The calcareous sand used in this study was collected from a reclaimed island-reef in the South China Sea (Figure 1). Calcareous coarse particles and fine particles are mixed together at the sampling site, and the particle size distribution is extremely non-uniform (Figure 1C).

Considering the weak water-holding capacity of coarse-grained calcareous sand (Hu et al., 2019), only calcareous sand with a particle size less than 0.25 mm was used to observe the water desorption phenomenon in this study. The calcareous soil particles in the sample are classified as coarse particles or fine particles. In this paper, particles 0.075–0.25 mm in size served as the skeleton of the sample and are called coarse particles; particle less than 0.075 in size are called fine particles. To study the effect of fines content on the soil-water characteristics of calcareous sand, calcareous sand samples with fines contents (C _F ) of 0, 10, 20, 30, 40, and 50% were prepared by adjusting the mass percentage of the fines. According to the Chinese National Standard of Soil Test Method (SL237, 2019), samples in which the mass of coarse particles above 0.075 mm in size accounts for no less than 50% of the total sample are defined as silty sand. Therefore, all the samples used in this study, with fines contents of 0, 10, 20, 30, 40, and 50%, were calcareous silty sand. For convenience, calcareous silty sand samples with fines contents of 0, 10, 20, 30, 40, and 50% were named CSS1, CSS2, CSS3, CSS4, CSS5, and CSS6, respectively. According to the Chinese National Standard of Soil Test Method (SL237, 2019), the hydrometer method was used to measure the size distribution of particles less than 0.075 mm in size. The particle size distributions of the samples are shown in Figure 2, and physical parameters of calcareous silty sand with different fines contents are listed in Table 1.

According to Figure 2 and Table 1, it is found that 1) Only the non-uniformity coefficient (C _u ) and curvature coefficient (C _c ) of CSS5 can simultaneously satisfy C _u > 5.0 and 1.0 < C _c < 3.0; therefore, the particle size distribution of CSS5 is good, while the particle size distributions of the other samples are poor. 2) The specific gravities of calcareous silty sand samples are uniformly 2.73. The samples were collected from the same marine environment with identical mineral compositions and proportions; hence, they have the same specific gravity.

A 1500F1 15 bar diaphragm pressure plate apparatus, manufactured by Soilmoisture Equipment Corp., was used in this study to measure the change in volumetric water content of a sample with matric suction ranging from 0 to 1,500 kPa (Tao et al., 2018). The pressure plate apparatus is primarily composed of six parts, i.e., nitrogen cylinder, pressure reducing valve, regulating valve, pressure chamber, porous ceramic plate, and water container (Figure 3). The nitrogen cylinder is used to continuously provide air compression stress to the pressure chamber, and the pressure reducing valve and the regulating valve are used to adjust the air compression stress to the designed value. The size of the pressure chamber is 10 cm in internal depth and 30 cm in diameter. Within the pressure chamber is a porous ceramic plate where the sample is placed during the test. Small pores are densely distributed on the ceramic plates. After the ceramic plate is immersed in water and saturated, a layer of shrinking film emerges on the surface of small pores due to surface tension, allowing for the passage of water but preventing air to enter the small pores. A stress difference forms between the inside and outside of the shrink film, and the water in the sample penetrates the pores in the ceramic plate and converges into the water container. When the mass of the water drained from the sample changes at a rate less than 0.1 g/24 h (Pham, 2005), the matric suction in the sample and the air compression stress in the pressure chamber reach an equilibrium state, and the matric suction in the sample is equal to the air compression stress. This means that, under equilibrium conditions, the change in the matric suction of a sample during water desorption can be assessed through the real-time monitoring of the air compression stress in the pressure chamber. In this study, calcareous silty sands with different fines contents took 4–5 days to reach an equilibrium state under different air compression stresses. During the test, the water drained by the sample is completely collected by the water container, and the sample volume is assumed to remain constant during the test. By monitoring the amount of water in the container under each level of air compression stress in a real-time, this study obtained the relationship between volumetric water content and matric suction, i.e., SWCC.

Water desorption tests were carried out on the samples with different fines contents. In general, the interlocking strength of calcareous silty sand is low, and under a low dry density, it is difficult for samples to take shape when saturated. In this study, the samples with different fines contents were tested at a dry density of 1.52 g/cm³ (As shown in the Table 2). To clarify the effect of dry density on the SWCC of calcareous silty sand, calcareous silty sands with dry densities of 1.44 g/cm³, 1.57 g/cm³, and 1.63 g/cm³ were tested with a fines content of 20%, and the samples were named as CSS7, CSS8, and CSS9, respectively (As shown in the Table 2).

The calcareous sand collected from the sampling site was first dried at 105°C and then cooled to room temperature (i.e., 25°C). Sieving was carried out to obtain the fine particles (d＜0.075 mm) and coarse particles (d = 0.075–0.25 mm). The sample mass was determined based on the dry density and size of the sample (i.e., diameter 61.8 mm, height 20 mm). Combined with the sample mass and the fines content (C _F ), the mass of coarse particles (d = 0.075–0.25 mm) and fine particles (d＜0.075 mm) in the sample was determined, respectively. The coarse and fine particles in the sample were evenly mixed, and the sample was formed into cutting ring shape through compression. Samples with different fines contents (C _F ) and dry densities (ρ _d ) were prepared, and the individual samples were 61.8 mm in diameter by 20 mm in height. Samples were fixed on a saturator and saturated via immersion saturation and vacuum saturation (i.e., saturation degree ≥95%). Each saturated sample was weighed. The mass water content (ω) and volumetric water content (θ) of a saturated sample can be calculated using Eqs 1, 2: ω = m ω m d = m s − m d m d , (1) θ = ω ρ d ρ ω , (2) where m _ω is the mass of water in the saturated sample; m _d and m _s are the mass of the dry and saturated sample, respectively; ρ _d and ρ _ω are the dry density of the sample and the density of water at 4°C.

The porous ceramic plate was immersed in distilled water and vacuum saturated for at least 3 h. The plate was then horizontally placed in the pressure chamber after the water on the surface was dried and the drainage line was connected. Then, the saturated sample was placed on the porous ceramic plate, and the pressure chamber was sealed after the sample was placed in close contact with the porous ceramic plate.

During testing, the air compression stress in the pressure chamber was regulated to be 0, 5, 10, 20, 35, 70, 120, 300, and 750 kPa using the axial translation technique (Li et al., 2016). Due to the air compression stress, the water in the saturated sample was drained into the water container, and the mass of the water in the water container was measured using a balance with precision of 0.001 g, three to five times a day. When the daily change in the mass of the drained water was less than 0.1 g, the sample was determined to have reached an equilibrium state under the current air compression stress. The air pressure stress was gradually applied to the sample until the sample reached equilibrium under the air pressure stress of 750 kPa.

After testing, the sample was taken out of the pressure chamber, and the water content of the sample was measured to obtain the mass of water in the sample. The mass of water drained from the sample under each air compression stress level is subtracted from the mass of water in the saturated sample before testing (m _ω ). The calculated value for the mass of water in sample after test is obtained and compared with the corresponding measured value, and the correctness of test data is rechecked. The mass of the saturated sample before testing (m _s ) and the mass of the water drained from the sample under each air compression stress level were used to calculate the total mass of the sample under each air compression stress level (m _l ). The volumetric water content of the sample under each air compression stress level (θ _l ) can be calculated using Eq. 3: θ l = m l − m d m d × ρ d ρ ω . (3)

Many SWCC models have been proposed based on experimental results and various hypotheses. Among them, the Fredlund-Xing model, Van Genutchen model, and Zhong Fangjie model have been widely applied in geotechnical engineering. The Fredlund-Xing model (Fredlund and Xing, 1994) was developed based on the relationship between SWCC, the capillary model, and the pore distribution function, which is expressed as follows: θ = θ s { ln [ e x p + ( ψ a ) b ] } c , (4) where θ _s is the saturated volumetric water content; ψ is the matric suction; exp is a constant with a value of 2.718; a, b, and c, are fitting parameters, where a is related to air entry value, and the greater the a value, the greater the air entry value; b is a parameter related to the slope of SWCC in the transition stage. As b increases, the turning near the point corresponding to the air entry value on SWCC increases as well. c is a parameter related to the matric suction in the residual stage, and the smaller the c value, the gentler the SWCC in the residual stage (Fredlund and Xing, 1994).

The SWCC model proposed by Van Genutchen (1980) based on Mualem’s theory is expressed as: θ = θ r + θ s − θ r [ 1 + ( a 1 ψ ) b 1 ] c 1 , (5) where θ _r is the residual volumetric water content; a ₁, b ₁, and c ₁ are fitting parameters.

Zhong (2007) proposed a SWCC model based on the results of soil-water characteristic tests on silica sand, expressed as follows: θ = a 2 + b 2 1 + ( ψ / c 2 ) d , (6) where a ₂, b ₂, c ₂, and d are fitting parameters.

In this study, the SWCC of calcareous silty sand was fit using these classical theories to check their applicability for describing the soil-water characteristics of calcareous sand.

As shown in Figure 4, the SWCCs of calcareous silty sands with varying fines content show the same trend as those of SWCCs of terrigenous unsaturated fine-grained soil (Rao and Singh, 2010; Ma et al., 2015; Han et al., 2017). Depending on the air entry value and residual volumetric water content, the SWCC can be divided into three stages, i.e., the boundary effect stage (ψ ≈ 0–5 kPa), transition stage (ψ ≈ 5–10 kPa), and residual stage (ψ ≈ 10–750 kPa) (Figure 5).

The SWCCs of calcareous silty sands with different fines contents varied significantly from each other in the boundary effect stage (ψ ≈ 0–5 kPa). A statistical quantity δ _i-j was defined to characterize the declining amplitude of the volumetric water content of calcareous silty sand during the gradual increase of matric suction from i to j, as expressed below: δ i − j = θ i − θ j j − i P a × 100 % , (7) where θ _i and θ _j are the volumetric water contents corresponding to matric suctions i and j, respectively. It’s worth noting that volumetric water content (θ _i or θ _j ) and statistical quantity (δ _i-j ) are dimensionless. In order to realize the dimensionless formula, the matric suction (i or j) is divided by standard atmospheric pressure stress. P _a is the standard atmospheric pressure stress, i.e., 101.3 kPa. Through statistical analysis of δ _0-5 of the samples with varying fines contents (CSS1, CSS2, CSS3, CSS4, CSS5, and CSS6) in the boundary effect stage, the δ _0-5 of the samples were determined to be 122.13, 107.12, 96.33, 71.42, 37.05, and 19.63%, respectively. δ i − j of the sample decreased with increasing fines content. In the boundary effect stage, the reduction of volumetric water content of calcareous silty sand with increasing matric suction further decreased with increasing fines content. The boundary effect stage corresponded to the low matric suction state, and the quantity and distribution characteristics of large pores containing free water were primary factors controlling the sample water-holding capacity within this stage. The large pores with higher fines content were effectively filled by the fine particles, causing the δ i − j of the sample to decrease.

In the transition stage (ψ ≈ 5–10 kPa), the volumetric water contents of calcareous silty sands with varying fines contents decreased abruptly with increasing matric suction, showing a high sensitivity. When the SWCC in the transition state was extended upwards to intersect with the horizontal line corresponding to the initial volumetric water content, the abscissa (i.e., matric suction) of the intersection point represents the air entry value of the sample (Figure 5). The air entry values of the samples were obtained in this way, i.e., 7.42, 7.80, 7.74, 7.85, 7.87, and 7.97 kPa, respectively. The air entry value of calcareous silty sand increased with increasing fines content except at certain discrete points. For samples with a fines content not exceeding 50%, the air entry values ranged from 7.42 to 7.97 kPa. Because air entry value characterizes the matric suction taken for the sample transforming from a quasi-saturated state to an unsaturated state (Pasha et al., 2015), a higher air entry value of samples with higher fines content suggests that the water-holding capacity of the samples improved by the increasing fines content, making the transition of the sample from a quasi-saturated state to an unsaturated state more difficult. However, the difference in the air entry values of the samples is limited, and the contribution of a higher fines content to the improvement of water-holding capacity is limited.

In the early residual stage (ψ ≈ 10–70 kPa), the SWCCs of the samples varied significantly from each other, and the higher the fines content, the higher the volumetric water content of the sample. For instance, under a matric suction of 20 kPa, the volumetric water contents of CSS1 and CSS6 were 13.25 and 24.27%, respectively. In the residual stage, the samples have been already transitioned from an unsaturated state to a quasi-dry state, and the water drained by the sample was capillary water and bound water (Wu et al., 2020). The fines particles in the samples improved the water-holding capacity, especially the capacity to adsorb capillary water (or weakly bonded water) within the calcareous silty sand. During the early residual stage, the water drained from the sample was predominantly capillary water; however, the porous structure of the samples could provide sufficient adsorption space for bound water even in the absence of fine particles, such that bound water can be firmly adsorbed onto particle surfaces (Wu et al., 2020). Therefore, the SWCCs of the samples approximately overlapped with each other in the late residual stage (ψ ≈ 70–750 kPa). The residual volumetric water contents of CSS1, CSS2, CSS3, CSS4, CSS5, and CSS6 were 16.80, 17.83, 17.96, 18.19, 18.22, and 16.48%, respectively. Except for some individual scattered points, the residual volumetric water content of calcareous silty sand increased with increasing fines content, indicating that the quantity of small pores containing capillary water and bound water inside the samples increased with increasing fines content and the water-holding capacity of the calcareous silty sand increased. However, for a fines content not exceeding 50%, the differences in residual volumetric water content between the samples were extremely small, and the maximum deviation among them is 1.74%, indicating that increasing fines content could only improve the water-holding capacity of calcareous silty sand to a limited degree, which is consistent with the conclusions of existing studies (Al-Badran and Schanz, 2009; Dolinar, 2015; Jiang et al., 2020). Additionally, the small size deviation between coarse and fine particles may also be the main reason for this phenomenon.

Figure 4 also shows that the initial volume water contents of calcareous silty sands with varying fines content have little difference; however, with the increasing fines content, the initial volume water content of calcareous silty sand presents a downward trend. This experimental phenomenon can be explained by the microscopic structure of calcareous silty sand. Soil is made up of solid, liquid and gas. When calcareous silty sand is in a saturated state, there are only solid and liquid two phase material inside. At this time, pore water exists in the pores of calcareous silty sands in the form of bound water, capillary water and free water. Among them, free water accounts for a large proportion in pore water and mainly exists in large pores; the proportion of bound water and capillary water in pore water is small, which mainly exists in small pores (Wu et al., 2020). With the increasing fines content, the large pores decrease and small pores increase in calcareous silty sand. Accordingly, the free water content decreases, while the combined water and capillary water content increases. With the increasing fines content, the decrease of free water content is greater than the increase of the content of bound water and capillary water, resulting in the volume water content of calcareous silty sand in saturated state (i.e., initial volume water contents) does not increase with the increasing fines content.

Figure 6 shows the SWCCs of calcareous silty sand with a fines content of 20% under different dry densities (CSS3, CSS7, CSS8, and CSS9). The SWCCs of the samples differ from each other primarily in the boundary effect stage (ψ ≈ 0–5 kPa) and the residual stage (ψ ≈ 10–750 kPa). In contrast, in the transition stage (ψ ≈ 5–10 kPa), the SWCCs of the samples approximately overlapped with each other.

In the boundary effect stage (ψ ≈ 0–5 kPa), the δ _0-5 of CSS7, CSS3, CSS8, and CSS9 were 145.79%, 96,33, 48.28, and 36.15%, respectively. The δ _i-j of calcareous silty sand with the same fines content decreased with increasing dry density, suggesting that the declining amplitude of the volumetric water content of calcareous silty sand with increasing matric suction further decreased with increasing dry density due to the amount of large pores containing free water being reduced with increasing dry density.

In the transition stage (ψ ≈ 5–10 kPa), the air entry values of CSS7, CSS3, CSS8, and CSS9, were 7.43, 7.74, 7.86, and 7.92 kPa, respectively. The higher the dry density of calcareous silty sand, the greater the air entry value of the sample. Samples with higher dry density experience more difficulty evolving from a quasi-saturated state to an unsaturated state. The quantity of the larger pores containing free water in the samples with higher dry density is fewer, and the water connectivity is poorer; hence, it is more difficult for water to drain out of the sample. As a result, a larger matric suction would be required for the sample to evolve from a quasi-saturated state to an unsaturated state.

In the residual stage (ψ ≈ 10–750 kPa), the residual volumetric water contents of the samples were 16.19, 17.96, 17.82, and 19.89%, respectively, indicating that the residual volumetric water content of the sample increases with increasing dry density. As discussed above, the water drained by the sample in the residual stage was capillary water and bound water. There are more soil particles in the samples with higher dry density, which could provide more adsorption space (such as pores on the surface of particles and small pores between particles) for capillary water and bound water (Wu et al., 2020).

The SWCC of calcareous silty sand under different test conditions was fit using the models described above. It is found that these models are all applicable to describe the SWCC of calcareous silty sand in terms of the general trend. Among them, the fitting result of the Fredlund-Xing model to the SWCC of the samples is the best. When the Zhong Fangjie model was used to fit the experimental results of the samples, the SWCC was approximately horizontal in the residual stage, which is contradictory to the measured result that the volumetric water content of calcareous silty sand presented a decreasing trend with increasing matric suction in the residual stage (Figure 7). The developmental trend of SWCC is related to the size and quantity of pores in the sample (Alves et al., 2020; Daneshian et al., 2021). The Zhong Fangjie model (Zhong, 2007) is proposed based on the measured results of soil-water characteristic of silica sand. There are almost no inner pores in silica sand particles and only a few outer pores on the surface of silica sand particles, so the change in the SWCC of silica sand in the residual stage is subject to the pores between particles. In contrast, for calcareous sand, both the outer pores distributed on the surfaces of the particles and the inner pores inside the particles are abundant, and these pores, in which water could reside, improve the water-holding capacity of calcareous sand. In addition, due to the decomposition of calcium carbonate, calcareous sand also carries a certain quantity of mineral ions, which have a strong ability to adsorb the water within the sample in the residual stage (Wu et al., 2020). Therefore, the soil-water characteristics of calcareous sand differs from that of silica sand in the residual stage, and the fitting result of Zhong Fangjie model (Zhong, 2007) is not satisfactory compared to the other models.

Table 3 provides the values of the parameters of the Fredlund-Xing model fitting for the SWCCs of calcareous silty sands with different fines contents. According to Table 3, 1) a increased with increasing fines content, which is consistent with the above measured air entry values of calcareous silty sands with different fines contents. 2) b decreased abruptly with increasing fines content, indicating that with increasing fines content, the slope of the SWCC of calcareous silty sand declined gradually, and the SWCC became gentler in the transition stage. The range of matric suction corresponding to the SWCC of the transition stage widened due to increasing fines content, indicating that the matric suction for the sample transit to the residual stage increased. 3) c increased with increasing fines content, and the SWCCs of the samples in the residual stage became steeper, which conforms to the change of the SWCCs of the samples in the early residual stage (Figure 4). The number of small pores and the water-holding capacity of the sample increases with increasing fines content; hence, making it more difficult for calcareous silty sand to reach an absolutely dry state. 4) In the transition stage and late residual stage, the SWCCs of the samples approximately overlapped with each other, suggesting that the values of b and c were highly sensitive to variations in the SWCC (Figure 4). The variation ranges of the three parameters also indicated that b was more sensitive to the change in fines content.

Table 4 shows the parameter values of the Fredlund-Xing model when used to fit the SWCCs of calcareous silty sands with different dry densities. It is found that 1) a increased with increasing dry density, which is consistent with the above measured result that the air entry value of calcareous silty sand increased with increasing dry density. 2) b did not show any obvious regularity with increasing dry density. However, among the three parameters, b changed most significantly with increasing dry density. 3) c changed only slightly with the change in dry density, which conforms to the observation that the SWCCs of calcareous silty sands with different dry densities were approximately parallel in the residual stage (Figure 6).

By analyzing the data in Table 3, the relationship between a, b, and c and fines content (C _F ) is shown in Figure 8 and can be expressed as follows: y = y 1 + A 1 e x p B 1 C F , (8) where y ₁, A ₁, and B ₁ are fitting parameters.

Figure 9 shows the relationship between dry density (ρ _d ) and the parameter a, which is expressed as follows: y = y 2 + A 2 ρ d + B 2 ρ d 2 , (9) where y ₂, A ₂, and B ₂ are fitting parameters.

By combining Eqs 8, 9, the following parameter calculation formulas are obtained for the Fredlund-Xing model, which is suitable for assessing the soil-water characteristics of calcareous silty sands with different fines contents and dry densities: a = [ ( 4.968 + 0.00146 e x p 0.152 C F ) × ( 21.96 − 24.67 ρ d + 8.88 ρ d 2 ) ] 0.5 , (10) b = − 4.322 + 76.777 e x p − 0.05 C F ,     (11) c = 0.325 + 0.00145 e x p 0.112 C F . (12)

By fitting the saturated volumetric water contents (θ _s ) and fines contents (C _F ) of calcareous silty sands with different fines contents (Figure 10A), the following equation is acquired to calculate the saturated volumetric water content (θ _s ) of calcareous silty sand based on fines content: θ s = 0.48 − 0.00855 e x p C F 24.37 . (13)

Similarly, by fitting the saturated volumetric water contents (θ _s ) and dry densities (ρ _d ) of calcareous silty sands with different dry densities (Figure 10B), the following equation is acquired to calculate the saturated volumetric water content (θ _s ) of calcareous silty sand based on dry density: θ s = 0.956 − 0.328 ρ d . (14)

Combining Eqs 13, 14 give the relational expression among saturated volumetric water content (θ _s ), fines content (C _F ), and dry density (ρ _d ): θ s = [ ( 0.956 − 0.328 ρ d )   × ( 0.48 − 0.00855 e x p C F 24.37 ) ] 0.5 . (15)

By substituting Eqs 10–12, 15 into Eq. 5, the following formula is acquired to calculate the volumetric water content (θ) of calcareous silty sands with different matric suctions (ψ) based on fines content and dry density: θ = θ s ( ρ d ,     C F ) { ln [ e x p + ( ψ a ( ρ d ,     C F ) ) b (   C F )     ] } c (   C F ) , (16) where θ s ( ρ d ,     C F ) and a ( ρ d , C F ) are the expressions containing dry density (ρ _d ) and fine content (C _F ), respectively. b (   C F ) and c (   C F ) are the expressions containing fine content (C _F ), respectively.

The proposed SWCC model was used to calculate the volumetric water content of calcareous silty sands with different fines contents and dry densities under different matric suctions, and the applicability of the model to characterize the SWCC of calcareous silty sand was assessed using the ratio between the calculated and measured values (Table 5). The values calculated from Eq. 16 were close to the measured results, and the ratio between the calculated and measured results ranges from 0.43 to 1.22, indicating that proposed model could fit the SWCC of calcareous sand. Particle shape, pressure loading mode, and numerous other factors also affect the soil-water characteristics of calcareous silty sand. Regarding the mathematical basis on which the equations are developed to calculate the volumetric water content of calcareous silty sand, it is also necessary to explore its change laws with various parameters as well as their mathematical correlations. Related research should be carried out to lay a solid theoretical foundation to develop an empirical formula universally applicable to assessing the SWCCs for various types of calcareous sands.