Investigation of the Water-Retention Characteristics and Mechanical Behavior of Fibre-Reinforced Unsaturated Sand

Abstract

The introduction of fibres into soil can effectively improve its engineering properties. Systematically understanding the unsaturated mechanical properties of fibre-reinforced soil is highly significant. Moreover, there is currently no suitable model for describing the water-retention characteristics of unsaturated fibre-reinforced soil. The purpose of this study is to propose a model for the soil–water characteristic curve (SWCC) that can accurately describe unsaturated fibre-reinforced soil. The research focuses on unsaturated sand reinforced with PP fibres. A series of compression and direct shear tests were performed to investigate the mechanical behaviour. In addition, the SWCC was measured using the axis-translation technique. Based on the van Genuchten (VG) model, a modified model considering fibre reinforcement (VG-CFR) is developed for quantitatively analysing the influence of fibre content on the SWCC. The results showed that the established VG-CFR model can reflect the water-retention characteristics of fibre-reinforced sand.

1. Introduction

In recent years, landslides, foundation settlements, and geotechnical structural damage have frequently occurred, causing considerable losses. Natural soils are generally considered lacking in the mechanical and geotechnical properties required for construction projects [1]. For example, silty sand, as a kind of widely distributed soil in the world, has low bearing capacity and strength. Disasters frequently occur due to a decrease in soil strength caused by an increase in moisture content; excessive moisture can result in large soil deformation under external loads [2]. Therefore, improving the resistance of sandy soil to deformation and water infiltration is necessary. Soil improvement can be achieved by adding binders (such as cement [3,4], fly ash [5], and lime [6,7,8]) and tension-resisting materials (such as fibres) to the soil to improve its engineering properties. The addition of fibres to soil is an advantageous approach in geotechnical engineering. The incorporation of fibres can improve soil liquefaction resistance [9,10], erosion resistance [11,12,13], freeze–thaw properties [14], deformation resistance [15], tensile strength (which delays crack formation) [16,17], engineering properties [18], strain-hardening characteristics [19], dynamic characteristics [20], and strength [21].

The mechanical properties and water-retention characteristics of soil are key to improving engineering properties. The mechanisms governing the shear strength and deformation properties of the fibre–soil mixture have not been fully defined [22]. Fibre reinforcement was observed to improve the mechanical response, shear strength, and ductility by providing tensile resistance at the intersection of shear-failure surfaces in soil [23,24]. The use of fibres was observed to improve the shear strength and cohesion of soil significantly [21]. The use of 6 mm-long fibres evenly distributed in soil was found sufficient to form a soil network to bear external loads. Based on a modified hyperbolic model, a nonlinear regression model was established to predict the effective shear-stress ratio. However, most of the studies have focused on saturated or dry soil; soils are usually in an unsaturated state in actual engineering. In the study of Nataraj and Mcmanis [19], soil strength was proposed to be a function of moisture and fibre contents. Moreover, the addition of fibre to sand and clay was suggested to increase the peak-friction angle and improve cohesion. Kumar et al. [25] reported that adding fibres to soil could increase the peak-friction angle of a specimen; however, this observation differs from that of other research [26]. Despite extensive research on the mechanical properties of fibre-reinforced soils, including fibre type [20,27,28], content [21,29,30,31], and distribution direction [32,33,34,35], studies on fibre-reinforced soil considering moisture content are few (Nataraj and Mcmanis [19] and Bao et al. [26]). As mentioned, most studies have focused on dry or saturated soils; however, the cases are different from the actual conditions. Moreover, the relationship between fibre reinforcement and moisture content is not fully understood. Therefore, investigating the effects of fibre length, fibre content, and moisture content on the shear resistance of soil is necessary.

On another hand, previous research has also shown that the addition of fibres can affect the water-retention characteristics of soil. For example, the effect of polypropylene (PP) fibre inclusions on the soil–water characteristic curve (SWCC) of expansive soils was investigated [36]. The fibres were found to enhance the bond among soil particles. Moreover, an increase in air entry was observed during soil desaturation. This agrees with the findings reported in the study of Soleimani [37] that the increasing amounts of fibre in fine-grained soils lead to higher desaturation degrees. The addition of fibres was shown to increase the water-retention capacity of soil because of the additional contact zone created between the soil and fibres [38]. The residual moisture content and water-retention capacity of fibre-reinforced soil were reported to increase with the fibre content [11]. These studies have shown that soil reinforcement has a significant effect on soil-engineering properties. The models commonly used to describe the SWCC include the models of van Genuchten (VG) [39], Gardner–Russo [40], McKee and Bumb [41], and Fredlund and Xing [42]. The mixture of fibres to sand significantly influences the water-retention characteristics of the soil; however, existing models do not consider the effect of fibres on soil water-retention characteristics.

In summary, previous studies on fibre-reinforced soil mostly focused on dry and saturated soil. The unsaturated mechanical properties and water-retention properties of fibre-reinforced soils require further investigation, particularly the lack of tools capable of predicting the hydraulic properties of fibre-reinforced soils. This poses numerous challenges to simulating fibre-reinforced soil structures, thereby limiting the practical application of fibres in soil reinforcement.

In this study, the mechanical properties and water-retention characteristics of unsaturated fibre-reinforced sand are investigated. The mechanical properties are examined through compression and direct shear tests. Furthermore, the effects of fibre content, fibre length, and moisture content on the shear strength, internal friction angle, and cohesion of soil are analysed. The SWCC of fibre-reinforced sand was obtained using the axial translation method. Subsequently, based on the VG model, a modified model considering fibre reinforcement (VG-CFR) is derived for quantitatively analysing the influence of fibre content on the SWCC. The research results can provide a scientific basis for improving the engineering properties of soil and gaining a deeper understanding of the water-retention characteristics of unsaturated fibre-reinforced soil. The proposed model can be used as a valuable tool for predicting the hydraulic properties of unsaturated fibre-reinforced soils.

2. Materials and Method

2.1. Materials and Test Preparation

The research object of this study is silty sand. The sand sample used in this paper is standard sand (produced in Fujian, China). It is necessary to mix the sand with quartz powder in a ratio of 9:1. The soil contains 10% fine particles (0.005–0.075 mm in size) and behaves like silty sand; the particle-size distribution is summarised in

Table 1. Particle-size distribution of sand.

Particle size (mm)	1	0.5	0.25	0.1	0.075	0.06	0.04	0.025	0.01
Cumulative percentage of fines (%)	100	88.3	22.1	11.2	10.2	9.3	6.1	5.5	1.9

. The physical properties are obtained through a series of laboratory tests; the parameters are listed in

Table 2. Physical properties of sand.

Parameters	Value	Parameters	Value
Specific gravity, $G_s$	2.62	Maximum dry density, ${\rho }_{dmax}$ (g/cm3)	4.70
Average particle size, $d_{50}$ (mm)	0.33	Minimum dry density, ${\rho }_{dmin}$ (g/cm3)	1.46
Uniformity coefficient, $C_u$	3.70	Internal friction angle, $\phi$ (°)	34.07
Curvature coefficient, $C_m$	1.97	Cohesion, $c$ (kPa)	2.94

. PP fibre is selected to reinforce silty sand, and its physical and mechanical parameters are listed in

Table 3. Material properties of PP fibre.

Length (mm)	Diameter (μm)	Specific Gravity (g/cm3)	Elasticity Modulus (GPa)	Tensile Strength (MPa)	Melting Point (°C)
6/12	30	0.91	1.0	350	165

. Based on the previous study [43], four types of fibre-reinforced sands with different fibre lengths and contents (PP-6 mm-0.25%; PP-6 mm-0.50%; PP-12 mm-0.25%; and PP-12 mm-0.50%), which were confirmed to have a significant effect on the improvement of shear strength, were used for the tests to investigate the water-retention characteristic. The PP-6 mm-0.25% represents sand reinforced with 6 mm-long fibres at a volume content of 0.25%. The steps of configuring fibre-reinforced sand are as follows: (1) configuring silt in proportion to standard sand and quartz powder; (2) add water and PP fibre to the configured silt, and configure a certain moisture content of PP fibre-reinforced sand. The specific production process of PP fibre-reinforced sand is shown in Figure 1.

2.2. Test Conditions

2.2.1. Consolidation Tests

The consolidation instrument used in the test was a single-lever consolidation instrument produced by the Nanjing Soil Instrument Factory (Figure 2). The dry density, moisture content, diameter, and height of the sand samples were 1.54 g/cm³, 5%, 61.8 mm, and 20 mm, respectively. To prepare the sample, a small amount of Vaseline was first applied to the inner wall of a ring knife to reduce friction during consolidation. The samples were prepared with five 4 cm-thick layers using the stratified static pressing method. Levelling and prestress tests were performed before starting the test. The details of tests were referred to the standard for soil test methods, GB/T 50123-2019 [44]. Each sample was loaded in several steps (50→100→200→300→400 kPa). A 24 h interval between two adjacent steps was set to obtain a stable state for each step. After the last step of consolidation at 400 kPa, a rebound test was conducted. The rebound deformation was measured 24 h after decompression in each step. Moisture content was measured after unloading; each soil sample was measured three times, and the average value was calculated. As shown in Figure 2, a sealing bag is placed in a consolidation container to reduce water evaporation and prevent water loss during the tests.

2.2.2. Direct Shear Tests

A stress-controlled direct shear instrument is used for the tests, as shown in Figure 3. Samples were prepared at the target moisture contents of 2.5%, 5%, 7.5%, 10%, 12.5%, and 15%, respectively. Rapid shearing was performed with a shear rate of 0.8 mm/min, and all samples were sheared for 3–5 min according to GB/T 50123-2019 [44]. The soil was sheared and measured four times at each moisture content, and the average value was calculated.

2.2.3. Water-Retention Tests

Water-retention capacity is the basis of mechanical studies on unsaturated soils [45,46]. The unsaturated permeability coefficient and unsaturated shear strength can be deduced from the SWCC. Because of the small suction capacity of sand, the axis-translation method was chosen to measure the SWCC of fibre-reinforced sand. The effect of fibre lengths and contents on the water-retention capacity of sand was investigated.

The axis-translation method elevates the pore air pressure to increase the pore water pressure, keeping it positive and preventing cavitation in water-drainage systems [47]. When pore air pressure rises, total stress on the sample must be increased in such a way that the net stress remains constant according to Equation (1) [48].

$${\sigma }_n=\sigma -u_a$$

(1)

where $\sigma$ is the total stress, ${\sigma }_n$ is the net stress, and $u_a$ is the pore air pressure.

The water-retention test system is shown in Figure 4. Pore air pressure was applied through a pressure chamber. Pore water pressure was applied through the ceramic disk, with the magnitude of the pore water pressure in this system equalling the atmospheric pressure. Sample saturation was achieved through the vacuum-saturation method. Clean sand without reinforcement and fibre-reinforced sand (PP-6 mm-0.25%; PP-6 mm-0.50%; PP-12 mm-0.25%; and PP-12 mm-0.50%) were used for testing. The quality of the samples was controlled according to the dry density ${\rho }^d=1.54$ g/cm³.

3. Results and Discussions

3.1. Mechanical Property Test Results

3.1.1. Consolidation Test Results

The void ratio changes of the clean sand and fibre-reinforced sand under compression and unloading tests are shown in Figure 5. For the same fibre length, the compressibility decreases with the higher fibre content. Notably, the effect of fibre length dominated that of fibre content. With the same fibre content, the compressibility of the sample with 12 mm-long fibres is lower than that of the sample with 6 mm-long fibres. Hence, both fibre length and fibre content affect the compressibility of soil. In contrast, an increase in fibre length and content increases the initial porosity ratio of the samples, resulting in reduced compressibility. Nevertheless, they improve the ability of soil to resist deformation. To further analyse the influence of fibre length and content on the soil quantitatively, the compression modulus ($E_s$) and compression index ($C_c$) are calculated by the following equations:

$$E_s=\frac{1000(p_{i+1}-p_i)(1+e_i)}{(S_{i+1}-S_i)(1+e_0)}$$

(2)

$$C_c=\frac{e_i-e_{i+1}}{\mathit{lg}{p}_{i+1}-\mathit{lg}{p}_i}$$

(3)

where $e_i$ is the porosity ratio of each step, which can be calculated as:

$$e_i=e_0-\frac{1+e_0}{h_0}\mathsf{\Delta }{h}_i$$

(4)

where $e_0$ is the initial porosity ratio; $p_i$ is the load value; $S_i$ is the unit settlement; $h_0$ is the initial height; and $\mathsf{\Delta }{h}_i$ is the deformation of the sample under a certain load level.

Table 4. Compression modulus ($E_s$), compression index ($C_c$), and rebound index ($C_s$) of soil reinforced with different fibre lengths and contents.

Fibre Length (mm)	Fibre Content (%)	Es (MPa−1)	Cc	Cs
/	/	8.360	0.078	0.0071
6	0.25	12.22	0.0534	0.0042
6	0.50	14.59	0.0449	0.003
12	0.25	15.06	0.0438	0.0021
12	0.50	17.27	0.0381	0.0031

summarises the values of $E_s$, $C_c$, and $C_s$. Fibre length and content affect the volumetric compression parameters of soil. Their most considerable influence is observed in PP-12 mm-0.5%, in which $E_s$ increases by 106% and $C_c$ decreases by 51% when compared with those of clean sand. With the increase in fibre content and length, $E_s$ increases while $C_c$ decreases, respectively. The results indicate that fibre addition can improve deformation resistance, increase the initial void ratio, reduce compressibility, enhance integrity, and increase the strength of sand. The friction force and inconsistent deformation of fibres in different directions during the compression process pull the fibres, producing interfacial forces. The magnitude of the interfacial force depends on interfacial friction and adhesion [49]. The force between the fibres and soil particles limits the relative sliding of the soil during the compression process, further reducing soil compressibility.

3.1.2. Direct Shear Test Results

The shear strength of the reinforced sand clearly exceeds that of the unreinforced soil under different levels of vertical stress, as shown in Figure 6. The shear strength increases to different degrees with increasing fibre length and content. Previous studies have shown that a critical fibre content exists with respect to the contribution of fibres to the shear strength or other strength indices of soil. When the fibre content exceeded 0.75%, the fibre contribution to the shear strength of soil was no longer positive or even became negative [50]. In the current study, only two contents, 0.25% and 0.5%, were used, and both were within the critical fibre-content range. Consequently, the influence of the critical fibre content is not observed in this study. Results show that the shear strength of soil with 0.5% fibre content was better than that with 0.25% fibre content. Therefore, the addition of fibres has improved the shear strength of sand. In addition, moisture content is also an important factor affecting the shear strength. The shear strength of clean and reinforced sands first increases and then decreases with increasing moisture content. By comparing the shear strengths of clean and reinforced sands at different moisture contents, it is found that, when the moisture content of the soil is 10%, the shear strength increases more rapidly with increasing vertical stress compared with those of soils with other moisture contents. The soil with 10% moisture content and reinforced with 6 mm-long fibres exhibits the highest shear strength. This indicates that 10% is the optimal moisture content for this sample type, and the shear strength decreases when the moisture content exceeds 10%. An increase in moisture content was found to lead to the capillary tension of pore water [26], increasing apparent cohesion, which could prevent the movement of soil particles. However, other studies reported contradictory conclusions. According to the study of Tang et al. [51], the interfacial peak strength and interfacial residual strength of fibre-reinforced soil decrease with increasing moisture content.

To further analyse the reinforcement mechanism of fibres, the effects of fibre length, fibre content, and moisture content on the shear strength parameters are analysed. As shown in Figure 7, the addition of fibres did not show the obvious effect on the internal friction angle of sand. This is similar to the results reported in the study of Lovisa et al. [26] in which the addition of fibres to sand does not significantly change the final friction angle. Notably, the addition of a small amount of fibre to the sample with 2.5% moisture content can increase the internal friction angle of sand; however, a further increase in fibre content decreases the internal friction angle. In Figure 7a,b, the increase in fibre length is observed to reduce the effect of water content on the friction angle of the soil. This causes the internal friction angle to fluctuate within a small range with different moisture contents. In general, the 12 mm fibres have a negligible effect on the internal friction angle of unsaturated silty sand compared to the 6 mm fibres. On another hand, distinct differences in the contribution of moisture content to the internal friction angle at different intervals are observed. For moisture contents ranging from 10% to 12.5%, a significant decrease in the internal friction angle occurs. In contrast, for moisture contents ranging from 12.5% to 15%, the internal friction angle does not change significantly. Hence, the influence of moisture content on the internal friction angle of soil does not show a simple linear relationship. Adding a small amount of water to sand can fill the pores and increase the friction angle. After reaching a critical value, a continuous increase in moisture content can decrease the soil suction and internal friction angle.

The influence of moisture content on the cohesion of reinforced sand is shown in Figure 8. The soil with 10% moisture content exhibits the highest cohesion. When the moisture content increased from 10% to 12.5%, the cohesion of the sample decreased. This indicates that, when the critical moisture content is around 10%, the highest cohesion is achieved. In addition, when the moisture content was larger than the critical content, the soil cohesion decreased faster than when the moisture content was less than the critical content. The fibre content and length also affected the cohesion of reinforced sand. The cohesion of reinforced sand increases with the fibre content and length. Comparing Figure 8a,b, with increasing moisture content, the cohesion of the sample with 12 mm-long fibres exceeds that of the sample with 6 mm-long fibres. This indicates that the sensitivity of sand to moisture content increases with the fibre length.

3.2. Determination of SWCC

The SWCC obtained using the axis-translation method is shown in Figure 9. It indicates that the water-retention curves of the clean sand and fibre-reinforced sand have three stages. In the first stage, the soil-mass saturation level is high (saturated state), and the soil pores are almost filled with water. In the second stage, the soil-mass saturation gradually decreases. Pore water in the soil mass is squeezed out, the unsaturated property gradually increases, and the matric suction rapidly increases. In the third stage, the moisture content of the soil mass slightly changes, a small amount of water remains in the discontinuous pores, and the moisture content slightly changes because of the increase in matric suction. The change in saturation under the same matric suction is shown in Figure 10. It indicates that, under each condition, the saturation of the fibre-reinforced sand is higher than that of clean sand. The saturation increases with the fibre content and decreases with the increasing fibre length. However, the influence of fibre content on saturation is more significant. The main reason is that the fibre and soil-particle crystals of sand are interwoven into a porous and dense network structure, providing additional contact area. Therefore, the water retention of the soil mass improves. The higher the fibre content, the better the water-retention performance. When the fibre content increases, the dense network structure formed by the soil and fibre increases, enlarging the contact area.

4. SWCC Model Considering Fibre Reinforcement

Among the commonly used SWCC models are VG (van Genuchten) [39], Gardner–Russo [40], Mckee and Bumb [41], and Fredlund and Xing [42]. The SWCCs fitted by these four models are shown in Figure 11. The VG model fits well on the SWCC of clean sand. It fully represents the saturated, transition, and dry sections of the curve. In the region close to saturation, the continuity of the function is maintained, and the fitting effect is satisfactory. The fitting curve of the Gardner–Russo model for sand is not smooth, and the fitting effect is inadequate in the saturated section. The relative accuracy of the Fredlund and Xing model in fitting parameters exceeds 0.97. However, the form of the model is complicated; consequently, its application is affected detrimentally to a certain extent. The coefficients of determination of the McKee and Bumb model for fitting the curves of five different samples all exceed 0.96. The McKee and Bumb model has a simpler form than the Fredlund and Xing model; however, its water-retention fitting curves also appear discontinuous and nonconvergent in the saturated section.

The relevant parameters of the VG model (including air-entry value (AEV), VG model parameter (α, n), and residual saturation (S_rr) are listed in

Table 5. Parameter values used in the VG model.

Parameter	Clean Sand	PP-6 mm-0.25%	PP-6 mm-0.50%	PP-12 mm-0.25%	PP-12 mm-0.50%
α	0.52	0.37	0.34	0.33	0.47
AEV (kPa)	1.92	2.70	2.94	3.03	2.13
n	1.95	2.20	2.16	2.32	2.03
Srr (%)	0.014	0.021	0.031	0.028	0.032

. The addition of fibre can reduce the α value of sand. Such reduction is more distinct, especially for samples with low contents of 6 mm-long and 12 mm-long fibres. The fibres have a slight effect on the value of n. The S_rr value of the fibre-reinforced sand is significantly higher than that of clean sand; it increases with the fibre content and length. This indicates that the addition of fibres improves the water-retention capacity of soil. This is mainly because numerous randomly distributed fibres interweave, form a spatial network structure in the soil, interlock with soil particles, and form a fibrous soil-wrapped ball. The presence of fibres provides an additional contact area for water molecules. With a wide surface area of soil particles, more water is required to satisfy the energy requirements for the interaction between particles and pore water, corresponding to high suction. Consequently, the water-retention characteristics improve [52].

The main factors affecting the total soil suction are capillary action [53,54], short-range adsorption (interaction between particles and pore water) [55], and infiltration [56]. The foregoing analysis indicates that fibres affect the water-retention characteristics of soil. Moreover, the additional contact area between the fibre and soil affects pore structure and compactness. The main factors affecting capillary action are the changes in soil-pore structure and pore-size distribution, which are inevitably affected by the addition of fibres. Therefore, the fibres affect the capillary action of soil and soil water-retention characteristics. Because the fine soil particles used in this study only account for approximately 10% of the dry weight of the soil, the short-range adsorption is low. Moreover, hydration cannot be quantified; hence, it is ignored. Infiltration is the result of the dissolution of solutes in pore water. However, the fibres do not change the chemical properties of the soil; therefore, the effect of fibres on infiltration is negligible. The addition of fibres increases the contact area between soil and water molecules, influencing soil parameters, such as pore-structure size and compaction state. The large surface area of soil particles requires more water to satisfy the high matric suction. The energy requirement for the interaction with pore water affects the water-retention characteristic and improves the water-retention capacity of the soil. Therefore, based on the VG model, a modified model considering fibre (VG-CFR) is derived and verified.

The VG model [39] expressed by Equation (5) describes a relatively simple water content–hydraulic pressure head relationship:

$$\left\{\begin{array}{c}\frac{\theta -{\theta }_r}{{\theta }_s-{\theta }_r}={\left[1+{\left|\alpha h\right|}^n\right]}^{-m},h<0\\ \theta \left(h\right)={\theta }_s,h\ge 0\end{array}\right.$$

(5)

where $\theta$ is the water content; ${\theta }_s$ is the saturated water content; ${\theta }_r$ is the residual water content; $h$ is the pressure head; $\alpha$ is the parameter associated with the inlet value; and $m$ and $n$ are the curve shape parameters ($m=1-n$ or $m=1-2n$) based on the VG model. The model of Mualem [57] is used to predict the hydraulic conductivity, $K_r$, as follows:

$$K_r={\Theta }^{1/2}{\left[{\int }_0^{\Theta }\frac{1}{h\left(x\right)}dx/{\int }_0^1\frac{1}{h\left(x\right)}dx\right]}^2$$

(6)

The dimensionless water content of fibre-reinforced soil is expressed by Equation (7):

$$\Theta =\frac{\theta -{\theta }_r}{{\theta }_s-{\theta }_r}$$

(7)

The dimensionless expression of the pressure head is given by Equation (8):

$$\Theta ={\left[\frac{1}{1+(\alpha h{)}^n}\right]}^m$$

(8)

where $a$, $n$, and $m$ are three unknown parameters. To simplify the equation in the VG model, parameters $m$ and $n$ in Equation (8) are limited; $m=1$ has been verified [58]. A simple closed-form expression for $K_r(\Theta )$ is obtained. Solve $h=h(\Theta )$, and substitute the resulting expression into Equation (6):

$$K_r(\Theta )={\Theta }^{1/2}{\left[\frac{f\left(\Theta \right)}{f\left(1\right)}\right]}^2$$

(9)

$$f(\Theta )={{\int }_0^{\Theta }\left[\frac{x^{1/m}}{1-x^{1/m}}\right]}^{1/n}dx$$

(10)

Substitute $x=y^m$ into Equation (10) as follows:

$$f(\Theta )={\int }_0^{{\Theta }^{1/m}}{y}^{m-1+1/n}(1-y{)}^{-1/n}dy$$

(11)

In Equation (11), it is easily shown that, for all integer values of $k=m-1+1/n$, the integration can be carried out without difficulties. In a specific case, $k=0$ and $m=1-1/n$. Accordingly, the closed-form expression can be obtained as

$$f\left(\Theta \right)=1-(1-{\Theta }^{1/m}{)}^m,m=1-1/n$$

(12)

Because fibre content has a more significant effect on the hydraulic characteristics of fibre-reinforced soil, the change in the water-retention capacity of the soil is only related to the change in fibre content, and the influence of fibre length is ignored. Let $k=ac$ (where $a$ is the parameter of the change in water-retention property caused by an increase in fibre content, and $c$ is the fibre content). In the VG model, $m$ is related to the symmetry of the SWCC. After adding the fibres, the symmetry of the SWCC changed. Note that if $k=ac$, then $ac=m-1+1/n$. Parameter $m$ in the formula is related to $n$ and the fibre content $c$. Substitute $f(1)=1$ into Equation (9):

$$K_r(\Theta )={\Theta }^{1/2}{\left[1-(1-{\Theta }^{1/m}{)}^m\right]}^2,m=ac+1-1/n$$

(13)

The hydraulic conductivity (Equation (5)), represented by the pressure head, is substituted into Equation (11):

$$K_r(\Theta )=\frac{{\left\{1-(\alpha h{)}^{n-1}{\left[1+(\alpha h{)}^n\right]}^{-m}\right\}}^2}{{\left[1+(\alpha h{)}^n\right]}^{m/2}},m=ac+1-1/n$$

(14)

From the soil–water retention curve and hydraulic conductivity, an expression for soil–water diffusivity can be derived [39]:

$$D(\theta )=K(\theta )\left|\frac{dh}{d\theta }\right|$$

(15)

The foregoing expression leads to the following equation for $D(\theta )$:

$$D(\theta )=\frac{(1-m)K_s}{\alpha m({\theta }_s-{\theta }_r)}{\Theta }^{1/2-1/m}\left[(1-{\Theta }^{1/m}{)}^{-m}+(m-1\left)\right(1-{\Theta }^{1/m})-2\right]$$

(16)

where $K_{\mathrm{s}}=K/K_r$ denotes the saturated hydraulic conductivity. In summary, the derived VG-CFR model can be expressed by Equation (15):

$$\left\{\begin{array}{c}\frac{\theta -{\theta }_r}{{\theta }_s-{\theta }_r}={\left[1+{\left|\alpha h\right|}^n\right]}^{(ac+1-1/n)},h<0\\ \theta \left(h\right)={\theta }_s,h\ge 0\end{array}\right.$$

(17)

where $m=ac+1-1/n$; $c$ is the fibre content; and $a$ is the parameter of the variation caused by the fibre content. The value range is $ac-1/n>1$.

The modified VG model, considering fibre reinforcement (VG-CFR), is used to fit the test results of the SWCC for both the test results of this study and in previous studies [38]; the calculation results are shown in Figure 12. In the figure, R² is the coefficient of determination (a high R² value indicates that a measured value approximates the fitted value). The average R² value for the low matric suction range is 0.986. The average R² value for the high matric suction range is 0.987. The test results and calculation results from the VG-CFR fit well, proving the accuracy of the model.

5. Conclusions

This study investigated the mechanical properties and water-retention characteristics of fibre-reinforced unsaturated sand and developed an improved VG-CFR model. The main conclusions of this study are summarised as follows.

(1)

The compressibility of soil decreases with increasing fibre length and content. The soil’s compression modulus increases with the fibre content and length, whereas the compressibility index exhibits an opposite trend. The addition of fibre increases the initial void ratio. The interface force at the fibre–soil interface can reduce the compressibility of sand;

(2)

Under the fibre content in this study, the influence of fibres on the internal friction angle is minor, while enhancing the cohesive forces to elevate the shear strength of the soil. Before reaching the critical water content, the internal friction angle and cohesion of the soil increase with increasing water content but decrease beyond the critical water content. Water can fill the soil pores, increasing the internal friction angle and cohesion. The main reason for the decrease in the internal friction angle and cohesion is the significant reduction in soil suction after exceeding the critical water content;

(3)

The test results of the axis-translation method show that the addition of fibres can improve the water retention of sand. The compression test shows that the addition of fibres can strengthen the initial void ratio of the soil, affect the capillarity of the soil, provide additional contact area between the soil and water molecules, and improve the water-retention characteristics of the soil.

The proposed model can be used in the design and evaluation of engineering projects related to fibre-reinforced soil, such as slopes and roadbeds Furthermore, this model can be utilized in numerical methods to determine the parameters of water-retention properties for unsaturated fibre-reinforced soil. The lengths of fibres need to be considered to further demonstrate the applicability of the proposed model in this study in future work.