Experimental investigation on the influence of scale effects on the permeability coefficient of coarse-grained soil

The test was conducted using a ST30-2A Vertical Penetrometer for coarse-grained soils, the test instrument and test process are shown in Figure 1. The equipment is mainly composed of three parts: specimen container, head manometer, and water tank. The water tank was mainly used to control the position of the head and adjust the head difference. The test container was of metal cylinder type, with a metal perforated, permeable plate and water outlet and overflow at the top and bottom of the container. The side of the specimen container was connected to a head manometer, and the central head manometer expressed the pressure mainly based on the height of the liquid column. Water inlets and overflows were provided at different heights on the sides to regulate and control the water level in the container. The permeation instrument allowed for the head permeation pressure to be set in the range of 0 to 50 kPa. This equipment could determine the vertical permeability coefficient of seepage water as it passed through a coarse-grained soil specimen, as well as the critical slope drop when fine particles were gradually lost with the seepage water (pipe surge) and the destructive slope drop when the soil was uplifted (runoff soil). The maximum applicable particle diameter was 60 mm, and the relative size of the test specimen is Φ300×300 mm for coarse-grained soils.

Figure 1.

ST30-2A large-scale vertical infiltration apparatus for coarse-grained soil and test process

The test material was selected from the rockfill materials for the dam construction of Lincangdaqiaopo Reservoir in Yunnan Province. This material is artificially blasted and crushed rock with weakly weathered granite as its parent rock. The rock possesses an average saturated uniaxial compressive strength of 50 MPa, a softening coefficient of 0.79, and a specific gravity of 2.70.The specimens with original gradations were subjected to particle sieving using a vibratory sieving machine, and groups of specimens with diameters of 60 to 40 mm, 40 to 20 mm, 20 to 10 mm, 10 to 5 mm, 5 to 2 mm, 2 to 1 mm, 1 to 0.5 mm, and less than 2 mm were retained as specimen soils, as shown in Figure 2.

Figure 2.

Grit and gravel of different grain size groups

In order to quantitatively analyze the effect of maximum grain size d_max and gradation area S on the permeability coefficient, as well as to accurately determine the relationship between the gradation structure and the permeability coefficient, the test scheme is based on the continuous gradation equation put forward by Zhu et al.[20], which is applicable to all types of coarse-grained soils: 1 P=1(1−b)(dmax/d)m+b×100% where P is the content of particles smaller than a certain particle diameter; d represents the particle diameter, d_max is the maximum particle diameter; b and m are gradation parameters, which determine the shape and degree of inclination of the gradation curve, respectively (collectively referred to as the gradation structure). This equation can accurately characterize the morphology of the particle gradation curve based on the maximum particle diameter and specific gradation parameters.

According to the findings of Jiang et al. [21] on the scale effect, there are no obvious patterns between b, m, and d_max, but a quantifiable pattern exists for the gradation area S and d_max in the gradation characterization. The gradation area is the region bounded by the gradation curve and the horizontal coordinate axis with d=d_max and d=d_max0, as shown in Figure 3. It can reflect the uniformity of particle diameter distribution and indicate the variation of gradation structure. The S can be calculated by the following equation: 2 S=ln(1−kb)−ln(1−b)mbln10

Figure 3.

Gradation area diagram

The k can be calculated as follows: 3 k=1(1−b)(dmax/dmax0)m+b

According to D. Gu et al. [22] and W. Guo et al. [23], the criterion for dividing gravel and sand is a particle diameter of 5 mm. Therefore, the minimum particle diameter d_max0 for calculating the gradation area was taken as 5 mm. In this research, a total of 18 sets of specimens were devised by establishing various values for both m and b. The gradation parameters for each group of specimens are shown in Table 1, and the gradation curves are shown in Figure 4.

Figure 4.

Sample gradation curve. (a) Area of different gradations S with the same maximum particle size d_max and (b) Maximum particle size d_max for different particles of the same grading area S

A constant head indoor test was performed to evaluate the permeability characteristics by applying a certain water head to the specimen. The permeation velocity was observed, and the permeability coefficient and other parameters were calculated. The particle size intervals were allocated according to the designed gradation curves and mixed homogeneously. The instrument was then checked for leaks at all piping connections. After confirming no water leakage, the specimens were loaded into the cylinder in layers. Each layer was 2 to 3 cm thick and was gently compacted to a certain thickness with a wooden hammer to control its porosity. Once the sample preparation was completed, the bolts were tightened, and the saturation was performed to ensure that the head difference was less than 2 cm until the water flowed continuously from the outlet pipe. The test was started after the saturation, and the water level in the pressure measuring tube was recorded after stabilization to obtain the difference between each pressure measuring tube.

By analyzing the results obtained from tests 1 to 8, the relationship between gradation area and permeability coefficient can be derived. The effect of S on k can be quantitatively represented by the following equation: 4 k=1+a1b1+Sc1 where a₁, b₁, and c₁ are fitting parameters.

Equation (4) was used to fit the test results of specimens 1 to 6. The results of this analysis are presented in Table 2, and the corresponding fitted curves are illustrated in Figure 6.

Figure 6.

Fitting relationship between gradation area and permeability coefficient

It can be seen from the figure that the gradation area increases from 0.271 to 0.475, and the permeability coefficient of the sample decreases from 3.244 to 2.474, a decrease of 23.74 %. The grading area increased from 0.475 to 0.754, and the permeability coefficient of the sample decreased from 2.474 to 0.422, a decrease of 82.94 %. The grading area increased from 0.754 to 0.874, and the permeability coefficient of the sample decreased from 0.422 to 0.374, a decrease of 11.37 %. That is, with the increase of the grading area S, the permeability coefficient k gradually decreases, and with the further increase of the grading area, the change of the permeability coefficient gradually tends to be stable. At a constant maximum particle diameter, fine particles content (d<5 mm) gradually increases from 0.5% to 66.8% as the gradation area increases. As a result, the relative content of coarse particles is reduced, and the voids among coarse particles can be filled by fine particles more readily, thus impeding water penetration. The decreasing water flow rate further causes a lower permeability coefficient.

At the same time, within the gradation area exceeding 0.4 and less than 0.7, there is a notable decrease in the permeability coefficient as the gradation area increases. When the gradation area surpasses 0.7, the permeability coefficient continues to decrease with an increased gradation area, although the rate of decrease is significantly reduced. This suggests that fine particles exhibit a more effective filling effect, and the permeability coefficient is more influenced by the content of fine particles, aligning with the observations of Bao [24].

As depicted in Figure 7, the fitted curves for the permeability coefficient (k) using Equation (5) exhibit a strong agreement with the measured values. Moreover, all the coefficients of determination are above 0.97. Consequently, Equation (4) can accurately and quantitatively describe the influence of coarse-grained soil gradation area S on the permeability coefficient k.

Figure 7.

Fitting relationship between maximum particle size and permeability coefficient

Based on the test data analysis of specimens 9 to 18, the d_max and permeability coefficient k can be expressed by the following equations: (5) k=a2(dmaxdmax0)2+b2(dmaxdmax0)+k0 where d_max0 is the value distinguishing sand and gravel, which is taken as 5 mm; k₀ is the permeability coefficient of the specimen when d_max is equal to 5 mm, cm/s; a₂ and b₂ are the fitting parameters.

The maximum particle size and permeability coefficients for specimens 9 to 18 were fitted by Equation (5). The fitting results are presented in Table 3, and the fitting curves are illustrated in Figure 7.

Under the identical gradation area S, the permeability coefficient k rises steadily as the maximum particle diameter d_max increases. In contrast, the permeability coefficient k varies differently with the maximum particle diameter d_max undeerent gradation areas S. The permeability coefficient k exhibits a tendency of increasing initially and then decreasing as the gradation area S grows, namely, one of the three gradation areas is an optimum gradation that leads to the highest permeability coefficient k for coarse-grained soils. A larger maximum particle diameter in coarse-grained soils can result in greater porosity in the soil, forming larger pores between particles. With a gradually increasing maximum particle diameter, a larger pore space is formed, increasing the connection between neighboring pores. As a result, water penetrates more easily through the pores, leading to an increase in the permeability coefficient with increasing maximum particle diameter. The permeability coefficient can be reduced when the fine particle content is properly filled into the interparticle spaces.

It is observable from Figure 7, the fitted curves for the permeability coefficient k through Equation (5) are in good agreement with the measured values. In addition, all the coefficients of determination are above 0.97. Therefore, Equation (5) can accurately describe the relationship between the maximum particle size dmax and the permeability coefficient k for coarse-grained soils.

Building upon the aforementioned analysis, the connection between the permeability coefficient k and the gradation area S of coarse-grained soils at a constant maximum particle size (d_max) can be represented by Equation (4). In addition, k₀ (the value of the permeability coefficient at d_max = 5 mm) in Equation (5), i.e., the relationship between k₀ and S, can also be expressed by Equation (4). Therefore, the coupled effects of gradation structure and maximum particle diameter on the permeability coefficient can be quantitatively described by the following equation: 6 k=a(dmaxdmax0)2+b(dmaxdmax0)+(1+cf+Se) where a, b, c, e, and f are fitting parameters.

For verification, Equation (6) was used to fit the test results of all specimens, and the fitting results are summarized in Table 4. The predicted permeability coefficients obtained from the fitting were contrasted with the measured values, as illustrated in Figure 8.

Figure 8.

Measured and calculated values of permeability coefficient for coarse-grained soil

Notably, the predicted points for each group closely align with the measured points. While one point exhibits a significant error, likely due to a systematic error stemming from test operation issues, the majority of predicted values have an error of no more than 9% from the measured values. The coefficient of determination is 0.833. Hence, it can be inferred that the empirical formula accurately describes the influence of gradations on the permeability coefficients for coarse-grained soils.

To validate the applicability of the aforementioned empirical formula, Equation (6) was used to fit the results of the permeability tests on coarse-grained soils from the literature [25]. The fitted and actual values are shown in Figure 9, and the computational error values obtained from the fitting are listed in Table 5.

Figure 9.

The relationship between the measured and fitted values of the permeability coefficient in literature [25] and the gradation area

It is evident that the disparities between the calculated and tested values of Equation (6) are minimal, with a maximum error of less than 5%. Hence, Equation (6) can precisely and quantitatively portray the relationship between the permeability coefficient and gradation for coarse-grained soils.

Based on the above analysis, a method for determining the permeability coefficient of coarse-grained soils in actual projects using the indoor test results can be obtained: First, the coarse-grained soil in the field is scaled to obtain four or more sets of different gradation specimens satisfying the required size. Then, indoor tests are carried out on all the downsized specimens to obtain the permeability coefficients for each group of specimens. After incorporating the gradation parameters and permeability coefficients of specimens into Equation (6), the corresponding parameter values a, b, c, e, and f can be obtained. Finally, the permeability coefficient of the soil material with the original gradation can be predicted by substituting the gradation parameters of the on-site soil material into Equation (6), thus providing a reference for the safe design of the project.