship for log(dV_m/dP) and logP was not a well-fitting straight line, suggesting that the fractal dimension is not unique over the entire aperture, and possibly 3~4 intervals fit the straight lines.

3.3. Thermodynamic Law-Based Model

In the MIP tests, the intrusion of mercury pressure P increased continuously, leading to a corresponding increase in the surface energy of the system with the increase of mercury volume V_i inside the pore structure. The relationship between the applied pressure on mercury and the increase in surface energy can be expressed as Equation (4). Zhang and Li derived a method for calculating the fractal dimension based on the thermodynamic law-based model [33].

$${\displaystyle \sum _{i=1}^n\overline{P_i}\Delta V_i=K{r}_n^2}{\left(V_n^{1/3}/r_n\right)}^{D_{\mathrm{n}}}$$

(4)

where, n refers to the number of intrusions of mercury; P_i is the pressure applied in the ith intrusion; V_i and r_n denote the intrusion volume and pore radius corresponding to n, respectively; V_n represents the cumulative intrusion volume of nth intrusion, and k is a constant.

The equations below were assumed:

$$W_n={\displaystyle \sum _{i=1}^n{P}_i\Delta V_i}$$

(5)

$$Q_n=V_n^{1/3}/r_n$$

(6)

The equation below could be obtained by substituting Equations (5) and (6) into (4).

$$W_n=K{r}_n^2{Q}_n^{D_n}$$

(7)

The logarithm on both sides of Equation (7) can be obtained. The fractal dimension, D_n, can be obtained, as expressed in Equation (8):

$$\mathrm{In}\left(\raisebox{1ex}{$W_n$}\!\left/ \!\raisebox{-1ex}{$r_n^2$}\right.\right)=D_n\mathrm{In}{Q}_n+C$$

(8)

where, W_n is the cumulative surface energy, Q_n is the function of r_n and V_n, and C is a constant that is associated with σ and $\theta$.

4. Results and Analysis

4.1. Pore Distribution Characteristics

Loess undergoes various mechanical, hydraulic, chemical, and thermal changes, leading to microstructure evolution. The most significant forms of this evolution involve compression and redistribution of tiny pores after consolidation and hydraulic penetration [25,26,34]. The micropore structure experienced multiple phases of change in different states: the initial state, under varied vertical pressures, before soaking, and after soaking (Figure 4). Although C0 and I0 exhibited the consistent dry density, the PSD curve revealed differences in their dominant macropore sizes (i.e., inter-aggregate pores). In the intact loess, the dominant macropore size was 7.4 μm, while in the compacted loess, it was 45.13 μm. This discrepancy is likely due to the fact that compacted loess experiences stress only during the compaction, while intact loess accumulates secondary calcium carbonate and undergoes clayization over the long geological process [35]. The dominant macropore sizes of intact and compacted specimens gradually decreased under different vertical loads. Specifically, after consolidation under 1000 kPa, the dominant macropore sizes decreased to 6.63 μm for intact specimens and 30.04 μm for compacted specimens. The reduced dominant macropore sizes become more pronounced after load immersion. The greater the force during immersion, the greater the deformation of the specimen, and the more reduction in the corresponding microporosity dominant macropore sizes. The macropore size of the intact specimens was reduced to 2.9~0.82 μm after collapsing at 200~1000 kPa, while that of the compacted specimens was 11.4~0.52 μm. Notably, the dominant macropore sizes of compacted loess decreased more after collapse, which was consistent with the micropore evolution trend obtained from the collapsibility test [8].

The criteria to classify the micropores have gradually attracted attention and discussion with the continuous development and application of soil characterization technologies [36,37,38,39,40]. Based on the PSD measured by MIP tests, 0.1 μm, 2 μm, and 10 μm could serve as the classification limits, and the pores were classified as ultra-micropores, micropores, small pores, and large pores [41]. Among them, ultra-micropores were mainly intra-aggregate pores, whereas micropores, small pores, and macropores were predominantly inter-aggregate pores.

The microscopic pore classification results are depicted in Figure 5 and Figure 6. When the intact loess was only consolidated by force, there was a noticeable reduction in macropores. After the load was increased from 200 kPa to 1000 kPa, the proportion of large pores decreased from 25% to 14%. In contrast, small pores, micropores, and ultra-micropores increased by 3%, 3%, and 5%, respectively. Correspondingly, dramatic changes were observed in the proportions of different pore classifications in the specimens after immersion in water under a series of loads. The proportion of large pores sharply decreased to 12% after water immersion with a vertical load of 200 kPa, which was greater than that of specimens with only the maximum load action. For specimens with a vertical load of 1000 kPa, the proportion of large pores reduced further to 5%. In contrast, the proportion of micropores of the specimen after collapse became greater, up to 59%. The pore structure was changed more significantly due to collapsing deformation compared to mechanical force. The decreased macropores and increased micropores after collapse is strongly associated with the sort of intergranular contact between pores and intergranular contacts in agglomerated pores. The pores are mainly in contact through point-contact and face-cementation. Pores with the point contact are more prone to collapse due to bare particles and fewer bonds. When wet or loaded, the point contact is highly susceptible to failure. The face-cementation contact can still be supported during the initial loading phase because of the large number of bonds. However, pore collapse will occur when water molecules dissolve the cemented substance, and the overhanging particles may fill the bracket pores [4].

The microscopic pore classification of compacted loess specimens is demonstrated in Figure 7 and Figure 8. The proportion of large pores in compacted loess was larger when compared to those that were only subjected to a load, as illustrated in Figure 5 and Figure 7. Specifically, the proportions of large pores in the intact and compacted loess were 14~25% and 42~48%, respectively. Due to the higher proportion of large pores in compacted loess, the proportions of small pores, micropores, and ultra-micropores after loading were not as obvious and regular as those in intact loess. Similar to intact loess, the number of micropores in compacted loess increased significantly after wetting under a series of loads. At 1000 kPa, the proportion of micropores in the collapsed specimens reached 52%, and that of ultra-micropores peaked at 36% among all specimens. The pore structure in intact loess is significantly supported by the presence of clay and carbonate cementation in the intact loess [14,42]. However, if these connections are broken, the proportions of ultra-micropores and micropores of compacted loess will increase after wetting and loading.

4.2. Fractal Dimension (D_s) Determined by the Capillary Pressure Model

The logarithm of the capillary pressure in porous media reveals a linear relationship with the corresponding logarithmic saturation of the aqueous phase. Figure 9 illustrates the pore fractal characteristics of intact and compacted loess based on the MIP tests. In the lg S-lg P coordinate system, this linear relationship could be divided into three intervals (D_s1, D_s2, and D_s3), forming a multi-fractal [43], as summarized in

Table 3. Fractal dimension (D_s) of loess specimens.

Sample ID	Ds1	Ds2	Ds3
I0	2.9387	2.6848	1.7285
I200	2.9469	2.6912	1.2503
IS200	2.9670	2.6928	1.3691
I400	2.9439	2.6920	1.7784
IS400	2.9722	2.7013	1.7362
I600	2.9631	2.6907	1.5874
IS600	2.9528	2.6415	1.7083
I800	2.9573	2.6981	1.4838
IS800	2.9722	2.6326	1.4943
I1000	2.9651	2.7030	1.7780
IS1000	2.9775	2.6031	1.4504
C0	2.9613	2.7635	1.6216
C200	2.9527	2.7924	1.7644
CS200	2.9763	2.7664	1.5822
C400	2.9631	2.7882	1.0226
CS400	2.9850	2.7602	0.7546
C600	2.9612	2.7861	0.8536
CS600	2.9789	2.7553	1.6744
C800	2.9591	2.7942	1.6897
CS800	2.9858	2.6820	0.8210
C1000	2.9736	2.7855	1.5240
CS1000	2.9843	2.6274	1.2374
Range	2.93~2.99	2.60~2.80	0.75~1.78

. This phenomenon has been discussed in several studies focusing on different soils and their fractals [20,44]. In the one-dimensional (1D) compression state of loess, the boundary of the fractal interval boundary coincides with the pore classification discussed in the previous section. However, some combined intervals differ for various specimens and loads. For example, the uncollapsed specimens with micropores and small pores in their initial state up to 1000 kPa presented a fractal interval, and the specimens with large and small pores after collapse at 1000 kPa presented a fractal interval. The fractal interval of compacted loess before and after collapse also behaves differently from the intact loess. The dry specimens did not form a fractal boundary at 0~1000 kPa at 10 μm. The first boundary of the fractal interval of the compacted specimen after collapse at 1000 kPa was observed at 2 μm. It demonstrates that the consolidation, collapse, and compaction of sample preparation change the fractal properties of pores, representing different evolutionary levels. The similarity is that the boundary of the second and third fractal intervals of all specimens was observed around 0.1 μm. As illustrated in Figure 4, the intra-aggregate and inter-aggregate pores differed significantly after about 0.1 μm. Whether consolidated or collapsed, the pore distribution below 0.1 μm did not change much. As a result, loess, like other soils, developed multifractal characteristics over time. The change in fractal dimension reflects the evolution of the soil.

The microstructure of loess will alter after compression and collapse, and it is crucial to note how the fractal dimensions vary with the loading stress. The plot in Figure 10 illustrates the relationship between the fractal dimension and vertical pressure for the first two fractal intervals. In fact, the fractal dimension is not particularly related to the loading stress, potentially due to the relatively insignificant compression time compared to the geological history spanning hundreds of millions of years. In this research, the fractal properties of the pore structure changed slightly due to brief compression. Specifically, the D_s1 of the specimens after soaking slightly increased, while the D_s2, conversely, decreased significantly with the increase of compressive stress. This effect was more pronounced in compacted loess, where the distribution of micropores and small pores presented the most significant changes before and during the collapse, Consequently, the D_s2 of compacted loess decreased the most after collapse.

4.3. The Fractal Dimension (D_m) Determined by the Menger Sponge Model

The fractal dimension determined by the Menger sponge model is reasonably distributed at 2~3. The medium is smooth at D_m = 2 and it becomes completely rough at D_m = 3. Figure 11 illustrates the fractal characteristics of the pore throat of the specimens before and after soaking measured by the MIP tests under 1D compression. In the lg (dV_m/dP)-lg P coordinate system, only the initial state of intact loess was linear, with a coefficient of determination of 0.9453, falling within a reasonable interval. Although the specimen at 600 kPa was also linear, its fractal dimension was 4.1156, deviating geometrically from the reasonable definition of fractal dimension geometrically. The remaining specimens showed a multi-segment linear relationship across the entire pore throat distribution interval and were not 2~3. For example, IS1000 was segmented at 2 μm and 0.1 μm. The fractal dimensions based on the Menger sponge model of the specimens are listed in

Table 4. D_m of intact loess specimens.

Sample ID	Dm1	R2	Dm2	R2	Dm3	R2
I0	2.9641	0.9453	-	-	-	-
I200	3.8745	0.1789	4.3543	0.8162	3.6105	0.5959
IS200	3.048	0.9481	4.2958	0.7607	2.5563	0.9985
I400	3.5407	0.6963	3.4152	0.2828	4.2185	0.5820
IS400	4.2503	0.3256	3.1154	0.8835	4.3400	0.4647
I600	4.1156	0.3908	-	-	-	-
IS600	4.3366	0.4608	3.6901	0.4478	4.4249	0.4734
I800	3.9102	0.0438	4.3851	0.8131	3.5975	0.6311
IS800	3.9985	0.0061	4.0125	0.2845	4.0012	0.0009
I1000	4.1974	0.4852	3.7488	0.2766	3.9248	0.0256
IS1000	4.5509	0.9119	3.6861	0.6411	4.4377	0.7187
Range	2.96~4.56	0.17~0.95	2.60~2.80	0.28~0.82	0.75~1.78	0.0009~0.72

. All specimens except for I0 did not present a reasonable fractal dimension, which may be attributed to two aspects. (1) The intact loess after compression and wetting did not have fractal behavior [31,45]; and (2) the Menger sponge model performed poorly in characterizing the fractal features of such specimens [44,46].

The results of fractal analysis for results of compacted specimens and intact specimens differ greatly. None of the compacted specimens exhibited an ideal linear relationship in the lg (dV_m/dP)-lg P coordinate system. Therefore, the fractal dimension of the compacted specimen could not be statistically determined. It demonstrates that the Menger sponge model performs poorly in analyzing the fractal features of compacted loess.

4.4. The Fractal Dimension (D_n) Determined by the Thermodynamic Law-Based Model

Porous media like coal, rock, and clay have been used to evaluate fractal features using the thermodynamic law-based model [32,47]. According to the thermodynamic law, the results of fractal analysis of the specimens before and after collapsibility also exhibited a strong correlation. As demonstrated in Figure 12, both the undisturbed and compacted samples exhibited a linear relationship in the ln (Wn/r_n²)-ln (Q_n) coordinate system, with a correlation coefficient of above 0.998. The fractal dimensions of all specimens except for CS1000 were suitable. The evolution characteristics of PSD during compression and collapse revealed that the reasons for abnormal fractal dimension of CS1000 (out of 2~3) could be attributed into two aspects. Firstly, the compacted loess was under 1000 kPa after soaking and the dominant macropore size decreased sharply from large pores to micropores. Secondly, its microscopic pore structure was significantly impacted by loading and collapsing, leading to variations in its fractal properties. The thermodynamic law-based model indicated that there is no reasonable fractal dimension in its PSD.

The relationship between the fractal dimension D_n derived by the thermodynamic law and the loading stress of the specimens before and after soaking was plotted (Figure 13) to study its correlation with loading and collapsing. IS1000 was not included in the figure because its fractal dimension was outside the reasonable range. No significant relationship between D_n and the loading was observed. It was demonstrated that consolidation stress slightly affects the fractal dimension of loess. This is inconsistent with the result that the fractal dimension increases linearly with the consolidation pressure after triaxial consolidation [48]. This discrepancy may be attributed to the fact that 1D consolidation exerts a smaller destructive effect on the specimen than the three-axis isotropic consolidation. The dominant macropore sizes of the specimens were reduced greatly. Similarly, the fractal dimension after soaking increased significantly in general. In addition, the microscopic pores of loess became coarser and more complex after the collapse. Figure 13 revealed that the maximum growth rates of fractal dimension of the intact loess and compacted loess were 1.8% and 15.8%, respectively, after collapsibility. Such large shift may be intimately connected to PSD reintegration of them after the collapse. The dominant macropore size of the specimens drastically reduced after collapsibility. The dominant macropore size of C0 was 5.09 times greater than that of I0, and that of IS1000 and CS1000 showed the same decrease magnitude order. The dominant pore sizes of I0 and C0 were eventually reduced to 0.83 μm and 0.52 μm, respectively. This same PSD evolution law aligns with the findings of Wang et al. [26], indicating that collapse exerts a much greater effect on the reduction of the dominant macropore size of the loess specimens than loading.

5. Discussion

The most commonly discussed aspects of loess include collapsibility and structural behavior, as well as the relationship between them [49,50]. Collapsibility is mainly caused by the diverse structural peculiarities of loess in different regions [12]. Structural characteristics can be analyzed from both macrostructural and microstructural perspectives. The macrostructural properties of loess are affected by the particle contact types and microscopic pore features, whether it is in its native state or has been compacted. The results of MIP tests are often applied to porous media materials such as coal, rock, and clay. This research objectively evaluated the microscopic pore composition of loess in a 3D space based on the accumulated mercury intrusion. For instance, Prof. S.W. Sloan recently focused on the development of microscopic characterization techniques in geotechnics and confirmed that these techniques can support the fractal analysis, providing a new mathematical perspective on the pore self-similarity of this porous medium [51].

The evolution characteristics of PSD of intact loess and compacted loess were compared before and after water immersion under different load conditions. The results suggested that loading and water saturation will change the PSD of both intact and compacted loess, with the impact of moisture being particularly pronounced. This is also indicative of loess collapsibility at the microscopic level. To describe the pore changes more clearly, the microscopic pores of clay were classified into inter-aggregate pores and intra-aggregate pores in the bimodal separation line of the PSD curve, also defined as micropores and macropores [52]. However, this classification method is somewhat singular in more accurately expressing the contrasting features before and after soaking because the intra-aggregate pores change slightly after loading or collapsing. Therefore, there is a need to divide the macroscopic pore component to more precisely express the pore compression. The fractal dimensions determined by the capillary pressure model and the thermodynamic law-based model mostly fall within the range of 2~3. All specimens in the capillary pressure model exhibited multifractal features. The dividing lines of the fractal region were marked at 0.1 μm and 10 μm in I0, while those for IS1000 were plotted at 0.1 μm and 2 μm. Thus, three distinctive fractal dividing lines were selected as the microscopic pore classification limits of loess to provide a clearer pore classification criterion based on the fractal characteristics for collapsible loess.

However, only the capillary pressure model and the thermodynamic law-based model were discussed in this research, because much of the fractal dimension D_m was not within a suitable range and the linear correlation was inadequate for the Menger sponge model. The multifractal dimensions D_s1 and D_s2 determined by the capillary pressure model did not show an apparent relationship with consolidation stress. However, it was more obvious that the D_s1 and D_s2 of the intact loess were smaller than those of the compacted loess, suggesting that the pore structure of compacted loess is more complicated based on the capillary pressure model. The thermodynamic law-based model clearly obtained a larger fractal dimension of the intact loess. The conclusions of the two models contradict each other, but the similarity is that the fractal dimensions of specimens increase after collapse. That is, the specimens after collapse present a greater minute pore self-similarity. The thermodynamic law-based model is favored if it is preferable to represent the fractal properties of pores in a fractal dimension. According to the evolution law of D_n before and after loading and collapse from Figure 13, it depicts how the microfractal properties and pore recombination of loess change under loading and hydraulic disturbance. Although some useful conclusions have been obtained from the fractal analysis, how to apply the fractal parameters to practical applications in subsequent studies needs further discussion.

6. Conclusions

The microscopic pores and fractal characteristics of intact loess and compacted loess before and after soaking were compared based on the 1D oedometer tests and MIP tests. In addition, this research examined the macromechanics and microstructure of the intact and compacted loess before and after collapse from the perspectives of structure and fractal properties. The primary conclusions are as follows:

(1) The compacted loess and the intact loess showed identical dry density in initial states but greatly different dominant macropore sizes (45.13 μm for the former and 7.4 μm for the latter). Consolidation and hydraulic penetration resulted in the compression and redistribution of loess pores. The dominant macropore sizes of intact loess and compacted loess progressively shrank after the application of vertical loads, with the minimum decreasing to 6.63 μm and 30.04 μm, respectively. This phenomenon was more significant after the soaking. The greater the force applied during soaking, the greater the deformation of the specimens, and the greater shrinking of corresponding microscopic pore dominant macropore size. In this research, the minimum values of the intact and compacted loess were reduced to 0.82 μm and 0.52 μm, respectively.

(2) The soil formation method of compacted loess was different from the intact loess, which is the fundamental reason for different microstructures between them. The pores were classified into ultra-micropores (d < 0.1 μm), micropores (0.1 μm < d < 2 μm), small pores (2 μm < d < 10 μm), and large pores (d > 10 μm). Among them, ultra-micropores were intra-aggregate pores, while the other three were mainly inter-aggregate pores. The compacted loess only subjected to 1D loads processed a higher proportion of large pores than intact loess (42~48% vs. 14~25%). The micropores of all specimens increased the most after soaking under a sequence of loads.

(3) There were three fractal intervals (or fractal dimensions) in both the intact and compacted loess, which were determined by the capillary pressure model, but only D_s1 and D_s2 met the definition of fractal dimension. D_s1 and D_s2 were not specifically related to vertical stress. After collapse, the D_s1 of all specimens increased significantly, while the D_s2 decreased.

(4) The Menger sponge model determined one or three fractal intervals (or fractal dimensions) in both the intact and compacted loess specimens, but only the intact loess was linear in the initial state, and the D_m was 2.9641. The remaining fractal dimensions were not within a reasonable range, and the coefficient of determination of linear correlation was less than 0.8.

(5) The thermodynamic law-based model provided the best linear fit in a single fractal interval. Except for CS1000, all specimens showed reasonable fractal dimensions (2~3). The D_n of the remaining specimens did not appreciably rise or fall with an increase in loading but increased significantly as a whole after soaking. The maximum growth rates of fractal dimension of the intact loess and compacted loess were 1.8% and 15.8%. It shows that the microscopic pores of compacted loess after collapse are coarser and more complex.