Discussing the Negative Pressure Distribution Mode in Vacuum-Preloaded Soft Foundation Drainage Structures: A Numerical Study

Abstract

The aim of this paper is to clarify the negative pressure distribution in drainage structures of soft foundations reinforced by vacuum preloading. The focus of this study was an actual engineering project, the Beijing–Shanghai high-speed railway; four different soil consolidation models were established using FLAC^3D to consider various loading conditions. The consolidation process of the soft foundation was calculated and analyzed in detail. The results show that (1) the settlement developed rapidly within the first 30 days, slowed during the period between 20 and 30 days, and finally stabilized. (2) The settlement curves obtained from the four different models were highly consistent with the site monitoring curve for the first 5 days, after which point significant differences appeared. (3) During the first 20 days, the pore water pressure decreased noticeably within the depth range of 0–18 m. Between days 20 and 30, the rate of pore water pressure decrease slowed down, and after the 30th day, the pore water pressure remained constant at all depths. (4) Vacuum preloading affected the soil to a depth of approximately 16 m. A concave or linear distribution of negative pressure in the drainage structure was found to be a reasonable assumption, providing a reference for the numerical analysis of vacuum preloading.

1. Introduction

Thick, soft soil layers are widely distributed in coastal areas of China. These soil layers are characterized by a high void ratio, high water content, low permeability coefficient, and low strength. To build structures on such soft soil, it is necessary to reinforce the foundation.

The vacuum preloading method is a mature and low-cost technique used to strengthen soft foundations in the field of drain consolidation. This method was first proposed by W. Kjellman in 1952 [1], and since then, civil engineering scholars at home and abroad have conducted practical applications, tests, and theoretical research on this method. In terms of theoretical analysis, various researchers, such as Barron [2], Horne [3], Hansbo [4], Yoshikuni and Nakanodo [5], Onoue [6], Dong [7], Lin et al. [8], Liu et al. [9], etc., have developed sand drain consolidation theory with reference to the consolidation calculation method of surcharge preloading, changing the load boundary conditions to derive accurate analytical solutions of consolidation suitable for vacuum preloading. However, these methods are not convenient for practical application and promotion due to their simplification of assumptions, obscure and complex derivation processes, large numbers of parameters, and extensive requirements. In terms of numerical calculations, many researchers, such as Cheung et al. [10], Hird et al. [11], Indraratna and Redana [12], Zeng et al. [13], Chai et al. [14], Sha et al. [15], Bergado et al. [16], Nguyen et al. [17], and Wang et al. [18], have conducted extensive research, including two-dimensional simulation, three-dimensional simulation, and sand drain simplification, considering smearing and other factors. However, few studies have been conducted on the negative pressure distribution mode of the vertical drain body under vacuum preloading. The plane strain model is usually used to represent the actual three-dimensional situation in the concrete implementation process. Although many engineering examples have been used to obtain relatively ideal results, the seepage and mechanical effects of soft foundations under vacuum preloading are three-dimensional, so a three-dimensional model should be applied in research. In the process of theoretical exploration and practical application, the mechanical and fluid disturbances of soft foundations strengthened by vacuum preloading are significant. Therefore, it is essential to analyze the settlement and pore pressure of soft foundations under this construction method using the fluid–solid coupling method, as suggested by Zhao et al. [19] and Liu et al. [20].

FLAC3D finite-difference method software based on the LaGrangian continuum method is a suitable option for conducting such analysis. This software has a strong analytical ability, making it suitable for complex engineering problems involving the mechanical seepage coupling of geotechnical materials. Compared to other finite element software, FLAC3D has the following advantages. (1) FLAC3D adopts the “mixed discrete method”, which is more accurate and reasonable than the “discrete integration method” commonly used in the finite element method. (2) There are no numerical barriers to simulating physically unstable processes with FLAC3D. (3) FLAC3D uses an explicit difference method to solve differential equations. It can conveniently calculate stress increments and unbalanced forces and track the evolution process of the system. (4) FLAC3D can simulate a large number of units with less memory, making it particularly suitable for operation on microcomputers.

In the present study, fluid–solid coupling analysis of soft soil under the combined action of vacuum and surcharge was simulated Using FLAC3D.

We utilized the FLAC3D finite-difference method software to model and calculate the soil consolidation process under vacuum preloading for specific projects while varying the negative pressure distributions of the drainage body and keeping other conditions constant. The calculated results were compared with measured values to assess the impact of different negative pressure distributions on soil consolidation. This study is significant, given its aim of improving the accuracy of soft foundation settlement and pore pressure calculation.

2. Engineering Background

The soft foundation reinforcement test section for the Beijing–Shanghai high-speed railway was located in Kunshan, China. Specifically, the section from k0 + 276.51 to k0 + 515 was strengthened using the vacuum preloading + plastic drain board drainage consolidation method, with a reinforcement depth of 14.5~18.5 m. The plastic drain boards were arranged in a quincunx shape with a spacing of 1.2 m, and a 0.8 m thick sand cushion layer with a geogrid was placed on top of the plastic drain board. The vacuum pressure under the membrane was maintained at a minimum of 80 kPa. The consolidated foundation soil layer primarily consisted of muddy silty clay with high compressibility and low strength. Figure 1 displays the calculation section.

The settlement of the foundation surface was measured using settlement plates and an N4 leveling instrument. Specifically, one settlement plate was buried at the midline of the road bed, with another at the midline of the left lane and a third at the shoulder of the right lane. The layered settlement of the foundation was tested using settlement tubes, magnetic rings, and an R40 settlement monitoring instrument. The settlement tube was buried near the centerline of the right lane close to the centerline of the road bed. The pore water pressure was measured using MSY pore water pressure gauges and a frequency instrument. Ten water pressure gauges were buried at intervals of 2 m, beginning 2 m below the surface near the center of the section. These test components are shown in Figure 2. Vacuum preloading lasted for 56 days, from 26 April to 20 June, after which time the embankment was formally filled.

3. Numerical Analysis Calculation Mode

FLAC (Fast Lagrangian Analysis of Continua) software, which uses the LaGrangian continuum method, was utilized for finite-difference numerical calculations. Initially, FLAC was developed for geotechnical and mining engineering applications. The strong analytical capabilities of the FLAC software for complex engineering problems make it adaptable to civil engineering, transportation, water conservancy, and other fields. FLAC^3D can be used to simulate the consolidation process of geotechnical materials, making it suitable for the study of the consolidation behavior of soil under vacuum preloading.

3.1. Constitutive Model and Seepage Model

Among numerous constitutive models of rock and soil, the Mohr Coulomb plastic constitutive model is the most universal. The Mohr Coulomb plastic constitutive model is suitable for materials that yield under shear stress, such as loose or cemented granular soil [21,22,23,24,25]. Geotechnical engineering problems often involve the action of pore water pressure, such as groundwater in the foundation, the seepage of earth dams, and the dewatering of foundation pits. When analyzing problems involving pore water pressure, FLAC^3D has two calculation modes that can be applied depending on whether fluid calculation is set up, namely, the seepage mode and non-seepage mode. Therefore, the use of the Mohr Coulomb plastic constitutive model to simulate the stress–strain relationship of soil and the anisotropic seepage model to simulate the anisotropic seepage characteristics of soil has practical significance. The calculation parameters are presented in

Table 1. Calculation parameters of the constitutive model.

Series	Clay	Smear Layer of Clay	Muddy Silty Clay	Smear Layer of Muddy Silty Clay	Silty Clay	Sand Drain
Compressive modulus, $E_s$/MPa	4.61	4.61	4.35	4.35	8.77	11.66
Poisson’s ratio	0.47	0.47	0.55	0.55	0.49	0.3
Cohesion, $c$/kPa	14	14	3.7	3.7	4	0
Internal friction angle, $\phi$/${}^{\circ }$	15.5	15.5	18.9	18.9	26.7	36
Bulk density, $\gamma$/kN·m−3	19.2	19.2	17.8	17.8	18.8	19
Water content, $\omega$/%	31.9	31.9	44.4	44.4	35	/

and

Table 2. Calculation parameters of the seepage model.

Series	Clay	Smear Layer of Clay	Muddy Silty Clay	Smear Layer of Muddy Silty Clay	Silty Clay	Sand Drain
Horizontal permeability coefficient, $k_{h100–200}$/cm/s	0.40 × 10−7	0.35 × 10−7	1.44 × 10−7	1.30 × 10−7	0.41 × 10−7	3 × 10−2
Vertical permeability coefficient, $k_{v100–200}$/cm/s	0.52 × 10−7	0.53 × 10−7	0.68 × 10−7	0.69 × 10−7	0.57 × 10−7	3 × 10−2

Density of water: 1000 kg/m³; Biot modulus: 4 × 10⁹ Pa.

.

3.2. Gridding

Based on the soil layer distribution map of the calculation section, a single sand drain consolidation model grid was established. To convert the plastic drain board into a sand drain with a radius of 0.05 m, the formula $r_w=\alpha \left(a+b\right)/4$ (where $a$ is the width of the drain board, $b$ is the thickness of the drain board, and $\alpha$ is the conversion factor) was used. The smear layer radius was taken as 3 times 0.15 m, and the affected area radius was taken as 7 times 0.35 m. The thickness of the clay layer is 1 m. The thickness of the muddy silty clay layer is 17 m. The bottom of the sand drain is flush with the bottom of the muddy silty clay layer, and the depth of the sand drain is 18 m. The thickness of the silty clay layer is 5 m.

This paper defines each soil layer as a group, the sand drain as a group, and the smear layer near the sand drain in different soil layers as a group. A total of six groups are defined: clay, muddy silty clay, silty clay, smear layer of clay, smear layer of muddy silty clay, and sand drain.

As the single sand drain foundation was axisymmetric, only one-quarter was taken for calculation. A cylindrical peripheral gradient radial grid was applied in the clay layer and muddy silty clay layer. A cylindrical peripheral gradient radial grid, cylindrical shell grid, and cylindrical grid were applied in the silty clay layer. A cylindrical grid was applied in the sand drain. A cylindrical shell grid was applied in the smear layers. The calculation model was 1.4 m wide and 23 m deep, divided into 2576 units and 3346 nodes, as depicted in Figure 3.

3.3. Boundary and Initial Conditions

The model had a free boundary on the top surface and a fixed boundary on the bottom surface, with no displacement in any direction. The four sides of the model had no horizontal displacement but allowed for vertical displacement due to the constraints of the surrounding soil. All sides of the hexahedron model were considered permeable boundaries.

The groundwater level was assumed to be flush with the ground, and the static pore water pressure at the ground node was set to 0, which increased linearly along the depth with a gradient of 10 kPa. The initial stress state was set as the gravity field. The density of each group was assigned, and the gravity acceleration was set to calculate the initial stress distribution for each group. After the calculation reached equilibrium, the deformation and the rate of the entire model node were assigned a value of 0.

A coordinate system was established for pore water pressure and sand drain height, with the center of the sand drain bottom as the origin, as shown in Figure 4.

In the fluid–solid coupling numerical analysis, the pore water pressure at the top node of the sand drain was assumed to be −80 kPa, as the vacuum degree under the membrane in the field exceeded 80 kPa within 8 h. The influence range of vacuum action was set at the bottom of the sand drain, where there was hydrostatic pressure but no negative excess pore pressure caused by vacuum, i.e., 0 kPa. Therefore, four models of pore water pressure distribution were assumed along the axial direction of the sand drain, as shown in Figure 3:

(1)

“Concave” parabolic distribution: According to the standard parabolic equation ($x^2=-2py$($p>0$)) and coordinate points (18, −80), $p=2.025$, so $y=-x^2/4.05$. Because the distribution was concave relative to the vertical axis of the sand well, it is referred to as a “concave” parabola.

(2)

“Convex” parabolic distribution: $y={\left(x-18\right)}^2/4.05-80$ was calculated according to the translation transformation of the parabolic equation obtained in (1). Because the distribution was convex relative to the vertical axis of the sand drain, it is referred to as a “convex” parabola.

(3)

Linear distribution: The linear equation across the origin was $y=-40x/9$.

(4)

Uniform distribution: The equation was $y=-80$.

The subcycle command flow was compiled to realize the application of a vacuum load under the above four working conditions. The unbalanced force ratio was set to 10⁻⁴. The master–slave program method was adopted to solve the problem. The number of mechanical substeps was subordinate to the number of seepage substeps, and the seepage time was set as 4.8384 × 10⁶ s.

4. Analysis of Calculation Results

4.1. Settlement Analysis

Figure 5, Figure 6, Figure 7 and Figure 8 display the settlement–time calculation curves for the four negative pore water pressure distribution modes along the sand drain, namely, the “concave” parabola, “convex” parabola, linear distribution, and uniform distribution. The legend represents the depth from the surface. The results demonstrate that the settlement of the soft foundation increased rapidly during the first 30 days and then gradually slowed down, finally reaching a stable state. The surface soil was subject to the most settlements, which decreased with increasing depth. Settlement changes were faster and more significant closer to the surface and slower and less noticeable further away from the surface. On the 56th day, the surface settlement reached 82.40 cm under the “concave” parabolic distribution of pore water pressure, 93.95 cm under the “convex” parabolic distribution of pore water pressure, 81.69 cm under the linear distribution of pore water pressure, and 109.33 cm under the uniform distribution of pore water pressure. The settlement–time curve at a depth of 22 m was almost horizontal. On the 56th day, the settlement deformation at a depth of 22 m reached 2.74 cm, 2.83 cm, 2.75 cm, and 3.27 cm for each of the four conditions, respectively, indicating that the vacuum effect was minimal at a depth of 22 m.

After vacuuming, within 8 h, the vacuum under the membrane reached over 80 kPa. Negative pore water pressure was transmitted vertically from the sand cushion layer under the membrane to the sand drains, causing a change in pore pressure in the sand drains. Then, the negative pore pressure in the sand drains was transmitted to the surrounding soil. Finally, the soil underwent consolidation under the negative pore pressure. The vertical distribution pattern of the pore pressure in the sand drains determined the horizontal transmission strength of the pore pressure in the surrounding soil. From the assumptions of the four types of sand drain pore pressure distribution modes, it could be known that under the “concave” parabolic distribution mode, the average pore pressure was the smallest, while under the uniformly distributed model, the average pore pressure was the largest. So, it could be inferred that the consolidation settlement of soil was in descending order: uniform distribution mode, “convex” parabolic distribution mode, linear distribution mode, and “concave” parabolic distribution mode.

Figure 9 and Figure 10 depict the settlement–time calculation curves and measured value curves at depths of 0 m and 4 m under the four different pore water pressure distribution modes. The qualitative development trend of the four calculation curves was consistent with the measured values. During the first 5 days, the four calculated curves were highly consistent with the curves of the measured values. However, a large gap between the four calculated curves and the measured values was observed between days 5 and 30. The maximum difference between the calculated settlement value and the measured settlement value under the concave parabolic and linear distributions occurred around the 10th day, reaching 8.95 cm. The maximum difference between the calculated settlement value and the measured settlement value under the “convex” parabolic distribution occurred around the 30th day, reaching 12.53 cm. The maximum difference between the calculated settlement value and the measured settlement value under uniform distribution also occurred around the 30th day, reaching 24.87 cm. After 30 days, the calculated settlement curve and the measured settlement curve under the “concave” parabolic and linear distributions had a high degree of coincidence, with differences of 0.724–2.4 cm and 0.01–1.65 cm, respectively. However, the difference between the calculated settlement value and the measured settlement value under the “convex” parabolic and uniform distributions was consistent with the difference around the 30th day.

Many simplifications and assumptions were applied in the calculation model, such as assuming that the three soil layers were horizontal and homogeneous with the sand drain and smear layer and that the soil parameters remained constant. However, in reality, the soil layer was not horizontal or homogeneous due to its complex geological origin, and the soil parameters changed gradually during stress deformation, which resulted in the calculated curve being smooth, whereas the measured curve was more variable. During the first 5 days, the vacuum effect had little impact on the soil mass, and the settlement at each depth was minimal, resulting in highly consistent values between the four calculated curves and the curve of measured values. Between days 5 and 30, the vacuum negative pore water pressure was gradually transferred from the sand drain to a wider range of soil layers, resulting in rapid settlement of the soil mass at each depth. However, due to the complex structure of the soil mass and the anisotropy of negative pore water pressure transmission, the distribution of pore water pressure at each point in the soil mass was extremely unbalanced, resulting in a large deviation between the calculated value curve and the measured value curve. After 30 days, the calculated settlement values under the “concave” parabola and linear distributions gradually became consistent with the measured settlement values, whereas the difference between the calculated and measured settlement values under the “convex” parabola and uniform distributions remained at the same level as between days 5 and 30. After 30 days, the unbalanced pore water pressure in the soil gradually reached equilibrium and stabilized at a constant value. The pore water pressure at the same depth within the influence range of vacuum action was basically the same, indicating that the isotropic pressure values at each point at the same depth were consistent. The pore water pressure at different depths tended to have different constant values, and the isotropic pressure values at various points at different depths also differed.

From the settlement calculation curve, using the two setting modes: “concave” parabolic and linear pore water pressure modes, the calculation results were significantly better than those yielded using “convex” parabolic and uniform distribution modes. The linear setting mode was slightly better than the “concave” parabolic setting mode.

In the numerical calculation of the vacuum preloading reinforcement of soft foundation, the pore pressure in the sand cushion and sand drains under the membrane can rapidly decrease to a constant value in a short period of time, so the sand cushion and sand drains can be regarded as negative pressure boundaries. We can assign and adjust the node pore pressure of the sand cushion and sand drain area to achieve the loading effect of vacuum load. Based on the calculation results of the four negative pressure distribution modes of sand drains in the article, the calculation results using smaller pore pressure distribution modes (linear distribution mode and “concave” parabolic distribution mode) are in line with the actual situation on site. Since the consolidation settlement curve in the linear mode is very close to the measured curve (especially the final consolidation settlement), we selected the surface settlement time curve in this mode for asymptotic fitting and obtained the calculation formula and coefficients, as shown in Figure 11.

4.2. Pore Water Pressure Analysis

Figure 12 and Figure 13 show the change curve of pore water pressure at each depth of the soil mass over time under the concave parabola and convex parabola distribution modes.

During the calculation, we assumed that the initial value of pore water pressure at each point in the soil was hydrostatic pressure, i.e., a positive value, as the groundwater level was flush with the top surface of the model. After applying negative pore water pressure due to vacuum preloading to the top surface of the model and sand drains, the pore water pressure at each point in the soil decreased to varying degrees. Between 0 and 20 days, the pore water pressure decreased most obviously within a depth of 18 m. Between 20 and 30 days, the rate of pore water pressure decrease was significantly reduced and gradually became stable. After 30 days, the pore water pressure at each depth was basically unchanged and maintained at a constant value.

The pore water pressure at a depth of 2 m under the concave parabolic distribution was −69.17 kPa, that under the convex parabolic distribution was −79.59 kPa, that under the uniform distribution was −80.28 kPa, and that under the linear distribution was −72.20 kPa. The pore water pressure at a 16 m depth under the concave parabolic distribution was 1.79 kPa, and that under the linear distribution was 5.84 kPa. The pore water pressure at an 18 m depth under the convex parabolic distribution was 10.52 kPa, and that under the uniform distribution was −5.38 kPa, indicating that the vacuum preloading influence depth was approximately 16 m under the “concave” parabola and linear distributions and approximately 18 m under the “convex” parabola and uniform distributions. Combining the comparison and analysis results between the settlement calculation and the measured values proved that the influence depth of vacuum preloading was approximately 16 m.

Vacuum preloading causes a decrease in pore pressure in the soil. The greater the absolute value of negative pore pressure is, the better the consolidation effect of the soil is. However, the impact range of vacuum preloading on soil is limited. The maximum negative pore pressure is one atmospheric pressure that is impossible to achieve. The distribution pattern of pore pressure in sand drains for numerical calculation should be consistent with the actual situation. From the above settlement analysis, it can be seen that the linear distribution mode and the “concave” parabolic distribution mode are more in line with the actual situation. Due to the different mean values of the four sand drain pore pressure distribution modes, we can also infer that the decreasing values of pore pressure in soil are in descending order: uniform distribution mode, “convex” parabolic distribution mode, linear distribution mode, and “concave” parabolic distribution mode.

Figure 14 shows the pore water pressure–time change curve in soil under the four distribution modes at a 2 m depth.

Figure 15 and Figure 16 show the pore water pressure–time curves of the four models and measured values at 4 m and 6 m depths.

The measured pore water pressure curve had a large dispersion, which may have been caused by various factors, such as equipment error, construction activity, and human errors during monitoring. However, this curve can qualitatively explain some of the phenomena. The pore water pressure–time curve obtained by the logarithmic fitting of the measured values was found to be closest to the calculated value of the linear distribution model.

5. Conclusions

The Beijing–Shanghai high-speed railway test section project used a fluid–solid coupling calculation model in FLAC^3D software to establish four modes of negative pore water pressure distribution along the sand drain: concave parabola, convex parabola, linear, and uniform distribution. The soil consolidation process was simulated and calculated, and the results were compared with the measured values of the project. The following conclusions were drawn.

The settlement of the soft foundation developed rapidly in the first 30 days, slowed down after 30 days, and eventually tended to stabilize. The settlement was most significant near the surface and gradually decreased with depth. The settlement–time curve at a depth of 22 m was nearly horizontal, indicating a weak vacuum effect at that depth.

Within the first 5 days of vacuum preloading, the settlement calculation curves for all four distribution modes were highly consistent with the measured settlement curves. Between days 5 and 30, there was a significant gap between the calculated and measured values. After 30 days, the settlement curves for the concave parabolic and linear distribution modes coincided closely with the measured settlement curves, with differences ranging from 0.724 to 2.4 cm and from 0.01 to 1.65 cm, respectively. However, the differences between the calculated and measured settlement values for the convex parabolic and uniform distribution modes were consistent with the differences observed around day 30.

The calculation results for the concave parabolic and linear pore water pressure modes were better than those for the convex parabolic and uniform distribution modes. The linear setting mode was slightly better than the concave parabolic setting mode.

During the first 20 days, the pore water pressure dropped most significantly within the depth range of 0–18 m. Between days 20 and 30, the rate of the pore water pressure decrease was significantly reduced and eventually became stable. After 30 days, the pore water pressure at each depth was essentially constant.

The influence depth of vacuum preloading for the concave parabolic and linear distribution modes was approximately 16 m, whereas the influence depth for the convex parabolic and uniform distribution modes was approximately 18 m. A comparison of the calculated settlement and the measured settlement showed that the influence depth of vacuum preloading was approximately 16 m.

This article did not consider the impact of group sand drains. Therefore, we will consider this impact in future research.