Buoyancy Force Acting on Underground Structures considering Seepage of Confined Water

1 School of Human Settlements and Civil Engineering, Xi’an Jiaotong University, Xi’an, Shanxi 710049, China 2China JIKAN Research Institute of Engineering Investigations and Design, Co, Ltd, Xi’an, Shanxi 710043, China 3Shaanxi Key Laboratory of Engineering Behaviour and Foundation Treatment for Special Soil, Xi’an, Shaanxi 710043, China 4Key Laboratory of Geotechnical and Underground Engineering of the Ministry of Education, Tongji University, Shanghai 200092, China 5Department of Geotechnical Engineering, Tongji University, Shanghai 200092, China


Introduction
The antifloating design of underground structures is an important aspect in underground engineering.Large underground structures, such as underground garage of highrise buildings and underground pipe galleries, in high-level underground water level are exposed to high buoyancy forces, which may uplift underground structures [1][2][3], particularly those located in confined water.The reasonable estimation of the buoyancy forces acting on underground structures in complex geological and hydrological conditions is important to the safety of such structures.
The buoyancy force without considering the effect of seepage can be calculated in two steps: (1) reasonably determining the underground water level and (2) calculating the buoyancy force based on Archimedes' principle.However, many researches showed that the measured buoyancy forces acting on underground structures are lower than those obtained based on the Archimedes' principle [4][5][6][7].The buoyancy force acting on an underground structure depends on many factors; e.g., the liquefaction of sand during an earthquake usually leads to a sudden increase in the buoyancy force [8][9][10].For cohesive soils, the buoyancy force would reduce because of the interaction of the soil particles and water at a microscopic level [7,11,12].Seepage flow is found to influence the buoyancy force acting on underground structures [13].The pore water pressure in soil, which changes the magnitude of the buoyancy force acting on underground structures, is significantly influenced by the seepage flow.Many studies have investigated seepage flow and the corresponding pore pressure induced by seepage [14][15][16].
Seepage leads to a change in the force acting on underground structures because of the change in the pore pressure.The complex foundation-soil-water interaction behaviour is not fully understood [17][18][19][20][21][22][23][24].Studies on the effect of seepage on the buoyancy force acting on underground structures are limited.
In some urban areas of China, such as Beijing, Shanghai, and Wuhan, the underground hydrological conditions are complex.Multilayer confined water with a high confined water head is observed [25].The seepage of confined water changes the pore pressure distribution and the buoyancy force acting on the structures.Studies have investigated the seepage of confined water in layered soil [26][27][28].The force acting on structures induced by seepage in a homogeneous soil is usually horizontal, whereas the force acting on structures induced by the seepage of confined water in layered soil is vertical.The floating potential of underground structures induced by vertical seepage of confined water is significantly greater than that induced by normal seepage in one soil layer.The influence of confined-water seepage on the buoyancy force acting on underground structures is more complicated than that of the horizontal seepage.Unfortunately, no study has been carried out to understand the influence of vertical seepage of confined water on the buoyancy force acting on underground structures.To estimate the buoyancy force acting on underground structures located in layered soil with confined water, we need to first study the influence of seepage of confined water on the buoyancy force acting on such structures.
In this paper, we study the influence of vertical seepage of confined water on the buoyancy force acting on underground structures in different soil configurations and hydraulic gradients through model tests.The relationship between the pore pressure, buoyancy force, and hydraulic gradient under vertical seepage in homogeneous sand and layered soils is investigated.Simplified equations are proposed to calculate the buoyancy force acting on the underground structures considering the effect of vertical seepage.Further, a series of FEM studies are carried out to validate the experimental findings and the rationality of the equations.

Experimental Setup and Procedure
2.1.Experimental Setup. Figure 1 shows the experimental apparatus used in this study.The apparatus comprises four main parts: a tank, a water supply circulation system, a model of the underground structure, and a measuring system.The inner size of the tank is 600 mm × 600 mm × 1000 mm (width × length × depth), as shown in Figure 2. The water supply circulation system includes a vertical steel frame with a height of 1.5 m, two suspended water tanks, which control the water levels at the upper and lower bored pipes in the tank, a sink for overflows during the test, and a submersible pump that pumps the water to the upper water tank.The upper bored pipe is 120 mm from the top of the tank, and the lower bored pipe is 80 mm from the bottom of the tank.The difference in the water levels between the upper and lower bored pipes can cause a steady vertical seepage field in the soil.
Figure 3 shows the model of the underground structure with dimensions of 200 mm × 200 mm × 400 mm (width × length × depth).It is made of 10 mm thick plastic plates.To accurately measure the buoyancy force acting on the bottom of the model, the base of the model is separated from the sides of the model.The base is connected to the sides of the model using waterproof rubber and can move freely in the vertical direction when subjected to buoyancy forces.Therefore, the soil-structure friction can be neglected when measuring the buoyancy force acting on the bottom of the structure during the test.The buoyancy measuring system comprises a reaction frame fixed to the tank, a force-measuring ring with a range of 0.6 kN, a dial indicator, and a force-transmitting rod with four legs, which transfer the force acting on the base of the structure to the force-measuring ring.Six micro-porepressure sensors are located around the model structure, as shown in Figure 4.The measuring range and the accuracy of the sensors are 20 and 0.04 kPa, respectively.Sensor 'A' is used to measure the pore pressure at the centre of the model underneath the base, and sensors 'B' , 'C' , 'D' , 'E' , and 'F' are used to measure the pore pressure around the corners of the structure at depths of 200, 300, 400, 500, and 600 mm, respectively.The data are recorded using the DataTaker DT85.
Fine sand and clay obtained from Shanghai were used to form the foundation in the test.Table 1 lists the wet density, dry density, void ratios, plasticity index, liquid index, and permeability coefficient of the soil compacted in the tank.The ratio of the permeability coefficient of fine sand to that of clay is 75.25.Therefore, fine sand is used to form the aquifer, and clay is used to form the overlying aquiclude.

Procedure.
Seven tests, divided into three series, were carried out, as listed in Table 2.The embedment depth of the structure in the soil is 300 mm.The structure is placed in pure water.

Homogeneous sand H1
The structure is embedded in fine sand, and the water head difference is 0. H2 The structure is embedded in fine sand, and the water head difference is 10 cm.H3 The structure is embedded in fine sand, and the water head difference is 20 cm.
Layered soil with confined water

O4
The structure is embedded in overlying aquiclude, and the water head difference is 0.

O5
The structure is embedded in overlying aquiclude, and the water head difference is 10 cm.O6 The structure is embedded in overlying aquiclude, and the water head difference is 20 cm.The case S0 corresponds to 'system calibration.' In this case, the underground structure model was placed in pure water.Four steps were carried out in this test.In the first step, the structure was slowly immersed in water until the embedment depth reached 300 mm.In the second step, the water level was lowered to the bottom of the structure.In the third step, the water level was raised until the embedment depth of the structure reached 300 mm.In the fourth step, the water level was lowered to the bottom of the structure again.The water level was adjusted by adjusting the height of the water tanks.
The cases H1-H3 correspond to 'homogeneous sand' and were considered to investigate the effects of the hydraulic gradient on the pore pressure and buoyancy force acting on the underground structure placed on homogeneous fine sand.The total depth of the sand is 900 mm.The embedment depth of the structure is 300 mm.The water head height of the lower bored pipe is higher than that of the upper bored pipe.The corresponding hydraulic gradients are 0, 0.125, and 0.25, respectively.
The cases O4-O6 correspond to 'layered soil with confined water' and were considered to investigate the influence of the overlying aquiclude on the buoyancy force acting on the structure.Here, the total depth of the soil is 900 mm.A 400 mm thick clay layer was placed over a 500 mm thick sand layer.The embedment depth of the structure is 300 mm.The corresponding hydraulic gradients for cases O4-O6 are 0, 0.125, and 0.25, respectively.

System Calibration.
The pore pressure and buoyancy force acting on the model bucket can be calculated from Archimedes' principle using (1) and (2), respectively.Figure 5 shows the relationship between the calculated buoyancy force and the measured deformation of the base of the structure.
The relationships between the buoyancy force and the measured deformation obtained from steps 2-4 are the same, whereas that obtained from step 1 is different.This may be because the system filled the initial gap between the various components of the mechanism in the first step.The system becomes more accurate and stable, making the test repeatable.The buoyancy force acting on the structure can be calculated from the measured deformation of the base using (3).
where  is the density of water,  is the acceleration due to gravity, h is the hydrostatic water head, A is the crosssectional area of the model bucket, and s is the deformation measured using the dial indicator.

Structure in Homogeneous
Soil. Figure 6 shows the pore pressure around the corners of the structure at different depths in terms of the measured pore pressure (P) and the raising ratio of pore pressure (P/P 0 −1).P 0 is the theoretical pore pressure calculated using (1).The measured pore pressure of test H1 is consistent with the theoretical hydrostatic pressure.The measured pore pressures from H2 and H3 are significantly greater than the theoretical hydrostatic pressure.The pore pressure is significantly influenced by the vertical seepage of the underground water.The pore pressure distribution along the depth remains linear.The raising ratio of the pore pressure along the depth is approximately constant, and the fluctuated amplitude is within 0.05, probably because of sensor fluctuation.The raising ratios of the pore pressure in tests H2 and H3 are approximately 0.14 and 0.29, respectively.The raising ratio is approximately equal to the vertical hydraulic gradient.The buoyancy force acting on the structure under seepage should be calculated from the highest water head in the hydraulically connected area and not from the water head height around the structure.Table 3 presents the pore pressure at the centre of the base slab under the structure (P c ) and the buoyancy force acting on the base slab (F).The pore pressure raising ratio (P c /P 0 −1) at the centre of the slab is slightly greater than the hydraulic gradient.Compared with the pore pressure measured using sensor 'b' , as shown in Figure 6(a), the pore pressure raising ratio at the centre of the structure is generally 6 Complexity   greater than that around the corner of the structure under the same condition.Similarly, the raising ratio of the buoyancy force (F/ F 0 −1) is greater than the hydraulic gradient.The raising ratio of the buoyancy force is consistent with the raising ratio of the pore pressure at the centre of the structure.Nevertheless, we believe it is acceptable from an engineering standpoint to use the hydraulic gradient to calculate the buoyancy force, as shown in (4).
where i is the hydraulic gradient.

Structure in Overlying
Aquiclude.Figure 7 shows the pore pressure in the 'layered soil with confined water' test, in which the underground structure is placed on the overlying aquiclude.The same conclusion can be drawn from hydrostatic test O1; i.e., the measured pore pressure is consistent with the theoretical hydrostatic pressure in the absence of seepage.The measured pore pressures are significantly greater than the theoretical hydrostatic pressure in the presence of a vertical seepage field in the tests.Unlike the results obtained when considering a homogeneous soil layer, the pore pressure curves along the depth in tests O2 and O3 become fold lines.The slope of the curve for the overlying aquiclude is greater than that for the deeper sand layer.In other words, the pore pressure in aquiclude reduces with depth much more quickly than that in sand.For tests O2 and O3, the raising ratio of the pore pressure is largely consistent when the depth is less than 0.4 m; this is similar to the results obtained under homogeneous sand.However, the raising ratio of the pore pressure decreases with the increase in the depth when the depth is greater than 0.4 m, which is located in the sand layer.The pore pressures induced by seepage in homogeneous sand and in layered soil with confined water are different.The maximum raising ratios of the pore pressure at the soil interface are 0.23 and 0.52 for O2 and O3, respectively.The maximum raising ratio of the pore pressure is approximately twice the hydraulic gradient.The pore pressure distribution along the depth is no longer linear as expected, making it unsafe to calculate the buoyancy force acting on the structure from the hydrostatic water level based on the current design method.Table 4 lists the pore pressure at the centre of the structure and the buoyancy force acting on the base of the structure (F).The raising ratio of the pore pressure is approximately twice the hydraulic gradient and is slightly greater than the raising ratio of the pore pressure around the corners of   the structure at the same depth (300 mm), as shown in Figure 7(b).The raising ratio of the buoyancy force acting on the base is similar to that of the pore pressure at the centre of the structure.The buoyancy force acting on the structure is significantly increased because of the effect of the overlying aquiclude or confined water.Compared with the structure placed on homogeneous sand, the increment in the buoyancy force acting on the structure is twice as that obtained from hydraulic gradient calculation when the structure is subjected to overflowing confined water.The buoyancy acting on the structure can be calculated using (5).

Numerical Simulation
4.1.Finite Element Model.ABAQUS/Standard was used to carry out a 3D simulation of the tests.The seepage calculation module of ABAQUS was employed.The soil was set as porous material.Table 5 lists the parameters of the soil.Figure 8 shows the meshed model.The soil element was set as the pore pressure element, i.e., C3D8P.The dimension of the mesh is the same as that of the model test.The side and bottom boundaries were set as impermeable boundaries.The top boundary was set as a free boundary.The interfaces between the structure and the soil were set as impermeable boundaries.This ensures that the water will not flow into the structure.The water head heights at the bottom and top were set according to the test cases H1, H2, and H3 and O1, O2, and O3.The buoyancy force was calculated by integrating the pore pressures at the lower surface of the base.The other conditions were the same as those in the model tests.

Results of FEM.
Figure 9 presents the pore pressure around the corners of the structure at different depths.
Figure 10 shows the pore pressure at the centre of the structure under the base.The results of the model tests and FEM are in good agreement.The effect of the overlying aquiclude on the pore pressure in an overflow field of confined water obtained from FEM is the same as that obtained from the model tests.
The buoyancy forces obtained from the model tests, FEM, and proposed equations, as shown in Figure 11, are in good agreement.Therefore, it is feasible to use the steady-state seepage module of the ABAQUS/standard to calculate the pore pressure and buoyancy force.Equations ( 3) and ( 4) are simplified solutions to calculate the buoyancy force acting on structures located in a confined water stratum subjected to seepage.

Conclusions
In this study, the buoyancy force and pore pressure acting on an underground structure considering vertical seepage  were investigated by conducting model tests and numerical simulation.The test results demonstrated that the pore pressure around the underground structure and the buoyancy force acting on it considering the vertical seepage (of the overflowing confined water through the overlying aquiclude) are significantly greater than those obtained under the hydrostatic condition.For the structure placed on a homogeneous permeable soil layer, the raising ratios of the buoyancy force and pore pressure were approximately equal to the vertical hydraulic gradient.The buoyancy force acting on the structure placed on an overlying aquiclude was greater than that acting on the structure placed on homogeneous sand.
The pore pressure raising ratio was maximum at the interface of the two soil layers and was approximately twice the hydraulic gradient.Two equations were proposed to calculate the buoyancy force acting on the underground structures considering the seepage of overflowing confined water.The results of the model tests and the rationality of the equations were verified by conducting FEM simulation.The equations were found to be feasible to calculate the buoyancy force acting on underground structures considering the vertical seepage of overflowing confined water.Unfortunately, the seepage is not always vertical in practice, and the leakage of the confined water which may cause the reduction on friction stress along the underground structure is not considered.The proposed equations are not enough for antifloating design although they are helpful to calculate the buoyancy force acting on the base of the underground structure considering the vertical seepage.

Figure 2 :
Figure 2: Image of the tank.

Figure 3 :
Figure 3: Model of the underground structure.

Figure 4 :
Figure 4: Layout of the pore pressure sensor.

Figure 5 :
Figure 5: Calibration result of the buoyancy force.
Raising ratios of the pore pressure

Figure 6 :
Figure 6: Measured pore pressure at different depths in structural corner position.
Raising ratio of the pore pressure

Figure 8 :
Figure 8: Schematic of the finite element mesh.

Figure 9 :
Figure 9: Comparison between measured pore pressure values and finite element results at corner positions.

Figure 10 :Figure 11 :
Figure 10: Comparison between measured pore pressure values and finite element results at base slab centre.

2 Complexity Table 1 :
Engineering properties of the foundation soil.

Table 2 :
Summary of system calibration, homogeneous soil layer, and overlying aquiclude tests.

Table 3 :
Measured and theoretical pore pressures and buoyancy forces acting on the structure placed on homogeneous soil layer.

Table 4 :
Measured and theoretical pore pressures and buoyancy forces acting on the structure placed on overlying aquiclude.

Table 5 :
Material parameters used in FEM simulation.