An analytical model to predict water retention curves for granular materials using the grain-size distribution curve

Ji-Peng et al. [35] have defined based on the classical van Genuchten model (van Genuchten [62]), the effective degree of saturation equation by (1) Se=(1+(sα)n)1n−1 {S_e} = {\left({1 + {{\left({{s \over \alpha}} \right)}^n}} \right)^{{1 \over n} - 1}}

Knowing that the effective degree of saturation is defined as describing the volume of water partially filling the soil microporosity. This effective degree of saturation defines the proportion of the prevailing suction that actually contributes to the effective stress. where s is the soil suction, α is a parameter related to the air entry value, and n is a parameter linked to the slope of the water retention curve.

It was revealed that soils with big pores desaturate at smaller suction values and hold back less water for specific matrix suction in comparison with soils with smaller pores, as illustrated in Figure 1a. In this context, Hillel [30] found out that the moisture content of soils with a wide pore size distribution changes progressively as suction changes (Figure 1b). In addition, soils with large pores have a larger parameter n, while those with a wide pore size distribution show a smaller n. Similarly, Tinjum et al. [56] indicated that soils containing minerals, which possess a small surface charge and a small affinity for hydration, possess reduced water content at specific matrix suction.

Figure 1

As is generally known, materials containing small amounts of fines, like clean sand and some other coarse materials, exhibit a close relationship between the particle size distribution, the pore size distribution, and the soil water characteristic curve (Yang et al. [57]; Aubertin et al. [2]; Jaafar and Likos. [33]). In addition, Juang and Holtz [36] found out that as the particle size becomes finer, the pore size becomes smaller. Likewise, it was revealed that, in general, sands with a broader particle distribution possess thinner pores because of the high packing efficiency, which engenders a broader particle size distribution. Based on the above results, and considering Young-Laplace's equation, one may therefore assert that high suction levels are needed to desaturate the pores. Consequently, it can be concluded that, for a given suction of the soil matrix, sands having a broad particle size distribution hold back more water. It is useful to indicate that the pore size diversity augments with the particle size. Moreover, as the distribution becomes wider, the soil-water characteristic curve (SWCC) shows a more progressively varied slope.

Consequently, the parameters α and n must change in a consistent manner with the median particle size and the breadth of particle sizes in sand. As is clearly illustrated in Figure 2a, the parameter α should decrease as the median particle size diminishes and the width of particle sizes rises, while the parameter n decreases as the breadth of particle sizes goes up, as is shown in figure 2b.

Figure 2

Terzaghi et al. [58] revealed that the three particle sizes d_10, d₃₀, and d₆₀ relating, respectively, to 10%, 30%, and 60% of passing through the corresponding sieves are the most important particle sizes commonly needed for the description and classification of soils. Moreover, it is well-known that the shape of the granular distribution can be determined using the coefficient of curvature (C_c=d²₃₀ / (d₆₀ d₁₀)), the coefficient of uniformity (C_u=d₆₀ / d₁₀), in addition to a measure of the mean particle size.

Furthermore, it must be emphasized that dimensional analysis, which is based on the Pi-theorem of Buckingham [10], is extensively utilized in technological analysis and in solving applied science problems for the purpose of identifying the essential factors and making the physical relationships simpler. Regarding the water retention curves, it is important to know that the effective degree of saturation (S_e) depends on the particle size distribution, on suction. In addition, any possible alterations in the void ratio of sandy soils may be considered as quite small. Therefore, the effective degree of saturation (S_e) can be expressed as a function of the dimensionless quantities C_u, C_c, d₆₀, and γ witch γ is a parameter of the model. In addition, Buckingham's Pi theorem can then be applied and S_e can be made simpler and written in the following form as (2) Se=f(Cu,d60,ID,s,α,ϒ) {S_e} = f({C_u},{d_{60}},{I_D},s,\alpha,\Upsilon)

Furthermore, according to Ji-Peng Wang et al., [35], the parameter C_c depends directly on C_u. It is worth recalling that C_u is a physical parameter that depends on d₆₀. Consequently, the previous expression may be expressed in a simpler form as (3) Se≈f′(Cu,SCuϒ) {S_e} \approx {f^{'}}\left({{C_u},{{{SC}_u} \over \Upsilon}} \right)

On the other hand, using the normalized suction (s*= s.C_u / γ) and the normalized α (α*= α.C_u / γ), van Genuchten's equation may be rewritten as (4) Se=(1+(s*α*)n)1n−1 {S_e} = {\left({1 + {{\left({{{s*} \over {\alpha *}}} \right)}^n}} \right)^{{1 \over n} - 1}}

It was shown on the drying path that the degree of saturation remains constant and equals to 1 until the suction reaches the air entry value (AEV). Then, when it reaches the air entry value (AEV), air bubbles enter the water phase. Afterward, the degree of saturation drops considerably as the suction increases further, and the large pores begin to flow while the small ones remain still filled with water, in such a way that the water and air phases become continuous. In addition, it was found that the water phase exists in the form of isolated water bridges and adsorption layers when the suction value gets very large. It should be mentioned that the change in suction does not have much effect on the degree of saturation, which corresponds to the residual state. Moreover, the soil-water retention curve (SWRC) may flatten for very small suction values, with the corresponding residual water content. Additionally, any alterations in the volumetric water content would engender an increase in suction. It is ought to be mentioned that inside the residual zone and beyond the residual suction, water is retained as water that was adsorbed on soil particles (McQueen and Miller. [41]) rather than as capillary water maintained by the action of capillary forces created by water menisci, as is clearly displayed in figure 3.

Figure 3

Based on the above findings, it was deemed interesting to introduce the residual saturation degree in the proposed model in order to develop an expression for the effective degree of saturation S_e that can be used to describe the water retention behavior as (5) Se=(1+(s*α*)n)1n−1−Ser {S_e} = {\left({1 + {{\left({{{s*} \over {\alpha *}}} \right)}^n}} \right)^{{1 \over n} - 1}} - {S_{er}} where s* is the normalized suction, with s* = s.c_u ; α* is the normalized α, with α* = α.c_u.

Note that α and n are the parameters of the model.

Also, γ is a parameter of the model which is equal to 1, and S_er is the degree of residual saturation.

It is worth noting that the model is ought to be fitted to experimental data in order to assess the model parameters which are based on the data obtained from various tests that were carried out using the tensiometric method previously established by Feia et al. [25], and Della and Feia. [47].

The flow model under consideration adopts a pore network with parallel cylindrical tubes in which the water flows along the direction of the cylinder axis. This assumption corresponds to the drying path when the pores are initially filled with liquid water. Regarding the humidification path, the condensation of the water vapor contained in pores takes place through the formation of cylindrical menisci along the pore walls, as presented in figure 4.

Figure 4

The soil water retention curve can be evaluated through the measurement of the matric suction using an oedometer cell (Fredlund and Rahardjo. [24]; Delage and Cui. [16]). This is a method that allows determining low water pressures, near the atmospheric pressure. If a saturated porous ceramic having pores thinner than those of soil is used, then the water phase continuity with the unsaturated soil is ascertained until the water phase in the soil specimen under consideration becomes discontinuous. It should be mentioned that the maximum suction that can be estimated is controlled by the air-entry pressure of the porous ceramic. In the present study, this pressure is set at 50 kPa. The highest soil water pressure head is approximately equal to 10 kPa (1 m of water column); this value is quite suitable for the study of the water retention characteristics of most sands. Moreover, it is ought to be noted that, during the test, no loading is applied. In this case, the use of the oedometer cell is highly encouraged by the fact that it is easy to install the porous ceramic that was provided for the cell.

A schematic view of the test apparatus is clearly illustrated in figure 5. This system consists of an odometer cell that has a diameter of 70 mm and height of 37 mm, supplied with a porous ceramic, and a graduated tube with a 7-mm inner diameter connected to a reservoir of water (Feia et al. [25]). As for the specimen, it is 27 mm thick and was prepared by compaction, in three layers, under dry conditions. It should be specified that the thickness was chosen smaller than the height of the odometer cell for the purpose of minimizing the potential errors that are associated with the matric suction along the specimen height. This would obviously introduce an uncertainty linked to both measurement and specimen thickness. An error analysis related to this uncertainty is presented later in the paper. Note also that diverse preparation methods, such as the pluviation method, can be used as well in order to achieve the needed relative density. The cell was then installed in a room where the temperature was set at 20°C, and the sample was wrapped in a film of cellophane in order to limit water evaporation. Since the preparation of the sample was done under dry conditions, the first step of the experiment started with the wetting path. Once the maximum suction level was attained, a gradual suction reduction was performed and the associated water content variations were measured and recorded. It is interesting to note that suctioning was done by positioning the level of water in the graduated tube lower than the sample center, as presented in Figure 5. Once the equilibrium was reached for the sands under study, after about 30 min, it was observed that the level of water inside the graduated tube changed. This made it possible to calculate the suction applied in each step by considering the vertical separation between the water level inside the graduated tube and the sample center. It should be noted that the reading accuracy for the water level in the tube was 1 mm, which represents a water volume of 38 mm³. This volume is satisfactorily smaller than that of the pore volume of the sample. In addition, the impact of the suction distribution along the height of the sample can be considered as negligible because the thickness of the specimen (27 mm) is relatively small. Bearing in mind the small area of the tube, one can say that the amount of water that was evaporated during the time required to reach equilibrium was also negligible. During the last phase of the wetting path, the water level in the graduated tube was positioned at the center of the sample after equilibrium; then, the drying path was applied by gradually increasing the suction to eventually reach the maximum level. It is worth noting that the drying path can be maintained uninterrupted to a point where the water phase is not continuous anymore. Consequently, the water content will not be reduced even when the suction is increased further. Therefore, the water retention curves for the wetting and drying paths can be derived through the application of the aforementioned approach using several steps of suction for each path.

Figure 5

Based on the water-retention curve, the Young-Laplace law can therefore be employed to estimate the pore size corresponding to each suction level. This law gives the suction as a function of the pore-access diameter as (6) s=ua−uw=4σscosθDp s = {u_a} - {u_w} = {{4{\sigma _s}cos \theta} \over {{D_p}}} where s is the suction, D_p is the pore-access diameter, u_a and u_w are, respectively, the air and water pressures. Also, σ_s= 72.75 × 10⁻³ N × m⁻¹ at 20°C refers to the tension at the surface between water and air, and θ is the angle of contact between the meniscus and solid. Note that θ depends on the solid mineral, surface cleanliness and roughness, menisci's movement (advancing or receding), and viscous effects (Decker et al. [17]; Espinoza and Santamarina [21]; Lourenço et al. [40]).

In the literature, the approximation cosθ ≈ 1 is a commonly accepted supposition for the estimation of the pore-size distribution (Innocentini and Pandolfelli. [32]; Mitchell and Soga. [43]; Hao et al. [29]).

The model proposed here is applied to the experimental results of the tensiometric tests carried out by Feia et al. [25]. It was therefore decided to select three tests that were performed at different density indexes, i.e., 0.5, 0.7, and 0.9 (Figure 6).

Figure 6

It ought to be noted here that the density index is expressed by the equation (7) as (7) ID=emax−eemax−emin {I_D} = {{{e_{max}} - e} \over {{e_{max}} - {e_{min}}}}

In the course of a tensiometric test, and more exactly during the drying path, a certain quantity of water still remains inside the sample. This amount of water that persists within the specimen, which corresponds to the degree of residual saturation, is mainly attributed to the water discontinuity between the tube and the porous medium. In addition, as reported in the literature, the value of the degree of residual saturation depends on several physical parameters of soil. One of these parameters is the density index I_D.

Figure 7 depicts the evolution of the degree of residual saturation against the density index, based on the data obtained from the tensiometric tests previously performed by Feia et al. [25]. One may easily note that this parameter varies linearly in accordance with the relationship expressed by equation (8) that is given below as (8) Ser=0.12ID+0.26 {S_{er}} = 0.12{I_D} + 0.26

Figure 7

Equation (8) is in the form: y = ax + b

Hence, the proposed model can be expressed as follows: (9) Se=(1+(s*α*)n)1n−1−(0.12ID+0.26) {S_e} = {\left({1 + {{\left({{{s*} \over {\alpha *}}} \right)}^n}} \right)^{{1 \over n} - 1}} - (0.12{I_D} + 0.26)

Once the calibration process was completed, the proposed model was then validated through the simulation of a test that was not used during calibration. To do this, it was decided to carry out a water retention test with soil type HN34 that had a uniformity coefficient C_U = 1.6 and a density index I_D=0.9. Based on the calibration results presented in figure (8), and according to the data presented in figures (9 and 10), it can be stated that parameters n and α both increased as the density index I_D augmented. Then, using the linear interpolation, presented by formulas (10) and (11), it was found that n = 8.51 and α=4.37, α^*= α × Cu =6.992. In addition, the simulation results displayed in figure (11) were then compared with the test results. This simple comparison showed that the simulation results were in good agreement with the test results, which validates the proposed model.

Figure 11

The model was applied to four types of sand possessing the characteristics illustrated in Table 3 and represented by photo 1:

Photo 1

The grain-size distributions of all four types of sand obtained by sieve analysis are shown in figure 12 below:

Figure 12

Figure 13 presents the water retention curves of all four types of sand, obtained by applying the proposed model for a density index equal to 0.90. It was noted that the degree of saturation started to diminish as the soil suction increased. One can clearly see that the different water retention properties of unsaturated soils are explicitly visible on these curves.

Figure 13

Figure 14 shows the water retention curves for type 3 sand, obtained by applying the proposed model for different density index values. This same figure shows a comparison between the water retention curves of sand of type 3 for four different density indices (0.3, 0.50, 0.70, and 0.90). The results revealed that the suction increased as the density index went up, which is in very good agreement with the experimental findings. In addition, it is clearly noted that the different water retention properties of unsaturated soils can be observed on these curves.

Figure 14

Similarly, figure 15 shows a typical result of the pore-access size distribution curves for the four types of sand with a density index I_D = 0.50. A simple comparison between these pore-access size distribution curves and the particle size distribution curves indicates that the uniformity of the particle size curves induces a good uniformity of the pore-access size distribution curves where D_p50 is the diameter of the pores at 50⁒ degree of saturation obtained from the results obtained by estimating the size of the pores using the model proposed.

Figure 15

And the pore-access size distribution curves were obtained by the application of young-Laplace's law where the diameter of the pore access size can be estimated for each suction value. This law gives the suction as a function of the pore-access diameter as: s=ua−uw=4σscosθDp s = {u_a} - {u_w} = {{4{\sigma _s}cos \theta} \over {{D_p}}}

Then Dp=4σscosθs {D_p} = {{4{\sigma _s}cos \theta} \over s}

Figure 16 compares the pore-access size distributions for type 3 sand, with four different density indices (0.3, 0.50, 0.70, and 0.90). A very good consistency was then found between the pore-access size distributions and the density indexes. Moreover, it was found that the looser the sand, the larger the median pore size. It was also revealed that the pore-access size distribution is also more spread out at lower density index values.

Figure 16

It was shown that for loose sand, the pore size becomes larger, and therefore, larger suctions are required to desaturate the pores. However, sands with higher density index values (dense sand) generally have finer pores.

Likewise, figure 17 shows a comparison between the pore sizes of sands with different particle distributions. It was observed that when the particle size becomes finer (the mean particle size D_g50 is smaller), the pore size becomes smaller, and therefore, a direct relationship is found between D_g50 (D_g50 mean grain size) and D_p50 (D_p50 mean pore size). In addition, the pore size diversity increases as the particle size grows.

Figure 17