Introduction

China is one of the countries most affected by landslide disasters. Over the past decade, more than 10,000 landslides have occurred in China every year, on average. In southern China, landslides usually occur in gravelly soils during or shortly after rainfall (Shen et al. 2018). Technically, gravelly soils are referred to as a soil with 50% or more of coarse fraction retained with 4.75-mm (no. 4) sieves, according to the standard D2487-11 (ASTM International 2011). These soils are well-graded mixtures composed of gravel, sand, and fines. Natural gravelly soils are formed from the weathering products of a protolith affected by erosion, transportation, deposition, and other geological processes since the Quaternary (Chang et al. 2014; Shen et al. 2018). The complex composition, formation history, and multiscale structure of gravelly soils create hydraulic characteristics that greatly differ from those of any other geomaterial. The saturated coefficient of permeability (k) is one of the most important hydraulic characteristics of soil. This parameter determines the water infiltration rate, affects pore pressure development, and, thus, plays a significant role in slope stability (Rahimi et al. 2010).

During the past two decades, numerous experiments (e.g., the constant-head test, the falling-head test, the indirect consolidation test, and the triaxial permeability test) have been conducted to measure the values of k of granular soils in the laboratory (Hatanaka et al. 2001; Côté et al. 2011; Chapuis 2012; Dong et al. 2017a). Previous studies have indicated that the value of k basically depends on the pore structural parameters [i.e., porosity (n), void ratio (e), pore size distribution, pore connectivity, pore shape parameters, and fractal dimension (Df)] (Vervoort and Cattle 2003; Nishiyama and Yokoyama 2017). It has also been reported that k is related to the characteristics of the grain size distribution, including the grain size, the coefficient of uniformity (Cu), the coefficient of curvature (Cc), coarse grain content, and fines content (Trani and Indraratna 2010; Feia et al. 2016). Generally, the presence of fine grains in a coarse-grained soil leads to a value of k that is significantly smaller than that for the same soil without fines. Furthermore, the soil permeability is affected by the geometric properties of grains (i.e., circularity, elongation, specific surface area, and Df) (Zheng and Tannant 2016; Cabalar and Akbulut 2016). In addition, k usually increases with increasing temperature due to the decrease in the viscosity of water (Ye et al. 2012; Wang et al. 2017).

Although laboratory testing is a direct and the most reliable way to determine the soil permeability, it is challenging to perform a permeability test on gravelly soils. With respect to a coarse-grained material, special sampling tools and large-scale testing apparatus are required. Moreover, the experimental results of gravelly soils are usually highly diverse and, hence, several repeated tests are necessary. These necessities inevitably increase the labor power and experimental costs. In this case, the application of an empirical equation becomes an important alternative approach to estimating the soil permeability. To date, many empirical or semi-empirical equations for granular materials have been developed based on easy-to-measure parameters, such as the void ratio (or porosity) and grain size distribution parameters (Hazen 1892; Slichter 1899; Terzaghi 1925; Taylor 1948; Kozeny 1953; Carman 1956; NAVFAC 1974; Shahabi et al. 1984; Chapuis 2004; Zhu et al. 2005; Zhou et al. 2006).

Among the previous empirical permeability formulas, the Hazen equation (Hazen 1892) is frequently used for estimating the permeability of clean sand or gravel. This equation calculates k from the square of the effective grain size. The formula proposed by Taylor (1948) is also popular since it requires only one variable, i.e., the void ratio. However, both the Hazen equation and the Taylor formula have obvious limitations because they neglect the influences of the void ratio or effective grain size and many other factors. For this reason, a coupled equation (hereafter referred to as the extended Hazen equation) based on these two formulas was proposed by Chapuis (2004, 2012). The Terzaghi equation (Terzaghi 1925) of soil permeability expressed by porosity and effective grain size is also widely used by many engineers. This equation is usually considered to be sufficient to predict the permeability of granular soils. Another commonly referenced permeability model for porous media is the Kozeny–Carman (K-C) equation (Kozeny 1953; Carman 1956). In one popular form of this model, k is expressed by the specific surface, porosity, and a constant that considers the geometry of the porous space. The reliabilities of 45 existing permeability equations were evaluated by Chapuis (2012). He noted that the extended Hazen equation, the Terzaghi equation, the K-C equation, the NAVFAC formula (NAVFAC 1974), the Shahabi et al. equation (Shahabi et al. 1984), and the Chapuis equation (Chapuis 2004) are sufficient models to predict the permeability of non-plastic soils.

However, most of the above equations were proposed primarily for narrowly graded materials (e.g., spheres, sand, and gravel). Consequently, the representative grain sizes (i.e., di, the grain size at which i% of the grains are finer) without considering the entire range of the grain size distribution are widely adopted to predict the soil permeability. In light of well-graded materials such as gravelly soil containing a fines fraction, the reliability of such a permeability estimation remains an open problem. Moreover, it is necessary to note that small voids between the fine grains are too small to permit the free flow of water (Boadu 2000). This means that, unlike sand and gravel, not all voids in gravelly soils contribute to the permeability. Therefore, these empirical equations are somewhat inappropriate for the calculation of the permeability of gravelly soils. X-ray computed tomography (CT) and optical microscopy are commonly used techniques for the characterization of soil structures (Viggiani et al. 2015; Zimbardo et al. 2016; Dong et al. 2017a). With these techniques, it may be possible to identify the pore structure at different scales, thus providing new insights into the precise modeling of the permeability of gravelly soils.

The aim of this study was to examine the structural properties of compacted gravelly soils and then quantify the relationship between the structural properties and hydraulic conductivity. To this end, large gravelly soil samples and small fines-sand mixture samples were prepared using compaction methods. These samples were then characterized using CT scans and optical microscopy. The pore structural characteristics at the gravel scale (≥ 1.0 mm) and sand scale (0.01–1.0 mm) were subsequently analyzed with image processing techniques. Constant-head tests were performed on the large gravelly soil samples after the CT scans to measure k. Moreover, the reliabilities of ten existing permeability equations were evaluated by comparing the estimated values with the measured data. These equations were then improved to make them applicable to gravelly soils. In addition, a permeability equation taking the structural characteristics of both gravel-scale pores and sand-scale pores into account was developed based on the Kozeny equation. Meanwhile, novel empirical permeability equations were proposed for gravelly soils based on the concepts of the effective porosity and the effective grain size.

Description of the material

Study area

Southern China, including the south-central and south-western parts, is mainly hilly and mountainous, with the exception of some plain areas in the middle and lower reaches of the rivers. Southern China is a subtropical monsoon climate, with abundant rainfall and distinctive seasons. Gravelly soils occur extensively in this region, where shallow slope instability frequently occurs due to the excavation during highway and railway constructions.

The Shuizhuwan Tunnel (GPS coordinates: 27°58′N, 113°01′E) is a key part of the intercity railway between Changsha City and Xiangtan City of Hunan Province, in south-central China (Fig. 1). The tunnel is located in a hilly area with an elevation of 50–180 m. The Cretaceous–Paleogene strata, composed of purple-red argillaceous siltstone, purple-red conglomerate, and gravelly sandstone, are widely distributed in this area.

Fig. 1
figure 1

Geographical location of the soil collection site

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The soil selected for study is an eluvial-colluvial gravelly soil collected from a slope near the entrance of the Shuizhuwan Tunnel. The mean gradient of the slope is approximately 24°. The natural surface of the slope is covered by low shrubs. Three soil collection positions spaced at equal intervals along the slope surface below the rooting zone were selected. These positions were respectively marked “Position #1”, “Position #2”, and “Position #3” from the top to the bottom of the slope.

Material properties

The geotechnical tests showed that the physical properties and the grain size distributions of the raw material spatially vary along the slope surface (see Table 1 and Fig. 2). The maximum grain size of the material is approximately 80 mm at every position. On average, the raw material is composed of 77% gravel, 21% sand, and nearly 2% fines, and may be characterized as sandy gravel (GW). Note that, here, gravel is referred to as grains greater than 2.0 mm; sand is defined as grains ranging between 0.075 mm and 2.0 mm; and fines are grains less than 0.075 mm, according to the Chinese national standard GB/T 50145 (Ministry of Housing and Urban-Rural Construction of the People’s Republic of China 2007).

Table 1 Physical properties of the raw material
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Fig. 2
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Grain size distribution of the raw material

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Experiments and analysis methods

Experimental program

CT scans and permeability tests on large samples

In this section, the gravel-scale pore structures (whose sizes are of the same order of magnitude as that of gravel, i.e., ≥ 1.0 mm) of the gravelly soils were identified using X-ray CT scans and the values of k were measured via the constant-head tests. Fourteen tests (Table 2) that consider different gravel contents, dry densities, and soil sources (i.e., Positions #1–#3) were proposed. Note that the soil collected from a certain position has a given gravel content. To obtain the various levels of gravel content, we adjusted the grain size distribution of the gravelly soils using the method shown in Fig. 3.

Table 2 Reconstituted samples for the X-ray computed tomography (CT) scans and permeability tests
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Fig. 3
figure 3

Adjustment of the grain size distribution according to the gravel content

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The soil samples were prepared in a cylindrical permeability cell with 200 mm internal diameter and 400 mm height. The maximum grain size of gravelly soils was reduced from 80 mm to 40 mm to facilitate the permeability cell. The oversized grains (> 40 mm) were replaced with grains that were 2.0–40 mm in size using the equivalent weight replacement method (Lin et al. 2004). A compaction scheme was followed during sample preparation: (i) the de-aired water was added to the reconstituted material (with the desired gravel content) and then the sample was mixed thoroughly to bring the water content to the natural value; and (ii) the wet material was poured into the permeability cell and then compacted in several layers to meet the desired dry density.

The prepared samples were sealed inside the permeability cell and cured at room temperature for 24 h. Thereafter, the soil samples together with the permeability cell were scanned by a medical CT system (SOMATOM Sensation 16, Siemens, Forchheim, Germany) with a 10-mm slice thickness and a 20-mm slice interval operated at 100–130 kV and 200–300 mA (Viggiani et al. 2015; Dong et al. 2017a). The reconstruction matrix consisted of 512 × 512 pixels, where the size of each pixel was approximately 0.5 mm.

After the completion of the X-ray CT scans, the permeability cell that contained the soils was assembled with a circulating water supply device and a measurement system to form an integrated permeability testing system. The soil samples were then fully saturated by applying a de-aired water flow from the lower base to the upper side of the sample. The value of k of each sample was thereafter measured using the constant-head method.

Optical microscopy observation of small samples

The soil structure identified using X-ray CT is mainly at the gravel scale due to the use of large samples. To quantify the soil pore structure at a smaller scale, more precisely the sand-scale pores (whose sizes are of the same order of magnitude as that of sand, i.e., between 0.01 and 1.0 mm), small samples of the gravel-removed soil (i.e., fines-sand mixture) were prepared and then observed using an optical microscope.

In general, sand-scale pores are mainly present between two sandy grains. It is noted that the ratios of the fines fraction to the sand fraction remained nearly constant (F/S = 7.3–7.5%) for soils with different gravel contents and soil sources (Tables 1 and 2). Therefore, it is reasonable to assume that the sand-scale pore structure depends only on the dry density, which is among the three factors considered previously. Moreover, pore characteristics such as n and e can be readily calculated from the dry density using theoretical equations. For this reason, all of the small samples were prepared under identical dry density (i.e., 1.75 g/cm3). In addition, to explore the influence of water content on the sand-scale pore structures, five levels of initial water content (w) (i.e., 5%, 7.5%, 10%, 12.5%, and 15%) were used during sample preparation. A description of the small samples for optical microscopy observation is summarized in Table 3.

Table 3 Small samples for optical microscopy observations
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To perform the experiments, gravelly soils collected from Position #3 were oven-dried and sieved. The sand and fines fractions (smaller than 2.0 mm) were then used for sample preparation. The de-aired water was added to the reconstituted material and mixed thoroughly to bring the initial water content to the desired value. Subsequently, the material was poured into a split mold of 39.1 mm in internal diameter and 40 mm in height, and then compacted in three layers to reach the desired dry density. For each level of water content, five parallel samples were proposed. Afterwards, a high-power optical microscope (Olympus BX51M, Tokyo, Japan) was employed to characterize the soil structure. More than 20 images with a magnification of 200 were randomly taken from small zones on the top (or bottom) surface of each cylindrical sample. Each image consisted of 1280 × 1024 pixels, where the size of each pixel was approximately 1.0 μm.

Note that the small samples with a high water content (e.g., the saturated water content, wsat) were also prepared. However, the pore structural observation of these samples by optical microscopy was unsuccessful because of a film of water on the sample surface. Therefore, the tests and corresponding results were not presented.

Image processing methods

Image analyses on the CT slices

ImageJ software (Schindelin et al. 2015) was adopted to analyze the pore structure of the soil from the CT slices. The image processing procedures are illustrated in Fig. 4. The brightness and contrast of the CT images were optimized using the window/level technique, which involves two parameters, i.e., window width (W) and window level (L). The values of W and L were approximately determined according to the following two cases: (i) when examining the morphology of soil solids, we considered the value of W to be 200 and the value of L to be in the ranges of 900–1100, 1100–1300, and 1300–1500 for samples with dry densities of 1.65, 1.75, and 1.85 g/cm3, respectively; and (ii) when identifying the morphology of voids, we regarded W to be 0 and the value of L to be in the ranges of 400–600, 600–800, and 800–1000 for samples with dry densities of 1.65, 1.75, and 1.85 g/cm3, respectively. The abovementioned values of W and L were tentatively used to process the images.

Fig. 4
figure 4

Image analysis method on the computed tomography (CT) slices: a raw image; bW/L adjustment; c threshold; d watershed; and e circles fitting

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The optimal L for every CT image was determined based on the concept of “minimum cumulative difference”, which was proposed by Fan and Liang (2012) for the determination of threshold values. According to this concept, there is an L value that minimizes the distance between two curves of the cumulative area versus the radius of pores (or particles) derived from two adjacent L values. The distance between these two curves is quantified by the cumulative difference. When the minimum cumulative difference is achieved, the influence of L values on the image analysis results is the smallest. Therefore, the processed image can be considered to reflect the actual morphology of soil, and the optimal L that corresponds to the minimum cumulative difference is obtained. Afterwards, the optimal L value is used to reprocess the CT images. The remaining image processing steps, including “threshold”, “watershed”, and “circles fitting”, are the same as that for analyzing the optical micrographs and are explained in the following section.

Image analyses on the optical micrographs

The image processing technique was adopted to analyze the optical micrographs (Liu et al. 2011). The main procedures were as follows: (i) the measurement scale was set, and the brightness and contrast of the raw images (Fig. 5a) were optimized; (ii) the images were converted to 8-bit grayscales (Fig. 5b); (iii) the grayscale images were processed using appropriate filters; (iv) the grayscale images were converted to binary images (Fig. 5c) by threshold; thus, the soil grains and voids were represented in white and black, respectively; (v) the connected voids were separated using the watershed algorithm method (Fig. 5d) and the void edges were recognized (Fig. 5e); and (vi) the segmented voids were fitted with circles (Fig. 5f) and the geometric parameters were measured.

Fig. 5
figure 5

Image analysis method on the optical micrographs: a raw image; b grayscale image; c threshold; d watershed; e find edges; and f circles fitting

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During the image processing, the threshold procedure is critical since it controls the analysis accuracy. In this study, the threshold operation was performed several times independently for each image and, every time, the threshold was determined according to the maximum similarity between the raw image and the binarized image. The average threshold of each image was then used for the final analyses. The separation of connected voids is another important step in order to obtain reasonable results. Because the watershed algorithm method sometimes over-segmented the images, a manual merging operation was performed by comparing the segmented images to the original images. Note that pores with a size less than 0.01 mm were ignored due to limited pixels. Therefore, the pores observed by optical microscopy were mainly sand-scale pores (from 0.01 mm up to 1.0 mm).

Characterization of the soil structures

  1. (1)

    Gravel area ratio

The gravel area ratio is a measure of the gravel content in the 2D cases and is defined as the ratio of the total gravel area to the total soil area:

$$ {\chi}_{\mathrm{G}}=\frac{S_{\mathrm{G}}}{S_{\mathrm{T}}}\times 100\% $$
(1)

where χG is the gravel area ratio, SG is the total area of the gravels, and ST is the total cross-sectional area of a gravelly soil sample.

  1. (2)

    Pore equivalent size

The shape of a pore in gravelly soils is complicated; thus, a parameter referred to as the equivalent size is proposed to characterize the pore size. The equivalent size is defined as the diameter of a circle that has the same area as a pore:

$$ {d}_{\mathrm{e}}=\sqrt{\frac{4A}{\pi }} $$
(2)

where de is the pore equivalent size and A is the actual area of a pore.

  1. (3)

    2D porosity

The 2D porosity of gravelly soil samples at the gravel scale (de ≥ 1.0 mm) is calculated by:

$$ {n}_{2\mathrm{G}}=\frac{S_{\mathrm{VG}}}{S_{\mathrm{T}}}\times 100\% $$
(3)

where n2G is the 2D gravel-scale porosity and SVG is the total area of the pores at the gravel scale.

The 2D porosity of the fines-sand mixture samples at the sand scale (1.0 mm > de ≥ 0.01 mm) is expressed as follows:

$$ {n}_{2\mathrm{s}}=\frac{S_{\mathrm{vs}}}{S_{\mathrm{t}}}\times 100\% $$
(4)

where n2s is the 2D sand-scale porosity, Svs is the total area of the pores at the sand scale, and St is the total imaging area of the fines-sand mixture sample.

  1. (4)

    Porosity variation

The variation in the gravel-scale porosity along the sample height is characterized by the coefficient of variation (CV) of the porosities measured from the various CT slices. Alternately, the variation in the sand-scale porosity along the sample height under compaction is characterized by the coefficient of porosity variation, which is expressed by:

$$ E=\frac{2\times \left({n}_{2\mathrm{st}}-{n}_{2\mathrm{sb}}\right)}{n_{2\mathrm{st}}+{n}_{2\mathrm{sb}}}\times 100\% $$
(5)

where E is the coefficient of porosity variation, and n2st and n2sb are the 2D sand-scale porosities on the top surface and the bottom surface of a sample, respectively.

  1. (5)

    Fractal dimension

Gravelly soil is a typical porous medium that has a wide pore size distribution with irregular pore shapes and self-similarity. Therefore, the pore structure of gravelly soils can be characterized using fractal dimensions. In the 2D cases, the fractal dimension has the following relationship with the area and perimeter of the structure (Liu et al. 2011):

$$ \log P=\frac{D_{\mathrm{f}}}{2}\log A+\varDelta $$
(6)

where P is the perimeter of a pore, Δ is a constant, and Df is the fractal dimension, which varies from 1.0 to 2.0. Specifically, the greater the value of Df, the more complicated the pore structure.

Results of gravel-scale and sand-scale soil structures

In this section, the gravel-scale structures of compacted gravelly soils with identical dry densities (ρd = 1.85 g/cm3) and gradations (#3) but different gravel contents (G = 30–70%) (i.e., CT tests 3 and 10–13) are specifically analyzed. In addition, the sand-scale pore structures of the fines-sand mixture at various water contents and constant dry densities (i.e., optical microscopy tests 1–5) are investigated. It should be noted that all the structural characteristics mentioned below are statistical data obtained by analyzing numerous images.

Gravel number and gravel area ratio

Figure 6 depicts some representative CT images of the gravelly soil samples with different gravel contents. As expected, the gravel number and total area of the gravel in a sample cross-section increase significantly as the gravel content increases from 30% to 70%.

Fig. 6
figure 6

Several representative CT images of sample cross-sections

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The average value and coefficient of variation of the gravel numbers (NG, the number of gravel particles in a sample cross-section) obtained from the various CT slices (corresponding to different sample heights) of every sample are presented in Fig. 7a. The results indicate that, unlike the average gravel number, the coefficient of variation slightly decreases with increasing gravel content. In general, the average gravel number increases by 26 and the coefficient of variation decreases by 0.7% when the gravel content increases by 10%.

Fig. 7
figure 7

Two-dimensional gravel properties: a gravel number and b gravel area ratio

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Figure 7b shows the average value and coefficient of variation of the gravel area ratio of each gravelly soil sample. It is observed that the average gravel area ratio linearly increases with increasing gravel content. In contrast, the coefficient of variation shows a tendency to decrease as the gravel content increases. The average value of the gravel area ratio increases by 6.1% and the coefficient of variation decreases by 4.5% as the gravel content increases by 10%. The linear relationship between the gravel area ratio and the gravel content of gravelly soils with a dry density of 1.85 g/cm3 originated from Position #3 can be expressed by the following equation (R2 = 0.964):

$$ {\chi}_{\mathrm{G}}=0.608\cdot G $$
(7)

Porosity and pore size distribution

Figure 8a presents the average value and coefficient of variation of the gravel-scale porosities of different samples. Note that the average gravel-scale porosity has a relatively small value and increases from 0.7% to 4.7% as the gravel content increases from 30% to 70%. This effect is likely because the gravels are dispersed in the fines-sand mixture, and, thus, less inter-gravel voids are present at a low gravel content, whereas at a high gravel content, ample gravel is directly in contact with each other, forming high inter-gravel voids, as seen in Fig. 6. In contrast, the coefficient of variation of the gravel-scale porosities linearly decreases with increasing gravel content. The relationship between the gravel-scale porosity and gravel content of gravelly soils with a dry density of 1.85 g/cm3 originated from Position #3 can be expressed by a quadratic function (R2 = 0.893):

$$ {n}_{2\mathrm{G}}=0.138\cdot {G}^2-0.031\cdot G $$
(8)
Fig. 8
figure 8

Two-dimensional porosity: a gravel-scale pores and b sand-scale pores

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The porosities and coefficients of porosity variation of the compacted fines-sand mixture samples are presented in Fig. 8b. The results demonstrate that the sand-scale porosity is between 25% and 29%. The n2s value slightly increases as the water content increases from 5% to 10%, and then decreases as the water content further increases. The relationship between the sand-scale porosity and water content of the compacted fines-sand mixture with a dry density of 1.75 g/cm3 can be expressed by (R2 = 0.931):

$$ {n}_{2\mathrm{s}}=-11.190\cdot {w}^2+2.331\cdot w+0.177 $$
(9)

According to Eq. (9), the sand-scale porosity of the saturated fines-sand mixtures (wsat = 20.4%) is approximately 18.8%. The variation in the porosities on the top surface and the bottom surface of a sample is evaluated by the coefficient of porosity variation, which slightly decreases from 5.3% to 4.6% with increasing water content. The value of the coefficient of porosity variation is positive, indicating that the porosity on the top is larger than that on the bottom. This variation is reasonable because the soil near the sample bottom received larger compaction energy during sample preparation.

Figure 9a, b illustrates the size distributions of the gravel-scale pores (de ≥ 1.0 mm) in compacted gravelly soils and the sand-scale pores (1.0 mm > de ≥ 0.01 mm) in compacted fines-sand mixtures, respectively. Note that both the gravel-scale pores and the sand-scale pores exhibit clear unimodal size distributions. A large number of gravel-scale pores have sizes between 2.0 mm and 20.0 mm, while the equivalent sizes of the sand-scale pores are mainly distributed from 0.03 mm to 0.4 mm. It is also observed that the dominant size (deG) of the gravel-scale pores increases from 3.16 mm to 6.31 mm as the gravel content rises from 30% to 70%. The reason for the increase in the gravel-scale porosity may also account for this phenomenon. By contrast, the dominant size (des) of the sand-scale pores remains at 0.06 mm at various water contents. This is reasonable because all of the fines-sand mixture samples have identical gradations and dry densities (see Table 3).

Fig. 9
figure 9

Two-dimensional pore size distribution: a gravel-scale pores and b sand-scale pores

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Fractal analysis of the pore structures

Figure 10a presents the relationship between the fractal dimension (DfG) of the gravel-scale pore structure of the compacted gravelly soils and gravel content. Note that the value of the fractal dimension of the gravel-scale pores visibly decreases with increasing gravel content, which is consistent with the findings of Xu et al. (2013). The result suggests that the gravel-scale pore structure of a sample with a lower gravel content is more complicated than that of a sample with a higher gravel content.

Fig. 10
figure 10

Two-dimensional fractal dimension: a gravel-scale pores and b sand-scale pores

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Figure 10b shows the fractal dimension (Dfs) of the sand-scale pore structure of the compacted fines-sand mixtures. The result demonstrates that the fractal dimension slightly decreases from 1.675 to 1.625 as the water content increases from 5% to 15%. This decrease in fractal dimension means that the geometry of the sand-scale pores becomes less complicated with a higher water content. Comparing the fractal dimension of the sand-scale pores with that of the gravel-scale pores, it is evident that the fractal dimension of the sand-scale pores is significantly greater than that of the gravel-scale pores. This result suggests that the gravel-scale pore structure is simpler than the sand-scale pore structure and, thus, is more conducive to the free flow of water. On the other hand, one can infer that the fines-scale pore structure would be even more complicated and, thus, less permeable than the sand-scale pore structure.

Hydraulic conductivity expressed by structural parameters

In this section, we aim to assess the applicability of ten existing permeability equations to the studied well-graded gravelly soil and improve these equations if the calculation results are not satisfactory. Afterwards, we attempt to link the k values to the structural characteristics of gravelly soils through empirical equations.

Permeability estimation using existing equations

  1. (1)

    Estimation of k of gravelly soils

Table 4 lists ten commonly used previous empirical or semi-empirical permeability equations for granular materials. These equations estimate k based on the pore parameters (e.g., n and e) and the grain size distribution parameters (e.g., di, Cu, and Cc). The permeability (kc) of the well-graded gravelly soils is calculated using the parameters shown in Table 2 and then compared with the measured data (km), as illustrated in Fig. 11. Note that none of these equations offers satisfactory results. The Kozeny equation may be the best among these formulas. However, this equation overestimates the permeability when k < 5 × 10− 4 cm/s and underestimates the permeability when k > 5 × 10− 4 cm/s. The Zhu et al. equation is also a relatively good model to estimate the permeability. Nevertheless, it has similar limitations as the Kozeny equation. This model overestimates the permeability when k < 2 × 10− 4 cm/s and underestimates the permeability when k > 2 × 10− 4 cm/s. All of the remaining empirical models overestimate the permeability, and the deviation increases with a decrease in the permeability.

Table 4 Summary of the existing permeability equations for granular materials
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Fig. 11
figure 11

Permeability estimation of well-graded gravelly soils using existing equations

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Such discrepancies can be attributed to several factors. One important factor is that these empirical equations were developed based on more or less uniform-sized materials (e.g., spheres, sands, and gravels). Thus, these equations are not able to appropriately consider the effect of the entire grain size distribution. Actually, among these models, only the Shahabi et al. equation and the Zhu et al. equation consider the influence of the shape of the grain size distribution by introducing Cu and/or Cc. The material used in this study is well graded (Cu = 12.36–56.6) and totally different from the fines, sand, or gravel alone. The presence of the fines fraction greatly decreases the permeability by blocking the sand-scale pores and the gravel-scale pores. Another factor that should be considered is that not all the pores contribute to the permeability. The small pores in fines are usually too small to permit the free flow of water. Therefore, to better estimate the values of k of well-graded gravelly soils, the previous equations must be improved or new models must be developed.

  1. (2)

    Modification of the existing equations

In Fig. 11, the permeability calculated using the existing equations has an approximately linear relationship with the measured data on a log-log scale. Therefore, the following equation is proposed:

$$ \log {k}_{\mathrm{m}}=M\cdot \log {k}_{\mathrm{c}}+N $$
(10)

where M and N are two fitting constants.

Replacing km with k and replacing kc with the right side of the equations listed in Table 4, we produce the modified form of the previous permeability equations:

$$ k={10}^N\cdot {\left(k\left({d}_i,e,n,{C}_{\mathrm{u}},C{}_{\mathrm{c}}\right)\right)}^M $$
(11)

where k (di, e, n, Cu, Cc) is the existing permeability equation that is a function of the soil physical characteristics.

Using the modified form [Eq. (11)] of the existing equations, we recalculated the values of k of the well-graded gravelly soils. The results are plotted in Fig. 12. These data indicate that, after modification, most of the previous permeability equations can provide an acceptably reliable prediction of the k values of the studied material.

Fig. 12
figure 12

Permeability estimation of well-graded gravelly soils using the modified equations

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Permeability model based on the pore structural parameters

The above analyses have shown that the Kozeny equation may be the best among the existing permeability models for well-graded gravelly soils before modification (Fig. 11). Thus, we attempt to develop a permeability formula that takes the structural characteristics of both gravel-scale pores and sand-scale pores into account based on the Kozeny equation. The permeability of gravelly soils is assumed to be composed of two parts: one is related to gravel-scale pores and the other is related to sand-scale pores. Therefore, the total coefficients of permeability of gravelly soils may be calculated by the following equation:

$$ k=X\left[Y\cdot \frac{n_{2\mathrm{G}}^3}{{\left(1-{n}_{2\mathrm{G}}\right)}^2}{d}_{\mathrm{eG}}^2+\left(1-Y\right)\frac{n_{2\mathrm{s}}^3}{{\left(1-{n}_{2\mathrm{s}}\right)}^2}{d}_{\mathrm{es}}^2\right] $$
(12)

where k is expressed in cm/s, X = 1.6 and Y = 0.1 are fitting constants for the studied material, and deG and des are the dominant equivalent pore sizes, expressed in mm.

The comparison between the calculated results by Eq. (12) and the measured data is presented in Table 5 and Fig. 13. One can note that the calculated results by Eq. (12) are acceptably in agreement with the measured data, indicating that the pore structural parameters are well correlated to the k values through this equation. However, Eq. (12) is inconvenient for practical use because of the difficulties and large costs in determining the input parameters (i.e., n2G, n2s, deG, and des). Among these parameters, deG and des are measured by laboratory experiments such as CT scans and optical microscopy observations, while n2G and n2s can either be measured by laboratory experiments or calculated from easy-to-measure parameters (i.e., G and w) with Eqs. (8) and (9). Note that the calculated values of n2G and n2s should be further corrected considering the influences of the dry density and gradation because Eqs. (8) and (9) are proposed based on the given conditions. Correcting the influence of gradation on 2D porosities is difficult to achieve based on the limited results of this study, while the influence of dry density may be corrected by the following equation:

$$ {n}_{2\mathrm{s}}=\frac{G_{\mathrm{s}}{\rho}_{\mathrm{w}}-{\rho}_{\mathrm{d}}}{G_{\mathrm{s}}{\rho}_{\mathrm{w}}-{\rho}_{\mathrm{d},\mathrm{re}}}{n}_{2\mathrm{s},\mathrm{re}} $$
(13)

where ρd,re is the reference dry density and n2s,re is the reference n2s of the fines-sand mixtures with a dry density of ρd,re.

Table 5 Soil structural characteristics for the permeability calculation
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Fig. 13
figure 13

Comparison between the calculated k via Eq. (12) and the measured data

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Permeability model based on the effective parameters

Considering the limitations of the existing equations, we herein attempt to develop new permeability models based on the concepts of the 2D effective porosity and the effective grain size.

  1. (1)

    Two-dimensional effective porosity

The pores in gravelly soils are divided into the fines-scale pores (de < 0.01 mm), the sand-scale pores (0.01 mm ≤ de < 1.0 mm), and the gravel-scale pores (de ≥ 1.0 mm). Since the fines-scale pores are too small, they are assumed to not contribute to the permeability (Boadu 2000). Therefore, the 2D effective porosity of gravelly soils is expressed by:

$$ {n}_{2\mathrm{eff}}=\frac{S_{\mathrm{VS}}+{S}_{\mathrm{VG}}}{S_{\mathrm{T}}}={n}_{2\mathrm{S}}+{n}_{2\mathrm{G}} $$
(14)

where n2eff is the 2D effective porosity, n2S is the 2D sand-scale porosity of gravelly soils, and SVS is the area of the sand-scale pores in gravelly soils.

Considering a gravel-removed gravelly soil (i.e., fines-sand mixture), the 2D sand-scale porosity is approximately calculated by:

$$ {n}_{2\mathrm{s}}=\frac{S_{\mathrm{VS}}}{S_{\mathrm{T}}-\left({S}_{\mathrm{G}}+{S}_{\mathrm{VG}}\right)}=\frac{S_{\mathrm{VS}}}{S_{\mathrm{T}}\left[1-\left(\frac{S_{\mathrm{G}}}{S_{\mathrm{T}}}+\frac{S_{\mathrm{VG}}}{S_{\mathrm{T}}}\right)\right]}=\frac{n_{2\mathrm{S}}}{1-\left({\chi}_{\mathrm{G}}+{n}_{2\mathrm{G}}\right)} $$
(15)

Therefore, the 2D sand-scale porosity of gravelly soils can be calculated from that of fines-sand mixtures:

$$ {n}_{2\mathrm{S}}={n}_{2\mathrm{s}}\left[1-\left({\chi}_{\mathrm{G}}+{n}_{2\mathrm{G}}\right)\right] $$
(16)

Substituting Eq. (16) into Eq. (14), the final expression of the 2D effective porosity is described as:

$$ {n}_{2\mathrm{eff}}={n}_{2\mathrm{s}}\left[1-\left({\chi}_{\mathrm{G}}+{n}_{2\mathrm{G}}\right)\right]+{n}_{2\mathrm{G}} $$
(17)

In Eq. (17), χG, n2G, and n2s can either be measured by laboratory experiments or calculated from G and w with Eqs. (7)–(9). However, further corrections of the influences of dry density and gradation should be conducted when using the calculation method. Therefore, the need to develop an inexpensive and more convenient method for the determination of n2eff may arise.

As already mentioned, the effective porosity is significantly affected by the proportion of the fines fraction, F. On the other hand, one can easily imagine that the 2D effective porosity has a positive correlation with n. Thus, the 2D effective porosity may be directly calculated from easy-to-measure parameters such as F and n by the following equation:

$$ {n}_{2\mathrm{eff}}=a\frac{1}{F^c}n $$
(18)

where a = 0.315 and c = 0.2 are two constants regarding the studied gravelly soil (Fig. 14).

Fig. 14
figure 14

Relationship between n2eff/n and F of the studied gravelly soils

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  1. (2)

    Effective grain size

The choice of the representative grain size is critical to the successful prediction of the hydraulic conductivity from the grain size distribution. However, the commonly used grain sizes (e.g., d10, d20, and d50) obtained from sieve analysis are unable to represent the entire grain size distribution. In this section, an effective grain size calculated from the entire grain size distribution is used, as referred by Trani and Indraratna (2010).

If a non-uniform soil material is composed of m discrete sizes, i.e., \( {d}_1^{\ast }=0,{d}_2^{\ast },{d}_3^{\ast },...,{d}_m^{\ast } \) (maximum grain size) and their corresponding mass percentage that is given by P1 = 0, P2, P3,..., Pm = 100%, then the effective grain size (deff) can be calculated as follows (Trani and Indraratna 2010):

$$ {d}_{\mathrm{eff}}=\frac{1}{\sum \limits_{j=2}^{\mathrm{m}}\frac{2{P}_j}{d_j^{\ast }+{d}_{j-1}^{\ast }}} $$
(19)
  1. (3)

    Novel permeability models

Using the measured structural parameters, the 2D effective porosity and the effective grain size are calculated with Eqs. (17) and (19), respectively (Table 5). These two parameters are then applied to fit the measured k of gravelly soils. The fitting equation is expressed as follows:

$$ k=8.568\times {10}^3\cdot {\left({n}_{2e\mathrm{ff}}\right)}^{5.85}\cdot {\left({d}_{\mathrm{eff}}\right)}^{5.06} $$
(20)

where k is expressed in cm/s and deff is expressed in mm.

Replacing n2eff in Eq. (20) with the right side of Eq. (18) and improving the fitting constants, one can derive another form of the permeability equation:

$$ k=3.2\cdot {\left(\frac{n}{F^{0.2}}\right)}^{6.75}\cdot {\left({d}_{\mathrm{eff}}\right)}^{3.53} $$
(21)

The estimated saturated coefficients of permeability using Eqs. (20) and (21) are presented in Table 5 and Fig. 15. The results indicate a reasonably good fit between the calculated values and the measured data. In addition, both of these two new equations can offer improved results compared with those calculated by Eq. (12) (Fig. 13) and the modified existing equations (Fig. 12) with respect to well-graded gravelly soils.

Fig. 15
figure 15

Comparison between the calculated k via Eqs. (20) and (21) and the measured data

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The latter permeability model [Eq. (21)] is further verified briefly by the in situ saturated coefficients of permeability (kin-situ) reported in the literature (Dong et al. 2017b). The verification results are listed in Table 6. Note that the kin-situ values were measured from 24 positions of the same slope as that examined in this study using the double-ring infiltrometers. The values of kc are calculated by Eq. (21) based on the parameters of gravelly soils in Positions #1–#3 (Table 1). From Table 6, it is noted that kc in Position #3 falls exactly in the range of kin-situ and kc in Position #2 is approximately near the upper limit of kin-situ, while kc in Position #1 is significantly larger than the upper limit of kin-situ. This is because the model constants of Eq. (21) are determined based on the analyses of the gravelly soils with F = 2.1–4.8% (Table 5). This raises the limitation of the proposed model. However, the form of Eq. (21) may be applicable to a wider range of gravelly soils.

Table 6 Verification of the permeability model [Eq. (21)] against the in situ data
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Conclusions

In this study, 14 large gravelly soil samples and five groups of small gravel-removed soil samples were prepared using compaction methods. These samples were then respectively characterized using X-ray computed tomography (CT) and optical microscopy. Consequently, the structures of the gravelly soil at the gravel scale and sand scale were quantified by image analysis. The results show that both the gravel area ratio and the gravel-scale porosity significantly increase as the gravel content rises from 30% to 70%. The sand-scale porosity exhibits a slight increase followed by a decrease with increasing water content at a given dry density. Moreover, a small gravel-scale porosity and a relatively large sand-scale porosity are observed. The dominant size of the gravel-scale pores in gravelly soils varies by several millimeters depending on the gravel content, whereas the dominant size of the sand-scale pores is approximately 0.06 mm and remains constant as the water content varies. In addition, both a decrease in the fractal dimension of the gravel-scale pores with increasing gravel content and a decrease in the fractal dimension of the sand-scale pores with increasing water content are noted. The results also indicate that the gravel-scale pore structure is simpler and more conducive to the free flow of water compared to that of the sand-scale pore structure.

The saturated coefficients of permeability, k, of the studied gravelly soil were measured by performing constant-head tests on large cylindrical samples that were previously CT scanned. Thereafter, ten existing empirical and semi-empirical equations were used to estimate k based on pore parameters (e.g., e and n) and grain size distribution parameters (e.g., grain size, Cu, and Cc). The analyses of the estimated results indicate that none of these existing equations offer satisfactory results for well-graded gravelly soils. This is likely because these equations were basically developed for more or less uniformly graded materials. However, in this paper, the materials studied are well graded and significantly different from spheres, sand, or gravel alone. For this reason, the existing permeability equations were improved [with the form in Eq. (11)] to make them applicable to gravelly soils. In addition, three novel empirical permeability models [Eqs. (12), (20), and (21)] were proposed based on the structural parameters of pores, as well as the concepts of the effective porosity and the effective grain size.

The results presented in this paper provide new insight regarding the relationship between the structural characteristics and hydraulic conductivities of gravelly soils. However, they were obtained based on limited soil samples and experiments. Further work should be conducted to verify and enrich these findings.