Experimental and Theoretical Evaluation of Buoyancy Reduction in Saturated Clay Soils

Abstract

The rational calculation of groundwater buoyancy directly impacts the safety of underground engineering. However, there is still no consensus on whether the reduction of groundwater buoyancy should be considered, and a theoretical explanation and quantification of buoyancy reduction in clayey soils is lacking. Based on laboratory engineering model tests, this study observed and analyzed the phenomenon of buoyancy reduction in saturated clayey soils. The contact area ratio of gravity water, calculated from geotechnical test data, was compared with the reduction slope. The experimental results indicated that the reduction slope of the fitted line between the static water head in the silty clay layer and the buoyancy water head was 0.8692. And theoretical analysis showed that the distribution of interparticle pore water pressure tends to attenuate from the pore center to the soil particle surface, suggesting a reduction in buoyancy head compared to the groundwater level. The reduction slope is theoretically equal to the contact area ratio of gravity water. Additionally, since limitations in current techniques for generalizing the soil–water constitutive models affect the reduction slope, this study proposes a method for determining the buoyancy reduction slope in saturated clayey soil based on the theory that interparticle pore water pressure distribution attenuates from the pore center to the soil particle surface. This method could potentially change the existing conceptual framework for buoyancy design in underground structures.

1. Introduction

Groundwater, soil, and underground structures together form a complex interacting system. This interaction can lead to numerous geotechnical challenges, particularly in calculating the buoyancy of groundwater in clayey soil [1]. The rational calculation of groundwater buoyancy has a direct impact on the safety and security of underground engineering [2].

When calculating the buoyancy of groundwater in high-permeability layers such as pebble and sand layers, researchers have demonstrated through numerous tests and theoretical analyses that the measured values of groundwater buoyancy are nearly equivalent to the theoretical values based on the Archimedes’ principle, and the calculation of groundwater buoyancy does not require consideration of reduction [3,4,5].

Scholars have compiled numerous test results and engaged in heated discussions regarding whether the buoyancy of groundwater in low-permeability layers (aquitards), such as clay and silty clay, requires reduction. Regarding the issue of buoyancy reduction caused by vertical groundwater seepage, Zhang et al. [6] proposed the concept of “seepage pore water pressure”, observing that leakage occurs between multilayer aquifers and that hydraulic head is lost along the seepage direction, which in turn leads to a decrease in groundwater buoyancy. Shen et al. [7], Li et al. [8], Zhang [9], and Li et al. [10] conducted field tests and modeling tests to support the above research results. However, there is no clear explanation as to whether the pore water pressure under hydrostatic conditions should be discounted. Regarding the smaller water buoyancy caused by binding water, Cui et al. [11] and Zhang et al. [12] contended that this pore water pressure reduction was not significant in saturated clayey soil through extensive laboratory tests. Xiang et al. [13] conducted model tests and concluded that under long-term stable conditions, saturated clayey soil can fully transfer pore water pressure without requiring consideration of the influence of bound water. It can be seen that water buoyancy is reduced in all of the above research tests, but is ignored by the authors due to the lack of theoretical support. Some researchers, through experimental tests and weakly permeable connectivity analysis, have concluded that even in the absence of seepage, the pore water pressure is reduced because of the poor connectivity of the aquitard due to bound water [14,15,16,17,18,19,20]. Zhou et al. [21] used model tests to determine that the reduction factor for groundwater buoyancy acting on underground structure models in clayey soil at different burial depths ranges from 0.25 to 0.52. Based on the semi-interval search method, Song et al. [22] conducted an experimental study on groundwater buoyancy acting on shallow foundations in saturated clayey soils. Their test results showed that the measured buoyancy acting on a foundation in clay is less than the theoretical buoyancy. Zhang et al. [18] evaluated the water buoyancy effect and groundwater buoyancy reduction of underground silos in sand and clay. By considering the side friction of the underground silos in the experiment, the buoyancy reduction factor for the sand layer and saturated clayey soil was determined as 0.95 and 0.79, respectively. Zhang et al. [23] conducted centrifugal model tests to analyze the effect of burial depth on the pore water pressure in the sand and remolded soft soil. Their results showed that the pore water pressure in sand equals the hydrostatic pressure, whereas in soft clay, it is significantly lower than the theoretical value. The pore water pressure reduction factor of soft clay varies with depth and stabilizes at 0.68 when the depth exceeds 10 m. Zhou [17] designed multiple-scale model tests and conducted field tests at selected sites. The results revealed that the measured groundwater buoyancy in the silty clay layer is only 75% of the traditional theoretical value. Fang [19] proposed the concept of a water pressure rate, noting that groundwater buoyancy calculations should be multiplied by this rate, though no validation was provided. It can be seen that it is feasible to determine the relationship between groundwater level and water buoyancy by using the method of indoor modeling tests, and the calculation of groundwater buoyancy is related to the combined water. In addition, there is a lack of theoretical support for the discounting of groundwater buoyancy in weakly permeable layers such as clay and silty clay.

Currently, research findings on the calculation of groundwater buoyancy in saturated clayey soil show significant variation. There is no consensus on whether a reduction should be considered, and studies advocating lack sufficient theoretical derivation. To address this issue, the present study employs model and geotechnical test methods, selecting fine sand and silty clay from representative high-permeability and low-permeability layers, respectively, as the test soils. Stress analysis was performed on an underground engineering module within these saturated soils to comparatively analyze the relationships between groundwater level and buoyancy at the module’s base in high- and low-permeability layers. The existence of a reduction slope in low-permeability layers is confirmed, and a theoretical derivation for this reduction is developed.

2. Research Methodology

2.1. Model Test

Field measurement of buoyancy in actual underground engineering is highly challenging, as the boundary conditions cannot be artificially controlled. In contrast, laboratory model tests allow precise control of boundary conditions, including the surrounding soil layers, water level, and side resistance of an underground structure module (USM). USM is a physical model simulating an underground structure, composed of a rigid circular steel plate and a triangular cone. The rigid circular steel plate is positioned at the bottom of the enclosure module, and the triangular cone is attached to it. The triangular cone serves as a support point for the tensile and compression transducers at the upper part of the underground engineering module. For this reason, laboratory model tests were selected for the experimental study of groundwater buoyancy in high- and low-permeability layers.

(1)

Test design

To determine the variation trend of buoyancy acting on the underground engineering module under different water supply conditions in the test soil layer, it is essential to establish an external environment simulating the soil layer and groundwater for the module and to perform continuous buoyancy monitoring tests via the model test box (MTB). MTB is a test chamber used for conducting experiments, which includes an underground engineering module, an enclosure module, a test soil layer, a water supply system, and a monitoring system. The underground engineering module consists of a rigid circular steel plate and a triangular cone; the enclosure module consists of a bottomless steel cylinder with a waterproof rubber base, and the test soil layer consists of silty clay and fine sand. The test box serves as a substitute for the test soil layer, allows adjustment of the test supply head, and enables real-time monitoring of the water level in the test soil layer and the buoyancy acting on the underground engineering module. The test box comprises an underground engineering module, an enclosure-structure module, a test soil layer, a water supply system, and a monitoring system.

The underground structure module USM was composed of a rigid circular steel plate and a triangular cone. The rigid circular steel plate was located at the bottom of the enclosure structure module, with the triangular cone connected to its upper side. The triangular cone provided a force-bearing fulcrum for the tension–compression sensors in the upper part of the underground structure module USM. The enclosure structure module, a physical model of the actual engineering structure, is composed of a bottomless steel cylinder with a waterproof rubber base. This module consisted of a steel cylinder with a waterproof rubber bottom, which ensured that the interior of the enclosure structure remained dry during the test. In addition, the internal underground structure module USM did not directly contact the enclosure wall, thereby eliminating the side friction on the underground structure module USM. The test soil layers were silty clay and fine sand. Silty clay is a representative low-permeability layer and was the focus of this study; fine sand is a representative high-was used to permeability layer, which verify the reliability of the test model and to perform a comparative analysis with the buoyancy monitoring test of the silty clay layer. The water supply system was composed of an external water tank, a water inlet pipe, and inverted filter layers. The external water tank could provide different water head conditions; the inlet pipe was a plastic hose that connected the external water tank and the model test box, and the inverted filter layers were composed of medium sand. Their main function was to provide a uniform water supply for the upper test soil layer. The monitoring system included head monitoring, soil layer water level monitoring, and buoyancy monitoring.

(2)

Composition of the test device

The main components of the test device included a model test box, test soil layer, triangular cone, rigid circular steel plate, enclosure structure, waterproof rubber, reaction beam, soil layer water level observation holes, water supply tank, and inverted filter layers. Figure 1 and Figure 2 show the details of the test device. This paper focuses on the study of the relationship between water level and water buoyancy in a single soil layer. The design principle is to reflect the water level and water buoyancy in a single stratum. In addition, the bottomless steel drum plays a role in isolating the test soil layer and underground structure module contact, to avoid the impact of sidewall friction on the test data.

The length, width, and height of the test box were 1000 mm, 1000 mm, and 1100 mm, respectively. The enclosure structure was a bottomless steel cylinder with a height of 400 mm and an inner diameter of 350 mm. The underground structure module consisted of a triangular cone and a rigid circular steel plate with an outer diameter of 335 mm. A lateral water supply from the bottom was used for the entire test. A 10-centimeter-thick medium sand layer was laid at the bottom of the test box, on top of which the test soil layer was placed.

A water supply tank was installed outside the test box to supply water to the 10-centimeter-thick medium sand layer at the bottom. The water supply tank was connected to a lift, and the water supply level was determined by adjusting the height of the tank. To facilitate subsequent discussion and analysis, the bottom of the rigid circular steel plate was used as the reference plane in each test, that is, the zero point of the head in the test. This method is only applicable to the determination of the relationship between the water table and water buoyancy in remodeled soils of any soil sample.

(3)

Test procedure

Figure 3 shows the details of the test procedure.

①

Preparation and filling of soil samples. The fine sand and silty clay were used in the test. The effect of structural soil on the test results was not considered in this test; therefore, remolded soil was used instead of undisturbed soil. The retrieved soil samples were manually crushed in the laboratory. During the soil sample filling process, a small amount of water was sprinkled while laying down the soil. The soil was compacted multiple times using manual tamping. The thickness of each filling layer was not more than 20 cm. The filling process for fine sand and silty clay samples was the same. The physical properties of the test soil samples after filling are shown in Table 1 and Table 2. First, a test was conducted on the fine sand layer to verify the reliability of the test system, and then a continuous buoyancy monitoring test was conducted on the silty clay layer.

②

Water supply from the external water tank. Water was continuously supplied to the model box through the external water tank, ensuring that the supply head remained aligned with the bottom of the rigid circular steel plate inside the model. The water level in the soil layer inside the box was continuously monitored. After the water level stabilized, the system remained at rest for 24 h, at which point the test soil layer below the underground engineering module was fully saturated.

③

Buoyancy monitoring test. The height of the water supply tank was adjusted to gradually increase the supply head. The increase in the supply head at each stage depended on the water level and buoyancy response rate of the soil layer in the model. At the beginning of each stage, water was added to the tank to maintain a stable supply head for a certain period; afterward, the water addition was stopped. After the water level in the test soil layer stabilized, the water supply level was further raised, and the next stage commenced. During the entire testing process, the water level and buoyancy were monitored in real time using automatic systems.

In the above tests, the water supply level and the groundwater level in the model box were observed manually. The tension–compression stress sensor test data were collected using a YSV8360 static strain gauge and an intelligent water and soil pressure monitoring data acquisition and analysis system (version 9.0).

Table 1. Physical properties of fine sand test samples.

Test Sample	Water Content w (%)	Wet Density ρ (g·cm−3)	Void Ratio e	Saturation Sr (%)	Dry Density ρd (g·cm−3)	Permeability Coefficient k (cm·s−1)
Sand	26.7	1.96	0.64	96.1	1.63	5.59 × 10−3

Table 1. Physical properties of fine sand test samples.

Test Sample	Water Content w (%)	Wet Density ρ (g·cm−3)	Void Ratio e	Saturation Sr (%)	Dry Density ρd (g·cm−3)	Permeability Coefficient k (cm·s−1)
Sand	26.7	1.96	0.64	96.1	1.63	5.59 × 10−3

Table 2. Physical properties of silty clay samples.

Test Sample	Water Content w (%)	Wet Density ρ (g·cm−3)	Porosity n (%)	Saturation Sr (%)	Dry Density ρd (g·cm−3)	Plastic Index Ip	Liquid Index IL	Permeability Coefficient
								kh (cm·s−1)	kv (cm·s−1)
Silty clay	21.1	1.71	44.6	95.9	1.35	13.4	0.46	1.45 × 10−5	1.16 × 10−5

Table 2. Physical properties of silty clay samples.

Test Sample	Water Content w (%)	Wet Density ρ (g·cm−3)	Porosity n (%)	Saturation Sr (%)	Dry Density ρd (g·cm−3)	Plastic Index Ip	Liquid Index IL	Permeability Coefficient
								kh (cm·s−1)	kv (cm·s−1)
Silty clay	21.1	1.71	44.6	95.9	1.35	13.4	0.46	1.45 × 10−5	1.16 × 10−5

(4)

Force analysis

In this test, the underground engineering module was not subjected to side friction, and only the vertical force analysis was considered. According to the static equilibrium principle, the force analysis of the underground engineering module is shown in Figure 4, and the equilibrium equation is as follows:

$$F_1+F_2=G_0+P_{\mathrm{s}}$$

(1)

where $G_0$ is the self-weight of the underground engineering module (N); $P_{\mathrm{s}}$ is the pressure monitored by a tension–compression sensor (N); $F_1$ is the vertical soil particle reaction (N); and $F_2$ is the groundwater buoyancy (N).

The initial water supply placed the water level of the soil layer in the test box exactly at the bottom of the structure. At this time, the groundwater buoyancy F₂ was 0, and the pressure monitored by the tension–compression sensor was a constant P_s0, which was 338 N for fine sand and 250 N for silty clay. According to Equation (1), the vertical soil particle reaction F₁ = P_s0 + G₀ (G₀ = 39.9 N). After the test started, the water level in the test soil layer increased, and the change in the readings of the tension–compression sensors represented the change in the total water–soil pressure caused by the combined effect of groundwater buoyancy. Therefore, assuming that the vertical reaction F₁ of the soil particle remained unchanged during the test, the change in total pressure was regarded as the groundwater buoyancy F₂. In a stable state, the relationship is expressed as

$$F_2=P_s-P_{s0}$$

(2)

The measured groundwater buoyancy is converted to a head, that is,

$$h_{buoy}={\displaystyle \frac{F_2}{Ag}}$$

(3)

where h_buoy is the buoyancy head converted from the measured water buoyancy, mm; A is the area of the bottom of the rigid circular steel plate, with a diameter of 335 mm and an area of 88,141.31 mm²; and g is the acceleration of gravity, taken as 10 N/kg.

2.2. Geotechnical Test

(1)

Mercury injection test

Silty clay has small pores. Four samples were tested via mercury injection to measure the pore sizes.

(2)

Measurement test for the volume ratios of soil particles and bound water

After the continuous buoyancy monitoring test, six ring-cutter soil samples were taken at the base of the underground engineering module. First, the samples were placed in a frame-saturator for vacuum saturation and removed after 24 h to measure the saturated density. Then, the saturated samples were suspended in a closed chamber for gravity drainage and removed after 24 h to measure their post-drainage density. Finally, the samples were dried at 105 °C, and their dry density and specific gravity were measured. The volume ratios of soil particles and bound water are calculated as follows:

$$V_1={\displaystyle \frac{{\rho }_d}{G_s\times {\rho }_w}}$$

(4)

$$V_2={\displaystyle \frac{\rho -{\rho }_d}{{\rho }_w}}$$

(5)

where $V_1$ is the volume ratio of soil particles (dimensionless); $V_2$ is the volume ratio of bound water (dimensionless); ${\rho }_d$ is the dry density (g/cm³); $\rho$ is the density after gravity drainage (g/cm³); ${\rho }_w$ is the density of water (g/cm³); and $G_s$ is the specific gravity of soil particles (dimensionless).

(3)

Particle size analysis test

The particle compositions of the above six silt clay samples were determined using the density meter method. First, a certain mass of sample was added to 10 mL of 4% (NaPO₃)₆ to form a 1000 mL suspension. Then, based on the principle that soil particles of different sizes in the suspension settle at different velocities, the diameters of different-sized particles were calculated using the Stokes law. Finally, the mass percentage of soil particles of different sizes was measured with a density meter.

3. Analysis of Experimental Results

The buoyancy monitoring tests for the fine sand and the silty clay layer were carried out separately. First, the dynamic process of buoyancy under instantaneous water supply conditions was analyzed. Then, the buoyancy reduction factor was calculated using the conventional method. Finally, the relationship between the stable water supply head and stable buoyancy was comparatively analyzed to demonstrate the existence of a reduction slope.

3.1. Analysis of the Dynamic Process of Buoyancy Under Instantaneous Water Supply Conditions

In accordance with the test procedure, the supply head $h_{supp}$ and the buoyancy head $h_{buoy}$ were obtained, and the dynamic process of the buoyancy head $h_{buoy}$ under the instantaneous water supply conditions in the fine sand layer and silty clay layer was plotted, as shown in Figure 5 and Figure 6.

As seen in Figure 5 and Figure 6, after the supply head $h_{supp}$ increased instantaneously at each stage, the water level in the water supply tank decreased over time, and the buoyancy head $h_{buoy}$ gradually increased. The response speed of the buoyancy head $h_{buoy}$ for the underground engineering module in the fine sand layer to the supply head $h_{supp}$ was fast, with a total test time of 10.5 h for the four stages. In each stage, the water supply continued for the first 0.5 h, resulting in a rapid increase in the buoyancy head $h_{buoy}$; it stabilized after 2 h, and the measured buoyancy head $h_{buoy}$ in the fine sand layer was almost equal to the supply head $h_{supp}$. In comparison, the response speed of the buoyancy head $h_{buoy}$ for the underground engineering module in the silty clay layer to the supply head $h_{supp}$ was slow, with a total test time of approximately 572 h for the seven stages. An average of 60 h was needed for each stage to reach a stable state. After stabilization, the difference between the two values was significant, with the buoyancy head $h_{buoy}$ markedly lower than the supply head $h_{supp}$.

As shown in Figure 5 and Figure 6, the measured buoyancy on the underground engineering module in the high-permeability layer is largely consistent with the theoretical value, showing no reduction phenomenon, which indicates the reliability of the test model system. However, the measured buoyancy on the underground module in the low-permeability layer is significantly lower than the theoretical value, demonstrating the existence of a reduction phenomenon.

3.2. Calculation of the Reduction Factor Under Hydrostatic Conditions

The measurement results of the stable supply head $h_{supp}$ and the buoyancy head $h_{buoy}$ for the fine sand layer and silty clay layer under hydrostatic conditions are shown in

Table 3. Measurement results of $h_{supp}$ and $h_{buoy}$ for the fine sand layer.

Stage	$\mathbf{Stable} {\mathit{h}}_{\mathit{s}\mathit{u}\mathit{p}\mathit{p}}$ (mm)	$\mathbf{Stable} {\mathit{h}}_{\mathit{b}\mathit{u}\mathit{o}\mathit{y}}$ (mm)	Reduction Factor
1	95	91	0.96
2	196	190	0.97
3	306	305	1.00
4	385	382	0.99

and

Table 4. Measurement results of the supply head and buoyancy head for the silty clay layer.

Stage	$\mathbf{Stable} {\mathit{h}}_{\mathit{s}\mathit{u}\mathit{p}\mathit{p}}$ (mm)	$\mathbf{Stable} {\mathit{h}}_{\mathit{b}\mathit{u}\mathit{o}\mathit{y}}$ (mm)	Reduction Factor
1	100	77	0.77
2	180	135	0.75
3	210	156	0.74
4	270	214	0.79
5	330	272	0.82
6	360	293	0.81
7	410	350	0.85

, respectively. The reduction factor of the buoyancy head $h_{buoy}$ in comparison with the supply head $h_{supp}$ in the fine sand layer was between 0.96 and 1.00, indicating almost no reduction. The reduction factor of the buoyancy head $h_{buoy}$ in comparison with the supply head $h_{supp}$ in the silty clay layer was in the range of 0.74 to 0.85, and the reduction factor varied under different supply heads $h_{supp}$, showing an increasing trend with higher supply head $h_{supp}$, a pattern consistent with previous research results [24].

3.3. Analysis of the Linear Relationship Between the Stable Supply Head and the Buoyancy Head

Linear regression was performed on the supply head $h_{supp}$ and the buoyancy head $h_{buoy}$ to obtain the fitting curves, as shown in Figure 7.

As shown in Figure 7, the slope of the linear fit between the stable supply head $h_{supp}$ and the buoyancy head $h_{buoy}$ in the fine sand layer is approximately 1, indicating that there is no head reduction in the high-permeability layer. Figure 7 shows that the supply head $h_{supp}$ and the buoyancy head $h_{buoy}$ in the silty clay layer under hydrostatic conditions are not proportional to each other but rather form a straight line that does not pass through the origin. This explains the phenomenon of inconsistent reduction factors under different water levels. The slope of the fitting curve is a constant value of 0.8692, and the correlation coefficient R2 = 0.9946. This slope is referred to as the “reduction slope” in the present study. The test results revealed that the slope of the fitting curve for the silty clay layer was pronounced, with a considerable impact on the calculation of groundwater buoyancy. It is inappropriate to use the conventional reduction factor to quantitatively study the relationship between the groundwater level and buoyancy, whereas the use of the reduction slope can more accurately reflect the relationship between the two. This test chamber can be reused to determine the slope of groundwater buoyancy reduction in various types of saturated soils, and this result can be used in the design of anti-floating for actual underground projects.

In this paper, remodeled soil was used for the test. Although the test results have guiding significance for this type of soil, the remodeled soil has been damaged compared to the original soil structure, which cannot completely and accurately reflect the situation of groundwater buoyancy in the original soil, so there are some limitations. In addition, due to the restricted test conditions, this paper uses a small model test and fails to take into account the effect of the size of the effect of results of the study, in the follow-up study. In the subsequent research, large-scale modeling tests and field tests can be carried out to reflect more realistically the relationship between groundwater level and water buoyancy in weakly permeable layers such as clay and silty clay.

4. Theoretical Derivation of the Reduction Slope

4.1. Theoretical Explanation

Underground engineering loads are borne jointly by soil particles and pore water in a soil layer. The buoyancy on an underground structure is calculated as the pore water pressure multiplied by the base area of the foundation. The key to calculating groundwater buoyancy lies in the contact relationship between the foundation and the two-phase water–soil media, as well as the types of pore water.

(1) The contact relationship between the foundation and the two-phase water–soil media. Li et al. [8] clarified the contact relationship between the foundation base and the two-phase water–soil media. They noted that in saturated loose granular materials, the foundation base cannot cut off the soil particles but instead forms an approximate point of contact with them. Under this contact relationship, the pore water at the foundation base surface is nearly 100% in contact with the foundation base. Currently, buoyancy calculations assume that all the pore water in the underground soil layer is gravity water, and the pore water pressure at the foundation base is entirely provided by gravity water (Figure 8). However, bound water and gravity water are gradually distributed among soil particles in the underground soil from the particle surface to the pore center. Bound water can be further divided into strongly bound water and weakly bound water. These three types of water all contact the foundation base. The strongly and weakly bound water provides low pore water pressure, whereas gravity water provides high pore water pressure, constituting the main source of groundwater buoyancy on the foundation base. Therefore, pore water pressure distribution from the pore center to the soil particle surface exhibits an attenuation trend (Figure 9). When the particle size is large, the interparticle pores are large; the proportion of strongly and weakly bound water is small; the gravity water proportion is an order of magnitude higher, and the pore water pressure attenuation is negligible. However, as the particle size decreases, the interparticle pores shrink, the proportion of gravity water decreases, and the strongly and weakly bound water proportion increases. In such cases, the pore water pressure attenuation becomes significant, representing the primary cause of pore water pressure reduction in fine-grained soils.

(2) The types of pore water. Yang et al. [25] studied the distributions of the above two types of water with different pore sizes. When the pore size φ < 0.06 µm, the pores mainly consist of intraparticle pores, and the entire pore water is bound to water. When 0.06 µm ≤ φ < 1.5 µm, the pores are primarily composed of interparticle pores, and the pore water gradually transitions from the dominance of bound water to the dominance of gravity water. When 1.5 μm ≤ φ < 10.0 μm, interparticle pores dominate, including some large pores compressed during transformation; these pores have a large diameter, high quantity, and improved connectivity. The pore water is dominated by gravity water, followed by bound water. When φ ≥ 10 μm, these pores include isolated pores and partially connected interparticle pores, which contain both bound water and gravity water, with the latter being overwhelmingly dominant. The results of mercury injection tests on the silt clay test layer are shown in

Table 5. Pore size distribution of clayey soil.

Sample No.	Pore Volume Distribution (%)
	>10 μm	10–1.0 μm	1.0–0.1 μm	<0.1 μm
1	7.11	8.75	70.16	13.98
2	0.15	11.31	74.90	13.64
3	15.47	13.54	39.43	31.56
4	14.30	10.23	41.11	34.37

. The pore sizes of pore water in the silty clay were generally less than 10 μm. Therefore, the pore water in the silty clay layer was dominated by gravity water, followed by bound water. It is assumed that bound water cannot provide buoyancy and that all buoyancy is provided by gravity water, which can flow naturally.

(3) The contact area ratio of gravity water. The interparticle pores of silty clay are filled with non-negligible bound water. In this case, the contact area ratio of the gravity water on the foundation base cannot be approximately 100%. This also explains the phenomenon of buoyancy reduction in saturated clayey soil. In theory, the contact area ratio of gravity water corresponds to the “reduction slope”. Therefore, it is necessary to determine the actual contact area ratio of gravity water.

4.2. Theoretical Assumptions

To explore the pore water pressure reduction induced by the above theory, the volume ratio of gravity water to strongly and weakly bound water, as well as the pore size and particle size of the test soil layer, should be measured, and the contact area ratio of gravity water should be calculated. For this purpose, it is necessary to generalize the water–soil constitutive model. The assumptions are as follows: ① Soil particles are regarded as rigid-body load-bearing units without deformation; ② The base of the foundation is in point contact with the soil particles; ③ The water in the soil attenuates from the boundary of gravity water to the particle surface (where the strongly and weakly bound water reside). In this case, the proportion of gravity water is less than 100%, the buoyancy is still provided based on the area of gravity water, and the buoyancy effect of bound water (i.e., the attenuation part) is ignored; ④ Soil particles are spherical.

4.3. Formula Derivation

The quantitative contact relationship between the pore water, soil particles, and the foundation base in the soil particle accumulation is established, as shown in Figure 10. Figure 10 illustrates that under the condition of point contact between the foundation base surface and soil particles, the contact area ratio of gravity water is equal to 1 minus the contact area ratio of bound water. A flowchart for solving the area ratio of gravity–water interaction is provided in Figure 11.

The calculation of the contact area ratio of bound water depends on the size and quantity of soil particles per unit area, as well as the thickness of bound water adhering to the soil particle surface. The size of soil particles can be measured through particle size analysis, and the number of soil particles of each particle size per unit area, the thickness of bound water adhering to the surface of soil particles, and the contact area ratio of gravity water are calculated using the following formulas.

(1)

The number of particles of each size per unit area is equal to the number of particles within the thickness of a single particle per unit volume. The calculation formulas are as follows:

$$M_i={\displaystyle \frac{N_1{r}_i}{\sqrt[3]{V}}}$$

(6)

$$N_i={\displaystyle \frac{V_i{x}_i}{\frac{400}{3}\pi r_i^3}}$$

(7)

where $M_i$ is the number of soil particles corresponding to each particle size per unit area, mm⁻²; $N_i$ represents the number of soil particles with different sizes per unit volume, mm⁻³; V is the unit volume, taken as 1 mm³; and $V_i$ is the volume ratio corresponding to different particle sizes per unit volume of the sample (dimensionless), determined via geotechnical tests; $r_i$ is the generalized particle size of the soil particles (mm), determined by particle size analysis; and $x_i$ is the percentage of particles of different particle sizes in the mass of soil particles (dimensionless), determined by particle size analysis.

(2)

Assuming that bound water covers the particle surface uniformly with a constant thickness, the sum of the volumes of all spheres formed by bound water and soil particles is equal to the sum of the volumes of the soil particles and bound water measured in the experiment. Based on this relationship, the adhesion thickness can be calculated as follows:

$$d=\sqrt{{\displaystyle \frac{3}{4\pi }}{\displaystyle \frac{V_1+V_2}{{\displaystyle {\sum }_{i=1}^n{N}_i{\left(r_i\right)}^2}}}-r_i}$$

(8)

where $d$ is the adhesion thickness of bound water on the particle surface (mm), and $V_2$ is the volume ratio of bound water in a unit volume of sample (dimensionless).

(3)

The contact area ratio of gravity water is calculated as follows:

$$F=1-{\displaystyle {\sum }_{i=1}^n{M}_i}\pi \left[{(r_i+d)}^2-r_i^2\right]$$

(9)

where $F$ is the contact area ratio of gravity water (dimensionless), and $n$ is the number of particle size groups (dimensionless).

5. Test Validation of the Reduction Slope

5.1. Verification Process

To accurately calculate the contact area ratio of the buoyancy of gravity water, tests on the six sets of silty clay samples were carried out to determine the volume ratios of soil particles and bound water, and the particle size distributions of the six sets of soil samples were also obtained. The test results are shown in

Table 6. Measurement results of the volume ratios of soil particles and bound water.

Sample No.	ρ (g/cm3)	ρd (g/cm3)	GS	V1	V2
1	1.63	1.32	2.74	0.48	0.31
2	1.65	1.37	2.74	0.50	0.28
3	1.52	1.25	2.74	0.46	0.27
4	1.68	1.40	2.74	0.51	0.28
5	1.69	1.44	2.74	0.53	0.25
6	1.66	1.34	2.74	0.49	0.32
Average	1.64	1.35	2.74	0.49	0.29
Dispersion coefficient	0.04	0.05	0.00	0.02	0.02

and Figure 12. As shown in Table 6, the average values of the soil particle volume ratio V₁ and bound water volume ratio V₂ are 0.49 and 0.29, respectively, with a small dispersion coefficient of 0.02. These results can represent the true pore water occurrence characteristics of the silty clay in the test soil layer. According to the results of the particle size analysis test, the soil samples were divided into medium sand (particle size from 0.50 to 0.25 mm), fine sand (particle size from 0.25 to 0.075 mm), silt (particle size from 0.075 to 0.05 mm), coarse silt (particle size from 0.05 to 0.025 mm), fine silt (particle size from 0.025 to 0.005 mm), and clay (particle size smaller than 0.005 mm). The measurement results showed that the sizes of all particles in the silty clay samples were less than 0.25 mm, primarily concentrated smaller than 0.05 mm.

To obtain the number of particles per unit area, the particle sizes of the five groups need to be generalized, and the arithmetic mean of the particle size threshold for each group is taken as the average size of the group. The average sizes of the five groups of particle sizes are 0.163 mm, 0.063 mm, 0.038 mm, 0.015 mm, and 0.003 mm. The number of soil particles corresponding to the average size of each group, calculated using Equation (7), is shown in Figure 13 and

Table 7. Calculation results of main parameters.

Sample No.	Particle Size Composition	${\mathit{r}}_{\mathit{i}}$ (mm)	${\mathit{M}}_{\mathit{i}}$ (mm−2)	$\mathit{d}$ (mm)	$\mathit{F}$
1	0.25~0.075	0.163	2.08	5.72 × 10−4	0.7448
	0.075~0.05	0.063	2.39 × 101
	0.05~0.025	0.038	1.65 × 102
	0.025~0.005	0.015	1.37 × 103
	<0.005	0.003	3.67 × 104
2	0.25~0.075	0.163	2.53	5.90 × 10−4	0.7612
	0.075~0.05	0.063	2.57 × 101
	0.05~0.025	0.038	1.92 × 102
	0.025~0.005	0.015	1.43 × 103
	<0.005	0.003	3.15 × 104
3	0.25~0.075	0.163	2.56	5.40 × 10−4	0.7754
	0.075~0.05	0.063	2.63 × 101
	0.05~0.025	0.038	1.18 × 102
	0.025~0.005	0.015	1.47 × 103
	<0.005	0.003	3.39 × 104
4	0.25~0.075	0.163	1.96	4.90 × 10−4	0.7623
	0.075~0.05	0.063	3.04 × 101
	0.05~0.025	0.038	1.43 × 102
	0.025~0.005	0.015	1.53 × 103
	<0.005	0.003	4.12 × 104
5	0.25~0.075	0.163	2.19	4.24 × 10−4	0.7888
	0.075~0.05	0.063	2.07 × 101
	0.05~0.025	0.038	1.66 × 102
	0.025~0.005	0.015	1.66 × 103
	<0.005	0.003	4.28 × 104
6	0.25~0.075	0.163	2.13	4.60 × 10−4	0.7779
	0.075~0.05	0.063	2.09 × 101
	0.05~0.025	0.038	1.46 × 102
	0.025~0.005	0.015	1.48 × 103
	<0.005	0.003	4.18 × 104
Average value	-	-	-	5.53 × 10−4	0.7684
Dispersion coefficient	-	-	-	0.00	0.01

. There was an order-of-magnitude difference in the number of particles across groups, and the number of soil particles increased as the particle size decreased; the number of soil particles per unit area corresponding to the average size of 0.163 mm was approximately 2.0 to 2.6 mm⁻², and the number corresponding to the average size of 0.003 mm was approximately 31,490.5 to 42,778.1 mm⁻².

To obtain the bound water adhesion thickness d, the adhesion thickness d for each set of soil samples was assumed to be a constant value and calculated, and the results were statistically analyzed, as shown in Table 7. The data indicate that the bound water adhesion thickness d of range from 4.24 × 10⁻⁴ to 5.90 × 10⁻⁴ mm, with an average of 5.13 × 10⁻⁴ mm and a dispersion coefficient of 0.00, indicating almost no dispersion. Furthermore, the measured adhesion thickness was approximately 1000 times the thickness of a water molecule (4.00 × 10⁻⁷ mm), consistent with previous research findings [26], and represents the actual adhesion thickness d of bound water in the silt clay of the test soil layer.

Based on the number of soil particles of each particle size per unit area $M_i$ and the bound water adhesion thickness d, the contact area ratio of the buoyancy of the gravity water $F$ was calculated using Equation (9). As shown in Table 5, $F$ of the test soil samples ranged from 0.7448 to 0.7888, with a very small dispersion coefficient of 0.01. Its average value (0.7684) is less than the reduction slope (0.8629) obtained from the continuous buoyancy measurement tests, with a percentage difference of 10.9%. Although there is a difference between $F$ and the measured reduction slope of buoyancy, the discrepancy is not significant. Moreover, the theoretical derivation process can fully explain the causes of the reduction phenomenon. These results are of great value for calculating the reduction slope of pore water pressure.

5.2. Error Analysis

The reduction slope of buoyancy in saturated clayey soil is the experimentally measured value, while the contact area ratio of gravity water is a theoretically derived value. In theory, the two values should be nearly equal, but the results showed a significant difference between them. There are three possible reasons: ① The soil particles are assumed to be spherical, but remolded silty clay particles are irregular in shape, which may introduce errors. ② It is assumed that bound water does not provide pore water pressure and that the pore water pressure is entirely provided by gravity water. However, in the water–soil constitutive model, pore water pressure distribution from the pore center to the soil particle surfaces should exhibit an attenuation process (Figure 4). However, the assumption defaults this attenuation to 0, ignoring the residual pore water pressure in the attenuation zone. As a result, $F$ is smaller than the actual effective pressure area, leading to a theoretical value smaller than the experimentally measured reduction slope. Using the currently available technical means. ③ It is difficult to analyze and measure $F$ between irregular particles, and the pore water pressure in the attenuation process cannot be accurately measured. Although there is an error between the test and theoretical results, the derivation provides a scientific framework for understanding and estimating pore water pressure reduction in underground rock formations.

Although the calculation results of gravity water action area ratio and buoyancy measurement test determine the slope of the discount, the difference is not large, and its theoretical derivation process can fully explain the theoretical reasons for the phenomenon of discount; the follow-up can be a further study of flocculent or flaky soil particles’ gravity water action area ratio and the distribution of pore water pressure in part of the inter-particle attenuation process or based on the results of the research to carry out the gravity water action area ratio and the slope of the discount of the linear relationship between the two, and revises a coefficient to guide the engineering investigation. The linear relationship between the area ratio of gravity water action and the slope revises a coefficient, and is then realized in the investigation of the actual project; the use of geotechnical tests can be determined in saturated clayey soil pore water pressure slope, guiding the purpose of engineering design.

6. Conclusions

To address the problem of inaccurate buoyancy calculations in engineering practice, buoyancy observation tests of an underground engineering module in subsurface rock formations were conducted. The main conclusions are as follows:

①

Through laboratory buoyancy observation tests, the reduction slope of the buoyancy head $h_{bouy}$ in comparison with the supply head $h_{supp}$ in the silty clay layer was 0.8629. The phenomenon of the reduction slope in groundwater buoyancy in saturated clay was discovered and confirmed.

②

Using geotechnical test data, the theoretical basis for the buoyancy reduction slope was derived, demonstrating that the reduction slope is theoretically equal to the contact area ratio of gravity water. Its test value (0.7684) is less than the reduction slope (0.8629) obtained from the continuous buoyancy measurement tests, with a percentage difference of 10.9%. Although there is a difference between $F$ and the measured reduction slope of buoyancy, the discrepancy is not significant. Finally, the main reasons for the differences between the reduction slope and the gravity water contact area ratio were analyzed.

③

Based on the theory that the interparticle pore water pressure distribution tends to attenuate from the pore center to the soil particle surface, this study provides a method for measuring the reduction slope, thereby offering a scientific method for addressing buoyancy challenges in underground engineering projects.

Engineers can take soil samples from the actual project site and use the geotechnical test in “Section 2.2” and the mathematical method in “Section 4.3” to calculate the ratio of the area under the action of gravity water. The “slope reduction” is then derived to guide the anti-floating design of the project. The area ratio of gravity water interaction between flocculent or flaky soil particles and the distribution of pore water pressure during the attenuation process between particles can be further investigated in the future, so as to continuously accumulate the number of samples and optimize the research method.