Prediction of Soil–Water Retention Curves of Road Base Aggregate with Various Clay Fine Contents

Abstract

The invasion of clay fines can change the water retention properties of base layers significantly. There is no soil–water retention curve prediction equation that is capable of considering the effects of fines. In this study, the soil–water retention curves (SWRC) of road base aggregate with different fine contents (FC) were measured. The variations in inter-aggregate pores (formed by coarse grains) and intra-aggregate pores (formed by fines) with FC were deduced from the SWRC at different FCs. It was found that the inclusion of fines shifted and scaled the inter-aggregate pore distribution curves and scaled the intra-aggregate pore distribution curves. Based on this finding, a prediction equation for SWRC with consideration of the effects of FC was proposed. The feasibility of this equation was verified by comparing the predicted and measured results. The proposed model can be used to predict the SWRC of soil with different FCs.

1. Introduction

Interlayers are created by the interpenetration of a base layer and subgrade layer, of which fine content (FC) varies with depth [1,2,3,4]. Variations in FC results in changes in water retention properties [5,6] and soil properties [7,8,9,10,11,12]. In addition, the soil water retention curve (SWRC) has been successfully used in the prediction of the mechanical properties of soil in the unsaturated state, i.e., shear behaviour [13,14,15] and resilient behaviour [16,17,18]. Therefore, it is necessary to study soil water retention properties in depth.

With an increase in FC, the microstructure and water retention properties change [19,20,21]. Generally, FC was found to increase along with the depth, and the interlayer is made up two parts: the upper part, characterized by a coarse-grain skeleton, and the lower part, characterized by a fine matrix [1,22]. In the case of soil characterized by a fine matrix, Su et al. [6] pointed out that the SWRC of the mixture is only dependent on the dry density of fines; the coarse-grain content has a limited effect on the water retention properties of the mixture. Therefore, to obtain the SWRC of the mixture with different FCs, the prediction curve built for fines grain soil at different dry densities can be adopted, as established by previous researchers [23,24,25,26,27]. In the case of coarse-grain, a limited SWRC prediction equation has been proposed, accounting for the effects of fine content. Previously established equations are focused on capturing the bimodal feature of SWRC, which is caused by the discontinuity of the pore size distribution (PSD) between inter-aggregate pores (formed by coarse grain) and intra-aggregate pores (formed by fines) [11,28,29,30,31]. Since the effects of matric suction on the mechanical behaviour of soils can be intensified by fines [2,4,11], it appears important to propose a prediction equation to address the fine content, which is essential for the prediction of the dynamic behaviour as well as shear behaviour of the base layer soils.

In the study, the water retention properties of soils with different FCs were tested by a large-scale SWRC test apparatus which was designed by the authors. Then, the evolution of the pore size distribution (PSD) with the increase in FC was emphatically analysed. After this, the SWRC prediction equation, which is able to consider the effect of FC, was proposed and validated by comparing the measured and predicted SWRCs.

2. Materials and Methods

The studied material is a kind of well-graded gravel (crushed tuff material, CTA), with a minimum void ratio (e_min) of 0.220 and maximum void ratio (e_max) of 0.386 [32]. The commercial Kaolin was used to mix the coarse-grain soil at different FCs (FC = m_f/m_c, where m_f and m_c are the dry mass of fines and coarse-grain soil) to mimic the coarse grain fouled by subgrade soil. The gradation curves of CTA and Kaolin clay are shown in Figure 1. The specimens with various FCs were prepared, having the identical dry densities of ${\rho }_d=2.12\mathrm{Mg}/{\mathrm{m}}^3$, with a compaction degree D_c = 95.1% for the specimen with FC = 0%.

This study aims to reveal the effects of FC on SWRC when soil is the coarse grain type. Therefore, the threshold fine content (FCth), which distinguishes the ranges for the soil characterized by being coarse grain and fines, needed to be determined first. According to the method proposed by Thevanayagam [33], $F{C}_{\mathrm{th}}=\frac{\left(1{+e}_{\max }\right)\cdot G_s}{{\rho }_d}-1$ (where $e_{\max }$ refers to the maximum void ratio of the CTA; $G_s$ refers to the specific density of CTA; and ${\rho }_d$ refers to the total dry density of the specimen). For the tested soil, FCth is 8%. In the tests, the CTA was mixed with fine contents of 0%, 2%, 4%, 6%, and 8%. For specimens with FC = 0%, 4% and 8%, the test results were used to study the evolution of PSD (including pores formed by coarse-grain soil and fines) when FC increases. Using three FCs allowed us to see the evolution of PSD with FC, since we include the minimum amount of fines (FC = 0%), medium amount of fines (FC = 4%) and maximum amount of fines (FC = 8%). In addition, the specimens with FCs of 2% and 6% were used to manifest the validation of the proposed SWRC model.

In the water retention tests, a large-scale SWRC test apparatus (LSSTA), which was designed especially for coarse-grain soils, was employed to test the SWRC of soils with different FCs. A sketch diagram of the LSSTA is shown in Figure 2. The LSSTA is composed of the following parts: pneumatic controllers, a back pressure controller and a ceramic plate (with an air entry value of 300 kPa), which are used to initialize the matric suction in specimens; an axial displacement transducer, axial stress transducer and air pressure transducer, used to measure the variation of volumetric strain, air pressure and axial stress during the test; an air flushing device, used to flush the pore air generated during the test and improve the accuracy of the measurements of the water volume variation in soil; and a data logger and computer, used to record the data generated during the test. Detailed descriptions of the LSSTA can be found in the work of Cao et al. [34].

The cylindrical specimen for water retention tests was 150 mm in diameter and 75 mm in height, prepared by the moisture tamping method in three layers. For specimens with different fine contents, the dry density of the soil was kept constant, ${\rho }_d=2.12\mathrm{Mg}/{\mathrm{m}}^3$. After transferring the specimens into the LSSTA, the specimens were submerged in de-air water for 48 h to achieve a saturated state (s = 0 kPa). A net axial stress of 3 kPa was applied to the top of the specimen to ensure a good contact between the loading cap and specimen. This was carried out to accurately measure the volumetric strain in the test. The net axial stress was kept constant during the test. Then, we increased the matric suction to the prescribed value based on the axis-translation method, s = u_a − u_w. u_a is the pore air pressure which is initialized by the pneumatic controller; u_w is the pore water pressure which is initialized by the back pressure controller. After the water volume change rate Δv < 50 mm³/h, the specimen was deemed to have achieved its suction equilibrium state [34], and then the water volume variation and the axial displacement were recorded. After this, flushing the air pores generated during the test to improve the accurate measurement of the water volume variation in soil. The tests ended after s = 300 kPa (the air entry values of the ceramic disk used in the present test), and then the moisture content was measured by the direct heating method [35].

3. Effect of Fine Content on Soil Water Retention Curve and Pore Size Distribution

Figure 3a presents the SWRCs with different FCs. The SWRC shifts upward with the increase in FC, indicating that the water retention properties increase with the increase in FC. Additionally, the increase in FC results in SWRC changing from unimodal (FC = 0%) to bimodal (FC = 4% and 8%). Bimodal SWRCs are caused by the discontinuity of pore size distribution (PSD) between inter-aggregate pores (due to coarse grain) and intra-aggregate pores (formed by fines) [11,29]. Furthermore, the bimodal SWRC can be divided into four stages (Figure 3b), which are “stage a”, “stage b”, “stage c”, and “stage d”. In stage a, the pores stay saturated, with no water exchange; in stage b, the bulk water is drained from the inter-aggregate pores; in stage c, the meniscus water is drained from the intra-aggregate pores, and a large increase in matric suction leads to minor variations in Sr; in “stage d”, the water is drained from the intra-aggregate pores [27,29].

To investigate the effects of FC on the SWRCs formed by inter-aggregate pores and intra-aggregate pores, the SWRCs formed by coarse-grain and fine-grain soil need to be distinguished first. This separation is based on the characteristics of “stage c”, in which a large increase in matric suction results in a minor variation in Sr; the average value of the S_r at the start and end points of “stage c” is deemed as the end point of the SWRC formed by inter-aggregate pores and the start point of the SWRC formed by intra-aggregate pores [27,29]. The SWRCs formed by the coarse-grain soil and fine-grain soil are presented in Figure 4a,b, respectively. It can be observed that an increase in FC leads to the water retention properties of the SWRC formed by inter-aggregate pores increased and the increase in the air entry values (the boundary between “stage a” and “stage b”). Moreover, the SWRC formed by intra-aggregate pores seemed to be affected by the addition of fines.

Before analysing the evolution of PSD with the increase in FC, two basic assumptions should be made. The first one is that the pores in soils are a bunch of connecting tubes that are randomly distributed with different diameters and heights [27,36,37]; thus, the relationship between matric suction and the radius of the pore tube can be established by the Laplace equation:

$$s=\frac{2T_{\mathrm{s}}cos\alpha }{r}$$

(1)

where T_s is the surface tension of water and soil and $\alpha$ is the contact angle.

The second assumption is for a given soil, the pores with a radius smaller than $2T_{\mathrm{s}}cos\alpha /r$ are potentially occupied by water and the rest by air, which is termed as the local equilibrium assumption [23,36,38]. The volumetric water content of soil can be obtained:

$$\theta \left(r\right)={{\displaystyle \int }}_{C/s_{max}}^{C/s}f\left(lnr\right)dlnr$$

(2)

where $f\left(lnr\right)$ represents the PSD function.

For the evolution of PSD, a shifting value $ln{\chi }_i$ and a scaling factor ${\eta }_i$ were employed [23]. The scaling factor is unable to change the feature of the SWRC, due to V_v and V_w scales at identical times. In comparison, the shifting factor can change the feature of SWRC. The shifting value $ln{\chi }_i$ and scaling factor ${\eta }_i$ at a given FC can be calculated from:

$$\begin{array}{c}ln{\chi }_i=ln{\overline{R}}_i-ln{\overline{R}}_0\\ {\eta }_i=\frac{f_i^{*}}{f_0^{*}}\\ i=1,2,3,4,\cdots \end{array}$$

(3)

where ${\overline{R}}_i$ and ${\overline{R}}_0$ are the mean pore radius at a target FC and FC = 0%, respectively.

For inter-aggregate pores, the increase in FC leads to a reduction in AEV and an increase in water retention properties. This implies that the PSD of inter-aggregate pores was shifted and scaled with the addition of fines. For intra-aggregate pores, the PSD was unaffected by the increase in fine contents, indicating that the PSD was only scaled. In doing this, a conceptual model accounting for the effects of FC on the evolution of inter-aggregate pores and intra-aggregate is presented in Figure 5.

There may be some concerns raised about the PSD of intra-aggregate pores being unaffected by the addition of fines. With the increase in FC, the dry density of fines (${\rho }_{d\text{-}f}$) indeed increases, from 0.97 Mg/m³ to 0.98 Mg/m³, when FC increases from 4% to 8%. The variation in ${\rho }_{d\text{-}f}$ seems minor. Furthermore, this investigation seeks a simpler approach that accounts for the effects of FC on the evolution of SWRC, and therefore the effect of ${\rho }_{d\text{-}f}$ on the SWRC formed by intra-aggregate pore was not considered; in other words, with the addition of fines, the shifting effect on PSD was omitted and only the scaling effect was considered.

4. SWRC Model

Since the inclusion of fines leads to the different evolution patterns of inter-aggregate pores and intra-aggregate pores, two PSD functions were employed to account for the change in PSD. This method was adopted to capture the bimodal feature of the SWRC [27,28,31]:

$$f\left(lnr\right)=f_c\left(lnr\right)+f_f\left(lnr\right)$$

(4)

where $f_c\left(lnr\right)$ represents the PSD of inter-aggregate pores and $f_f\left(lnr\right)$ represents the PSD of intra-aggregate pores.

For inter-aggregate pores, the PSD function considering the variation of FC can be expressed as:

$$\begin{array}{cc}{f}_c\left(lnr\right)\stackrel{FC\mathrm{increases}}{\to }\hfill & {\eta }_i\cdot f_c(lnr+ln{\chi }_i)\hfill \\ & ={\eta }_i\cdot f_c\left[ln\left(r{\chi }_i\right)\right]\hfill \end{array}$$

(5)

For intra-aggregate pores, the PSD function considering the variation of FC can be expressed as:

$$f_f\left(lnr\right)\stackrel{FC\mathrm{increases}}{\to }{\eta }_i\cdot f_f(lnr+ln{\chi }_i)$$

(6)

By incorporating Equations (5) and (6) into Equation (4), the PSD function changes to:

$$f\left(lnr\right)={\eta }_c\cdot f_c\left[ln\left(r\cdot {\chi }_c\right)\right]+{\eta }_f\cdot f_f\left[ln\left(r\right)\right]$$

(7)

where ${\eta }_c$ and ${\chi }_c$ refer to the scaling and shifting factor for inter-aggregate pores; ${\eta }_f$ refers to the scaling factor for intra-aggregate pores.

By putting Equation (7) into Equation (2), the volumetric water content of soil can be rewritten as:

$$\begin{array}{cc}\theta \left(r\right)\hfill & ={{\displaystyle \int }}_{R_{min}}^{R}f\left(lnr\right)dlnr\hfill \\ & ={{\displaystyle \int }}_{R_{min}}^R{\eta }_c\cdot f_c\left[ln\left(r\cdot {\chi }_c\right)\right]dlnr+{{\displaystyle \int }}_{R_{min}}^R{\eta }_f\cdot f_f\left[ln\left(r\right)\right]dlnr\hfill \\ & ={\eta }_c\cdot {\theta }_{\mathrm{sc}}{S}_{\mathrm{ec}}\left(r\cdot {\chi }_c\right)+{\theta }_{\mathrm{sf}}\cdot {\eta }_f{S}_{\mathrm{ef}}\left(r\right)\hfill \end{array}$$

(8)

where $S_{\mathrm{ec}}\left(r\cdot {\chi }_c\right)$ refers to the effective saturation of inter-aggregate pore and $S_{\mathrm{ef}}\left(r\right)$ refers to the effective saturation of intra-aggregate pore; ${\theta }_{\mathrm{sc}}$ and ${\theta }_{\mathrm{sf}}$ refers to the saturated volumetric water content of the inter-aggregate pores and intra-aggregate pores.

The effective saturation is calculated by $S_e=\frac{\theta \left(r\right)-{\theta }_r}{{\theta }_s-{\theta }_r}$, where ${\theta }_s$ is the volumetric water content at the saturated state and ${\theta }_r$ is the volumetric water content at the residual state, which is assumed to be zero [28,31]. Thus, $\theta \left(r\right)$ can be rewritten as:

$$\theta \left(r\right)={\eta }_c\cdot {\theta }_{\mathrm{sc}}\cdot S_{\mathrm{ec}}\left(r\cdot {\chi }_c\right)+{\theta }_{\mathrm{sf}}\cdot {\eta }_f\cdot S_{\mathrm{ef}}\left(r\right)$$

(9)

Furthermore, the saturation of the bimodal SWRC at a given pore radius/suction can be obtained as:

$$S_e\left(r\right)=\frac{\theta \left(r\right)}{{\theta }_s}={\eta }_c\cdot \frac{{\theta }_{\mathrm{sc}}}{{\theta }_s}\cdot S_{\mathrm{ec}}\left(r\cdot {\chi }_c\right)+{\eta }_f\cdot \frac{{\theta }_{\mathrm{sf}}}{{\theta }_s}\cdot S_{\mathrm{ef}}\left(r\right)$$

(10)

where ${\theta }_s$ is the saturated volumetric water content of the soils; ${\theta }_s={\theta }_{\mathrm{sc}}+{\theta }_{\mathrm{sf}}$.

For the SWRC with the bimodal feature, $\eta$ affects the volumetric water content in the saturated state. For a given suction located in “stage c”, $S_{\mathrm{ec}}\left(r\cdot {\chi }_c\right)$ is assumed to be zero and $S_{\mathrm{ef}}\left(r\right)$ is assumed to be one; thus, the saturation of the bimodal SWRC is equal to ${\eta }_f\cdot \frac{{\theta }_{\mathrm{sf}}}{{\theta }_s}$.

Since the $S_e\left(r\right)$ increases linearly with the increase in FC (the saturation at “Stage c” is 24% for FC = 4% and 47% for FC = 8%), ${\eta }_f\cdot \frac{{\theta }_{\mathrm{sf}}}{{\theta }_s}=k\cdot FC$ is assumed, where k is a fitting parameter. Furthermore, if r approaches infinity (s approaches zero), $S_e\left(r\right)$, $S_{\mathrm{ec}}\left(r\cdot {\chi }_c\right)$ and $S_{\mathrm{ef}}\left(r\right)$ approaches one, then ${\eta }_c\cdot \frac{{\theta }_{\mathrm{sc}}}{{\theta }_s}=1-k\cdot FC$.

By putting ${\eta }_f\cdot \frac{{\theta }_{\mathrm{sf}}}{{\theta }_s}=k\cdot FC$ and ${\eta }_c\cdot \frac{{\theta }_{\mathrm{sc}}}{{\theta }_s}=1-k\cdot FC$ into Equation (9), $S_e\left(r\right)$ can be rewritten as:

$$S_e\left(r\right)=\left(1-k\cdot FC\right)\cdot S_{\mathrm{ec}}\left(r\cdot {\chi }_c\right)+k\cdot FC\cdot S_{\mathrm{ef}}\left(r\right)$$

(11)

In the present study, the van Genuchten model [39] was adopted to express the relationship between effective saturation, S_e and matric suction, s:

$$S_e\left(s\right)={\left[1+{\left(\alpha s\right)}^n\right]}^{-m}$$

(12)

By introducing Equation (12) into Equation (11), $S_e\left(r\right)$ turns into:

$$S_e\left(r\right)=\left(1-k\cdot FC\right)\cdot S_{\mathrm{ec}}\left(r\cdot {\chi }_c\right)+\left(k\cdot FC\right){\left[1+{\left({\alpha }_{f}s\right)}^{n_f}\right]}^{-m_f}$$

(13)

where ${\alpha }_f$, n_f and m_f are the fitting parameters of the SWRC formed by intra-aggregate pores.

For the SWRC of inter-aggregate pores, the formation of the SWRC is much more complicated, due to the PSD of inter-aggregate pores being shifted by the addition of fines. The SWRC considering the shifting factor needs to be determined from the integration of PSD function as Equation (2), due to the change in PSD. The PSD function is defined as follows [23,36,37]:

$$f\left(lnr\right)=-\left({\theta }_s-{\theta }_r\right)\frac{\partial S_e}{\partial s}\frac{C}{r}$$

(14)

By introducing the van Genuchten model [39] (Equation (12)) and Laplace equation (Equation (1)) into Equation (14), the PSD function can be obtained as:

$$f\left(lnr\right)={\theta }_{s}mn{\left[1+{\left(\frac{\alpha C}{r}\right)}^n\right]}^{-m-1}{\left(\frac{\alpha C}{r}\right)}^n$$

(15)

By putting Equation (5) into Equation (15), the PSD function of the inter-aggregate pores accounting for the variation of FC can be expressed as:

$$\begin{array}{cc}{f}_c\left(lnr\right)\hfill & ={\eta }_c{f}_c\left[ln(\chi r\right)]\hfill \\ & ={\eta }_c\left({\theta }_{\mathrm{sc}}-{\theta }_{\mathrm{cr}}\right)m_c{n}_c{\left[1+{\left(\frac{{\alpha }_{c}C}{{\chi }_{c}r}\right)}^n\right]}^{-m_c-1}{\left(\frac{{\alpha }_{c}C}{{\chi }_{c}r}\right)}^{n_c}\hfill \end{array}$$

(16)

where m_c, n_c, and ${\alpha }_c$ are the fitting parameters for the inter-aggregate pores and ${\chi }_c$ and ${\eta }_c$ are the shifting parameter and scaling parameter for inter-aggregate pores. ${\theta }_{\mathrm{sc}}$ is the saturated volumetric water content of inter-aggregate pores.

It can be observed from Figure 3a that AEV increases with the increase in FC, which implies that $\chi$ increases with the increase in FC. An exponential relationship was assumed to exist between $\chi$ and FC as:

$$\begin{array}{l}ln\left(\chi \right)=\beta \cdot ln\left(FC\right)\\ \chi ={\left(FC\right)}^{\beta }\end{array}$$

(17)

where $\beta$ is a fitting parameter.

By incorporating Equation (17) into (16), $f_c\left(lnr\right)$ can be rewritten as:

$$\begin{array}{cc}{f}_c\left(lnr\right)\hfill & ={\eta }_c{f}_c\left[ln(\chi r\right)]\hfill \\ & ={\eta }_c\left({\theta }_{\mathrm{cs}}-{\theta }_{\mathrm{cr}}\right)m_c{n}_c{\left[1+{\left(\frac{{\alpha }_{c}C}{{\left(FC\right)}^{\beta }r}\right)}^n\right]}^{-m_c-1}{\left(\frac{{\alpha }_{c}C}{{\left(FC\right)}^{\beta }r}\right)}^{n_c}\hfill \end{array}$$

(18)

Thus, at a given matric suction, the saturation of the SWRC formed by inter-aggregate pores is:

$$\begin{array}{cc}{S}_{\mathrm{ec}}\left(r\cdot {\chi }_c\right)\hfill & =\frac{{{\displaystyle \int }}_{r_{min}}^r{f}_c\left[ln\left(r\cdot {\left(FC\right)}^{\beta }\right)\right]dr}{{{\displaystyle \int }}_{r_{min}}^{r_{max}}{f}_c\left[ln\left(r\cdot {\left(FC\right)}^{\beta }\right)\right]dr}\hfill \\ & =\frac{{\left({\theta }_{\mathrm{cs}}-{\theta }_{\mathrm{cr}}\right){\left[1+{\left(\frac{{\alpha }_{c}s}{{\left(FC\right)}^{\beta }}\right)}^{n_c}\right]}^{-m_c}|}_0^r}{{\left({\theta }_{\mathrm{cs}}-{\theta }_{\mathrm{cr}}\right){\left[1+{\left(\frac{\alpha s}{{\left(FC\right)}^{\beta }}\right)}^{n_c}\right]}^{-m_c}|}_0^{+\infty }}\hfill \\ & ={\left[1+{\left(\frac{{\alpha }_{c}r}{{\left(FC\right)}^{\beta }}\right)}^{n_c}\right]}^{-m_c}\hfill \end{array}$$

(19)

By introducing Equation (19) into Equation (13), the function between S_e and FC that can account for the variation of FC is:

$$S_e=\left(1-k\cdot FC\right){\left[1+{\left(\frac{{\alpha }_{c}s}{{\left(FC\right)}^{\beta }}\right)}^{n_c}\right]}^{-m_c}+\left(k\cdot FC\right){\left[1+{\left({\alpha }_{f}s\right)}^{n_f}\right]}^{-m_f}$$

(20)

5. Validation of the Proposed SWRC Model

To verify the validation of the proposed model, SWRC with FC = 0%, 4%, and 8% were used to calculate the fitting parameters in Equation (20). The values of the fitting parameters are shown in

Table 1. The fitting parameters for the model.

${\mathit{\alpha }}_{\mathit{c}}$	nc	mc	${\mathit{\alpha }}_{\mathit{f}}$	nf	mf	$\mathit{\chi }$	k
1.1 × 10−5	1.174	2,781,837.05	0.015	3.48	0.054	2.36	0.0567

.

Figure 6 shows the comparison between the measured SWRC and predicted SWRC. It can be observed from Figure 6c that there is good agreement between the measured and predicted SWRCs, since the correlation coefficient (R²) is 0.9869 for FC = 0%, 0.9974 for FC = 4% and 0.9966 for FC = 8%, respectively. The large R² implies that the values of the fitting parameters are reasonable.

Figure 7 shows the predicted and measured SWRCs at FC = 2% and 6% by using the fitting parameters in Table 1. The correlation coefficient is 0.9926 for FC = 2% and 0.9986 for FC = 6%, respectively. The reasonably good agreement between the predicted and measured SWRCs indicates that the proposed model can predict SWRC satisfactorily at different FCs.

6. Conclusions

The SWRCs of soils with different FCs (0%, 2%, 4%, 6% and 8%) were measured using a large-scale SWRC testing apparatus. Based on the test results, the evolution of the PSD of the inter-aggregate pores and the intra-aggregate pores were deduced: with the addition of fine content, the PSD of the inter-aggregate pores was shifted and scaled, and the PSD of the intra-aggregate pores was only scaled. Then, an SWRC model was formulated considering the effect of fine content on the PSD of the inter-aggregate pores and intra-aggregate pores. The values of the fitting parameters were obtained from the SWRCs of FC = 0%, 4% and 8%, and then verified with two other FCs (2% and 6%). In the verification, the relatively large R² indicates that the agreement between the proposed model and testing results is satisfactory. The proposed model can predict the SWRC at different fine contents of the soil characterized by the coarse-grain skeleton.