Traditional models employed to predict the soil water retention curve (SWRC) from the particle size distribution (PSD) always underestimate the water content in the dry range of the SWRC. Using the measured physical parameters of 48 soil samples from the UNSODA unsaturated soil hydraulic property database, these errors were proven to originate from an inaccurate estimation of the pore size distribution. A method was therefore proposed to improve the estimation of the water content at high suction heads using a pore model comprising a circle-shaped central pore connected to slit-shaped spaces. In this model, the pore volume fraction of the minimum pore diameter range and the corresponding water content were accordingly increased. The predicted SWRCs using the improved method reasonably approximated the measured SWRCs, which were more accurate than those obtained using the traditional method and the scaling approach in the dry range of the SWRC.

The soil water retention curve (SWRC), which represents the relationship between the water pressure and water content, is fundamental to researching water flow and chemical transport in unsaturated media (Pollacco et al., 2017). Direct measurements of the SWRC consume both time and money (Arya and Paris, 1981; Mohammadi and Vanclooster, 2011), while estimating the SWRC from the particle size distribution (PSD) is both rapid and economical. Therefore, a number of associated conceptual and physical models have been proposed.

The first attempt to directly translate a PSD into an SWRC was made by Arya and Paris (1981) (hereafter referred to as the AP model). In this model, the PSD is divided into multiple size fractions and the bulk and particle densities of the natural-structure samples are uniformly applied to each particle size fraction, from which it follows that the pore fraction and the corresponding solid fraction are equal. Thus, the degree of saturation can be set equal to the cumulative PSD function. The soil suction head can be obtained using the capillary equation based on a “bundle of cylindrical tubes” model, and the pore size in the equation is determined by scaling the pore length and pore volume (Arya et al., 2008). Based on the principle of the AP model, many researchers have focused on improving the suction head calculations, which are commonly based on the capillary equation; but methods that are used to translate the particle diameter into the pore diameter are different (Haverkamp and Parlange, 1986; Zhuang et al., 2001; Mohammadi and Vanclooster, 2011; Jensen et al., 2015). Some models estimate the pore diameter based on the particle packing patterns (e.g., the MV model) (Meskini-Vishkaee et al., 2014), while others utilize the proportionality factor between the pore size and the associated particle diameter (e.g., the HP model and the two-stage approach) (Haverkamp and Parlange, 1986; Jensen et al., 2015). However, the scheme employed to estimate the water content has not been modified and follows the approach of the AP model. The SWRC prediction models which use the same scheme to predict the water content and only improve the suction head calculation are referred to as the traditional models in the following text.

However, these traditional models underestimate the water content in the dry range of the SWRC (Hwang and Powers, 2003; Meskini-Vishkaee et al., 2014). Therefore, some researchers have attempted to improve the water content calculation by attributing model errors to both a simplified pore geometry and an incomplete desorption of residual water in the soil pores within a high suction head range (Tuller et al., 1999; Mohammadi and Meskini-Vishkaee, 2012). Recent findings have revealed the existence of corner water, lens water and water films in soils at high matric suction heads (Tuller et al., 1999; Mohammadi and Meskini-Vishkaee, 2012; Or and Tuller, 1999; Shahraeeni and Or, 2010; Tuller and Or, 2005). Therefore, Mohammadi and Meskini-Vishkaee (2012) predicted an SWRC based on the PSD while considering the adsorbed water films and lens water between the soil particles, and slightly improved upon the traditional MV model. Tuller et al. (1999) proposed a pore space geometry containing slit-shaped spaces and derived a corresponding SWRC that considered both the water films and water inside the angular-shaped pores; however, the predicted SWRC failed to describe experimental data at an intermediate water content due to the limitations of the gamma distribution function used to characterize the pore size distribution (PoSD) (Lebeau and Konrad, 2010). Moreover, this model was mathematically complex. Mohammadi and Meskini-Vishkaee (2013) incorporated the residual water content into the MV model and consequently decreased the magnitude of the underestimation in the dry range of the SWRC. However, an accurate estimation of the residual water content remains a challenge. Meskini-Vishkaee et al. (2014) improved the traditional MV model by defining a soil particle packing scaling factor. This method could improve the estimation of the SWRC, and is particularly significant for fine- and medium-textured soils.

Many traditional models are based on a “bundle of cylindrical tubes” representation of the pore space geometry (Arya and Paris, 1981; Zhuang et al., 2001), which results in intrinsic errors when predicting the water flow in variably saturated soils. Consequently, some researchers have considered pore networks as bundles of triangular tubes, which could incorporate the contribution of water in pore corners to the water content (Helland and Skjæveland, 2007). A new pore geometry model comprised of a polygon-shaped central pore connected to slit-shaped spaces was proposed by Tuller et al. (1999) to provide a more realistic representation of natural pore spaces (Tuller et al., 1999; Or and Tuller, 1999; Tuller and Or, 2001). This pore model could represent a foundation for accurately describing the water status in natural soils, particularly in arid environments.

Therefore, the objectives of this study were to evaluate the leading factors that lead to an underestimation of the water content in the dry range of the predicted SWRC using traditional methods and to furthermore propose a method for accurately estimating the water content using a pore space geometry containing slit-shaped spaces to improve the prediction of the SWRC.

The relationship between the PSD and the PoSD is a fundamental element when predicting the SWRC from the PSD. Hwang and Powers (2003) found that the nonlinear relationship between the PSD and the PoSD is more appropriate than the linear relationship applied in the AP model and therefore described both the PSD and the PoSD as lognormal distributions. However, since the PSD and PoSD of soils do not strongly follow a lognormal distribution, this model performed very poorly for moderately fine-textured soils (Hwang and Choi, 2006). Obtaining an accurate PoSD from the PSD of a soil is highly difficult, and the errors that arise from this approach could cause inevitable errors in the predicted SWRC. However, the underestimation of the water content in the dry range of an SWRC has not been comprehensively evaluated from this perspective.

In this study, the measured PoSDs of 48 soil samples were compared with the PoSDs calculated using a traditional model (they were actually the corresponding PSDs) to identify the origins of the errors and their effects on the accuracy of the predicted SWRC. The provided 48 soil samples exhibited a wide range of physical properties (Table 1) and were selected from the UNSODA unsaturated soil hydraulic property database, which contains 790 soil samples with general unsaturated soil hydraulic properties and basic soil properties (e.g., water retention, hydraulic conductivity, soil water diffusivity, PSD, bulk density, and organic matter content) (Nemes et al., 2001). The maximum, minimum and mean values of the soil bulk density and the percentages of clay and sand of the used soil samples for the calibration stage are presented in Table 2.

Codes and texture classes of the 48 soils selected from UNSODA.

Basic soil properties of 48 samples for the model calibration.

Measured vs. calculated pore volume fraction curves
using the traditional method for

Calculating the PoSD using a traditional model

Traditional models commonly assume that the pore volume fraction of each size
fraction can be set equal to the corresponding solid fraction (Arya and Paris,
1981). Thus, the cumulative pore volume fraction can take the following form:

The routine procedures employed among the several traditional models to
translate a particle diameter into a pore diameter are different. The
equivalent pore diameter can be derived from physical properties, including
the bulk density and the particle density, or from the proportionate
relationship between the pore size and associated particle diameter. Although
the former can logically characterize a pore, a complicated pattern can
slightly reduce the model performance, while the latter approach is easy to
use, and its rationality has been demonstrated by some researchers (Hamamoto
et al., 2011; Sakaki et al., 2014). Here, the latter technique is applied,
and it can be expressed as

Estimating the PoSD from the measured SWRC

It is generally difficult to measure the PoSD of a soil; however, the PoSD
can be indirectly obtained using the measured water content and suction head
(Jayakody et al., 2014). The cumulative pore volume fraction of the

Meanwhile, the corresponding pore diameters are derived on the basis of
Laplace's equation and Eq. (4):

The calculated and measured PoSD data were fitted using a modified logistic
growth model (Eq. 6) (Liu et al., 2003):

Pore model containing two slit-shaped spaces (

The measured pore volume fraction curves for the typical samples, namely,
sand (code: 3172) and clay (code: 2360), and their calculated curves using
the traditional model are presented in Fig. 1. The small maps embedded in
Fig. 1 exhibit the measured and calculated PoSD curves, which show that the
calculated PoSD curves approximately coincide with the measured curves in the
larger pore diameter range, while in the smaller range, which corresponds to
the higher suction head range on the SWRC, the calculated values are
obviously smaller than the measured values. The underestimation of the pore
volume fraction in the smaller pore diameter range can consequently lead to
an underestimation of the water content at high suction heads. In particular,
the calculated pore volume fraction associated with the smallest pore
diameter (

In this study, the soil pore structure was conceptualized within a pore
model in which the elementary unit cell is composed of a relatively larger
circle-shaped central pore connected to two slit-shaped spaces (see Fig. 2).
Relative to the polygonal central pore connected to the slit-shaped spaces
as described by Or and Tuller (1999), both the slit width and the slit
length are proportional to the diameter of the associated central pore

When estimating the pore volume fraction using the pore model described above, the volume fractions of the central pore and slit-shaped spaces are distinguished. Considering that the sizes of the slit-shaped spaces are smaller than that of the minimum central pore, the slit-shaped spaces are accordingly classified into it. Therefore, the pore volume fractions of the soil samples were simplified into those of the central pores, but the volume fraction of the minimum central pores included that of all slit-shaped spaces. Using the geometric relationship described in Fig. 2 and the traditional assumption that the volume fraction of each unit cell (i.e., the central pore connecting to two slit-shaped spaces) is equal to the corresponding particle mass fraction, the pore volume fractions with respect to different sizes can be readily obtained.

The procedure utilized to calculate the pore volume fractions is shown in
Fig. 3. Assuming that the soil pores are composed of numerous unit cells with
various sizes, the fraction of the

Schematic of the procedure used to calculate the pore volume fraction.

To obtain the values of

Consequently, the quantitative relationship between the parameters

The routine procedure for handling a soil sample involved the following
steps. First, given the initial value of

The

Therefore, the approach was simplified by setting

The estimated values of

The values of

Twenty-nine soil samples with a wide range of physical properties were also
selected from the UNSODA database to validate the model; the codes of the
samples are summarized in Table 4 and their detailed information is
presented in Table 5. For the soil samples that were not provided with a
saturated water content

Codes of the 29 soil samples selected from UNSODA for the model validation.

To generate a detailed PSD, a modified logistic growth model (Eq. 6) was
used to fit the measured PSD data. Here, the detailed PSD was generated at
diameter classes of 2, 5, 10, 15, 20, 30, 40, 50, 60, 70, 80, 90, 100, 200,
500, 1000 and 2000

The SWRC was also predicted using the traditional method presented in Sect. 2. In the traditional method, the predicted PoSD was equivalent to the PSD (Eq. 1) and was substituted into Eq. (3) to obtain the water content. The corresponding suction heads were predicted using Eqs. (2) and (5).

A scaling approach proposed by Meskini-Vishkaee et al. (2014) was used to compare with the proposed method to demonstrate its prediction performance. The detailed calculation procedures were described by Meskini-Vishkaee et al. (2014).

The van Genuchten equation (Eq. 13) was used to fit the predicted SWRC
calculated via the three models (van Genuchten, 1980):

For each set of predictions, the agreement between the predicted and
measured water contents was expressed in terms of the root mean square error
(

The predicted and measured SWRCs in Fig. 5 show that the improved method
exhibited good fits with the measured data in the entire range of the SWRC;
moreover, the improved method was clearly better than the traditional method
and the scaling approach, especially in the dry range (the other 25 samples
are listed in Fig. S3). In this study, the scaling approach, which improved
the performance of the original MV–VG model via scaling of the parameter

Measured and predicted SWRCs for clay (code: 1360), loam (code: 3190), loamy sand (code: 3160) and sand (code: 3144).

Table 6 shows the

Basic soil properties of 29 samples for the model validation.

The measured and predicted pore volume fraction curves using the improved method and traditional method for clay (code: 1360), loam (code: 3190), loamy sand (code: 3160) and sand (code: 3144).

The root mean square errors (

The accuracy of the predicted SWRC using the improved method depends on the
accuracy of the corresponding predicted pore volume fractions. The calculated
and measured pore volume fraction curves in Fig. 6 indicate that the
predicted curves using the improved method are more similar to the measured
data than those predicted using the traditional method, thereby showing that
the proposed method performed better. The errors in the predicted pore
fractions using the traditional method mainly occur at the minimum pore size
(

The effects of a change in the estimated

The calculated slit width

When capillary water coexists with adsorptive water in the narrow pores, the
capillary and surface forces, including ionic–electrostatic, molecular,
structural, and adsorption forces, contribute to the potential energy of
water in the slit-shaped pores (Tuller et al., 1999; Iwamatsu and Horii,
1996). When considering only the capillary forces, the drainage potential in
slit-shaped pores is given as Eq. (15) (Derjaguin and Churaev, 1992):

However, the applicability of this formula is limited by the width of the
slit. Tuller and Or (2001) defined a critical slit spacing (

In our study, the critical drainage suction head for the minimum central pore
calculated using Eq. (5) is 5000 cm

The

Previous work has shown that the

Since the central pore diameter

The traditional models employed to translate the PSD into the SWRC
underestimate the water content in the dry range of the SWRC. The errors
originate from a setting that the cumulative PoSD is equal to the
corresponding PSD, which resulted in an underestimate of the pore volume
fraction of the minimum pore diameter range and consequently the water
content in the dry range of the SWRC. If slit-shaped pore spaces are taken
into consideration when estimating the PoSD with a pore model comprising a
circle-shaped central pore connected to slit-shaped spaces, the pore volume
fraction of the minimum pore diameter range will be accordingly increased;
therefore, the SWRC can be more accurately predicted. The estimation of the

The unsaturated soil hydraulic properties and basic soil
properties of samples are available from the UNSODA database
(

The supplement related to this article is available online at:

CCC developed the method and wrote the paper. DHC performed the analysis and contributed ideas and comments on the method and writing.

The authors declare that they have no conflict of interest.

This research was partially supported by the Special Fund for Basic Scientific Research of Central Colleges (310829162015) and by the National Natural Science Foundation of China (41472220). The authors thank Kang Qian for providing the UNSODA unsaturated soil hydraulic property database. Edited by: Roberto Greco Reviewed by: Fatemeh Meskini-Vishkaee and one anonymous referee