#### Abstract

The soil–water characteristic curve (SWCC) and pore-size distribution (PSD) are fundamental characteristics of soils that determine many physical and mechanical properties. Recent studies demonstrate that both SWCC and PSD sometimes exhibit a bimodal feature. In this paper, soils with bimodal SWCCs are mainly divided into three categories: gap-graded soils, compacted clayey soils, and natural dual-porosity structural soils, from the perspective of microporosity structure. Based on the Fredlund and Xing unimodal SWCC equation, a bimodal SWCC equation is presented. The bimodal PSD equation for mercury intrusion porosimetry (MIP) is derived theoretically, according to the relationship between the mercury intrusion process in MIP and the water desorption process along the drying SWCC. These two associated equations have the same set of parameters, so the corresponding relationship between the bimodal SWCC and bimodal PSD can be directly shown. Three main presenting forms of PSD in MIP tests are summarized. Regression analysis results show that the proposed PSD equation can well fit bimodal PSD experimental data of various soils in the literature, and the SWCCs are predicted at the same time.

#### 1. Introduction

The soil-water characteristic curve (SWCC), which denotes the water retention character of the soil, is the most fundamental and important unsaturated soil property [1]. SWCC may be unimodal or bimodal. Compared with unimodal SWCC soils, the unsaturated shear strength, compressibility, and permeability of soils with bimodal SWCCs are different and more complex. Satyanaga and Rahardjo [2] and Zhao et al. [3] proposed equations for estimating the unsaturated shear strength of soils with bimodal SWCCs, based on the bimodal SWCC parameters. The estimated results had good agreement with their measured shear strength. Zhao et al. [3] indicated that the bimodal compressibility behavior of well-graded sand with silt and gravel (SW-SM with gravel) was related to the bimodal SWCC. Chen et al. [4] and Li et al. [5] proposed bimodal permeability equations based on the bimodal SWCCs. Therefore, deep research on the bimodal SWCC will contribute to understanding the unsaturated soil properties comprehensively and obtaining engineering characteristic parameters of soils more accurately.

Pores in soil are much studied in soil science and soil mechanics. On the one hand, many factors that may affect plant growth are controlled by pores in arable soils, such as water movement, solute transport, and aeration [6]. On the other hand, some physical and mechanical properties of soils are determined by pores, such as compressibility and hydraulic conductivity [7]. Pores are the ensemble of interconnected voids or spaces existing in a soil [6]. The microporosity structure of a soil involves the feature information of pores. Different soils may have different microporosity structures. The pore-size distribution (PSD) is usually used to quantitatively describe the soil microporosity structure [8–10]. Similar to the SWCC, the PSD is another important basic property of the soil and also has a unimodal and bimodal feature.

According to capillary theory, the SWCC can be uniquely determined by the PSD of the soil [11]. Recently, some bimodal equations of the SWCC and PSD have been proposed, but most of them were separately obtained from the law of experiment results and empirical equations, rather than theoretical derivation. Few of them can reflect the relationship between the bimodal SWCC and PSD. The main objective of this study is to establish the associated bimodal SWCC equation and bimodal PSD equation based on the theoretical connection between them.

#### 2. Soils with Bimodal SWCCs and Dual-Porosity Structure

Most of the early SWCC equations are suggested based on the shape of the SWCC, which is generally sigmoidal or symmetrical S-shaped [12]. Typical of these are the equations of van Genuchten [13]; Mckee and Bumb [14]; Fredlund and Xing [11]. Obviously, these sigmoid SWCCs are unimodal because the differential curves of these equations are unimodal or because these equations can be derived from unimodal PSDs by integration [11].

Then, it is found that SWCCs are not always unimodal but bimodal for some kinds of soils. Some experts pointed out that SWCCs of well-graded soils usually exhibit unimodal features, while gap-graded soils (soils with bimodal grain-size distribution) may have bimodal SWCCs [4, 5]. However, Satyanaga et al. [1, 15] and Rahardjo et al. [16] presented that some gap-graded soils can result in bimodal SWCCs, whereas others may result in unimodal SWCCs. Therefore, it can be said that the grain-size distribution may have a significant effect on the shape of the SWCC, but the influence is not decisive.

The cause of the bimodal feature of the SWCC is complex and may be well revealed from the perspective of the PSD and microporosity structure. Some experts concluded that the bimodal SWCC may be associated with any soil with a dual-porosity structure [5, 17, 18]. The dual-porosity structure includes macropore structure and micropore structure [4], which means two dominant distinct pore series: macropores with relatively large size and micropores with relatively small size. When summarizing the existing research results, there are roughly three categories of soils with bimodal SWCCs.(1)The first category is gap-graded soils, such as residual and colluvial soils [1, 2, 15, 19], coarse-grained soils [3, 20, 21], and so on. The typical feature of this kind of soil is that the particle size of the coarse grain is far larger than that of the fine grain [19]. Some pores formed by coarse grains and not completely filled by fine grains are macropores, and pores between fine grains are micropores. These two pore series constitute the dual-porosity structure, as shown in Figure 1(a). The dual-porosity structure of this kind of soil is mainly caused by its poor gradation.(2)The second category is compacted clayey soils [7, 22], especially soils compacted at the dry side of optimum [23]. In those soils, particles tend to bind together to form lumps or aggregates that behave as much larger particles [24, 25]. Pores between soil particles inside lumps or aggregates are called intra-aggregate pores. Many soil lumps or aggregates accumulate to form the soil body, and pores between soil lumps or aggregates are called interaggregate pores. The size of intra-aggregate pores is much smaller than that of interaggregate pores. Therefore, the dual-porosity structure of compacted clayey soil is composed of interaggregate pores and intra-aggregate pores, which can be referred to as macropores and micropores, respectively. The dual-porosity structure of compacted clayey soil is formed during soil preparation, and its schematic diagram is shown in Figure 1(b).(3)The third category is natural dual-porosity structural soils, such as lateritic soils [26–28], loam soils [29], and pelletized diatomaceous earth [30, 31]. These soils are naturally formed under special geological or climatic conditions, and some of them may have special mineral composition. Aggregates and dual-porosity structure develop during the soil formation process simultaneously. The dual-porosity structure of this kind of soil is similar to that of compacted clayey soil, as shown in Figure 1(b). Even after compaction, the dual-porosity structure of these soils may still exist.

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Although these three categories of soils all have dual-porosity structures, their formation mechanisms and structural features are different, and their physical and mechanical properties may also be different.

These soils with bimodal SWCCs and dual-porosity structure are common in geotechnical engineering. For example, coarse-grained soils are the main materials in natural colluvial slopes [3]; compacted soils are widely used in subgrades, seepage control layers of earth dams and landfills; lateritic soils are distributed in many areas of the earth’s surface [26]. These soils or part of them are often unsaturated, so it is of great significance to research their SWCCs and microporosity structures.

#### 3. Bimodal SWCC Equation

Typical curves of unimodal and bimodal SWCCs are shown in Figure 2. From the figure, it can be seen that the two curves are quite different. If the bimodal SWCC is divided at the inflection point in the middle gentle segment of the curve, as shown in Figure 2, the bimodal SWCC can be regarded as two connected unimodal SWCCs in two ranges: low-suction range and high-suction range.

A series of studies have been carried out on the bimodal SWCC, and many theoretical models have been established [1, 19, 22, 32–34]. Wijaya and Leong [18] categorized the development of these bimodal SWCC equations into three approaches, including the piecewise approach, the fraction of total volume approach, and the unique parameter approach.

Many of the continuous bimodal SWCC equations proposed previously have a similar feature, which is that they can be regarded as the superposition of two unimodal SWCC equations. Therefore, a simplified general equation of the bimodal SWCC can be expressed in a superposition form of two unimodal SWCC equations:where *t* is the weight factor; *θ*_{1}(*ψ*) and *θ*_{2}(*ψ*) are the unimodal SWCC equations in terms of volumetric water content.

In past studies, the SWCC equation proposed by Fredlund and Xing [11] was widely used and best fitted for unimodal SWCC experimental results. Therefore, this equation is also used as the expression of *θ*_{1}(*ψ*) and *θ*_{2}(*ψ*) in this paper. Then, bimodal SWCC equation *θ*(*ψ*) can be written aswhere *θ*_{s} is the volumetric water content in the saturation case and is equal to the soil porosity, that is, *e*_{0}/(1 + *e*_{0}), where *e*_{0} is the void ratio of the soil; *ψ* is matric suction in units of kPa; *a*_{1}, *n*_{1}, and *m*_{1} and *a*_{2}, *n*_{2}, and *m*_{2} are fitting parameters. The units of *a*_{1} and *a*_{2} are kPa, and *n*_{1}, *m*_{1}, *n*_{2}, *m*_{2}, and *t* are dimensionless.

Figures 3(a) and 3(b) show the effect of varying parameters *a*_{1}, *n*_{1}, *m*_{1}, *a*_{2}, *n*_{2}, and *m*_{2} in (2) on the shape of the bimodal SWCC. Parameters *a*_{1}, *n*_{1}, and *m*_{1} mainly affect the shape of the curve in the low-suction range but have little influence on the shape of the curve in the high-suction range. The parameters *a*_{2}, *n*_{2}, and *m*_{2} do just the opposite. Parameters *a*_{1} and *a*_{2} are approximately equal to the suction of the inflection point of the curve in the low-suction range and high-suction range, respectively.

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Figure 3(c) shows the effect of varying parameter *t* on the shape of the bimodal SWCC. It can be seen from the figure that *t* is roughly equal to the ratio of the volumetric water content of the curve in the low-suction range to the saturated volumetric water content of the soil.

Obviously, the bimodal SWCC equation in this paper (2) is based on the Fredlund and Xing [11] unimodal equation, but it can also be based on others, such as van Genuchten [13] model and Mckee and Bumb [14] model. Equation (1) provides a unified equation of the bimodal SWCC.

#### 4. Bimodal PSD Equation Derived from Bimodal SWCC

Fredlund and Xing [11] theoretically established the relationship between the PSD and SWCC. The soil can be considered to contain a set of interconnected pores that are randomly distributed. The volumetric water content can be obtained by integrating the PSD equation *f*(*r*) aswhere *θ*(*R*) is the volumetric water content when all pores with radii less than *R* are filled with water; *R*_{min} is the minimum pore radius in the soil.

Therefore, the PSD equation *f*(*r*) can be expressed as the derivative of *θ*(*r*):

In general, *θ* is expressed as a function of matric suction *ψ*, as shown in (2), so

According to the capillary law, there is an inverse relationship between matric suction and the pore radius:where *T*_{sw} is the surface tension of water, 72.8 × 10^{−3} N/m when the water temperature is 20°C; *φ* is the contact angle of the water-air interface to the soil, which is approximately zero. Therefore, 2*T*_{sw}cos*φ* is a constant that can be replaced by *C*. Thus, *ψ* = *C*/*r*. So, the following equation is obtained:

By substituting (2) and (7) into (5), the PSD function *f*(*r*) can be obtained:

Equation (8) is an entire expression of the bimodal PSD equation, with the form of a pore size density distribution. It has a strictly logical relationship with the bimodal SWCC equation in theory since it is derived from (2). It should be pointed out that the expression of *f*(*r*) will be different, if other unimodal SWCC equations are used in (1). That is to say, (8) will change with (2).

#### 5. MIP Method

Soil microstructure can be investigated by many experimental methods, such as scanning electron microscopy (SEM), X-ray computer tomography scanning (CT), and mercury intrusion porosimetry (MIP), among which the MIP method is the most widely used at present.

In the MIP test, the nonwetting fluid (mercury) intrudes into a porous medium under a constant pressure *P*. The pore that is intruded by mercury is assumed to be cylindrical, and its radius *r* can be calculated using the following equation:where *r* is the equivalent entrance pore radius; *T*_{sm} is the surface tension of nonwetting fluid (484 × 10^{−3} N/m for mercury);*α* is the contact angle of the mercury-air interface with the porous material, and its value is different in soils with different kinds of minerals [35, 36]. Here, an average value 140° is adopted for *α*.

From (9), we know that the larger the pressure is, the smaller the size of pores entered by mercury, so the mercury fluid will intrude into the soil from the larger pores to the smaller pores when the applied pressure increases steps by steps. The mercury intrusion volume at each pressure step is measured, and the cumulative mercury intrusion volume can be referred to as the cumulative pore volume , which means the volume of pores with radii larger than *r*. Correspondingly, the differential of cumulative mercury intrusion volume in the semilog coordinate system can be used to represent the PSD. Generally, for soil with a single dominant pore size, the PSD has only one peak [37], while for soil with a dual-porosity structure, the PSD has two peaks, called bimodal PSD.

#### 6. Relationship between SWCC and PSD Curve in MIP

As previously stated, in SWCC *θ*(*r*) is volumetric water content when all the pores with radii less than *r* are filled with water, and in MIP is the volume of pores with radii larger than *r*. The expression d*θ*(*r*)/d*r* and can both represent PSD, but they are not mathematically equivalent. Otalvaro et al. [28] pointed out that mercury intrusion in a porous medium is a process similar to air injection in saturated soil along the drying curve of a SWCC. Plot two schematic diagrams of the water–air interface and mercury–air interface illustrated by Aung et al. [38] on the same diagram, as shown in Figure 4. There exists a microspace between the air–water interface and mercury–air interface, since the contact angles (i.e., *α* and *φ*) of these two interfaces to soil particles are different. Mercury intrusion is the process of pores entered by mercury, while desorption SWCC is the process of water leaving pores. So, the following equation can be obtained:where and are the soil total volume and soil particle volume, respectively. is the volume of the space between the water–air interface and mercury–air interface. is the volume of mercury, which can be denoted as . is the volume of water, which can be expressed as .

Because *φ* is approximately equal to 0 and *α* is more than 90°, the following inequation can be obtained:

Therefore, is very small relative to the total volume, and

Then, (10) can be approximately expressed as

Take the derivative of both sides in (13) with respect to *r*:

MIP test results are always presented in terms of in semilogarithmic coordinates, and the following equation exists:

Substitute (15) into (16), and then, the following equation can be obtained:

Equation (17) gives the general expression of the PSD in the MIP, in which *f*(*r*) is the PSD for the SWCC. It directly shows the internal transformation relationship between these two PSDs, and on this basis, the relationship between the SWCC and PSD in the MIP test can also be established at the same time. The PSD equation (17) and SWCC equation (2) have the same set of parameters (i.e., *a*_{1}, *n*_{1}, *m*_{1}, *a*_{2}, *n*_{2}, *m*_{2}, and *t*), so if either of them is measured in the test for a soil, the other can be determined simultaneously. These two equations are both bimodal.

From (17), is negative, because decreases as pore radius *r* increases. However, sometimes for convenience, the PSD result in MIP may be expressed as a positive value [3, 20, 39–41]. In this paper, it is all expressed as a minus form.

Figure 5 gives typical curves of bimodal SWCC equation (2) and bimodal PSD equation (17) with the same parameter value. In the figure, the SWCC in black is plotted in the coordinate system formed by the black axes on the left and bottom, and the PSD curve in blue is plotted in the coordinate system formed by the blue axes on the right and top. The abscissa axis of suction at the bottom and the abscissa axis of pore radius at the top have conversion relations, which can be obtained by (6).

In Figure 5, the PSD curve has two dominant peak areas, corresponding to macropores and micropores of the dual-porosity structure. The size of macropores is relatively larger, so the suction required for water drainage in macropores is relatively lower, corresponding to the low-suction range of the bimodal SWCC. Micropores are smaller, and higher suction is required for water drainage, corresponding to the high-suction range of the bimodal SWCC. Therefore, the desorption process of the bimodal SWCC for soils with dual-porosity structures can be simply divided into two stages: the first is the drainage of the water stored in macropores, and the second is the drainage of the water stored in micropores. Figure 5 clearly shows the corresponding relationship between the bimodal SWCC and bimodal PSD curve, and both are determined by the dual-porosity structure of soils.

Hence, in (17), parameters *a*_{1}, *n*_{1}, and *m*_{1} are mainly related to macropores, while parameters *a*_{2}, *n*_{2}, and *m*_{2} are mainly related to micropores. Parameter *t* essentially represents the proportion of the volume of macropores in the total pore volume. If parameter *t* is equal to zero, bimodal PSD equation (17) will degenerate to a unimodal equation, and the same rule applies to bimodal SWCC equation (2).

#### 7. Evaluation of the Proposed PSD Equation

The bimodal PSD equation proposed in this paper (17) can be used to describe the dual-porosity structure quantitatively and predict the SWCC of the soil. Some MIP test results in the literature are used to evaluate the applicability of this equation.

The PSD in the MIP test is commonly represented by . However, other variations are also used in some studies. In summary, there are three main forms of PSD and they are essentially related to each other.(1). This is the basic presenting form of the MIP test results, and its expression is (17). is the cumulative volume of mercury intrusion per gram of dry soil sample. The total volume of the soil sample can be calculated as = 1/*ρ*_{d}, where *ρ*_{d} is the dry density of the soil sample, with a unit of g/cm^{3}. So, (17) can be rewritten as(2). This is the derivative of the cumulative mercury intrusion volume with respect to the log of pore diameter *D*. Sometimes it is written as log diff. intrusion [40] or incremental intrusion [27]. Since , it can be expressed as(3)d*e*_{MIP}/dlog *D* It is sometimes written as *δe*_{nw}*/δ* log *D* [36]. *e*_{MIP} or *e*_{nw} is the void ratio of pores intruded by mercury, that is, the ratio of the cumulative mercury intrusion volume to the soil particle volume . The regression analysis is conducted on some measured PSD data by the MIP test in the literature using (18) to (20) as shown in Figure 6. The parameter best fit values are listed in Table 1. In the table, soils no. 1 to 2 are gap-graded soils, soils no. 3 to 7 are compacted clayey soils, and soil no. 8 is natural dual-porosity structural soil. Regression analysis results show that the proposed PSD equation can well fit bimodal PSD data in MIP tests of various soils with dual-porosity structures. That is, the equation has good applicability.

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Figure 6 also gives the predicted SWCCs of these soils, which are plotted using (2) and the same parameter values in Table 1. Therefore, it is very convenient to predict the SWCC directly from the fitted PSD in the MIP test based on the proposed SWCC equation and PSD equation in this paper. This is the benefit of establishing the associated SWCC and PSD equation. In this regard, if the PSD is measured by the MIP test or other methods, the SWCC can be obtained at the same time, without the need for further SWCC test, which is time-consuming.

#### 8. Conclusions

Soils with bimodal SWCCs and dual-porosity structure are common in geotechnical engineering. They can be divided into three categories: gap-graded soils, compacted clayey soils, and natural dual-porosity structural soils. Their dual-porosity structure is analyzed theoretically.

A bimodal SWCC equation is presented based on the Fredlund and Xing unimodal SWCC equation, and the influence of parameter variation on the shape of the bimodal SWCC is analyzed.

According to capillary theory, a bimodal PSD equation *f*(*r*) is derived from the presented bimodal SWCC equation. The PSD equation , which can be used as a general PSD expression in MIP, is proposed by equation *f*(*r*) based on the relationship between the mercury intrusion process in MIP and the water desorption process along the drying SWCC.

The associated bimodal equations of the SWCC and PSD have been established with the same set of parameters. From these two equations, the corresponding relationship between the bimodal SWCC and PSD is directly shown, and both are determined by the dual-porosity structure of soils.

The regression analysis is conducted on some measured PSDs by the MIP test in the literature, and the results show that the proposed PSD equation can well fit bimodal PSD experimental data of various soils. The SWCCs are predicted at the same time. Therefore, if the PSD is measured by the MIP test or other methods, the SWCC can also be obtained, without the need for further SWCC test, which is time-consuming.

#### Data Availability

The data used to support the findings of this study are included within the article.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

This study was supported by the National Natural Science Foundation of China (grant no. 52178328), the Key Project from NSFC of the Yangtze River Water Science Research Joint Fund (grant no. U2040221), and the 111 Project (grant no. B13024).