Effect of the particle-size distribution variability on the SWCC predictions of coarse-grained materials

The particle-size distribution (PSD) is the key information required by several models for prediction of the soil-water characteristic curve (SWCC). The performance of these models has been extensively investigated in the literature; however, limited studies have been undertaken with respect to the uncertainty associated with the SWCC predictions resulting from the variability in the PSD. This study aims to investigate the influence of the variability of the PSD in the prediction of SWCCs using five different models applied to three different glass beads (GBs). The PSD curves were determined by sieve analysis, laser diffraction, and image analysis. The various testing procedures were statistically evaluated to understand the influence of variability of the PSD in terms of the coefficient of uniformity (CU) and de size of particles corresponding to 10% in the PSD (D10). For each prediction model, a combination of PSD curves and their coefficient of variation were used to estimate the SWCCs. Both the CU and D10 proved to have a strong relationship with the predicted SWCCs. The CU appears to influence more the residual suction prediction while the D10 seems to have a major role for the transition and residual stages.

The performance of the SWCC prediction models is influenced by variables such as the particle shape, mineralogical composition, and arrangement of the particles. Also, hysteresis between wetting and drying cycles are mainly due to the differences of the receding and advancing contact angles, the "ink-bottle" effect and entrapped air (Fredlund et al. 2012 [10], Lu & Likos 2004 [11]). The water may contain dissolved salts which affects its surface tension, which is also dependent on the temperature (Vargaftik et al. 1983 [12], Karagunduz et al. 2001 [13], Sghaier et al. 2006 [14]). In addition, the presence of organic matter influences the interaction forces mainly for the fine fractions (Gupta & Larson 1979 [15], Jong et al. 1983 [16], Liu et al. 2014 [17]). Thus, considering that these aspects act simultaneously to determine the soil suction, it is clear the difficulty involved in predicting the SWCC.
There are several studies that present investigation on the variability of the SWCC. Zapata et al. (2000) [18] investigated the uncertainty of direct suction measurements and that associated with the predicted SWCC. The authors found that the influence of the operator in the measurement of the SWCC was greater than that associated with prediction models based on index properties. Likos et al. (2013) [19] studied the uncertainty of the fitting parameters used for defining the relation between drying and wetting paths. The results of 25 cohesive and cohesionless soils indicated a coefficient of variation (CV) of 56% for the air-entry value ( b  ). Phoon et al. (2010) [20] presented a probabilistic analysis of the SWCC based on the log-normal and shifted lognormal distributions of soils with different textures from the Unsaturated Soil Database, UNSODA (Nemes et al. 1993 [21]   ) and also for the hydraulic conductivity function parameters of 186 soils, including clays, loams and sands. Mishra et al. (1989) [23] analyzed 250 soils and pointed out that the direct measurement of the saturated hydraulic conductivity is highly preferable than its estimation due to the uncertainty in the predictions.
Zhai & Rahardjo (2013) [24] quantified the uncertainty associated with fitting parameters of the Fredlund & Xing (1994) [3] equation. They observed a high variability in the water content in the transition zone for a silty sand. According to Zhai et al. (2016) [25], the variability of unsaturated properties is significantly greater than that of saturated properties for the 59 materials studied.
There are only a few studies in the literature that specifically discuss the assessment of the variability of the PSD. Dubé et al. (2020) [26] analyzed the influence of different sampling techniques on the variability of PSD. The authors found an average CV of 5.28% for particle sizes less than 2 mm and 10.54% for a sieve opening of 4.75 mm. Narizzano et al. (2008) [27] suggest that the mean variability of the PSD due to sampling is 4.77% for clays, gravels and sands.
Due to its central role as an input parameter of several SWCC prediction models, it is important to evaluate the variability of the PSD and how it contributes to uncertainties of the predicted SWCCs. Therefore, the key objective of this paper is to investigate the effects of the variability of the PSD in the SWCCs predicted using selected models from the literature; namely, Arya & Paris (1981)

Materials
Three glass beads (GBs) were selected in this study. Because these materials are spheres, the role of particle shape in the variability of the PSD curve can be considered relatively less complex. All the materials are relatively uniform, however, two of them are more uniform with a value of CU that is close to unity. The properties of these materials are shown in Table 1 and their physical characteristics are presented in Figure 1.

Determination of the PSD
Three different methods are used for the determination of PSDs in this study. The first method is sieve analysis (SA), which is widely used and a standard technique in the determination the size of particle. This technique is simple; however, it is time consuming. The SA results produce a discrete distribution of particle sizes because of the openings of sieves that are limited to specific sizes. In addition, the SA results do not offer any information about the shape of the particles. The second method used in the study is the laser diffraction (LD) which is based on diffraction patterns of a laser beam that is pointed to the particle. The method makes use of the intensity of the light scattered, which is directly proportional to the particle size (Beuselinck et al. 1998 [28], Yang et al. 2019 [29]).
The device used in the LD tests was the particle analyzer S3500 from Microtac. Ultrasonication was not necessary because the materials used in the study were glass beads with no aggregation between the particles. The accuracy and precision of the results are dependent on the specifications of the equipment. When ultrasound is not required, the tests usually take less than a minute. The sample mass was slightly wetted and homogenized prior to performing the tests to alleviate this limitation. The third method used for the determination of the PSD was based on image analysis (IA). This approach consists in taking a digital image of the sample and measure one or more geometrical characteristics of each particle. Some more complex image methods can provide information about the particles in three dimensions (Fernlund et al. 2007 [30], Wang et al. 2016 [31]). The images of the samples were obtained using a stereoscopic A (a) B 0 C device (Leica MZ12-5). The samples were placed into a metallic recipient covered by a black opaque paper to avoid light reflection. Samples were slightly wetted to prevent segregation. Glass beads are transparent, making it more difficult to identify the particles boundaries when the particles are stacked. This inconvenience was overcome by placing the particles in a single layer on the paper.
The stereoscopic images of the materials are shown in Figure 1. After the images were taken, they were inserted into a CAD (Computer-Aid Design) software and the shape of the particles were replicated. The ImageJ software (Schindelin et al. 2012 [32]) was used to identify and measure the shape of the particles. ImageJ is an image processor able to execute several tasks including edition, processing, and analyzing. ImageJ was used to calculate the area of each particle. Mass-volume relationships allowed the determination of volume and the weight of the particles.
For each material, three laser diffractions and two image analyses were carried out. In addition, four sieve analyses were performed on GB #1. The low CU (close to 1) of materials GB #2 and GB #3 prevented the use of sieve analysis because of the unavailability of sieves with intermediate openings.

Prediction methods
Five methods summarized in Table 1 were used in the present study to predict the SWCCs of the three glass beads. The models of Arya [5] use the capillary bundle approximation to estimate at least one parameter of the SWCC. These models were originally tested by the authors for soils with different textures, ranging from sands to clays. These models present one or more empirical input parameters determined using specific soil databases.  [9] model is comparatively more difficult to implement; however, all the input parameters have physical meaning.

Methods of data analysis
Each PSD method results in curves with different quantities of pairs values of diameter and percentage passing (d, %P). For instance, the SA of material GB #1 presents eight pairs (corresponding to eight sieve size openings) while a single IA analysis produced 434 pairs of d and %P. Thus, to avoid bias in the manner how the PSDs are used, compared, and combined, only two parameters of the PSD curve were analyzed, namely: D10 and CU. These two attributes were selected for the following reasons: i) CU (D60/D10) is used as an input variable for the models of Aubertin [8]). It is important to note that CU is independent of D10 because its definition involves both D10 and D60. However, CU and D10 may be statistically correlated.
The statistical analyses of the SWCC were performed using two different approaches, depending on how the PSD is used by the various prediction models. The first approach, used along with Arya & Paris (1981) [39]. The method combines the Taylor series approximation and an univariated point estimate method. The method provides the statistical moments of selected variables (i.e., b  , res  and θres) as a function of the frequency distributions of the input variables (D10 and CU), including their correlation. In this study, it was assumed that D10 and CU present a log-normal distribution.
The scenarios used for the analysis of the variability propagated to the SWCC consisted in varying the value of each input variable (i.e., CU and D10) around its average value (μ) by adding or subtracting one standard deviation The dispersion measures -(i.e.,  and CV) of material GB #1 were extended to the other materials because of the limited data available, allowing higher statistical significance. To prevent physically inadmissible combinations for GB #2 and #3 such as CU < 1, the logarithm of CV was used.  [11] resort in the discretization of the PSD in order to calculate the corresponding discretized SWCC. The latter model describes the interactions between particles of different sizes by a correction factor also dependent of the CU.
In summary, the evaluation of the predicted SWCC was based on three characteristics, namely, the AEV ( b  ), the residual suction ( res  ) and its volumetric water content (θres). These parameters were calculated using the two-point method presented by Pham (2005) [41]. A total of 30 predicted SWCCs (6 combinations for 5 models) were generated. Figure 2 shows the PSD curves obtained using the different techniques of measurement of the particle sizes. For GB #1, note that results of sieve analysis and laser diffraction are reasonably similar and both techniques underestimate the particle diameter in comparison to image analysis. Also, the two image analysis tests present differences under 50% of weight passing. Considering GB #2 and #3, the results produce low variability among the techniques and the curves of each technique intersects at three distinct diameters. In this case, the difference between the results are higher for GB #3. Table 3 presents the statistical measures of D10 and CU. For both parameters, the highest CV and standard error (SE) are observed for GB #1 and those corresponding values of GB #2 are approximately ten times lower. Table 4 contains the descriptive statistics of the materials for each SWCC prediction model. The average values of the θres is 0.0071 cm³/cm³, which is relatively small probably because the adsorption of coarse-grained materials is relatively low. Thus, the following analyses of the statistics showed in Table 4 are focused on b  and res  .

Results
From the results summarized in Table 4 [4] model. This implies that the first model is the most sensitive to the variability of the PSD parameters. A possible explanation is that the Arya & Paris (1981) [1] model required a significant discretization of the PSD curves because its equations produce incorrect values of matric suction in case of very narrow range between the pairs (d, %P).  For GB #1 and #3, the CVs of all models are quite similar (average of 20.5 and 9.6%, respectively), except for the Arya & Paris (1981) [1]. This means that the variability of the PSD affected in a similar manner the variability of the predicted values of b  , regardless of the prediction model. For GB #2, the same behavior is presented except for the Fredlund et al. (2002) [4] model. Zapata et al. (2000) [18] verified the variability of a sandy soil using three SWCC prediction models based on soil index properties. For the b  , the authors found a mean variability of 24.7% which is greater than the mean CV (16.6%) of the three materials used in this study. Nevertheless, this scenario is different for the residual suction. In this case, the authors obtained a mean CV of 2.7% against the 54.2% demonstrated in Table 4. Figure 3 depicts the variability of the predicted SWCCs for the models that rely on the entire SWCC (Arya & Paris 1981 [1], Fredlund et al. 2002 [4] and Alves et al. 2020 [9]) using a log-normal distribution. The confidence interval (CI) adopted was 95%. The plot in the semi-logarithmic scale shows that GB #1 presents a variability of b  lower than the variability of res  , (a) contrary to the behavior showed by GB #2 and #3. This trend was not observed in Table 4. However, considering that Gitirana Jr. & Fredlund (2016) [22] pointed out that the log-normal distribution is adequate to represent the b  and the res  , it is noticeable that different statistics approaches can lead to different conclusions. Close examination of the results summarized in Table 3 and Figure 3, suggests that the influence of D10 in b  decreases with an increase in CU. This argument is supported by comparing the average value of CU and the range corresponding to a 95% confidence interval of b  where it is possible to notice that the CU and the variability of b  are inversely correlated.  Table 5 demonstrates the statistical analyses for the models of Aubertin et al. (2003) [5] and Wang et al. (2017) [8] using the probabilistic method developed by Gitirana Jr. (2005) [40]. Similarly, as summarized in Table 4, the values of residual volumetric water content are relatively low and are going to be neglected in the analyses presented herein. In general, the CV values indicate that the b  is more sensitive to changes in the PSD curve than the res  . The probabilistic analysis results show greater variability for GB #2.  and the res  , respectively. These diagrams demonstrate the sensibility of the output parameters to each specific input variable. The closer the size of a bar is to that of the "full model", the less sensitive is the output variable to the corresponding input parameters. According to Figure 4, b  is more sensitive to D10, suggesting a better correlation between the b  and D10. The odd behavior presented by Figures 5(a) and (b) showing the CU bar wider than the "full-model" bar is caused by the negative correlation between D10 and CU (-0.952). Figure 5 also shows that the variability of D10 affects significantly the res  when compared to CU. Therefore, it seems that D10 has a major role for both the transition and residual stages of the predicted SWCCs.

Conclusions
The influence of the variability in the PSD on the predicted SWCCs was evaluated using the methods of Arya & Paris (1981) [9] for the PSD of three glass beads. Sieves analysis, laser diffraction and image analysis composed the techniques to obtain the PSD curves. Among the five models, the Arya & Paris (1981) [1] resulted in the greatest variability propagation, probably due to the influence of the discretization of the PSD curve.  The variability for the other models were, in general, reasonable akin. The results demonstrated that both b and res present variabilities with sensitivity to variations in D10 and CU. Overall, the variations in the SWCC parameters are usually similar in magnitude of those presented by D10 and CU. The CU seems to be more related to res while D10 sound to affect more significantly the b, however, the D10 appears to have an important role for all ranges of matric suction. This study demonstrates that the PSD is quite important for the predicted SWCCs which are related to the accuracy of the techniques for the PSD measurement. Studies using natural coarse-grained materials and soils with fines are necessary to complement the results and investigate the variability related to the adsorbed water.