1 Introduction

The relationship between the soil suction (s) and the water content (expressed in terms of gravimetric water content w, volumetric water content θ or degree of saturation Sr) is defined as soil–water retention curve (SWRC). As a crucial hydraulic property of unsaturated soils, SWRC significantly influences the other unsaturated soil parameters, such as effective stress [1, 2], shear resistance [3, 4], volume change [5], and hydraulic conductivity of gas and water [6, 7]. The knowledge of the SWRC is hence essential for understanding the hydromechanical behavior of unsaturated soils and for the practical applications in geotechnical and geological engineering, e.g., radioactive waste disposal [8, 9], slope stability analysis [10], and water movement in the vadose zone [11, 12].

Water in unsaturated soil can be attributed mainly to two binding mechanisms, capillary and adsorption forces (see Fig. 1) [13,14,15,16]. Capillary water (or free water) is retained in the pores by the surface tension of water and the contact angle between the soil grains and the water surface, forming a meniscus (air–water interface). Soil pore size distribution hence dominates the degree of saturation in the capillary regime. Adsorption water (or bound water) is strongly bonded on the surface of the soil particles as a thin film. It is caused by attractive forces between the soil particles and the water in the form of van der Waals forces and hydrogen bonds. Therefore, soil mineralogy (e.g., cation types on the mineral surface) and specific surface determine the maximum adsorptive water content in soil [8, 17].

Fig. 1
figure 1

Schematic for capillary and adsorption water in an unsaturated soil

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In recent decades, numerous empirical water retention expressions have been advanced and successfully applied to solve unsaturated soil problems (e.g., [18,19,20,21]). However, these expressions ignored the desorption process of adsorption water. The assumed constant residual water content contradicted the experimental observation on the water retention curve in the high suction range [22,23,24]. These traditional SWRC expressions are thus inappropriate to describe the dry end of the water retention curve [25].

Recently, extensive SWRC models defined on the entire suction range were developed in piecewise [25,26,27,28,29] or continuous form [13, 17, 30]. The adsorption water content in these models was assumed to be suction-dependent. For instance, Zhang [28] incorporated a piecewise suction-dependent function for adsorption water content into van Genuchten model (VGM) [19]. However, the adsorption curve in a piecewise form is not smooth at junction points, leading to inconvenience in model implementation. In contrast, Revil and Lu [17] [Revil and Lu model (RLM)] advanced a continuous adsorption equation based on the Freundlich isotherm [31] to replace the constant residual water content in VGM. The total water content was regarded as the superposition of capillary and adsorption components. Afterward, Lu [13] [Lu model (LM)] modified the adsorption equation to correct the overprediction of the maximum suction of the soil in a completely dry state by RLM. However, these full suction range functions employed a single best-fitting procedure to calibrate the model parameters, which might lead to the loss of the physical meaning of the parameters. On the other hand, considering the large number of parameters (e.g., six in RLM and eight in LM) and the intercorrelation between them, convergence problems might occur in the optimization process [32, 33]. Very recently, special attention has been paid to determining model parameters from the graphical properties of the SWRC data to overcome these problems, e.g., Alibrahim and Uygar [34].

Compared with clayey soils, only a few full suction range water retention data are available in the literature for granular soils. This paper is aimed to undertake an experimental study on the water retention behavior of a well-graded sand with clay on a broad range of water saturations. Based on the experimental observations, a continuous SWRC expression defined on the entire suction range will be developed, and a simple parameter calibration approach will be proposed.

2 Material and methods

2.1 Material and sample preparation

The soil used for the experimental investigations was taken from a railway embankment located in Northern Germany [35]. The soil was passed through a 4-mm sieve prior to testing. The grain size distribution is shown in Fig. 2, and the index properties of the soil are listed in Table 1. The liquid limit (wL) and plasticity index (Ip) of the fines are 22.8% and 10.5%, respectively. The soil is classified as well-graded sand with clay (SW-SC) according to the Unified Soil Classification System (USCS).

Fig. 2
figure 2

Grain size distribution curve of the soil (SW-SC) (data from Angerer [35])

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Table 1 Index properties of the soil (data from Angerer [35])
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The soil was first oven-dried and then mixed with distilled water at three target water contents, i.e., 3%, 6%, and 10%. After that, it was stored for several days in an airtight container for homogenization. For each water content, samples were statically compacted at three relative densities (Id), i.e., loose (Id ≈ 0.5), medium dense (Id ≈ 0.7), and dense (Id ≈ 0.9) to further investigate SWRC. More details on the sample preparation procedure can be found in Angerer [35].

2.2 Experimental procedure

Two laboratory techniques and apparatus were jointly employed to measure soil suction from saturated to nearly oven-dry conditions, which improved the accuracy and reliability of the SWRC data.

In the low suction regime (i.e., for suctions up to about 120 kPa), an evaporation test using the “hydraulic property analyzer (HYPROP)” was conducted. The samples had a diameter of 8 cm and a height of 5 cm and were statically compacted in five layers using a sample ring. The samples were first saturated in a water bath and then connected to the sensors. The test started with applying the evaporation from the top. The sample weight and the suction at two depths inside the sample were recorded. The evaluation of the test data, which was done automatically by the HYPROP device, is based on the findings of Peters and Durner [36], Schindler and Müller [37], and Schindler [38].

For suctions higher than 1 × 103 kPa, a chilled-mirror dewpoint hygrometer (WP4C) was used. The soil samples were statically compacted into sample cups with a diameter of 4 cm and a height of 1.2 cm. The samples were carefully wetted by adding water with a pipette and then homogenized for several days. Afterward, the specimens were stepwise dried under a heating lamp to the target water content and stored in closed containers for seven days to homogenize. Suction measurements were then undertaken for every drying step.

For the tested granular soil with low plasticity fines, osmotic suction was neglected [39]. Therefore, the total suction value measured by the chilled-mirror hygrometer (WP4C) can be regarded as a matric suction value. On the other hand, volumetric deformation could not be monitored when using the HYPROP device and the chilled-mirror hygrometer (WP4C). For deformable soils, a considerable suction-induced volume change may occur and markedly affect the shape of the SWRC in terms of degree of saturation or volumetric water content [40,41,42,43]. Nevertheless, it was assumed that volumetric deformation induced by hydraulic loading is negligible for the examined soil, considering the rigid structure formed by sand grains [35]. In this study, the water retention data (in terms of gravimetric water content) of the dense sample compacted at wcomp = 6% was used for data analysis and model development. The other samples compacted at wcomp = 6% and 10% to different relative densities were used to validate the proposed SWRC model. Since only the main drying curves were measured, the hysteretic effect was not taken into account.

2.3 Analysis of experimental data

Figure 3 demonstrates the water retention data of the samples compacted to different initial densities at a water content of 6%. The SWRC data shows similar characteristics in the low suction range (s < 120 kPa) (see Fig. 3a). When the suction is lower than the air entry value, the water content remains at a maximum value (\({w}_{{\text{max}}}\)) lower than the saturated water content (\({w}_{s}\)). Since a preceding carbon-dioxide-flush was not carried out, the lack of full saturation might be attributed to the entrapped air bubbles in the pore structure, although the saturation process in the water bath took up to three days [35]. The constant maximum gravimetric water content is due to the rigid sand grain skeleton in the tested soil. In contrast, deformable fine-grained soils undergo dewatering even before the air entry suction, resulting from the volume reduction with increasing suction, as pointed out by Pasha et al. [44].

Fig. 3
figure 3

Water retention data of the samples statically compacted at wcomp = 6% a water retention data in the low suction range measured with tensiometer b water retention data in the high suction range measured with chilled-mirror hygrometer c linearization capillary and adsorption components in log s − log Se plane (data from Angerer [35])

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Beyond the air entry suction, water content strongly decreases in a narrow suction range of several kilopascals, and the three drying curves merge at a turning point at s ≈ 3 kPa. The turning point signifies the drainage of the capillary water stored in the macro-voids sensitive to initial density (i.e., easily collapse with increasing compaction effort [45]). One also sees a slight reduction of the air entry suction with the decreasing initial density, which is physically reasonable and in line with the results in the literature, e.g., [46,47,48]. Subsequently, capillary water in the micro-voids (almost unaffected by initial density) slightly decreases with a further increase in suction. The slope of the drying curves gradually decreases nearly to zero at s ≈ 20 kPa and \({w}_{0}^{a}\) = 4% (denoted as “delimiting point”, for definition, refer to Sect. 3.4), indicating the complete drainage of capillary water.

Figure 3b highlights the water retention data measured by the chilled-mirror hygrometer technique in a high suction regime (s > 1 × 103 kPa), which shows a nonlinear characteristic in the semilogarithmic plot. This observation contradicts the assumption of a log-linear reduction of adsorption water in existing full suction range models, for instance, Campbell and Shiozawa [22] and Konrad and Lebeau [14]. It is also interesting to note that the nonlinear suction–water content relationship is identical for different densities, which might be ascribed to similar material properties (e.g., soil mineralogy and specific surface) and hence is independent of compaction procedure, which is in line with the results of Birle et al. [49].

Based on the data analysis (see Fig. 3a and b), it is not hard to find that the total water retention curve comprises capillary (for water content higher than \({w}_{0}^{a}\) = 4%) and adsorption components (for water content lower than \({w}_{0}^{a}\) = 4%). Since the mechanisms dominating these components are not related [30], the capillary (\({S}_{e}^{c}\)) and adsorptive (\({S}_{e}^{a}\)) curves in terms of effective degree of saturation can be recalculated as

$$\left\{ {\begin{array}{*{20}l} {S_{{\text{e}}}^{{\text{c}}} \left( s \right) = \frac{{w\left( s \right) - w_{0}^{{\text{a}}} }}{{w_{{{\text{max}}}} - w_{0}^{{\text{a}}} }}} \hfill & { \quad {\text{for}} \,w > w_{0}^{{\text{a}}} } \hfill \\ {S_{{\text{e}}}^{{\text{a}}} \left( s \right) = \frac{w\left( s \right)}{{w_{0}^{{\text{a}}} }}} \hfill & { \quad {\text{for}} \,w \le w_{0}^{{\text{a}}} } \hfill \\ \end{array} } \right.$$
(1)

Figure 3c shows the \({S}_{{\text{e}}}^{{\text{c}}}\left(s\right)\) and \({S}_{{\text{e}}}^{{\text{a}}}\left(s\right)\) curves in log–log plot. One sees that both the \({S}_{{\text{e}}}^{{\text{c}}}\left(s\right)\) and \({S}_{{\text{e}}}^{{\text{a}}}\left(s\right)\) relationships are characterized by linear segments. The linear segments of the capillary component translate with the initial density due to the change in air entry value. Meanwhile, the linear segments representing the adsorption mechanism are unique due to the identical material properties.

3 Theory

3.1 An equivalent form of VGM

At a given suction level, the water content held by a granular soil is generally expressed as

$$w = w_{0}^{{\text{c}}} \int_{0}^{2T\cos \theta /s} {f\left( r \right){\text{d}}r + w_{0}^{{\text{a}}} S_{{\text{e}}}^{{\text{a}}} }$$
(2)

The first term on the right-hand side of Eq. (2) represents the capillary component dominated by soil pore size distribution \(f\left(r\right)\) and capillary law (i.e., the Young—Laplace equation, \(r = 2T \,\cos \theta /s\)) [20, 50]. Here, \(T\) is water surface tension (\(T\) = 0.072 N/m at 25 °C), and \(\theta\) is the contact angle between soil and water (\(\theta\) = 0°). \({w}_{0}^{c}\) is the capillary water content at zero suction.

The second term \({w}_{0}^{{\text{a}}}{S}_{{\text{e}}}^{{\text{a}}}\left(s\right)\) represents the adsorption component governed by soil material property. Here, \({w}_{0}^{{\text{a}}}\) is defined as “adsorption capacity” following Lu [13], representing the maximum amount of water that can be held in soil via adsorption force. \({S}_{{\text{e}}}^{{\text{a}}}\) is the ratio of the adsorption water content to adsorption capacity. According to Revil and Lu [17] and Lu [13], the value of \({S}_{{\text{e}}}^{{\text{a}}}\) monotonically decreases with suction.

From Eq. (2), the maximum water content \({w}_{{\text{max}}}\) at zero suction gives

$$w_{{{\text{max}}}} = w_{0}^{{\text{c}}} + w_{0}^{{\text{a}}}$$
(3)

Substituting Eq. (3) into Eq. (2) and denoting the integration of the pore size distribution as \(S_{{\text{e}}}^{{\text{c}}}\) gives

$$w = \left( {w_{{{\text{max}}}} - w_{0}^{{\text{a}}} } \right)S_{{\text{e}}}^{{\text{c}}} + w_{0}^{{\text{a}}} S_{{\text{e}}}^{{\text{a}}}$$
(4)

Based on the experimental data in Fig. 3c, an equivalent form of the van Genuchten model [19] is proposed to describe the linear relationship in the log–log graph for capillary and adsorption water. VGM in terms of the effective degree of saturation gives:

$$S_{{\text{e}}} = \left[ {1 + \left( {\alpha s} \right)^{n} } \right]^{ - m}$$
(5)

here \(\alpha\), \(n,\) and \(m\) are fitting parameters. Taking the logarithm of both sides of Eq. (5), we have

$$\log \,S_{{\text{e}}} = - mn\,\log \alpha - mn\,\log \,s - m\,\log \left[ {1 + \left( {\alpha s} \right)^{ - n} } \right]$$
(6)

Considering that

$$m\,\log \left[ {1 + \left( {\alpha s} \right)^{ - n} } \right] > 0$$
(7)

and

$$\mathop {\lim }\limits_{s \to + \infty } m\,\log \left[ {1 + \left( {\alpha s} \right)^{ - n} } \right] = 0$$
(8)

We find an asymptote of VGM (denoted as \(\hat{S}_{{\text{e}}} { }\)) in the \(\log \,s\) − \(\log \,S_{{\text{e}}}\) plane

$$\log \,\hat{S}_{{\text{e}}} = - mn\,\log \alpha - mn\,\log s$$
(9)

or equivalently,

$$\hat{S}_{{\text{e}}} = \left( {\alpha s} \right)^{ - mn}$$
(10)

The suction value \(s_{{{\text{ae}}}}\) at the intersection of the asymptote and the line \(\hat{S}_{{\text{e}}}\) = 1 can be solved from Eq. (9):

$$s_{{{\text{ae}}}} = 1/\alpha$$
(11)

which is commonly denoted as air entry suction [13]. Since the value of mn in VGM represents the slope of the asymptote, let us introduce a simple mathematic transformation (\(\beta > 0\) and \(K > 0\))

$$\left\{ {\begin{array}{*{20}l} {m + \frac{1}{n} = \beta } \hfill \\ {mn = K} \hfill \\ \end{array} } \right.$$
(12)

Then, the parameters \(m\) and \(n\) in the original VGM can be solved from Eq. (12)

$$\left\{ {\begin{array}{*{20}l} {n = \frac{K + 1}{\beta }} \hfill \\ {m = \frac{\beta K}{{K + 1}}} \hfill \\ \end{array} } \right.$$
(13)

Substituting Eqs. (11) and (13) into Eq. (5) yields

$$S_{{\text{e}}} = \left[ {1 + \left( {\frac{s}{{s_{{{\text{ae}}}} }}} \right)^{{\frac{K + 1}{\beta }}} } \right]^{{ - \frac{\beta }{K + 1}K}}$$
(14)

Equation (14) is mathematically equivalent to the original van Genuchten model and referred to as MVGM in the following section. The asymptote of Eq. (14) gives

$$\hat{S}_{{\text{e}}} = \left( {\frac{s}{{s_{{{\text{ae}}}} }}} \right)^{ - K}$$
(15)

3.2 Assessment of the model parameter

In Fig. 4, a parametric analysis is conducted to highlight the physical meanings of the parameters in MVGM. Figure 4a shows MVGM curves for different values of \(K\) (= 0.5, 1.0 and 2.0), while β (= 1) and \({s}_{{\text{ae}}}\) (= 10 kPa) are constants. In this case, the asymptotes with different slopes pass through the identical air entry suction on the horizontal line \({S}_{{\text{e}}}=1\). Figure 4b presents the MVGM curves with fixed \(K\) (= 1.0) and \(\beta\) (= 1.0) and various \({s}_{{\text{ae}}}\) (= 1, 10, 100 kPa) values. One sees that the curves of the identical shape translate along the abscissa and the asymptotes are parallel.

Fig. 4
figure 4

Parametric analysis for MVGM: a influence of \(K\) b influence of \({s}_{ae}\) c influence of \(\beta\)

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Figure 4c illustrates the curves with fixed \(K\) (= 1.0) and \({s}_{{\text{ae}}}\) (= 10 kPa) and various \(\beta\) (= 0.5, 1.0 and 1.5) values, showing the influence of parameter \(\beta\) on the curve graphical properties. The three asymptotes merge into a single one because of the identical slopes \(K\) and air entry values \({s}_{{\text{ae}}}\). Moreover, one finds that the size of transition zone connecting the line \({S}_{{\text{e}}}=1\) and the asymptote, reduces with decreasing \(\beta\). For the extreme case of \(\beta \to 0\), the transition zone will degrade into a point. The curve becomes a combination of the horizontal line \({S}_{{\text{e}}}=1\) (for \(s\le {s}_{{\text{ae}}}\)) and the inclined line with the slope \(K\) (for \(s>{s}_{{\text{ae}}}\)).

The parametric study highlights the essential advantage of MVGM over other existing water retention models and the original VGM: the parameters are not intercorrelated and possess a clear physical and graphical meaning. \(K\) is the slope of the asymptote, which is dominated by pore size distribution. Air entry suction \({s}_{ae}\) is the suction value at the intersection between the asymptote and the horizontal line \({S}_{{\text{e}}}=1\), reflecting the maximum pore size in soil. \(\beta\) is a parameter that only affects the sharpness and size of the transition zone. For the aim of simplicity, it is sufficient to assume \(\beta\) = 1, which is consistent with the original VGM (i.e., \(m+1/n=1\)). Then, Eq. (14) becomes

$$S_{{\text{e}}} = \left[ {1 + \left( {\frac{s}{{s_{{{\text{ae}}}} }}} \right)^{K + 1} } \right]^{{ - \frac{K}{K + 1}}}$$
(16)

3.3 Modeling the capillary, adsorption, and total SWRC

Using MVGM to simulate the desorption process of adsorption water, we have

$$w^{{\text{a}}} = w_{0}^{{\text{a}}} S_{{\text{e}}}^{{\text{a}}} = w_{0}^{{\text{a}}} \left[ {1 + \left( {\frac{s}{{s^{{\text{a}}} }}} \right)^{{K_{{\text{a}}} + 1}} } \right]^{{ - \frac{{K_{{\text{a}}} }}{{K_{{\text{a}}} + 1}}}}$$
(17)

For granular soils with fine content, the clay component dominates the adsorption capacity because clayey minerals such as kaolinite, montmorillonite, and illite have a much larger specific surface area \({S}_{m}\) (in m2/g) than granular components, e.g., silica sand [51, 52]. \({K}_{{\text{a}}}\) is the slope of the linear segment controlling the desorption rate, referred to as “adsorption strength” according to Lu [13]. In the adsorption regime, van der Waals forces control the thickness of the water film adsorbed on the soil particle surface and inner-layer surface (expansive soils) [13, 53]. \({s}^{{\text{a}}}\) is the threshold value of suction denoting the onset of desorption process of adsorption water and the lower bound of the adsorption regime [13, 54].

Similarly, in the capillary regime, we obtain

$$w^{{\text{c}}} = \left( {w_{{{\text{max}}}} - w_{0}^{{\text{a}}} } \right)S_{{\text{e}}}^{{\text{c}}} = \left( {w_{{{\text{max}}}} - w_{0}^{{\text{a}}} } \right)\left[ {1 + \left( {\frac{s}{{s_{{{\text{ae}}}}^{{\text{c}}} }}} \right)^{{K_{{\text{c}}} + 1}} } \right]^{{ - \frac{{K_{{\text{c}}} }}{{K_{{\text{c}}} + 1}}}}$$
(18)

here \({K}_{{\text{c}}}\) is the slope of the linear segment for capillary water and represents the pore size spectrum number: the lower the \({K}_{{\text{c}}}\), the more dispersed the pore radius is [55]. \({s}_{{\text{ae}}}^{{\text{c}}}\) is the air entry suction, reflecting the maximum pore size in the soil.

Figure 5a shows the reproduced capillary and adsorptive curves in the log s − log Se plane for the dense sample compacted with a water content of 6% (\({K}_{{\text{c}}}\) = 2.5, \({K}_{{\text{a}}}\) = 0.4, \({s}_{{\text{ae}}}^{{\text{c}}}\) = 1.6 kPa and \({s}^{{\text{a}}}\) = 200 kPa). The model replicates the linear reduction of capillary and adsorption water content with suction and matches the experimental data well.

Fig. 5
figure 5

Modeling the water retention behavior of the dense sample compacted at w = 6% a linearization of capillary and adsorption curves in log–log graph b the total water retention curve over the entire suction range

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Since the capillary and adsorption water are in hydraulic equilibrium and have the same total suction [8], the total water content is conceptualized as the sum of the capillary and adsorption components:

$$w = \left( {w_{{{\text{max}}}} - w_{0}^{{\text{a}}} } \right)\left[ {1 + \left( {\frac{s}{{s_{{{\text{ae}}}}^{{\text{c}}} }}} \right)^{{K_{{\text{c}}} + 1}} } \right]^{{ - \frac{{K_{{\text{c}}} }}{{K_{{\text{c}}} + 1}}}} + w_{0}^{{\text{a}}} \left[ {1 + \left( {\frac{s}{{s^{{\text{a}}} }}} \right)^{{K_{{\text{a}}} + 1}} } \right]^{{ - \frac{{K_{{\text{a}}} }}{{K_{{\text{a}}} + 1}}}}$$
(19)

Figure 5b demonstrates the reproduced water retention curve using Eq. (19) (\({w}_{{\text{max}}}\) = 0.15 and \({w}_{0}^{{\text{a}}}\) = 0.04), showing a strong consistency with the SWRC data in both capillary and adsorption regimes. Simultaneously, the capillary and adsorption components are differentiated using Eqs. (17) and (18). The capillary curve is directly related to the contribution of suction to the effective stress [1] and the unsaturated hydraulic conductivity at high water content [6, 7]. The adsorption curve dominates the liquid film flow and the transport process in dry soils [53, 56].

3.4 Parameter calibration method

This section highlights the parameter calibration approach. The model parameters can be easily determined based on the graphical properties of the water retention data. The schematic in Fig. 6a demonstrates the concept of linearization of the water retention curve in a log–log plot. The water retention curve over the entire suction range is regarded as an assembly of two linear segments, LS1 and LS2, representing the capillary and adsorption water drainage process, respectively. A transition segment (TS), corresponding to the overlap of the capillary and adsorption curves, has a small slope close to zero between the two linear segments. For simplicity, the delimiting point “A” dividing the capillary and adsorption regimes is assumed in the middle of the transition segment.

Fig. 6
figure 6

a schematic for the linearization of water retention curve in log–log plot b parameter calibration procedure for the dense sample compacted at wcomp = 6%

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Figure 6b shows an example of the parameter calibration procedure using the dense sample (compacted at wcomp = 6%) data, which is summarized in six steps: (1) \({w}_{{\text{max}}}\) is the maximum value of the measured water content; (2) calculate the effective degree of saturation and present the data in the \({\text{log}}s\) − \({\text{log}}{S}_{{\text{e}}}\) plane; (3) set the delimiting point in the middle of the transition segment and calculate \({w}_{0}^{a}\); (4) calculate the \(s\) – \({S}_{{\text{e}}}^{{\text{c}}}\) and \(s\) − \({S}_{{\text{e}}}^{{\text{a}}}\) data using Eq. (1); (5) determine the capillary and adsorption asymptotes using simple linear regression; (6) determine the parameters \({K}_{{\text{c}}}\), \({K}_{{\text{a}}}\), \({s}_{{\text{ae}}}^{{\text{c}}}\), and \({s}^{{\text{a}}}\) based on the graphical properties of the capillary and adsorption asymptotes.

4 Model validation

4.1 Water retention data of the tested soil (SW-SC)

The proposed model is validated using the experimental results of the tested soil compacted at 6% and 10%. Figure 7 compares the reproduced water retention curves with the measurements of the medium dense (a) and loose (b) samples compacted at wcomp = 6%. The data points detected by the tensiometer are presented with gray solid points, and those by chilled-mirror hygrometer with black solid points. The parameters adopted in MVGM are given in Table 2. The reproduced water retention curves match the data well in low and high suction regimes. The capillary and adsorption curves are explicitly separated.

Fig. 7
figure 7

Simulation of the water retention behavior of the tested soil (SW-SC) compacted at wcomp = 6% a medium dense sample b loose sample

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Table 2 Parameters for MVGM
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The water retention behavior of the medium dense (Fig. 7a) and loose samples (Fig. 7b) shows similar characteristics. The capillary water comprises a substantial portion of void space at zero suction and is highly sensitive to the suction potential. In the suction range dominated by the capillary force (from about 1 kPa to 20 kPa), a slight increase in suction will lead to a drastic reduction in water content during the drying process. In contrast, adsorption water occupies a relatively small volumetric fraction at zero suction, e.g., 24% (= \({w}_{0}^{{\text{a}}}/{w}_{{\text{max}}}\) = 0.04/0.17) for the medium dense sample and 22% for the loose sample. In the suction range controlled by the adsorptive mechanism, a significant change in suction potential is required to remove water from the soil until an oven-dry state is reached at about 1 × 106 kPa.

Figure 8 shows the experimental results of the dense (a), medium dense (b), and loose (c) samples compacted with a water content of 10%. Table 2 lists the parameters adopted in MVGM. Again, a good agreement between the reproduced curves and the SWRC measurement was observed, especially in the adsorption regime. In the capillary regime, the model slightly underestimates the water content near the air entry suction. Compared with the samples compacted at wcomp = 6%, the capillary water was discharged in a relatively wider suction range of tens of kPa. This phenomenon indicated that a higher compaction water content leads to a more dispersed pore size distribution in the capillary regime [35].

Fig. 8
figure 8

Simulation of the water retention behavior of the tested soil (SW-SC) compacted at wcomp = 10% a loose sample b medium dense sample c dense sample

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4.2 Influence of initial void ratio and compaction water content

Numerous experimental results in the literature have shown that the soil–water retention behavior is far from “characteristic” but related to initial density [58,59,60,61] and sample preparation methods [62]. Khoshghalb et al. [63] interpreted the dependency of SWRC on void ratio using the fractal approach. Pasha et al. [64] proposed a method to quantify the volume change dependency of SWRC based on the form of SWRC at the reference void ratio. This section analyzes the influence of the initial void ratio (or density) and the compaction water content on the soil water retention behavior.

Figure 9a compares the water retention curves of the loose and dense samples compacted at wcomp = 6%. One sees that their adsorption parts overlap, indicating identical values of \({K}_{{\text{a}}}\) and \({s}^{{\text{a}}}\) (see Table 2). In contrast, the initial void ratio affects the capillary water twofold, i.e., as the void ratio increases, the maximum gravimetric water content increases and the air entry value slightly decreases.

Fig. 9
figure 9

a Influence of initial density on SWRC b Influence of compaction water content on SWRC c The unique absorption curve

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Figure 9b shows the water retention behavior of the dense samples compacted at wcomp = 6% and wcomp = 10%. Again, one observes an identical adsorption curve for both samples. In the capillary regime, the slopes of the curves vary significantly, which is ascribed to the changes in pore structure. With the compaction water content increasing from 6 to 10%, the value of \({K}_{{\text{c}}}\) decreases from 2.5 to 1.0, meaning that the pore size distribution becomes more dispersed. This result implies that water content remarkably affects the soil microstructure during compaction, which is in line with the findings in the literature, e.g., Delage et al. [65].

Figure 9c demonstrates the adsorption data of the six samples (wcomp = 6% and wcomp = 10%) measured with the chilled-mirror hygrometer, showing a unique nonlinear relationship between soil suction and water content. This observation testified that the adsorption mechanism depends only on soil material properties (e.g., mineralogy and specific surface area) and not on the morphological properties of pore structure (e.g., the total void space and pore size distribution). For this reason, sample preparation conditions, including initial density and compaction water content, do not affect the adsorption curve (in terms of gravimetric water content).

4.3 Water retention data from the literature

4.3.1 Shonai due sand

In this section, the proposed model is validated using SWRC data in the literature. MEHTA et al. [23] investigated the hydraulic properties (i.e., the water retention behavior and hydraulic conductivity function) of a sandy soil from saturation to very low water content. The soil tested was obtained from Shonai sand due in Japan, which comprised 93.3% sand, 1% silt, and 5.7% clay. The initial dry density was 1.4 g/cm3. The saturated volumetric water content \({\theta }_{{\text{s}}}\) was equal to 43%. The hanging water method, pressure plate apparatus, and thermocouple psychrometer were utilized to measure water retention data at low (from zero to 10 kPa), moderate (from 10 to 1 × 103 kPa), and high suction levels (greater than 1 × 103 kPa).

Figure 10a shows the water retention data (in filled symbols) of Shonai due sand in \({\text{log}}s\) − \({\text{log}}{S}_{e}\) diagram. One sees that the drying process is characterized by two distinguishable linear segments in \({\text{log}}s-{\text{log}}{S}_{e}\) diagram. The former (LS1) reflects the pore structure of the sand grain skeleton, and the latter (LS2) is governed by the adsorption mechanism mainly contributed by the clay component. Their intersection point (denoted as “A”) separates the data into capillary and adsorption parts (i.e., the transition segment degrades to a point in this case). The \(s-{S}_{e}^{c}\) and \(s-{S}_{e}^{a}\) data are calculated and replotted (in open symbols) to determine capillary and adsorption asymptotes.

Fig. 10
figure 10

Simulation of the water retention behavior of Shonai due sand in a log s − log Se diagram b log s − θ diagram (data from MEHTA et al. [23])

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Since the water retention data was measured in terms of volumetric water content in MEHTA et al. [23], \({w}_{{\text{max}}}\) and \({w}_{0}^{{\text{a}}}\) in Eq. (19) were replaced by \({\theta }_{s}\) (= 0.43, i.e., the maximum volumetric water content at zero suction) and \({\theta }_{0}^{{\text{a}}}\) (= 0.06, i.e., adsorption capacity expressed in volumetric water content), respectively. Besides \({\theta }_{s}\) and \({\theta }_{0}^{{\text{a}}}\), the other parameters (see Table 2) were attained based on the graphical properties of the asymptotes: \({K}_{{\text{c}}}\) (= 3.0) and \({K}_{{\text{a}}}\) (= 0.20) were the slopes of the capillary and adsorptive segments; \({s}_{{\text{ae}}}^{{\text{c}}}\) (= 2 kPa) and \({s}^{{\text{a}}}\) (= 10 kPa) were suction values at the intersection of the linear segments and the horizontal line for \({S}_{{\text{e}}}=1\). As shown in Fig. 10b, the reproduced curve matches the data over the entire range of suction very well. The capillary and adsorption components were estimated and could be incorporated into other unsaturated soil property formulations.

4.3.2 Arizona soils

Recently, Jensen et al. [24] conducted a series of laboratory tests to investigate the water retention behavior of Arizona silty soils (Specific Gravity Gs = 2.65) with different clay contents. The clay mineralogy included kaolinite, illite, and smectite. Table 3 gives the physical properties of the soils. The water retention data were obtained using Temp Cell (for suction lower than 100 kPa) and WP4-T Dewpoint Potentiometer (for suction higher than 1 × 103 kPa).

Table 3 Physical properties of Arizona silty soils
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Figure 11 demonstrates the water retention data of AZ1 (clay content = 2%, Fig. 11a and b), AZ4 (clay content = 9%, Fig. 11c and d), and AZ14 (clay content = 26%, Fig. 11e and f, which is used to validate the proposed approach. Table 2 gives the adopted parameters in MVGM for these soils. The proposed model reproduces continuous water retention curves, agreeing well with the measurements in low and high suction regimes.

Fig. 11
figure 11

Simulation of the water retention behavior of Arizona silty soils: a AZ1 in log s − θ plot b AZ1 in log s − log Se plot c AZ4 in log s − θ plot d AZ4 in log s − log Se plot e AZ14 in log s − θ plot f AZ14 in log s − log Se plot (data from Jensen et al. [24])

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Based on the adsorption curves in Fig. 11a–f, it is interesting to note that the adsorption capacity increases significantly with increasing clay content because clay has a much larger specific surface area than the granular components (i.e., silt and sand). In soil AZ1 with a low clay content (= 2%), capillary water occupies a much higher volume than adsorption water at zero suction (see Fig. 11a). In contrast, adsorption water comprises a substantial portion of void space in soil AZ14 as a consequence of the high clay content (= 26%) (see Fig. 11c).

Figure 12 depicts the evolution of the adsorption curve in terms of effective degree of saturation (i.e., \({S}_{{\text{e}}}^{{\text{a}}}\left(s\right)\)), which is obtained by normalizing the water retention data in the adsorption regime with adsorption capacity. Despite different clay contents, three Arizona silty soils exhibit a unique \({S}_{{\text{e}}}^{{\text{a}}}-s\) relationship. This phenomenon might be attributed to the adsorption mechanism dominated by the identical mineralogical properties of clay minerals in these soils (e.g., specific surface area and mineral composition). In MVGM, the adsorption mechanism is characterized by an identical \({s}^{{\text{a}}}\) value (= 100 kPa) and a mean \({K}_{{\text{a}}}\) value (= 0.3) (see Table 2).

Fig. 12
figure 12

The unique normalized adsorption curve (data from Jensen et al. [24])

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4.3.3 Pachapa loam

Jackson et al. [57] conducted experiments to investigate the relationship between the water retention behavior and hydraulic conductivity of unsaturated soils. The water retention data of Pachapa loam were measured using pressure plates (at high water content) and pressure membranes (at low water content). Figure 13a shows the data presented in \({\text{log}}s\) − \({\text{log}}{S}_{e}\) diagram (\({S}_{e}=\theta /{\theta }_{s}\) and \({\theta }_{s}\) = 0.46). The total water retention curve consists of two linear segments (LS1 and LS2) and a transition segment (TS) in the \({\text{log}}s\) − \({\text{log}}{S}_{e}\) plane. A delimiting point “A” is set in the middle of the transition segment. After dividing the SWRC into capillary and adsorption components, the model parameters (see Table 2) are determined graphically. Figure 13b shows the reproduced water retention curve as well as the predicted capillary and adsorption curves in traditional log s − θ plane. The simulation is consistent with the measurement.

Fig. 13
figure 13

Simulation of the water retention behavior of Pachapa loam in a log s − log Se diagram b log s − θ diagram (data from Jackson et al. [57])

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5 Discussion

The proposed SWRC model has a simple mathematical form and is capable of simulating the water content behavior of soils from saturated to oven-dry states. It also provides a simple approach for estimating adsorption capacity. This value is similar to the residual water content defined in traditional water retention models, representing the point at which the desorption of capillary water terminates and the water phase becomes discontinuous [58]. In unsaturated soil mechanics, distinguishing between capillary and adsorption components is essential for predicting effective stress. Soil suction in the capillary regime predominates the enhancement of the effective stress in unsaturated soil. When the water content is lower than the residual value, the effective stress degrades to net stress (the difference between total stress and air pressure) due to the vanishing capillary water. Therefore, the capillary curve should be considered in the effective stress formulations instead of the total water retention curve [1].

The authors suggest using the proposed SWRC expression MVGM for granular soils with rigid microstructure. For fine-grained soils, the total volume and the pore structure may change significantly during SWRC testing [62, 66, 67]. The single parameter \({K}_{{\text{c}}}\) is incapable of characterizing the complex interaction between the varying pore size distributions and soil suction [62, 67,68,69]. On the other hand, for swelling clay minerals such as montmorillonite and bentonite, the distance between clay platelets (i.e., the interlayer distance) depends on the hydraulic state, i.e., suction [8, 70], relative humidity [71, 72], or saturation [68]. Thus, the validity of the proposed approach for fine-grained soils needs further investigation.

6 Summary and conclusions

This study experimentally investigated and modeled the water retention behavior of low plastic granular soils over the entire suction range. Soil–water retention curve tests along the drying path were performed on well-graded sand with clay (SW-SC) samples prepared at different compaction water contents and initial densities. The water retention data at low (lower than 120 kPa) and high suction regimes (greater than 1 × 103 kPa) were measured using tensiometer and chilled-mirror hygrometer techniques, respectively. The capillary water was discharged in a narrow suction range. In contrast, desorption of adsorption water took place over a wide suction range of several magnitudes, i.e., from a threshold suction to the oven-dry state. The experimental results showed that sample preparation conditions, including initial density and compaction water content, affected the capillary component of the drying curve but not the adsorption component, which depends on soil mineralogical properties.

Based on the analysis of the experimental results, two linear segments characterizing the drainage process in the capillary and adsorption regimes could be distinguished when presenting the SWRC data in a log–log plot. A novel water retention model MVGM was developed by conceptualizing the total water content as the sum of capillary and adsorption components. The model has significant advantages over the existing models in the literature (e.g., RLM and LM). The model parameters possess a clear physical meaning and can be easily determined using the graphical properties of the water retention curve. The simple parameter calibration method ensures unique values in the parameter set. The proposed framework also provides a simple method to approximate the adsorption capacity of granular soils, which is critical for the predictive formulations of other unsaturated soil properties (e.g., the effective stress and hydraulic conductivity). The model was validated against the water retention data of the tested soil and the available data of granular soils in the literature. A strong agreement between the reproduced curves and the measurements was observed.